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BCS-BEC Crossover: From High Temperature Superconductors toUltracold Superfluids

Qijin Chen 1,∗ Jelena Stajic2,† Shina Tan2, and K. Levin2

1 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218 and2 James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637

We review the BCS to Bose Einstein condensation (BEC) crossover scenario which is based on the well knowncrossover generalization of the BCS ground state wavefunction Ψ0. While this ground state has been summarizedextensively in the literature, this Review is devoted to less widely discussed issues: understanding the effects offinite temperature, primarily belowTc, in a manner consistent withΨ0. Our emphasis is on the intersectionof two important problems: highTc superconductivity and superfluidity in ultracold fermionic atomic gases.We address the “pseudogap state” in the copper oxide superconductors from the vantage point of a BCS-BECcrossover scenario, although there is no consensus on the applicability of this scheme to highTc. We argue thatit also provides a useful basis for studying atomic gases near the unitary scattering regime; they are most likelyin the counterpart pseudogap phase. That is, superconductivity takes place out of a non-Fermi liquid state wherepreformed, metastable fermion pairs are present at the onset of their Bose condensation. As a microscopic basisfor this work, we summarize a variety ofT -matrix approaches, and assess their theoretical consistency. A closeconnection with conventional superconducting fluctuationtheories is emphasized and exploited.

PACS numbers: 03.75.-b, 74.20.-z Journal Ref: Physics Reports 412, 1-88 (2005).

Contents

I. Introduction to Qualitative Crossover Picture 2A. Fermionic Pseudogaps and Meta-stable Pairs: Two Sides ofthe

Same Coin 2B. Introduction to highTc Superconductivity: Pseudogap Effects 4C. Introduction to HighTc Superconductivity: Mott Physics and

Possible Ordered States 8D. Overview of the BCS-BEC Picture of HighTc Superconductors

in the Underdoped Regime 91. Searching for the Definitive HighTc Theory 11

E. Many Body Hamiltonian and Two Body Scattering Theory:Mostly Cold Atoms 111. Differences Between One and Two Channel Models: Physics

of Feshbach bosons 14F. Current Summary of Cold Atom Experiments: Crossover in the

Presence of Feshbach Resonances 14G. T-Matrix-Based Approaches to BCS-BEC Crossover in the

Absence of Feshbach Effects 151. Review of BCS Theory Using theT -matrix Approach 162. Three Choices for theT -matrix of the Normal State:

Problems with the Nozieres Schmitt-Rink Approach 17H. Superconducting Fluctuations: a type of Pre-formed Pairs 18

1. Dynamics of Pre-formed Pairs AboveTc: Time DependentGinzburg-Landau Theory 19

II. Quantitative Details of Crossover 20A. T = 0, BEC Limit without Feshbach Bosons 20B. Extending conventional Crossover Ground State toT 6= 0: BEC

limit without Feshbach Bosons 21C. Extending Conventional Crossover Ground State toT 6= 0:

T -matrix Scheme in the Presence of Feshbach Bosons 21D. Nature of the Pair Dispersion: Size and Lifetime of

noncondensed Pairs BelowTc 23E. Tc Calculations: Analytics and Numerics 24

III. Self Consistency Tests 24

∗Current address: James Franck Institute and Department of Physics, Univer-sity of Chicago, Chicago, Illinois 60637†Current address: Los Alamos National Laboratory, Los Alamos, NM 87545

A. Important check: behavior ofρs 25B. Collective Modes and Gauge Invariance 25C. Investigating the Applicability of a Nambu Matrix Green’s

Function Formulation 27

IV. Other Theoretical Approaches to the Crossover Problem 27A. T 6= 0 Theories 27B. Boson-Fermion Models for HighTc Superconductors 29

V. Physical Implications: Ultracold Atom Superfluidity 29A. Homogeneous case 29

1. Unitary Limit and Universality 312. Weak Feshbach Resonance Regime:|g0| = 0+ 33

B. Superfluidity in Traps 33C. Comparison Between Theory and Experiment: Cold Atoms 35

VI. Physical Implications: High Tc Superconductivity 36A. Phase Diagram and Superconductor-Insulator Transition: Boson

Localization Effects 36B. Superconducting Coherence Effects 37C. Electrodynamics and Thermal Conductivity 38D. Thermodynamics and Pair-breaking Effects 40E. Anomalous Normal State Transport: Nernst and Other Precursor

Transport Contributions 401. Quantum Extension of TDGL 402. Comparison with Transport Data 41

VII. Conclusions 42

Acknowledgments 43

A. Derivation of the T = 0 variational conditions 43

B. Proof of Ward Identity Below Tc 43

C. Quantitative Relation between BCS-BEC crossover andHartree-TDGL 44

D. Convention and Notation 451. Notation 452. Convention for units 453. Abbreviations 45

References 46

2

I. INTRODUCTION TO QUALITATIVE CROSSOVERPICTURE

This review addresses an extended form of the BCS theoryof superfluidity (or superconductivity) known as “BCS-BoseEinstein condensation (BEC) crossover theory”. Here we con-template a generalization of the standard mean field descrip-tion of superfluids now modified so that one allows attractiveinteractions of arbitrary strength. In this way one describeshow the system smoothly goes from being a superfluid sys-tem of the BCS type (where the attraction is arbitrarily weak)to a Bose Einstein condensate of diatomic molecules (wherethe attraction is arbitrarily strong). One, thereby, sees thatBCS theory is intimately connected to BEC. Attractive inter-actions between fermions are needed to form “bosonic-like”molecules (called Cooper pairs) which then are driven statisti-cally to Bose condense. BCS theory is a special case in whichthe condensation and pair formation temperatures coincide.

The importance of obtaining a generalization of BCS theorywhich addresses the crossover from BCS to BEC at generaltemperaturesT ≤ Tc cannot be overestimated. BCS theoryas originally postulated can be viewed as a paradigm amongtheories of condensed matter systems; it is complete, in manyways generic and model independent, and well verified ex-perimentally. The observation that a BCS-like approach goesbeyond strict BCS theory, suggests that there is a larger meanfield theory to be addressed. Equally exciting is the possibil-ity that this mean field theory can be discovered and simul-taneously tested in a very controlled fashion using ultracoldfermionic atoms. We presume here that it may also have ap-plicability to other short coherence length materials, such asthe high temperature superconductors.

This Review is written in order to convey important newdevelopments from one subfield of physics to another. Here itis hoped that we will communicate the recent excitement feltby the cold atom community to condensed matter physicists.Conversely we wish to communicate a comparable excitementin studies of high temperature superconductivity to atomicphysicists. There has been a revolution in our understandingof ultracold fermions in traps in the last few years. The mile-stones in this research were the creation of a degenerate Fermigas (1999), the formation of dimers of fermions (2003), Bose-Einstein condensation (BEC) of these dimers (late 2003) andfinally superfluidity of fermionic pairs (2004).

In high temperature superconductors the BCS-BECcrossover picture has been investigated for many years now.Itleads to a particular interpretation of a fascinating, but not wellunderstood phase, known as the “pseudogap state”. In the coldatom system, this crossover description is not just a scenario,but has been realized in the laboratory. This is because onehas the ability (via Feshbach resonances) to apply magneticfields in a controlled way to tune the strength of the attrac-tion. We note, finally, that research in this field has not beenlimited exclusively to the two communities. One has seenthe application of these crossover ideas to studies of excitonsin solids (Laiet al., 2004), to nuclear physics (Baldoet al.,1995; Heiselberg, 2004a) and to particle physics (Itakura,2003; Klinkhamer and Volovik, 2004; Kolb and Heinz, 2003)

as well. There are not many problems in physics which haveas great an overlap with different subfield communities as this“BCS-BEC crossover problem”.

In this Review, we begin with an introduction to the qual-itative picture of the BCS-BEC crossover scenario which isrepresented schematically in Fig. 1. We discuss this picturein the context of both highTc superconductivity and ultra-cold atomic superfluidity, summarizing experiments in both.In Section II, we present a more quantitative study of thiscrossover primarily at and below the superfluid transition tem-peratureTc. Section III contains a detailed discussion of thesuperfluid density and gauge invariance issues, in large partto emphasize self consistency checks for different theoreti-cal approaches to crossover. These alternative theories aresummarized and compared in Section IV. Sections V and VI,respectively, present the physical implications of crossoverphysics, in one particular rendition, in the context of ultracoldfermionic superfluids and high temperature superconductors.Our conclusions are presented in Section VII and a numberof Appendices are included for clarity. A reader interestedinonly the broad overview may choose to omit Sections II-IV.

The convention and notation as well as a list of abbrevia-tions used in this Review are listed in Appendix D at the end.

A. Fermionic Pseudogaps and Meta-stable Pairs: TwoSides of the Same Coin

A number of years ago Eagles (Eagles, 1969) and Leggett(Leggett, 1980) independently noted1 that the BCS groundstate wavefunction

Ψ0 = Πk(uk + vkc†k,↑c

†−k,↓)|0〉 (1)

had a greater applicability than had been appreciated at thetime of its original proposal by Bardeen, Cooper and Schrief-fer (BCS). As the strength of the attractive pairing interactionU (< 0) between fermions is increased, this wavefunction isalso capable of describing a continuous evolution from BCSlike behavior to a form of Bose Einstein condensation. Whatis essential is that the chemical potentialµ of the fermions beself consistently computed asU varies.

The variational parametersvk anduk which appear in Eq. 1are usually represented by the two more directly accessibleparameters∆sc(0) andµ, which characterize the fermionicsystem. Here∆sc(0) is the zero temperature superconduct-ing order parameter. These fermionic parameters are uniquelydetermined in terms ofU and the fermionic densityn. Thevariationally determined self consistency conditions aregivenby two BCS-like equations which we refer to as the “gap” and

1 Through private correspondence it appears that F. Dyson mayhave madesimilar observations even earlier.

3

BCS BEC

weak coupling strong coupling

large pair size small pair sizek-space pairing r -space pairing

strongly overlapping ideal gas ofCooper pairs preformed pairs

T ∗ = Tc T ∗ ≫ Tc

FIG. 1 Contrast between BCS- and BEC-based superfluids. HereT ∗ represents the temperature at which pairs form, whileTc is thatat which they condense.

“number” equations respectively.

∆sc(0) = −U∑

k

∆sc(0)1

2Ek

(2)

n =∑

k

[

1 − ǫk − µ

Ek

]

(3)

where2

Ek ≡√

(ǫk − µ)2 + ∆2sc(0) (4)

andǫk = k2/2m is the fermion energy dispersion.3 Withinthis ground state there have been extensive studies of collec-tive modes (Cote and Griffin, 1993; Micnaset al., 1990; Ran-deria, 1995), effects of two dimensionality (Randeria, 1995),and, more recently, extensions to atomic gases (Peraliet al.,2003; Viveritet al., 2004). Nozieres and Schmitt-Rink werethe first (Nozieres and Schmitt-Rink, 1985) to address nonzeroT . We will outline some of their conclusions later in this Re-view. Randeria and co-workers reformulated the approach ofNozieres and Schmitt-Rink (NSR) and moreover, raised theinteresting possibility that crossover physics might be relevantto high temperature superconductors (Randeria, 1995). Sub-sequently other workers have applied this picture to the high

2 In general, the order parameter∆sc and the excitation gap may have ak-dependence,∆k = ∆ϕk, except for a contact potential. To keepthe mathematical expressions simpler, however, we neglectϕk for thes-wave pairing under investigation of this Review. It can be easily rein-serted back when needed in actual calculations. In particular, we useϕk = exp(−k2/2k2

c ) with a large cutoffkc in most of our numericalcalculations.

3 Throughout this Review, we set~ = 1 andkB = 1.

0 1 2U/Uc

-1

0

1

µ/E

F

BCS PG BEC

FIG. 2 Typical behavior of theT = 0 chemical potential in the threeregimes. As the pairing strength increases from 0, the chemical po-tentialµ starts to decrease and then becomes negative. The characterof the system changes from fermionic (µ > 0) to bosonic (µ < 0).The PG (pseudogap) case corresponds to non-Fermi liquid based su-perconductivity, andUc corresponds to critical coupling for forminga two fermion bound state in vacuum, as in Eq. (17) below.

Tc cuprates (Chenet al., 1998; Micnas and Robaszkiewicz,1998; Ranninger and Robin, 1996) and ultracold fermions(Milstein et al., 2002; Ohashi and Griffin, 2002) as well as for-mulated alternative schemes (Griffin and Ohashi, 2003; Pieriet al., 2004) for addressingT 6= 0. Importantly, a number ofexperimentalists, most notably Uemura (Uemura, 1997), haveclaimed evidence in support (Deutscher, 1999; Junodet al.,1999; Renneret al., 1998) of the BCS-BEC crossover picturefor highTc materials.

Compared to work on the ground state, considerably lesshas been written on crossover effects at nonzero temperaturebased on Eq. (1). Because our understanding has increasedsubstantially since the pioneering work of NSR, and becausethey are the most interesting, this review is focussed on thesefinite T effects, at the level of a simple mean field theory.Mean field approaches are always approximate. We can as-cribe the simplicity and precision of BCS theory to the factthat in conventional superconductors the coherence lengthξ isextremely long. As a result, the kind of averaging procedureimplicit in mean field theory becomes nearly exact. Onceξbecomes small, BCS is not expected to work at the same levelof precision. Nevertheless even when they are not exact, meanfield approaches are excellent ways of building up intuition.And further progress is not likely to be made without investi-gating first the simplest of mean field approaches, associatedwith Eq. (1).

The effects of BEC-BCS crossover are most directly re-flected in the behavior of the fermionic chemical potentialµ.We plot the behavior ofµ in Fig. 2, which indicates the BCSand BEC regimes. In the weak coupling regimeµ = EF andordinary BCS theory results. However at sufficiently strongcoupling,µ begins to decrease, eventually crossing zero andthen ultimately becoming negative in the BEC regime, withincreasing|U |. We generally viewµ = 0 as a crossing point.For positiveµ the system has a remnant Fermi surface, and wesay that it is “fermionic”. For negativeµ, the Fermi surface isgone and the material is “bosonic”.

The new and largely unexplored physics of this problem

4

lies in the fact that once outside the BCS regime, but beforeBEC, superconductivity or superfluidity emerge out of a veryexotic, non-Fermi liquid normal state. Indeed, it should beclear that Fermi liquid theory must break down somewherein the intermediate regime, as one goes continuously fromfermionic to bosonic statistics. This intermediate case hasmany names: “resonant superfluidity”, the “unitary regime”or, as we prefer (because it is more descriptive of the manybody physics) the “pseudogap (PG) regime”. The PG regimeplotted in Fig. 2 is drawn so that its boundary on the left co-incides with the onset of an appreciable excitation gap atTc,called∆(Tc); its boundary on the right coincides with whereµ = 0. This non-Fermi liquid based superconductivity hasbeen extensively studied(Chenet al., 1998, 2000; Jankoet al.,1997; Malyet al., 1999a).

Fermi liquid theory is inappropriate here principally be-cause there is a gap for creating fermionic excitations. Thenon-Fermi liquid like behavior is reflected in that of thefermionic spectral function (Malyet al., 1999a) which has atwo-peak structure in the normal state, broadened somewhat,but rather similar to its counterpart in BCS theory. Impor-tantly, the onset of superconductivity occurs in the presence offermion pairs, which reflect this excitation gap. Unlike theircounterparts in the BEC limit, these pairs are not infinitelylong lived. Their presence is apparent even in the normalstate where an energy must be applied to create fermionic ex-citations. This energy cost derives from the breaking of themetastable pairs. Thus we say that there is a “pseudogap” atand aboveTc. But pseudogap effects are not restricted to thisnormal state. Pre-formed pairs have a natural counterpart inthe superfluid phase, as noncondensed pairs.

It will be stressed throughout this Review that gaps in thefermionic spectrum and bosonic degrees of freedom are twosides of the same coin. A particularly important observationto make is that the starting point for crossover physics is basedon the fermionic degrees of freedom. Bosonic degrees of free-dom are deduced from these; they are not primary. A nonzerovalue of the excitation gap∆ is equivalent to the presence ofmetastable or stable fermion pairs. And it is only in this indi-rect fashion that we can probe the presence of these “bosons”,within the framework of Eq. (1).

In many ways this crossover theory appears to represent amore generic form of superfluidity. Without doing any calcu-lations we can anticipate some of the effects of finite tempera-ture. Except for very weak coupling,pairs form and condenseat different temperatures. The BCS limit might be viewed asthe anomaly. Because the attractive interaction is presumedto be arbitrarily weak, in BCS the normal state is unaffectedby U and superfluidity appears precipitously, that is withoutwarning atTc. More generally, in the presence of a moder-ately strong attractive interaction it pays energeticallyto takesome advantage and to form pairs (say roughly at temperatureT ∗) within the normal state. Then, for statistical reasons thesebosonic degrees of freedom ultimately are driven to condenseatTc < T ∗, just as in BEC.

Just as there is a distinction betweenTc andT ∗, there mustbe a distinction between the superconducting order parame-ter ∆sc and the excitation gap∆. The presence of a normal

∆

∆∆(Τ)

cΤ T*

sc

FIG. 3 Contrasting behavior of the excitation gap∆(T ) and super-fluid order parameter∆sc(T ) versus temperature. The height of theshaded region roughly reflects the density of noncondensed pairs ateach temperature.

state excitation gap or pseudogap for fermions is inextricablyconnected to this generalized BCS wavefunction. In Figure 3we present a schematic plot of these two energy parameters.It may be seen that the order parameter vanishes atTc, as ina second order phase transition, while the excitation gap turnson smoothly belowT ∗. It should also be stressed that there isonly one gap energy scale in the ground state (Leggett, 1980)of Eq. (1). Thus at zero temperature

∆sc(0) = ∆(0) . (5)

In addition to the distinction between∆ and∆sc, anotherimportant way in whichbosonic degrees of freedom are re-vealed is indirectly through the temperature dependence of∆.In the BEC regime where fermion pairs are pre-formed,∆ isessentially constant for allT ≤ Tc (as isµ). By contrast inthe BCS regime it exhibits the well known temperature depen-dence of the superconducting order parameter. This is equiva-lent to the statement that bosonic degrees of freedom are onlypresent in the condensate for this latter case. This behavior isillustrated in Fig. 4.

Again, without doing any calculations we can make onemore inference about the nature of crossover physics at finiteT . The excitations of the system must smoothly evolve fromfermionic in the BCS regime to bosonic in the BEC regime.In the intermediate case, the excitations are a mix of fermionsand meta-stable pairs. Figure 5 characterizes the excitationsout of the condensate as well as in the normal phase. Thisschematic figure will play an important role in our thinkingthroughout this review. In the BCS and BEC regimes one isled to an excitation spectrum with a single component. For thePG case it is clear that the excitations now have two distinct(albeit, strongly interacting) components.

B. Introduction to high Tc Superconductivity: PseudogapEffects

This Review deals with the intersection of two fields andtwo important problems: high temperature superconductorsand ultracold fermionic atoms in which, through Feshbachresonance effects, the attractive interaction may be arbitrarilytuned by a magnetic field. Our focus is on the broken symme-try phase and how it evolves from the well known ground stateatT = 0 to T = Tc. We begin with a brief overview (Tallon

5

∆

Tc T TTc

∆

FIG. 4 Comparison of typical temperature dependences of theexci-tation gaps in the BCS (left) and BEC (right) limits. For the former,the gap is small and vanishes atTc; whereas for the latter, the gap isvery large and essentially temperature independent.

BCS Pseudogap (PG) BECc c c

FIG. 5 The character of the excitations in the BCS-BEC crossoverboth above and belowTc. The excitations are primarily fermionicBogoliubov quasiparticles in the BCS limit and bosonic pairs (or“Feshbach bosons”) in the BEC limit. In the PG case the “virtualmolecules” consist primarily of “Cooper” pairs of fermionic atoms.

and Loram, 2001; Timusk and Statt, 1999) of pseudogap ef-fects in the hole doped high temperature superconductors. Astudy of concrete data in these systems provides a rather nat-ural way of building intuition about non-Fermi liquid basedsuperfluidity, and this should, in turn, be useful for the coldatom community.

It has been argued by some (Chenet al., 1999; Micnas andRobaszkiewicz, 1998; Pieri and Strinati, 2000; Ranninger andRobin, 1996; Yanaseet al., 2003) that a BCS-BEC crossover-induced pseudogap is the origin of the mysterious normal stategap observed in high temperature superconductors. Although,this is a highly contentious subject, a principal supporting ra-tionale is that this above-Tc excitation gap seems to evolvesmoothly into the fermionic gap within the superconductingstate. This, in conjunction with a panoply of normal stateanomalies which appear to represent precursor superconduc-tivity effects, suggests that the pseudogap has more in com-mon with the superconducting phase than with the insulatingmagnetic phase associated with the parent compound.

The arguments in favor of this (albeit, controversial)crossover viewpoint rest on the following observations: (i) thetransition temperature is anomalously high and the coherencelength ξ for superconductivity anomalously short, around10A as compared more typically with1000A. (ii) These sys-tems are quasi-two dimensional so that one might expect pre-formed pairs or precursor superconductivity effects to be en-hanced. (iii) The normal state gap has the samed-wave sym-metry as the superconducting order parameter (Campuzanoet al., 2004; Damascelliet al., 2003), and (iv) to a good ap-proximation its onset temperature, calledT ∗, satisfies (Ku-

FIG. 1. This Uemura plot indicates how “exotic” su-FIG. 6 Uemura plot (Uemura, 1997) of the superconducting transi-tion temperature as a function of Fermi temperature (lower axis) andof the low T superfluid density (upper axis) for various supercon-ductors. The plot indicates how “exotic” superconductors appear tobe in a distinct category as compared with more conventionalmetalsuperconductors such as Nb, Sn, Al and Zn. Note the logarithmicscales.

gler et al., 2001; Odaet al., 1997)T ∗ ≈ 2∆(0)/4.3 whichsatisfies the BCS scaling relation. (v) There is widespreadevidence for pseudogap effects both above (Tallon and Lo-ram, 2001; Timusk and Statt, 1999) and below (Loramet al.,1994; Stajicet al., 2003b)Tc. Finally, it has also been arguedthat short coherence length superconductors may quite gener-ally exhibit a distinctive form of superconductivity (Uemura,1997). Figure 6 shows a plot of data collected by Uemurawhich seems to suggest that the traditional BCS superconduc-tors stand apart from other more exotic (and frequently shortξ) forms of superconductivity. From this plot one can infer,that, except for highTc, there is nothing special about the highTc superconductors; they are not alone in this distinctive classwhich includes the organics and heavy fermion superconduc-tors as well. Thus, to understand them, one might want tofocus on this simplest aspect (shortξ) of highTc superconduc-tors, rather than invoke more exotic and less generic features.

In Fig. 7 we show a sketch of the phase diagram for thecopper oxide superconductors. Herex represents the concen-tration of holes which can be controlled by adding Sr substi-tutionally, say, to La1−xSrxCuO4. At zero and smallx thesystem is an antiferromagnetic (AFM) insulator. Preciselyathalf filling (x = 0) we understand this insulator to derive fromMott effects. These Mott effects may or may not be the sourceof the other exotic phases indicated in the diagram, as “SC”for superconductivity and the pseudogap phase. Once AFMorder disappears the system remains insulating until a criti-cal concentration (typically around a few percent Sr) when

6

Hole concencentration x

Tem

pera

ture

T

AFM

Pseudo Gap

Tc

T*

SC

FIG. 7 Typical phase diagram of hole-doped highTc superconduc-tors. There exists a pseudogap phase aboveTc in the underdopedregime. Here SC denotes superconductor, andT ∗ is the temperatureat which the pseudogap smoothly turns on.

∆(Τc )

hole concentration xFIG. 8 Pseudogap magnitude atTc versus doping concentration inhole doped highTc superconductors measured with different experi-mental techniques (from Tallon and Loram, 2001). Units on verticalaxis are temperature in degrees K.

an insulator-superconductor transition is encountered. Herephotoemission studies (Campuzanoet al., 2004; Damascelliet al., 2003) suggest that once this line is crossed,µ appearsto be positive. Forx ≤ 0.2, the superconducting phase has anon-Fermi liquid (or pseudogapped) normal state (Tallon andLoram, 2001). We note an important aspect of this phase di-agram at lowx. As the pseudogap becomes stronger (T ∗ in-creases), superconductivity as reflected in the magnitude ofTc becomes weaker. One way to think about this competi-tion is through the effects of the pseudogap onTc. As thisgap increases, the density of fermionic states atEF decreases,so that there are fewer fermions around to participate in thesuperconductivity.

Importantly, there is a clear excitation gap present at theonset of superconductivity which is called∆(Tc) and whichvanishes atx ≈ 0.2. The magnitude of the pseudogap atTc

is shown in Fig. 8, from (Tallon and Loram, 2001). Quite re-markably, as indicated in the figure, a host of different probesseem to converge on the size of this gap.

Figure 9 indicates the temperature dependence of the exci-tation gap for three different hole stoichiometries. Thesedata

0

5

10

15

20

25

0 50 100 150 200 250 300

T (K)

Spe

ctra

l Gap

FIG. 9 (Color) Temperature dependence of the excitation gapfromARPES measurements for optimally doped (filled black circles), un-derdoped (red squares) and highly underdoped (blue inverted trian-gles) single-crystal BSCCO samples (taken from Dinget al., 1996).There exists a pseudogap phase aboveTc in the underdoped regime.

FIG. 10 STM measurements of thedI/dV characteristics of an un-derdoped BSCCO sample inside (solid) and outside (dashed) avortexcore at lowT from Renneret al., 1998.dI/dV is proportional to thedensity of states.

were taken (Dinget al., 1996) from angle resolved photoemis-sion spectroscopy (ARPES) measurement on single-crystalBi2Sr2CaCu2O8+δ (BSCCO) samples. For one sample shownas circles, (corresponding roughly to “optimal” doping) thegap vanishes roughly atTc as might be expected for a BCSsuperconductor. At the other extreme are the data indicatedby inverted triangles in which an excitation gap appears to bepresent up to room temperature, with very little temperaturedependence. This is what is referred to as a highly underdopedsample (smallx), which from the phase diagram can be seento have a rather lowTc. Moreover,Tc is not evident in thesedata on underdoped samples. This is a very remarkable fea-ture which indicates that from the fermionic perspective thereappears to be no profound sensitivity to the onset of supercon-ductivity.

While the highTc community has focussed on pseudogapeffects aboveTc, there is a good case to be made that these ef-fects also persist below. Shown in Fig. 10 are STM data (Ren-ner et al., 1998) taken belowTc within a vortex core (solidlines) and between vortices in the bulk (dashed lines). ThequantitydI/dV may be viewed as a measure of the fermionicdensity of states at energyE given by the voltageV . This

7

0

0.01

0.02

0.03

0.04

0.05C

v/(

T/T

c)

0 1 2T/Tc

0

0.02

0.04

S

0

0.1

0.2

0 1 2 3T/Tc

0

0.1

0.2

SCPGFL

BCS pseudogapx = 0.25 x = 0.125

(a)

(b) (d)

(c)

FIG. 11 Behavior of entropyS and specific heat coefficientγ ≡Cv/T as a function of temperature for (a)-(b) BCS and (c)-(d) pseu-dogapped superconductors. Dotted and dashed lines are extrapolatednormal states belowTc. The shaded areas were used to determinethe condensation energy.

figure shows that there is a clear depletion of the density ofstates around the Fermi energy (V = 0) in the normal phasewithin the core. Indeed the size of the inferred energy gap(or pseudogap) corresponds to the energy separation betweenthe two maxima indI/dV ; this can be seen to be the samefor both the normal and superconducting regions (solid anddashed curves). This figure emphasizes the fact that the ex-istence of an energy gap has little or nothing to do with theexistence of phase coherent superconductivity. It also sug-gests that pseudogap effects effectively persist belowTc; thenormal phase underlying superconductivity forT ≤ Tc is nota Fermi liquid.

Analysis of thermodynamical data (Loramet al., 1994; Tal-lon and Loram, 2001) has led to a similar inference. Figure 11presents a schematic plot of the entropyS and specific heatfor the case of a BCS superconductor, as contrasted with apseudogapped superconductor. Actual data are presented inFig. 38. Figure 11 makes it clear that in a BCS supercon-ductor, the normal state which underlies the superconductingphase,is a Fermi liquid; the entropy at high temperatures ex-trapolates into a physically meaningful entropy belowTc. Forthe PG case, the Fermi liquid-extrapolated entropy becomesnegative. In this way Loram and co-workers (Loramet al.,1994) deduced that the normal phase underlying the super-conducting state is not a Fermi liquid. Rather, they claimedtoobtain proper thermodynamics, it must be assumed that thisstate contains a persistent pseudogap. They argued for a dis-tinction between the excitation gap∆ and the superconduct-ing order parameter, within the superconducting phase. To fittheir data they presumed a modified fermionic dispersion

Ek =√

(ǫk − µ)2 + ∆2(T ) (6)

where

∆2(T ) = ∆2sc(T ) + ∆2

pg (7)

Here ∆pg is taken on phenomenological grounds to beT -independent. The authors argue that the pseudogap contribu-

tion may arise from physics unrelated to the superconductiv-ity. While others (Chakravartyet al., 2001; Nozieres and Pis-tolesi, 1999) have similarly postulated that∆pg may in factderive from another (hidden) order parameter or underlyingband structure effect, in general the fermionic dispersionrela-tion Ek will take on a different character from that assumedabove, which is very specific to a superconducting origin forthe pseudogap.

Finally, Fig. 12 makes the claim for a persistent pseudo-gap belowTc in an even more suggestive way. Figure 12(a)represents a schematic plot of excitation gap data such as areshown in Fig. 9. Here the focus is on temperatures belowTc.Most importantly, this figure indicates that theT dependencein ∆ varies dramatically as the stoichiometry changes. Thus,in the extreme underdoped regime, where PG effects are mostintense, there is very littleT dependence in∆ belowTc. Bycontrast at highx, when PG effects are less important, thebehavior of∆ follows that of BCS theory. What is most im-pressive however, is that these wide variations in∆(T ) arenot reflected in the superfluid densityρs(T ), which is propor-tional to the inverse square of the London magnetic penetra-tion depth and can be measured by, for example, muon spinrelaxation (µSR) experiments. Necessarily,ρs(T ) vanishesat Tc. What is plotted (Stajicet al., 2003b) in Fig. 12(b) isρs(T ) − ρs(0) versusT . That these data all seem to sit on arather universal curve is a key point. The envelope curve inρs(T ) − ρs(0) is associated with an “optimal” sample where∆(T ) essentially follows the BCS prediction. Figure 12 thenindicates that,despite the highly non-universal behavior for∆(T ), the superfluid density does not make large excursionsfrom its BCS- predicted form. This is difficult to understand ifthe fermionic degrees of freedom through∆(T ) are dominat-ing at allx. Rather this figure suggests that something otherthan fermionic excitations is responsible for the disappearanceof superconductivity, particularly in the regime where∆(T )is relatively constant inT . At the very least pseudogap effectsmust persist belowTc.

Driving the superconductivity away is another importantway to probe the pseudogap. This occurs naturally withtemperatures in excess ofTc, but it also occurs when suffi-cient pair breaking is present through impurities (Tallonet al.,1997; Tallon and Loram, 2001; Zhaoet al., 1993) or appliedmagnetic fields (Andoet al., 1995). An important effect oftemperature needs to be stressed. With increasingT > Tc,thed-wave shape of the excitation gap is rapidly washed out(Campuzanoet al., 2004; Damascelliet al., 2003); the nodesof the order parameter, in effect, begin to spread out, justaboveTc. The inverse of this effect should also be empha-sized: when approached from above,Tc is marked by theabrupt onset of long lived quasiparticles.

One frequently, and possibly universally, sees asuperconductor-insulator (SI) transition whenTc is drivento zero in the presence of a pseudogap. This suggests asimple scenario: thatthe pseudogap may survive whensuperconductivity is suppressed. In this way the ground stateis no longer a simple metal. This SI transition is seen uponZn doping (Tallonet al., 1997; Tallon and Loram, 2001; Zhaoet al., 1993), as well as in the presence of applied magnetic

8

0 40 80T(K)

−30

−15

0

ρ s(T)−

ρ s(0)(

meV

)

Tc=90 K

Tc=90 K

Tc=88 K

Tc=65 K

Tc=55 K

Tc=50 K

Tc=48 K

0 1T / Tc

0

1

∆(T

) / ∆

(0)

Tc = 48 K

Tc = 90 K

a) b)

FIG. 12 Temperature dependence of (a) the fermionic excitation gap∆ and (b) superfluid densityρs from experiment for various dopingconcentrations (from (Stajicet al., 2003b)). Figure (a) shows a schematic plot of∆ as seen in the ARPES and STM tunneling data. When∆(Tc) 6= 0, there is little correlation between∆(T ) andρs(T ); this figure suggests that something other than fermionic quasiparticles (e.g.,bosonic excitations) may be responsible for the disappearance of superconductivity with increasingT . Figure (b) shows a quasi-universalbehavior for the slopedρs/dT at different doping concentrations, despite the highly non-universal behavior for∆(T ).

fields (Andoet al., 1995). Moreover, the intrinsic change instoichiometry illustrated in the phase diagram of Fig. 7 alsoleads to an SI transition. One can thus deduce that the effectsof pair-breaking onTc and T ∗ are very different, with theformer being far more sensitive than the latter.

It is clear that pseudogapped fermions are in evidence in amultiplicity of experiments. Are there indications of bosonicdegrees of freedom in the normal state of highTc super-conductors? The answer is unequivocally yes:meta-stablebosons are observable as superconducting fluctuations. Thesebosons are intimately connected to the pseudogap; in effectthey are the other side of the same coin. These bosonic ef-fects are enhanced in the presence of the quasi-two dimen-sional lattice structure of these materials. Very detailedanaly-ses (Larkin and Varlamov, 2001) of thermodynamic and trans-port properties of the highestTc or “optimal” samples revealclearly these pre-formed pairs. Moreover they are responsi-ble (Kwon and Dorsey, 1999) for divergences atTc in the dcconductivityσ and in the transverse thermoelectric (Ussishkinet al., 2002) responseαxy. These transport coefficients aredefined more generally in terms of the electrical and heat cur-rents by

Jelec = σE + α( −∇T ) (8)

Jheat = αE + κ( −∇T ) (9)

Hereσ is the conductivity tensor,κ the thermal conductivitytensor, andα, α are thermoelectric tensors. Other coefficients,while not divergent, exhibit precursor effects, all of which arefound to be in good agreement (Larkin and Varlamov, 2001)with fluctuation theory. As pseudogap effects become morepronounced with underdoping much of the fluctuation behav-ior appears to set in at a higher temperature scale associatedwith T ∗, but often some fraction thereof (Tan and Levin, 2004;Xu et al., 2000).

Figures 13 and 14 make the important point that precur-sor effects in the transverse thermoelectric response (αxy) andconductivityσ appear at higher temperatures (∝ T ∗) as pseu-dogap effects become progressively more important; the dot-ted lines which have the strongest pseudogap continue to the

highest temperatures on both Figures. Moreover, both trans-port coefficients evolve smoothly from a regime where theyare presumed to be described by conventional fluctuations(Larkin and Varlamov, 2001; Ussishkinet al., 2002) as shownby the solid lines in the figures into a regime where their be-havior is associated with a pseudogap. A number of peoplehave argued (Corsonet al., 1999; Xuet al., 2000) that nor-mal state vortices are responsible for the so called anomaloustransport behavior of the pseudogap regime. We will argue inthis Review that these figures may alternatively be interpretedas suggesting that bosonic degrees of freedom, not vortices,are responsible for these anomalous transport effects.

C. Introduction to High Tc Superconductivity: MottPhysics and Possible Ordered States

Most workers in the field of highTc superconductivitywould agree that we have made enormous progress in char-acterizing these materials and in identifying key theoreticalquestions and constructs. Experimental progress, in largepart, comes from transport studies (Tallon and Loram, 2001;Timusk and Statt, 1999) in addition to three powerful spectro-scopies: photoemission (Campuzanoet al., 2004; Damascelliet al., 2003), neutron (Aeppliet al., 1997a,b; Cheonget al.,1991; Fonget al., 1995; Kastneret al., 1998; Mooket al.,1998; Rossat-Mignodet al., 1991; Tranquadaet al., 1992) andJosephson interferometry (Mathaiet al., 1995; Tsueiet al.,1994; Wollmanet al., 1995). These data have provided uswith important clues to address related theoretical challenges.Among the outstanding theoretical issues in the cuprates are(i) understanding the attractive “mechanism” that binds elec-trons into Cooper pairs, (ii) understanding the evolution of thenormal phase from Fermi liquid (in the “overdoped” regime)to marginal Fermi liquid (at “optimal” doping) to the pseudo-gap state, which is presumed to occur as doping concentrationx decreases, and (iii) understanding the nature of that super-conducting phase which evolves from each of these three nor-mal states.

The theoretical community has concentrated rather exten-sively on special regions andx-dependences in the phase dia-

9

(T-Tc)/Tc

0 1 2 3 4 50

5

10

more UDUDOP

ConventionalFluctuations

Pseudogap Region

αxy

FIG. 13 Transverse thermoelectric response (which relatesto theNernst coefficient) plotted here in fluctuation regime aboveTc. HereOP and UD correspond to optimal and underdoping, respectively.Data were taken from Xuet al., 2000.

gram which are presumed to be controlled by “Mott physics”.There are different viewpoints of precisely what constitutesMott physics in the metallic phases, and, for example, whetheror not this is necessarily associated with spin-charge separa-tion. For the most part, one presumes here that the insulatingphase of the parent compound introduces strong Coulomb cor-relations into the doped metallic states which may or may notalso be associated with strong antiferromagnetic correlations.There is a recent, rather complete review (Leeet al., 2004)of Mott physics within the scenario of spin-charge separation,which clarifies the physics it entails.

Concrete experimental signatures of Mott physics are theobservations that the superfluid densityρs(T = 0, x) → 0asx → 0, as if it were reflecting an order parameter for ametal insulator transition. More precisely it is deduced thatρs(0, x) ∝ x, at low x. Unusual effects associated withthis linear- in-x dependence also show up in other experi-ments, such as the weight of coherence features in photoemis-sion data (Campuzanoet al., 2004; Damascelliet al., 2003),as well as in thermodynamical signatures (Tallon and Loram,2001).

While there is no single line of reasoning associated withthese Mott constraints, the low value of the superfluid densityhas been argued (Emery and Kivelson, 1995) to be responsi-ble for soft phase fluctuations of the order parameter, whichmay be an important contributor to the pseudogap. However,recent concerns about this “phase fluctuation scenario” fortheorigin of the pseudogap have been raised (Lee, 2004). It isnow presumed by a number of groups that phase fluctuationsalone may not be adequate and an additional static or fluctu-ating order of one form or another needs to be incorporated.Related to a competing or co-existing order are conjectures(Chakravartyet al., 2001) that the disappearance of pseudo-gap effects aroundx ≈ 0.2 is an indication of a “quantumcritical point” associated with a hidden order parameter whichmay be responsible for the pseudogap. Others have associatedsmall (Vojtaet al., 2000)x or alternatively optimal (Varma,1999)x with quantum critical points of a different origin. Thenature of the other competing or fluctuating order has been

(T-Tc)/Tc

σ−b

ackg

roun

d

0 0.5 1.0 1.5 2.00

5

10

more UDUDOP

ConventionalFluctuations

Pseudogap Region

FIG. 14 Conductivity in fluctuation regime for optimally (OP) andunderdoped (UD) cuprate superconductors. Data from Watanabeet al., 1997.

conjectured to be RVB (Andersonet al., (2004), “d-densitywave” (Chakravartyet al., 2001), stripes (Castellaniet al.,1995; Zacharet al., 1997) or possibly antiferromagnetic spinfluctuations (Chubukovet al., 1996; Demler and Zhang, 1995;Pines, 1997). The latter is another residue of the insulatingphase.

What is known about the “pairing mechanism”? Somewould argue that this is an ill-defined question, and that su-perconductivity has to be understood through condensationenergy arguments based for example on the data generatedfrom the extrapolated normal state entropy (Loramet al.,1994; Tallon and Loram, 2001) discussed above in the con-text of Fig. 11. These condensation arguments are often builtaround the non-trivial assumption that there is a thermody-namically well behaved but meta-stable normal phase whichcoexists with the superconductivity. Others would argue thatCoulomb effects are responsible ford-wave pairing, either di-rectly (Leggett, 1999; Liu and Levin, 1997), or indirectly viamagnetic fluctuations (Pines, 1997; Scalapinoet al., 1986).Moreover, the extent to which the magnetism is presumedto persist into the metallic phase near optimal doping is con-troversial. Initially, NMR measurements were interpretedassuggesting (Milliset al., 1990) strong antiferromagnetic fluc-tuations, while neutron measurements, which are generallyviewed as the more conclusive, do not provide compellingevidence (Siet al., 1993) for their presence in the normalphase. Nevertheless, there are interesting neutron-measuredmagnetic signatures (Kaoet al., 2000; Liu et al., 1995; Zhaet al., 1993) belowTc associated withd-wave superconductiv-ity. There are also analogous STM effects which are currentlyof interest (Wang and Lee, 2003).

D. Overview of the BCS-BEC Picture of High Tc

Superconductors in the Underdoped Regime

We summarize here the physical picture associated withthe more controversial BCS-BEC crossover approach to highTc superconductors, based on the standard ground state ofEq. (1). One presumes that the normal state (pseudo)gap

10

arises from pre-formed pairs. This gap∆(T ) is thus presentat the onset of superconductivity, and reflects the fact thatCooper pairs form before they Bose condense, as a result ofsufficiently strong attraction. BelowTc, ∆ evolves smoothlyinto the excitation gap of the superconducting phase.

This behavior reflects the fact thatTc corresponds to thecondensation of theq = 0 pairs, while finite momentum pairspersist into the order phase as excitations of the condensate.As the temperature is progressively lowered belowTc, thereare fewer and fewer such bosonic excitations. Their numbervanishes precisely atT = 0, at which point the system reachesthe ground state of Eq. (1). Here the superconducting or-der parameter and the excitation gap become precisely equal.This approach is based on a mean field scheme and is thus notequivalent to treating phase fluctuations of the order parame-ter. In general, belowTc, the superconducting order parameter∆sc and the pseudogap contributions add in quadrature in thedispersion of Eq. (6) as

∆2(T ) = ∆2sc(T ) + ∆2

pg(T ) (10)

where∆pg → 0 asT → 0; this behavior should be contrastedwith that in Eq. (7).

Transport and thermodynamics in the normal and supercon-ducting phases are determined by two types of quasiparticles:fermionic excitations which exhibit a gap∆(T ), and noncon-densed pairs or bosonic excitations. Contributions from theformer are rather similar to their (lowT ) counterparts in BCStheory, except that they are present well aboveTc as well,with substantially larger∆ than in strict BCS theory. In thisway, experiments which probe the opening of an excitationgap (such as Pauli susceptibility in the spin channel, and re-sistivity in the charge channel) will exhibit precursor effects,that is, the gradual onset of superconductivity from aroundT ∗

to belowTc.The contributions to transport and thermodynamics of the

bosonic excitations are rather similar to those deduced fromthe time dependent Ginsburg-Landau theory of superconduct-ing fluctuations, provided that one properly extends this the-ory, in a quantum fashion, away from the critical regime. Itis these contributions which often exhibit divergences atTc,but unlike conventional superconductors, bosonic contribu-tions (pre-formed pairs) are present, up to high temperaturesof the order ofT ∗/2, providing the boson density is suffi-ciently high. Again, these bosons do not represent fluctua-tions of the order parameter, but they obey a generically sim-ilar dynamical equation of motion, and, in the more generalcase which goes beyond Eq. (1), they will be coupled to orderparameter fluctuations. Here, in an oversimplified sense thereis a form of spin-charge separation. The bosons are spin-less(singlet) with charge2e, while the fermions carry both spinand charge, but are non-Fermi-liquid like, as a consequenceof their energy gap.

In the years following the discovery of highTc supercon-ductors, attention focused on the so-called marginal Fermiliquid behavior of the cuprates. This phase is associatedwith near-optimal doping. The most salient signature of thismarginality is the dramatically linear in temperature depen-dence of the resistivity from just aboveTc to very high tem-

peratures, well above room temperature. Today, we know thatpseudogap effects are present at optimal doping, so that theprecise boundary between the marginal and pseudogap phasesis not clear. As a consequence, addressing this normal stateresistivity in the BCS-BEC crossover approach has not beendone in detail, although it is unlikely to yield a strictly linearTdependence at a particular doping. Nevertheless, the bosoniccontributions lead to adecreasein resistivity as temperatureis lowered, while the opening of ad-wave gap associatedwith the fermionic terms leads to anincrease, which tends tocompensate. These are somewhat analogous to Aslamazov-Larkin and Maki-Thompson fluctuation contributions (Larkinand Varlamov, 2001). The bosons generally dominate in thevicinity of Tc, but at higherT , in the vicinity ofT ∗, the gappedfermions will be the more important.

The vanishing ofTc with sufficiently largeT ∗ ( extremeunderdoping) occurs in an interesting way in this crossovertheory. This effect is associated with the localization of thebosonic degrees of freedom. Pauli principle effects in con-junction with the extended size ofd-wave pairs inhibit pairhopping, and thus destroy superconductivity. One can specu-late that, just as in conventional Bose systems, once the bosonsare localized, they will exhibit an alternative form of longrange order. Indeed, “pair density wave” models have beeninvoked (Chenet al., 2004) to explain recent STM data (Fuet al., 2004; Vershininet al., 2004). In contrast to the coldatom systems, for thed-wave case the BEC regime is neveraccessed. This superconductor-insulator transition takes placein the fermionic regime whereµ ≈ 0.8EF .

In the context of BCS-BEC crossover physics, it is not es-sential to establish the source of the attractive interaction. Forthe most part we skirt this issue in this Review by taking thepairing onset temperatureT ∗(x) as a phenomenological in-put parameter. At the simplest level one may argue that as thesystem approaches the Mott insulating limit (x→ 0) fermionsbecome less mobile; this serves to increase the effectivenessof the attractive interaction at low doping levels.

The variablex in the highTc problem should be viewedthen, as a counterpart to the magnetic field variable in thecold atom problem which tunes the system through a Fesh-bach resonance; bothx and field strength modulate the sizeof the attraction. However, for the cuprates, in contrast tothecold atom systems, one never reaches the BEC limit of thecrossover theory.

It is reasonable to presume based on the evidence to date,that pairing ultimately derives from Coulombic effects, notphonons, which are associated withl = 0 pairing. Whilethe widely used Hubbard model ignores these effects, longerranged screened Coulomb interactions have been found (Liuand Levin, 1997) to be attractive for electrons in ad-wavechannel. In this context, it is useful to note that, similarly,in He3 short range repulsion destroyss-wave pairing, butleads to attraction in a higher (l = 1) channel (Andersonand Brinkman, 1973). There is, however, no indication ofpseudogap phenomena inHe3, so that an Eliashberg extendedform of BCS theory appears to be adequate. Eliashberg the-ory is a very different form of “strong coupling” theory fromcrossover physics, which treats in detail the dynamics of the

11

mediating boson. Interestingly, there is an upper bound toTc

in both schemes. For Eliashberg theory this arises from theinduced effective mass corrections (Levin and Valls, 1983),whereas in the crossover problem this occurs because of thepresence of a pseudogap atTc.

Other analogies have also been invoked in comparing thecuprates with superfluidHe3, based on the propensity towardsmagnetism. In the case of the quantum liquid, however, thereis a positive correlation between short range magnetic order-ing (Levin and Valls, 1983) and superconductivity while thecorrelation appears negative for the cuprates ( and some oftheir heavy fermion “cousins”) where magnetism and super-conductivity appear to compete.

1. Searching for the Definitive High Tc Theory

In no sense should one argue that the BCS-BEC crossoverapproach precludes consideration of Mott physics. Nor shouldit be viewed as superior. Indeed, many of the experiments wewill address in this Review have alternatively been addressedwithin the Mott scenario. Presumably both components areimportant in any ultimate theory. We have chosen in this Re-view to focus on one component only, and it is hoped that acomparably detailed review of Mott physics will be forthcom-ing. In this way one could compare the strengths and weak-nesses of both schemes. A summary (Carlsonet al., 2002)of some aspects of the Mott picture with emphasis on one di-mensional stripe states should be referred to as a very usefulstarting point. Also available now is a new Review (Leeet al.,2004) of Mott physics, based on spin charge separation.

Besides those discussed in this Review, there is a litany ofother experiments which are viewed as requiring explanationin order for a highTc theory to be taken seriously. Amongthese are the important linear resistivity in the normal state(at optimal doping) and unusual temperature andx depen-dences in Hall data. While there has been little insight offeredfrom the crossover scheme to explain the former, aspects ofthe latter have been successfully addressed within pre-formedpair scenarios (Geshkenbeinet al., 1997). Using fermiologybased schemes the major features of photoemission(Normanand Pepin, 2003) and of neutron scattering and NMR datawere studied theoretically and rather early on (Kaoet al.,2000; Siet al., 1993; Zhaet al., 1993), along with a focuson the now famous41 meV feature (Liuet al., 1995). Theseand other related approaches to the magnetic data, make useof a microscopically derived (Siet al., 1992) random phaseapproximation scheme which incorporates the details of theFermi surface shape andd-wave gap. Among remaining ex-periments in this litany is the origin of incoherent transportalong thec-axis (Rojo and Levin, 1993).

On the basis of the above cited literature which addressedmagnetic andc-axis 4 data we are led to conclude that pseu-

4 Here theab-plane refers to the crystal plane in parallel with the layers of thecuprate compounds, and thec-axis refers to the axis normal to the layers.

Ene

rgy

ν δ (B)

S=1S=0

Internuclear distance

FIG. 15 A schematic representation of the Feshbach resonance for6Li. The effective interaction between the scattering of atoms in thetriplet state is greatly enhanced, when the energy level of the singletbound state approaches the continuum of theS = 1 channel.

dogap effects and related crossover physics are not centraltotheir understanding. The presence of a normal state gap willlead to some degree of precursor behavior aboveTc, but d-wave nesting related effects such as reflected in neutron data(Kao et al., 2000; Liuet al., 1995; Zhaet al., 1993) will begreatly weakened aboveTc by the blurring out of thed-wavegap nodes (Campuzanoet al., 2004; Damascelliet al., 2003).It should be said, however, thatc-axis optical data do appearto reflect pseudogap effects (Ioffe and Millis, 1999) aboveTc.(Theab-plane counterparts are discussed in Section VI.C). Insummary, we are far from the definitive highTc theory and thefield has much to gain by establishing a collaboration with thecold atom community where BCS-BEC crossover as well asMott physics (Hofstetteret al., 2002) can be further elucidatedand explored.

E. Many Body Hamiltonian and Two Body ScatteringTheory: Mostly Cold Atoms

The existence of Feshbach resonances in a gas of ultracoldfermionic atoms allows the interaction between atoms to betuned arbitrarily, via application of a magnetic field. Figure15 presents a characteristic plot of the two body interactionpotentials which are responsible for this resonance. Theseinteraction potentials represent an effective description of allCoulomb interactions between the electrons and nuclei. Fordefiniteness, we consider the case of6Li here. In the region ofmagnetic fields near the “wide” Feshbach resonance,≈ 837G,we may focus on the two lowest atomic levels characterizedby |F,mF 〉, where the two quantum numbers correspond tothe total spin and itsz component. We specify the two states|1〉 ≡ |12 , 1

2 〉 and |2〉 ≡ |12 ,− 12 〉. In place of these quantum

numbers one can alternatively characterize the two states interms of the projections of the electronic (ms) and nuclear(mi) spins along the magnetic field. In this way the states 1and 2 havems = − 1

2 andmi = 1 and0, respectively.The curve in Fig. 15 labelledS = 1 represents the inter-

action between these two states. This corresponds to the so-called “open” channel and is associated with a triplet (elec-tronic) spin state. A Feshbach resonance requires that therebe an otherwise unspecified near-by state which we refer to as

12

in the “closed” channel. The potential which gives rise to thissinglet (electronic) spin state is shown by theS = 0 curve inFig. 15. A Feshbach resonance arises when this bound statelies near the zero of energy corresponding to the continuum ofscattering states in the triplet or open channel. The electronicZeeman coupling to the triplet state makes is possible to shiftthe relative position of the two potentials by application of amagnetic field. This shift is indicated byδ in Fig. 15. In thisway, the resonance is magnetically tunable and the scatteringlength of the system will vary continuously with field, diverg-ing at that magnetic field which corresponds to the Feshbachresonance. We represent the deviation of this field from theresonance condition by the “detuning parameter”ν. Whenνis large and positive, the singlet state is substantially raisedrelative to the triplet– and it plays relatively little role. Bycontrast when the magnitude ofν is large, but with negativesign, then the physics is dominated by the singlet or closedchannel.

The most general form for the Hamiltonian contains twotypes of particles: fermions which represent the open chan-nel and “molecular bosons” which represent the two fermionbound state of the closed channel. We will also refer to thelatter as “Feshbach bosons”. These, in turn, lead to two typesof interaction effects: those associated with the direct attrac-tion between fermions, parameterized byU , and those asso-ciated with “fermion-boson” interactions, whose strengthisgoverned byg. The latter may be viewed as deriving from hy-perfine interactions which couple the (total) nuclear and elec-tronic spins. In the highTc applications, one generally, butnot always, ignores the bosonic degrees of freedom. Other-wise the Hamiltonians are the same, and can be written as

H − µN =∑

k,σ

(ǫk − µ)a†k,σak,σ +∑

q

(ǫmbq + ν − 2µ)b†qbq

+∑

q,k,k′

U(k,k′)a†q/2+k,↑a

†

q/2−k,↓aq/2−k′,↓aq/2+k′,↑

+∑

q,k

(

g(k)b†qaq/2−k,↓aq/2+k,↑ + h.c.)

(11)

Here the fermion and boson kinetic energies are given byǫk = k2/(2m), and ǫmb

q = q2/(2M), andν is an impor-tant parameter which represents the magnetic field “detun-ing”. M = 2m is the bare mass of the Feshbach bosons.For the cold atom problemσ =↑, ↓ represents the two dif-ferent hyperfine states (|1〉 and |2〉). One may ignore (Tim-mermanset al., 2001) the non-degeneracy of these two states,since there is no known way for state 2 to decay into state 1.In the two channel problem the ground state wavefunction isslightly modified and given by

Ψ0 = Ψ0 ⊗ ΨB0 (12)

where the normalized molecular boson contributionΨB0 is

ΨB0 = e−λ2/2+λb†

0 |0〉 . (13)

as discussed originally by Kostyrko and Ranninger, 1996.Hereλ is a variational parameter. We present the resultingvariational conditions for the ground state in Appendix A.

-150 -100 -50 0 50 100ν0

-3

-2

-1

0

1

2

3

k Fa s

BEC PG BCS

FIG. 16 Characteristic behavior of the scattering lengthas in thethree regimes.as diverges whenν0 = 2µ. Here, and in subsequentfiguresν0 is always in units ofEF and we estimate1 G ≈ 60EF fora typicalEF = 2 µK, for either6Li or 40K.

Whether both forms of interactions are needed in eitherthe highTc or cold atom systems is still under debate. Thebosons (b†k) of the cold atom problem (Hollandet al., 2001;Timmermanset al., 2001) will be referred to as Feshbachbosons. These represent a separate species, not to be con-fused with the fermion pair (a†ka

†−k) operators. Thus we call

this a “two channel” model. In this review we will discussthe behavior of crossover physics both with and without theseFeshbach bosons (FB). Previous studies of highTc supercon-ductors have invoked a similar bosonic term (Geshkenbeinet al., 1997; Micnaset al., 1990; Ranninger and Robin, 1996)as well, although less is known about its microscopic ori-gin. This fermion-boson coupling is not to be confused withthe coupling between fermions and a “pairing-mechanism”-related boson ([b + b†]a†a) such as phonons. The couplingb†aa and its conjugate represents a form of sink and source forcreating fermion pairs, inducing superconductivity in someways, as a by-product of Bose condensation. We will em-phasize throughout that crossover theory proceeds in a similarfashion with and without Feshbach bosons.

It is useful at this stage to introduce thes-wave scatteringlength,as. This parameter is associated with the scatteringamplitude in the low energy limit, for a two body scatteringprocess. This scattering length can, in principle, be deducedfrom Figure 15. It follows thatas is negative when there isno bound state, it tends to−∞ at the onset of the bound stateand to+∞ just as the bound state stabilizes. It remains posi-tive but decreases in value as the interaction becomes increas-ingly strong. The magnitude ofas is arbitrarily small in boththe extreme BEC and BCS limits, but with opposite sign (SeeFig. 16).The fundamental postulate of crossover theory is thateven though the two-body scattering length changes abruptlyat the unitary scattering condition (|as| = ∞), in the N-bodyproblem the superconductivity varies smoothly through thispoint. Throughout this Review, we presume an equivalencebetween the “pseudogap” and unitary scattering regimes.

An important goal is to understand how to incorporate allthe constraints of the two body physics into the superfluid gapequation. As a by-product, one avoids implicit divergenceswhich would also appear in the problem. To begin we ignore

13

Feshbach effects. The way in which the two body interactionU enters to characterize the scattering (in vacuum) is differentfrom the way in which it enters to characterize the N-bodyprocesses leading to superfluidity. In each case, however, oneuses aT -matrix formulation to sum an appropriately selectedbut infinite series of terms inU . For the two-body problem invacuum, we introduce the scattering length,as,

m

4πas≡ 1

U0, (14)

which is related toU via the Lippmann-Schwinger equation

m

4πas≡ 1

U+∑

k

1

2ǫk. (15)

In this way one may solve for the unknownU in terms ofas or U0. The two-bodyT -matrix equation [Eq. (15)] can berewritten as

U = ΓU0, Γ =

(

1 +U0

Uc

)−1

, (16)

where we define the quantityUc as

U−1c = −

∑

k

1

2ǫk. (17)

HereUc is the critical value of the potential associated with thebinding of a two particle state in vacuum. Specific evaluationof Uc requires that there be a cut-off imposed on the abovesummation, associated with the range of the potential.

Now let us turn to the Feshbach problem, where the super-fluidity is constrained by a more extended set of parametersU , g, ν and alsoµ. Provided we redefine the appropriate “twobody” scattering length, Equation (15) holds even in the pres-ence of Feshbach effects (Milsteinet al., 2002; Ohashi andGriffin, 2002). It has been shown thatU in the above equa-tions is replaced byUeff andas by a∗s

m

4πa∗s≡ 1

Ueff+∑

k

1

2ǫk. (18)

Here it is important to stress that many body effects, via thefermionic chemical potentialµ enter intoa∗s.

More precisely the effective interaction between twofermions isQ dependent. It arises from a second order pro-cess involving emission and absorption of a molecular boson.The net effect of the direct plus indirect interactions is givenby

Ueff (Q) ≡ U + g2D0(Q) , (19)

where

D0(Q) ≡ 1/[iΩn − ǫmbq − ν + 2µ] (20)

is the non-interacting molecular boson propagator. What ap-pears in the gap equation, however, is

Ueff (Q = 0) ≡ Ueff = U − g2

ν − 2µ. (21)

Experimentally, the two body scattering lengtha∗s varieswith magnetic fieldB. Near resonance, it can be parameter-ized (Kokkelmanset al., 2002) in terms of three parametersof the two body problemU0, g0 and the magnetic field whichat resonance is given byB0. One fits the scattering length toabg(1 − w

B−B0

) so that the widthw is related tog0 and thebackground scattering lengthabg is related toU0. Away fromresonance, the crossover picture requires that the scatteringlength vanish asymptotically as

a∗s ≈ 0 , in extreme BEC limit, (22)

so that the system represents an ideal Bose gas. Here it shouldbe noted that this BEC limit corresponds toν → µ+. Thecounterpart of Eq. (22) also holds for the one channel prob-lem. Note that the actual resonance of the many body problemwill be somewhat shifted from the atomic-fitted resonance pa-rameterB0.

It is convenient to define the parameterν0 which is directlyrelated to the difference in the applied magnetic fieldB andB0

ν0 = (B −B0)∆µ0, (23)

where∆µ0 is the difference in the magnetic moment of thesinglet and triplet paired states.We will use the parameterν0as a measure of magnetic field strength throughout this Re-view. The counterpart of Eq. (14) is then

m

4πa∗s≡ 1

U0 − g2

0

ν0−2µ

≡ 1

U∗, (24)

whereU0 =4πabg

m andg20 = U0∆µ

0w. For the purposes ofthis Review we will use the scattering lengtha∗s, although forsimplicity we often drop the asterisk and write it asas.

We can again invert the Lippmann-Schwinger orT -matrixscattering equation (Bruun and Pethick, (2004; Kokkelmanset al., 2002) to arrive at the appropriate parametersg andνwhich enter into the Hamiltonian and thus into the gap equa-tion. Just as in Eq. (16), one finds a connection between mea-surable parameters (with subscript0) and their counterparts inthe Hamiltonian (without the subscript):

U = ΓU0, g = Γg0, ν − ν0 = −Γg20

Uc. (25)

To connect the various energy scales, which appear in theproblem, typically1G ≈ 60EF for both 6Li and 40K. Hereit is assumed that the Fermi energy isEF ≈ 2µK.

In Fig. 16 and using Eq. (18), we plot a prototypical scat-tering lengthkF as ≡ kFa

∗s as a function of the magnetic field

dependent parameter,ν0. This figure indicates the BEC, BCSand PG regimes. Here, as earlier, the PG regime is boundedon the left byµ = 0 and on the right by∆(Tc) ≈ 0.

A key finding associated with this plot is thatthe PG regimebegins on the so-called “BEC side of resonance”, indepen-dent of whether Feshbach effects are included or not. That is,the fermionic chemical potential reaches zero while the scat-tering length is positive. This generic effect of the ground

14

state self consistent equations arises from the Pauli principle.For an isolated system, two particle resonant scattering occursat U = Uc, whereUc is defined in Eq. (17). In the manybody context, because of Pauli principle repulsion betweenfermions, it requires an effectively stronger attraction to bindparticles into molecules. Stated alternatively, the onsetof thefermionic regime (µ > 0) occurs for more strongly attractiveinteractions, than those required for two body resonant scat-tering. As a consequence ofµ > 0, there is very little conden-sate weight in the Feshbach boson channel near the unitarylimit, for values of the coupling parameterg0, appropriate tocurrently studied (and rather wide) Feshbach resonances.

Finally, it is useful to compare the two interaction termswhich we have defined above, (Ueff andU∗), and to con-trast their behavior in the BEC and unitary limits. The quan-tity U∗ is proportional to the scattering length; it reflects thetwo body physics and necessarily diverges at unitarity, whereUeff = Uc. By contrast, in the BEC regime,U∗, or a∗s ap-proaches zero, (under the usual presumption that the interac-tion is a contact interaction). This vanishing ofU∗ should becompared with the observation that the quantityUeff , whichreflects the many body physics, must diverge in this BEClimit.

1. Differences Between One and Two Channel Models:Physics of Feshbach bosons

The main effects associated with including the explicit Fes-hbach resonance are twofold. These Feshbach bosons providea physical mechanism for tuning the scattering length to bearbitrarily large, and in the BEC limit, they lead to differentphysics from the one channel problem. As will be illustratedlater via a comparison of Figures 23 and 25, in an artificialway one can capture most of the salient physics of the uni-tary regime via a one channel approach in which one drops allmolecular boson related terms in the Hamiltonian, but takesthe interactionU to be arbitrary. In this way there is very littledifference between fermion-only and fermion-boson models.There is an important proviso, though, that one is not dealingwith narrow resonances which behave rather differently nearunitarity, as will be seen in Fig. 32. In addition, as in the onechannel model a form of universality (Ho, 2004) is found inthe ground state at the unitary limit, providedg is sufficientlylarge. This is discussed later in the context of Fig. 30.

We emphasize that, within the BEC regime there are threeimportant effects associated with the Feshbach couplingg. Aswill become clear later, (i) in the extreme BEC limit whengis nonzero, there are no occupied fermionic states. The num-ber constraint can be satisfied entirely in terms of Feshbachbosons. (ii) The absence of fermions will, moreover, greatlyweaken the inter-boson interactions which are presumed tobe mediated by the fermions. In addition, as one decreases|Ueff | from very attractive to moderately attractive (i.e., in-creasesa∗s on the BEC side) the nature of the condensed pairschanges. Even in the absence of Feshbach bosons (FB), thesize of the pairs increases. But in their presence the admix-ture of bosonic (i.e., molecular bosons) and fermionic (i.e.,

Cooper pair) components in the condensate is continuouslyvaried from fully bosonic to fully fermionic. Finally (iii)therole of the condensate enters in two very different ways intothe self consistent gap and number equations, depending onwhether or not there are FB. The molecular boson condensateappears principally in the number equation, while the Cooperpair condensate appears principally in the gap equation. Tomake this point clearer we can refer ahead to Eq. (81) whichcontains the FB condensate contributionn0

b . By contrast theCooper condensate contribution∆sc enters into the total ex-citation gap∆ as in Eq. (10) and∆ is, in turn, constrainedvia Eq. (69). Points (i) and (iii), which may appear some-what technical, have essential physical implications, amongthe most important of these is point (ii) above.

We stress, however, thatfor the PG and BCS regimes thedifferences with and without FB are, however, considerablyless pronounced.

F. Current Summary of Cold Atom Experiments:Crossover in the Presence of Feshbach Resonances

There has been an exciting string of developments over thepast few years in studies of ultracold fermionic atoms, in par-ticular,6Li and40K, which have been trapped and cooled viamagnetic and optical means. Typically these traps contain∼ 105 atoms at very low densities≈ 1013 cm−3 when cooledto very lowT . Here the Fermi temperatureEF in a trap can beestimated to be of the order of a micro-Kelvin. It was arguedon the basis of BCS theory alone (Houbierset al., 1997), andrather early on (1997), that the temperatures associated withthe superfluid phases may be attainable in these trapped gases.This set off a search for the “holy grail” of fermionic super-fluidity. That a Fermi degenerate state could be reached atall is itself quite remarkable; this was was first reported (De-Marco and Jin, 1999) by Jin and deMarco in 1999. By late2002 reports of unusual hydrodynamics in a degenerate Fermigas indicated that strong interactions were present (O’Haraet al., 2002). This strongly interacting Fermi gas (associatedwith the unitary scattering regime) has attracted widespreadattention independent of the search for superfluidity, becauseit appears to be a prototype for analogous systems in nuclearphysics (Baker, 1999; Heiselberg, 2004a) and in quark-gluonplasmas (Itakura, 2003; Kolb and Heinz, 2003). Moreover,there has been a fairly extensive body of analytic work on theground state properties of this regime (Carlsonet al., 1999;Heiselberg, 2001), which goes beyond the simple mean fieldwave function ansatz.

As a consequence of attractives-wave interactions betweenfermionic atoms in different hyperfine states, it was antici-pated that dimers could also be made. Indeed, these moleculesformed rather efficiently (Cubizolleset al., 2003; Regalet al.,2003; Streckeret al., 2003) as reported in mid-2003 either viathree body recombination (Jochimet al., 2003a) or by sweep-ing the magnetic field across a Feshbach resonance. More-over, they are extremely long lived (Streckeret al., 2003).From this work it was relatively straightforward to anticipatethat a Bose condensate would also be achieved. Credit goes

15

FIG. 17 (Color) Spatial density profiles of a molecular cloudoftrapped40K atoms in the BEC regime in the transverse directionsafter 20 ms of free expansion (from Greineret al., 2003), showingthermal molecular cloud aboveTc (left) and a molecular condensate(right) belowTc. (a) shows the surface plots, and (b) shows the cross-sections through images (dots) with bimodal fits (lines).

to theorists such as Holland (Hollandet al., 2001) and to Tim-mermans (Timmermanset al., 2001) and their co-workers forrecognizing that the superfluidity need not be only associatedwith condensation of long lived bosons, but in fact could alsoderive, as in BCS, from fermion pairs. In this way, it wasargued that a suitable tuning of the attractive interactionviaFeshbach resonance effects would lead to a realization of aBCS-BEC crossover.

By late 2003 to early 2004, four groups (Bourdelet al.,2004; Greineret al., 2003; Jochimet al., 2003b; Zwierleinet al., 2003) had observed the “condensation of weakly boundmolecules” (that is, on theas > 0 side of resonance), andshortly thereafter a number had also reported evidence for su-perfluidity on the BCS side (Chinet al., 2004; Kinastet al.,2004; Regalet al., 2004; Zwierleinet al., 2004). The BECside is the more straightforward since the presence of the su-perfluid is reflected in a bi-modal distribution in the densityprofile. This is shown in Fig. 17 from (Greineret al., 2003),and is conceptually similar to the behavior for condensed Boseatoms (Dalfovoet al., 1999). On the BEC side but near reso-nance, the estimatedTc is of the order of 500 nK, with conden-sate fractions varying from 20% or so, to nearly 100%. Thecondensate lifetimes are relatively long in the vicinity ofres-onance, and fall off rapidly as one goes deeper into the BEC.However, foras < 0 there is no clear expectation that the den-sity profile will provide a signature of the superfluid phase.

These claims that superfluidity may have been achieved onthe BCS side (as < 0) of resonance were viewed as partic-ularly exciting. The atomic community, for the most part,felt the previous counterpart observations on the BEC sidewere expected and not significantly different from conden-sation in Bose atoms. The evidence for this new form of“fermionic superfluidity” rests on studies (Regalet al., 2004;Zwierlein et al., 2004) that perform fast sweeps from nega-

tive as to positiveas across the resonance. The field sweepsallow, in principle, a pairwise projection of fermionic atoms(on the BCS side) onto molecules (on the BEC side). It ispresumed that in this way one measures the momentum dis-tribution of fermionic atom pairs. The existence of a con-densate was thus inferred. Other experiments which sweepacross the Feshbach resonance adiabatically, measure the sizeof the cloud after release (Bourdelet al., 2004) or within atrap (Bartensteinet al., 2004b). Evidence for superfluidityon the BCS side, which does not rely on the sweep experi-ments, has also been deduced from collective excitations ofafermionic gas (Bartensteinet al., 2004a; Kinastet al., 2004).Pairing gap measurements with radio frequency (RF) probes(Chin et al., 2004) have similarly been interpreted (Kinnunenet al., 2004a) as evidence for superfluidity, although it appearsmore likely that these experiments establish only the existenceof fermionic pairs. Quite recently, evidence for a phase tran-sition has been presented via thermodynamic measurementsand accompanying theory (Kinastet al., 2005). The latter,like the theory (Kinnunenet al., 2004a) of RF experiments(Chin et al., 2004), is based on the formalism presented inthis Review.

Precisely what goes on during the sweep is not entirely un-derstood. It should be stressed again that where the scatter-ing length changes sign is not particularly important to theN-body physics. Thus starting on the “BCS side of reso-nance” and ending on the “BEC side of resonance” involves avery continuous sweep in which the physics is not qualitativelychanged. It has been speculated that for this sweep procedureto work a large percentage of the Cooper pair partners must becloser than the inter-atomic spacing. Others have alternativelyviewed the sweep as having little to do with the large Cooperpairs of BCS superconductivity.

It is useful to establish the various pair sizes here. In thePG regime the normal state consists of a significant numberof small (compared to BCS) pre-formed pairs. BelowTc, thecondensate pair size is also small. Finally, there are excitedpair states in the superfluid phase, with characteristic sizeξpg.The Cooper pair sizeξ as plotted in Fig. 18 was deduced fromthe condensate (Randeria, 1995) atT = 0, but this can beshown to be rather similar toξpg, corresponding to the size ofexcited pair states (Chen, 2000; Chenet al., 1998) both aboveand belowTc. It seems plausible that during the sweep thereis some rearrangement of excited states (both fermionic andbosonic) and condensate. In addition, all pairs (excited andcondensate) contract in size. When, at the end of a sweep,they are sufficiently small, then they are more “visible”. Whatappears to be essential, however, is that the sweep time scalesare rapid enough so that the existence of a condensate at theinitial starting point is a necessary and sufficient condition forobserving a condensate at the end of a sweep.

G. T -Matrix-Based Approaches to BCS-BEC Crossover inthe Absence of Feshbach Effects

In this subsection we introduce the concept of aT -matrixwithin the pairing channel. This is defined in Eq. (26) be-

16

-150 -100 -50 0 50 100ν0

0

1

2

3

k Fξ

BEC PG BCS

FIG. 18 Characteristic behavior of Cooper pair sizeξ in the con-densate in the three regimes. The dotted vertical lines are not sharptransitions but indicate whereµ = 0 (left line) and where∆(Tc) ≈ 0(right line). The dimensionlesskF ξ gives the ratio between pair sizeξ and the mean inter-particle spacing1/kF . Pairs shrink and becometightly bound in the deep BEC regime.

low and it represents an effectively screened interaction inwhich the appropriate polarizability (χ) is associated with theparticle-particle rather than the particle-hole channel,as in themore conventional dielectric susceptibility. TheT -matrix thusappears as an infinite series of two particle “ladder” diagrams.

Away from zero temperature, a variety of different manybody approaches have been invoked to address the physics ofBCS-BEC crossover. For the most part, these revolve aroundT -matrix schemes. Here one solves self consistently for thesingle fermion propagator (G) and the pair propagator (t).That one stops at this level without introducing higher orderGreen’s functions (involving three, and four particles, etc) isbelieved to be adequate for addressing a leading order meanfield theory such as that represented by Eq. (1). One can seethat pair-pair (boson-boson) interactions are only treated in amean field averaging procedure; they arise exclusively fromthe fermions and are presumed to be sufficiently weak so asnot to lead to any incomplete condensation in the ground state,as is compatible with Eq. (1).

One can view this approach as the first step beyond BCS ina hierarchy of mean field theories. In BCS, aboveTc one in-cludes only the bare fermion propagatorG0. BelowTc, pairsplay a role but only through the condensate. At the next levelone accounts for the interaction between particles and noncon-densed pairs in both the normal and superconducting states.The pairs introduce a self energyΣ into the particles, whichrepresents a correction to BCS theory. The pairs are treatedat an effective mean field level. By truncating the equationsof motion in this way, the effects of all higher order Green’sfunctions are subsumed intot in an averaged way.

Below we demonstrate that at thisT -matrix level there area number of distinct schemes which can be implemented toaddress BCS-BEC crossover physics. These same alternativeshave also entered into a discussion of pre-formed pairs as theyappear in treatments of superconducting fluctuations. AboveTc, quite generally one writes for theT -matrix

t(Q) =U

1 + Uχ(Q)(26)

and theories differ only on what is the nature of the pairsusceptibilityχ(Q), and the associated self energy of thefermions. Here and throughout we useQ to denote a four-vector and write

∑

Q ≡ T∑

iΩn

∑

q, whereΩn is a Mat-subara frequency. BelowTc one can also consider aT -matrixapproach to describe the particles and pairs in the condensate.For the most part we will defer extensions to the broken sym-metry phase to Section II.B.

1. Review of BCS Theory Using the T -matrix Approach

In order to assess alternative schemes, it is useful to re-view BCS theory within aT -matrix formalism. In BCS the-ory, pairs explicitly enter into the problem belowTc, but onlythrough the condensate. These condensed pairs are associatedwith aT -matrix given by

tsc(Q) = −∆2scδ(Q)/T (27)

with fermionic self energy

Σsc(K) =∑

Q

tsc(Q)G0(Q−K) (28)

so thatΣsc(K) = −∆2scG0(−K). Here, and throughout,G0

is the Green’s function of the non-interacting system. Thenumber of fermions in a BCS superconductor is given by

n = 2∑

K

G(K) (29)

and

G(K) = [G−10 (K) − Σsc(K)]−1 (30)

Doing the Matsubara summation in Eq. (29), one can thendeduce the usual BCS expression for the number of particles,which determines the fermionic chemical potential

n =∑

k

[

1 − ǫk − µ

Ek

+ 2ǫk − µ

Ek

f(Ek)

]

(31)

where

Ek =√

(ǫk − µ)2 + ∆2sc(T ) (32)

We need, however, an additional equation to determine∆sc(T ). The BCS gap equation can be written as

1 + UχBCS(0) = 0, T ≤ Tc (33)

where

χBCS(Q) =∑

K

G(K)G0(Q−K) (34)

This suggests that one consider the uncondensed or normalstate pair propagator to be of the form

t(Q) =U

1 + UχBCS(Q)(35)

17

then in the superconducting state we have a BEC like condi-tion on the pair chemical potentialµpair defined by

t−1(Q = 0) = µpair × const. (36)

where the overall constant is unimportant for the present pur-poses. Thus we say that the pair chemical potential satisfies

µpair = 0, T ≤ Tc . (37)

Thatµpair vanishes atall T ≤ Tc is a stronger condition thanthe usual Thouless condition forTc. Moreover, it should bestressed thatBCS theory is associated with a particular asym-metric form for the pair susceptibility. These uncondensedpairs play virtually no role in BCS superconductors but thestructure of this theory points to a particular choice for a pairsusceptibility. We can then write the self consistent conditionon ∆sc which follows from Eq. (33). Using Eqs. (27- 34),we find the expected form for the BCS gap equation. Thisrepresents a constraint on the order parameter∆sc:

∆sc(T ) = −U∑

k

∆sc(T )1 − 2f(Ek)

2Ek

. (38)

The above discussion was presented in a somewhat differ-ent way in a paper by Kadanoff and Martin (Kadanoff andMartin, 1961).

2. Three Choices for the T -matrix of the Normal State:Problems with the Nozieres Schmitt-Rink Approach

On general grounds we can say that there are three obvi-ous choices forχ(Q) which appears in the general definitionof theT -matrix in Eq. (26). Each of these has been studiedrather extensively in the literature. All of these introduce cor-rections to BCS theory and all were motivated by attempts toextend the crossover ground state to finiteT , or to understandwidespread pseudogap effects in the highTc superconductors.In analogy with Gaussian fluctuations, one can consider

χ0(Q) =∑

K

G0(K)G0(Q−K) (39)

with self energy

Σ0(K) =∑

Q

t(Q)G0(Q−K) (40)

which appears inG in the analogue of Eq. (30). The numberequation is then deduced by using Eq. (29).

This scheme was adopted by Nozieres and Schmitt-Rink(NSR), although these authors (Nozieres and Schmitt-Rink,1985; Randeria, 1995) approximated the number equation(Serene, 1989) by using a leading order series forG inEq. (30) with

G = G0 + G0Σ0G0 . (41)

It is straightforward, however, to avoid this approximation inDyson’s equation, and a number of groups (Jankoet al., 1997;Pieriet al., 2004) have extended NSR in this way.

Similarly one can consider

χ(Q) =∑

K

G(K)G(Q−K) (42)

with self energy

Σ(K) =∑

Q

t(Q)G(Q−K) . (43)

This latter scheme (sometimes known as fluctuation-exchangeor FLEX) has been also extensively discussed in the lit-erature, by among others, Haussmann (Haussmann, 1993),Tchernyshyov (Tchernyshyov, 1997), Micnas and colleagues(Micnaset al., 1995) and Yamada and Yanatse (Yanaseet al.,2003).

Finally, we can contemplate the asymmetric form (Chenet al., 1998; Giovannini and Berthod, 2001) for theT -matrix,so that the coupled equations fort(Q) andG(K) are based on

χ(Q) =∑

K

G(K)G0(Q−K) (44)

with self energy

Σ(K) =∑

Q

t(Q)G0(Q−K) (45)

Each of these three schemes determines the superfluid tran-sition temperature via the Thouless condition. Thusµpair = 0leads to a slightly different expression forTc, based on the dif-ferences in the choice ofT -matrix which appears in Eq. (36).

A central problem with the NSR scheme (χ ≈ G0G0) isthat it incorporates self energy effects only through the num-ber equation. The absence of self energy effects in the gapequation is equivalent to the statement that pseudogap effectsonly indirectly affectTc: the particles acquire a self energyfrom the pairs but these self energy effects are not fed backinto the propagator for the pairs. On physical grounds one an-ticipates that the pseudogap must have a direct effect onTc

as a consequence of the associated reduction in the density ofstates. Other problems related to the thermodynamics werepointed out (Sofo and Balseiro, 1992) when NSR was appliedto a two dimensional system. Finally, because only the num-ber equation contains self energy effects, it is not clear ifonecan arrive at a proper vanishing of the superfluid densityρs atTc within this approach. This requires a precise cancellationbetween the diamagnetic current contributions (which dependon the number equation) and the paramagnetic terms (whichreflect the gap equation).

One might be inclined, then, to prefer (Serene, 1989) theFLEX (χ ≈ GG) scheme since it isφ-derivable, in the senseof Baym (Baym, 1962). This means that it is possible to writedown a closed form expression for the thermodynamical po-tential, from which one derives other physical quantities via

18

functional derivatives of this potential. Theoretical consis-tency issues in this approach have been rather exhaustivelydiscussed by Haussmann (Haussmann, 1993) aboveTc. Weare not aware of a fully self consistent calculation ofρs belowTc, at the same level of completeness as Haussmann’s normalstate analysis (or, for that matter, of the counterpart discussionto Section III.A and accompanying Appendix B). However, inprinciple, it should be possible to arrive at a proper expressionfor the superfluid density, providing Ward identities are im-posed. There is some ambiguity (Benardet al., 1993; Micnaset al., 1995; Tchernyshyov, 1997) about whether pseudogapeffects are present in the FLEX approach; a consensus has notbeen reached at this time. Numerical work based on FLEX ismore extensive (Benardet al., 1993; Deiszet al., 1998) thanfor the other two alternative schemes.

It will be made clear in what follows that, if one’s goal is toextend the usual crossover ground state of Eq. (1) to finite tem-peratures, then one must choose the asymmetric form for thepair susceptibility. Hereχ ≈ GG0. This is different from theapproach of Nozieres and Schmitt-Rink in which there is noevident consistency between their finiteT treatment and thepresumed ground state. Some have claimed (Randeria, 1995)that an NSR-based approach, when extended belowTc, repro-duces Eqs. (69) and (70). But elsewhere in the literature ithas been shown that these ground state self consistency con-ditions are derived from a lower level theory (Kostyrko andRanninger, 1996). They arise from aT = 0 limit of what iscalled the saddle point approximation, or mean field approach.When pairing fluctuations beyond the saddle point (Nozieresand Schmitt-Rink, 1985) are included, the number equation ischanged. Then the ground state is no longer that of Eq. (1)(Pieri et al., 2004). In the cold atom literature, the distinctionbetween the NSR-extrapolated ground state and the groundstate of Eq. (1) has been pointed out in the context of atomicdensity profiles (Peraliet al., 2004b) within traps.

It should be evident from Eq. (34) and the surroundingdiscussion that the asymmetric form is uniquely associatedwith the BCS-like behavior implicit in Eq. (1). The othertwoT -matrix approaches lead to different ground states whichshould, however, be very interesting in their own right. Thesewill need to be characterized in future.

Additional support for the asymmetric form, which will beadvocated here, was provided by Kadanoff and Martin. Intheir famous paper they noted that several people had sur-mised that the form forχ(Q) involvingGGwould be more ac-curate. However, as claimed in (Kadanoff and Martin, 1961),“This surmise is not correct.” Aside from theoretical counter-arguments which they present, the more symmetric combina-tion of Green’s functions “can also be rejected experimentallysince they give rise to aT 2 specific heat.” Similar arguments insupport of the asymmetric form were introduced (Hassing andWilkins, 1973) into the literature in the context of addressingspecific heat jumps.

H. Superconducting Fluctuations: a type of Pre-formedPairs

While there are no indications of bosonic degrees of free-dom, (other than in the condensate), within strict BCS the-ory, it has been possible to access these bosons via probesof superconducting fluctuations. Quasi-one dimensional, orquasi-2D superconductors in the presence of significant dis-order exhibit fluctuation effects (Larkin and Varlamov, 2001)or precursor pairing as seen in “paraconductivity”, fluctuationdiamagnetism, as well as other unusual behavior, often con-sisting of divergent contributions to transport. One frequentlycomputes (Kwon and Dorsey, 1999) these bosonic contribu-tions to transport by use of a time dependent Ginzburg-Landau(TDGL) equation of motion. This is rather similar to a Gross-Pitaevskii (GP) formalism except that the “bosons” here arehighly damped by the fermions.

Alternatively T -matrix based approaches (involving allthree choices ofχ(Q)) have been extensively used to discussconventional superconducting fluctuations. The advantageofthese latter schemes is that one can address both the anoma-lous bosonic and fermionic contributions to transport throughthe famous Aslamazov-Larkin and Maki-Thompson diagrams(Larkin and Varlamov, 2001). We defer a discussion of theseissues until Appendix B.

It is useful to demonstrate first how conventional supercon-ducting fluctuations behave at the lowest level of self con-sistency, called the Hartree approximation. This scheme isclosely associated(Stajicet al., 2003a) with aGG0 T -matrix,as is discussed in more detail in Appendix C. It is also closelyassociated with BCS theory, for one can show that, in the spiritof Eq. (1), at this Hartree level the excitation gap∆(Tc) andTc lie on a specific BCS curve (specified byn andU ). What isdifferent from strict BCS theory is that the onset of supercon-ductivity takes place in the presence of a finite excitation gap(ie, pseudogap), just as shown in Fig. 3. This, then, reflectsthe fact that there are pre-formed pairs atTc. By contrast withhighTc superconductors, however, in conventional fluctuationeffects, the temperatureT ∗ at which pairs start to form is al-ways extremely close to their condensation temperatureTc.We thus say that there is a very narrow critical region.

In Hartree approximated Ginzburg Landau theory (Hassingand Wilkins, 1973) the free energy functional is given by

F [Ψ] = a0(T−T ∗)|Ψ|2+b|Ψ|4 ≈ a0(T−T ∗)|Ψ|2+2b∆2|Ψ|2(46)

Here∆2 plays the role of a pseudogap in the normal state. Itis responsible for a lowering ofTc relative to the mean fieldvalueT ∗. Collecting the quadratic terms in the above equa-tion, it follows that

Tc = T ∗ − 2b

a0∆2(Tc) . (47)

Moreover, self consistency imposes a constraint on the mag-nitude of∆(Tc) via

∆2(T ) =

∫

DΨe−βF [Ψ]Ψ2/

∫

DΨe−βF [Ψ] . (48)

19

FIG. 19 Pseudogap in the density of states aboveTc in conventionalsuperconductors, from Cohenet al., 1969. ν1 ≡ N(E) is the nor-malized density of states.

It should be noted that Eq. (47) is consistent with the state-ment that∆(Tc) andTc lie on the BCS curve, since for smallseparation betweenTc andT ∗, this curve is, belowTc, definedby

∆2BCS(T ) ≈ a0

2b(T ∗ − T ) . (49)

The primary effect of fluctuations at the Hartree level is thatpairing takes place in the presence of a finite excitation gap,reflecting the presence of very short lived, but pre-formedpairs. It should be clear that∆ is not to be confused with thesuperconducting order parameter∆sc, as should be evidentfrom Fig. 3. We refer the reader to a more detailed discussionwhich is presented in Appendix C.

Figure 19 illustrates how this pseudogap appears experi-mentally, via a plot of the normalized density of states (Cohenet al., 1969)ν1 ≡ N(E) in the normal state. This figure de-rives from tunneling measurements on Al-based films. Notethe depression ofN(EF ) at low energies. This is among thefirst indications for a pseudogap reported in the literature.

Alternatively, fluctuations have also been discussed at theGaussian (G0G0) level in which there is no need for self con-sistency, in contrast to the above picture. Here because thecalculations are so much more tractable there have been verydetailed applications (Larkin and Varlamov, 2001) of essen-tially all transport and thermodynamic measurements. ThisGaussian analysis led to unexpected divergences in the para-conductivity from the so-called Maki-Thompson term whichthen provided a motivation to go beyond leading order theory.Patton (Patton, 1971a,b) showed how this divergence could beremoved within a more self consistentGG0 scheme, equiva-

lent to self consistent Hartree theory. Others argued (Schmid,1969; Tucker and Halperin, 1971) that Hartree-Fock (GG)was more appropriate, although in this weak coupling, nar-row fluctuation regime, the differences between these lattertwo schemes are only associated with factors of 2. Neverthe-less with this factor of 2,∆(Tc) andTc are no longer on theBCS curve. It should be noted that a consensus was neverfully reached (Hassing and Wilkins, 1973) by the communityon this point.

1. Dynamics of Pre-formed Pairs Above Tc: Time DependentGinzburg-Landau Theory

We turn again to Fig. 5 which provides a useful startingpoint for addressing bosonic degrees of freedom and their im-plications for experiment. It is clear that transport in theun-usual normal state, associated with the pseudogap (PG) phase(as well as thermodynamics) contains contributions from bothfermionic and bosonic excitations. The bosons are not in-finitely long lived; their lifetime is governed by their inter-actions with the fermions. In some instances (Larkin and Var-lamov, 2001), the bosonic contributions to transport becomedominant or even singular atTc. Under these circumstancesone can ignore the fermionic contributions except insofar asthey lead to a lifetime for the pairs.

To address the transport properties of these more well es-tablished pre-formed pairs, one appeals to a standard way ofcharacterizing the dynamics of bosons – a phenomenologicalscheme known as time dependent Ginsburg- Landau (TDGL)theory. This theory has some microscopic foundations in dia-grammaticT -matrix approaches in which one considers onlythe Aslamazov-Larkin terms (which are introduced in Ap-pendix B). At the Gaussian level both TDGL and theT -matrixapproaches are tractable. At the Hartree level things rapidlybecome more complicated, and it is far easier to approach theproblem (Tan and Levin, 2004) by adopting a strictly phe-nomenological TDGL.

The generic Hartree-TDGL equation of motion for classicalcharged bosons interacting with electromagnetic fields (φ,A)is given by

γ

(

i∂

∂t−e∗φ(x, t)

)

ψ(x, t) =

(

−i∇− e∗A(x, t))2

2M∗ψ(x, t)

− µpair(T )ψ(x, t) +D(x, t), (50)

Here ψ(x, t) is a classical field variable representing thebosons which have vanishing chemical potentialµpair at Tc.The functionD(x, t) introduces a noise variable into the equa-tion of motion. Generallyγ is complex.

We will explore the implications of this equation in ourmore quantitative discussion of Section VI.E.

20

II. QUANTITATIVE DETAILS OF CROSSOVER

A. T = 0, BEC Limit without Feshbach Bosons

We begin by reviewingT = 0 crossover theory in the BEClimit. Our starting point is the ground state wavefunctionΨ0 of Eq. (1), along with the self consistency conditions ofEq. (2). For positive chemical potentialµ, the quantity∆sc(0)also corresponds to the energy gap for fermionic excitations.In the ground state the two energy scales∆(0) and∆sc(0) aredegenerate, just as they are (at all temperatures) in strictBCStheory.

It is convenient to rewrite these equations5 in terms of theinter-fermion scattering lengthas

m

4πas=∑

k

[

1

2ǫk− 1

2Ek

]

, (51)

n =∑

k

[

1 − ǫk − µ

Ek

]

, T = 0 . (52)

In the fermionic regime (µ > 0) these equations are essen-tially equivalent to those of BCS theory, although at weakcoupling appropriate to strict BCS, little attention is paid tothe number equation sinceµ = EF is always satisfied. Themore interesting regime corresponds toµ ≤ 0 where theseequations take on a new interpretation. Deep inside the BECregime it can be seen that

n = ∆2sc(0)

m2

4π√

2m|µ|, (53)

which, in conjunction with Eq. (51) (expanded in powers of∆2

sc(0)/µ2) :

m

4πas= (2m)3/2

√

|µ|8π

[

1 +1

16

∆2sc(0)

µ2

]

, (54)

yields

µ = − 1

2ma2s

+asπn

m. (55)

This last equation is equivalent (Peraliet al., 2003; Viveritet al., 2004) to its counterpart in Gross-Pitaevskii theory. Thistheory describes true bosons, and is associated with the wellknown equation of state

npairs =mB

4πaBµB . (56)

5 To make the equations simpler, in much of what we present in this re-view we will consider a contact potential interaction. The momentumsums are assumed to run to infinity. More generally, one can include acutoff function ϕk for the momentum integrals. The gap function thusbecomes∆k = ∆ϕk. For the cuprated-wave cuprate superconductors,ϕk = cos kxa − cos kya. For our numerical work on atomic gases we

use a Gaussian cutoffϕ2k

= e−k2/k2

0 with k0 taken to be as large as isnumerically feasible.

To see the equivalence we associate the number of bosonsnpairs = n/2, the boson massmB = 2m and the bosonicscattering lengthaB = 2as. Here the bosonic chemical po-tentialµB = 2µ + ǫ0 and we defineǫ0 = 1/(2ma2

s). Thisfactor of 2 inaB/as has received a fair amount of attentionin the literature. Although it will be discussed later in theReview, it should be noted now that this particular numericalvalue is particular to the one channel description(Stajicet al.,2005b), and different from what is observed experimentally(Petrovet al., 2004).

Despite these similarities with GP theory, the fundamentalparameters are the fermionic∆sc(0) and chemical potentialµ. It can be shown that in this deep BEC regime the numberof pairs is directly proportional to the superconducting orderparameter

npairs =n

2= Z0∆

2sc(0) (57)

where

Z0 ≈ m2as

8π. (58)

One may note from Eq. (57) that the “gap equation” now cor-responds to a number equation (for bosons). Similarly thenumber equation, or constraint on the fermionic chemical po-tential defines the excitation gap for fermions, once the chem-ical potential is negative. In this way the roles of the twoconstraints are inverted (Randeria, 1995) relative to the BCSregime.

That a Gross-Pitaevskii approach captures the leading or-der physics atT = 0 can also be simply seen by rewriting theground state wavefunction, as pointed out by Randeria (Ran-deria, 1995). Definevk/uk ≡ ηk

Ψ0 = const× Πk(1 + ηkc†kc

†−k)|0〉

= const× exp

(

∑

k

ηkc†kc

†−k

)

|0〉 . (59)

Projecting onto a state with fixed particle numberN yields

Ψ0 = const×(

∑

k

ηkc†kc

†−k

)N/2

|0〉 . (60)

This is a GP wavefunction of composite bosons with the im-portant proviso: that the characteristic size associated withthe internal wavefunctionηk is smaller than the inter-particlespacing. It should be stressed that, except in the BEC asymp-totic limit, (where the “bosons” are non-interacting) thereare essential differences between Eq. (60) and the Gross-Pitaevskii wavefunction. Away from this extreme limit, pair-pair interactions (arising indirectly via the fermions andthePauli principle), as well as the finite size of the pairs, willdifferentiate this system from that of true interacting bosons.This same issue arises more concretely in the context of thecollective mode spectrum of ultracold gases, as discussed inSection V.C.

21

B. Extending conventional Crossover Ground State toT 6= 0: BEC limit without Feshbach Bosons

How do we extend (Chenet al., 1998) this picture to fi-nite T? In the BEC limit, as shown in Fig. 4, fermion pairsare well established or “pre-formed” within the entire rangeof superconducting temperatures. The fundamental constraintassociated with the BEC regime is that:for all T ≤ Tc, thereshould, thus, be no temperature dependence in fermionic en-ergy scales. In this way Eqs. (51) and (52) must be imposedat all temperaturesT .

m

4πas=∑

k

[

1

2ǫk− 1

2Ek

]

, (61)

n =∑

k

[

1 − ǫk − µ

Ek

]

, T ≤ Tc . (62)

It follows that the number of pairs atT = 0 should be equalto the number of pairs atT = Tc. However, all pairs are con-densed atT = 0. Clearly, the character of these pairs changesso that atTc, all pairs are noncondensed. To implement thesephysical constraints (and to anticipate the results of a moremicroscopic theory) we write

npairs =n

2= Z0∆

2 (63)

npairs = ncondensedpairs (T ) + nnoncondensed

pairs (T ) (64)

so that we may decompose the excitation gap into two contri-butions

∆2 = ∆2sc(T ) + ∆2

pg(T ) (65)

where∆sc(T ) corresponds to condensed and∆pg(T ) to thenoncondensed gap component. Each of these are proportionalto the respective number of condensed and noncondensedpairs with proportionality constantZ0. At Tc,

nnoncondensedpairs =

n

2=∑

q

b(Ωq, Tc) , (66)

whereb(x) is the usual Bose-Einstein function andΩq is theunknown dispersion of the noncondensed pairs. Thus

∆2pg(Tc) = Z−1

0

∑

b(Ωq, Tc) =n

2Z−1

0 . (67)

We may deduce directly from Eq. (67) that∆2pg =

−∑Q t(Q), if we presume that belowTc, the noncondensedpairs have propagator

t(Q) =Z−1

0

Ω − Ωq. (68)

It is important to stress that the dispersion of the pairsΩq

cannot be put in by hand. It is not known a priori. Rather,it has be toderivedaccording to the constraints imposed byEqs. (61) and (62). We can only arrive at an evaluation ofΩq after establishing the nature of the appropriateT -matrix

theory. In what follows we derive this dispersion using thetwo channel model. We state the one channel results herefor completeness. In the absence of FB,Ωq ≡ B0q

2, whereB0 ≡ 1/(2M∗

0 ). As a consequence of a “bosonic” self en-ergy, which arises from fermion-boson (as distinguished fromboson-boson) interactions, the bosonic dispersion is found tobe quadratic. This quadratic dependence persists for the twochannel case as well.

C. Extending Conventional Crossover Ground State toT 6= 0: T -matrix Scheme in the Presence of FeshbachBosons

To arrive at the pair dispersion for the noncondensed pairs,Ωq, we need to formulate a generalizedT -matrix basedscheme which is consistent with the ground state conditions,and with theT dependence of strict BCS theory. It is usefulfrom a pedagogical point to now include the effects of Fes-hbach bosons (Stajicet al., 2004). Our intuition concerninghow true bosons condense is much better than our intuitionconcerning the condensation of fermion pairs, except in thevery limited BCS regime. We may assume that∆ and µevolve with temperature in such a way as to be compatiblewith both the temperature dependences of BCS and with theabove discussion for the BEC limit. We thus take

∆(T ) = −Ueff

∑

k

∆(T )1 − 2f(Ek)

2Ek

, (69)

n =∑

k

[

1 − ǫk − µ

Ek

+ 2ǫk − µ

Ek

f(Ek)

]

, (70)

whereEk =√

(ǫk − µ)2 + ∆2(T ). Note that in the pres-ence of Feshbach bosons, we usen to denote the density offermionic atoms which is to be distinguished from the con-stituents of the Feshbach bosons. The total density of particleswill be denoted byntot. Equations (69) and (70) will play acentral role in this review. They have been frequently invokedin the literature, albeit under the presumption that there is nodistinction between∆(T ) and∆sc(T ).

Alternatively one can rewrite Eq. (69) as

m

4πa∗s=∑

k

[

1

2ǫk− 1 − 2f(Ek)

2Ek

]

. (71)

Clearly Eqs. (69) and (70) are consistent with Eqs. (61) and(62) since Fermi functions are effectively zero in the BEClimit. Our task is to find aT -matrix formalism which is com-patible with Eqs. (69) and (70), and to do this we focus first onthenoncondensedmolecular bosons. Their propagator may bewritten as

D(Q) ≡ 1

iΩn − ǫmbq − ν + 2µ− ΣB(Q)

. (72)

We presume that the self energy of these molecules can bewritten in the form

ΣB(Q) ≡ −g2χ(Q)/[1 + Uχ(Q)] , (73)

22

whereχ(Q) is as yet unspecified. This RPA-like self energyarises from interactions between the molecular bosons and thefermion pairs, and this particular form is required for self-consistency.

noncondensed bosons in equilibrium with a condensatemust necessarily have zero chemical potential.

µboson(T ) = 0, T ≤ Tc. (74)

This is equivalent to the Hugenholtz-Pines condition that

D−1(0) = 0, T ≤ Tc. (75)

Using Eqs. (73) and (75) it can be seen that

U−1eff (0) + χ(0) = 0, T ≤ Tc . (76)

This equation can be made compatible with our fundamentalconstraint in Eq. (69) provided we take

χ(Q) =∑

K

G(K)G0(Q−K) , (77)

whereG(K)

G(K) ≡ [G−10 (K) − Σ(K)]−1 (78)

includes a self energy given by the BCS-form

Σ(K) = −G0(−K)∆2 , (79)

which now involves the quantity∆ to be distinguished fromthe order parameter∆sc. Note that with this form forΣ(K),the number of fermions is indeed given by Eq. (70).

More generally, we may write the constraint on the totalnumber of particles as follows. The number of noncondensedmolecular bosons is given directly by

nb(T ) = −∑

Q

D(Q) . (80)

For T ≤ Tc, the number of fermions is given via Eq. (70).Thetotal number (ntot) of particles is then

n+ 2nb + 2n0b = ntot , (81)

wheren0b = φ2

m is the number of molecular bosons in thecondensate, andφm is the order parameter for the molecularcondensate.

Thus far, we have shown that the condition that noncon-densed molecular bosons have zero chemical potential, canbe made consistent with Eqs. (69) and (70) provided we con-strainχ(Q) andΣ(K) as above. We now want to examinethe counterpart condition on the fermions and their conden-sate contribution. The analysis leading up to this point shouldmake it clearthat we have both condensed and noncondensedfermion pairs, just as we have both condensed and noncon-densed molecular bosons. Moreover, these fermion pairs andFeshbach bosons are strongly admixed, except in the very ex-treme BCS and BEC limits. Because of noncondensed pairs,

= +

++

U

G

D

G

g g0

0

t

t

t

FIG. 20 Diagrammatic scheme for presentT -matrix theory. HereUis the open-channel interaction,g is the coupling between open andclosed channels.G0, G andD0 are bare fermion, full fermion, andbare Feshbach boson Green’s functions, respectively.U andg2

0D0

yield an effective pairing interaction,Ueff .

we will see that the excitation gap is distinct from the super-conducting order parameter, although in the literature this dis-tinction has not been widely recognized.

Just as the noncondensed molecular bosons have zerochemical potential belowTc we have the same constraint onthe noncondensed fermion pairs which are in chemical equi-librium with the noncondensed bosons

µpair = 0, T ≤ Tc . (82)

TheT -matrix scheme in the presence of the Feshbach bosonsis shown in Fig. 20. Quite generally, theT -matrix consists oftwo contributions: from the condensed (sc) and noncondensedor “pseudogap”-associated (pg) pairs,

t = tpg + tsc , (83)

tpg(Q) =Ueff (Q)

1 + Ueff (Q)χ(Q), Q 6= 0 , (84)

tsc(Q) = − ∆2sc

Tδ(Q) , (85)

where we write ∆sc = ∆sc − gφm, with ∆sc =−U

∑

k〈a−k↓ak↑〉 andφm = 〈bq=0〉. Here, the order pa-rameter is a linear combination of both paired fermions andcondensed molecules. Similarly, we have two contributionsfor the fermion self energy

Σ(K) = Σsc(K) + Σpg(K) =∑

Q

t(Q)G0(Q−K) , (86)

where, as in BCS theory,

Σsc(K) =∑

Q

tsc(Q)G0(Q−K) . (87)

Without loss of generality, we choose order parameters∆sc

andφm to be real and positive withg < 0. Importantly, thesetwo components are connected (Kokkelmanset al., 2002) bythe relationφm = g∆sc/[(ν − 2µ)U ].

The vanishing of the pair chemical potential implies that

t−1pg (0) = U−1

eff (0) + χ(0) = 0, T ≤ Tc . (88)

This same equation was derived from consideration of thebosonic chemical potential. Most importantly, we argued

23

above that this was consistent with Eq. (69) provided thefermion self energy assumes the BCS form. We now verifythe assumption in Eq. (79). A vanishing chemical potentialmeans thattpg(Q) is strongly peaked aroundQ = 0. Thus,we may approximate (Malyet al., 1999a) Eq. (86) to yield

Σ(K) ≈ −G0(−K)∆2 , (89)

where

∆2(T ) ≡ ∆2sc(T ) + ∆2

pg(T ) , (90)

and we define the pseudogap∆pg

∆2pg ≡ −

∑

Q6=0

tpg(Q). (91)

This bears a close resemblance to Eq. (80), if we write thenumber of noncondensed fermion pairs and molecular bosonsas

np(T ) = Z∆2pg(T ) . (92)

Note that in the normal state (whereµpair is nonzero) onecannot make the approximation of Eq. (89). Referring back toour discussion of Hartree-approximated TDGL, a strong anal-ogy between Eq. (91) and (48) should be observed. There isa similar analogy between Eq. (69) and (49); more details areprovided in Appendix C. We thus have that Eqs. (51) and (89)with Eq. (78) are alternative ways of writing Eqs. (69) and(70). Along with Eq. (91) we now have a closed set of equa-tions for addressing the ordered phase. Moreover the propa-gator for noncondensed pairs can now be quantified, using theself consistently determined pair susceptibility. At moderatelystrong coupling and at small four-vectorQ, we may expand toobtain

tpg(Q) =Z−1

Ω − Ωq + µpair + iΓQ, (93)

Consequently, one can rewrite Eq. (91) as

∆2pg(T ) = Z−1

∑

b(Ωq, T ) . (94)

Here we introduce a simple notation for the smallQ expan-sions:

χ(Q) − χ(0) = Z0(iΩn −B0q2),

U−1eff (Q) − U−1

eff (0) = Zg(iΩn −Bgq2)

with

Z0 =1

2∆2

[

n− 2∑

k

f(ǫk − µ)

]

,

Zg =g2

[(2µ− ν)U + g2]2.

We have thatZ = Z0 + Zg, Zb = (Z0 + Zg)/Z0, and theq2

coefficientB ≡ 1/(2M∗) in Ωq ≡ Bq2 is such that

B =B0Z0 + 1

2MZg

Z.

The strong hybridization between the molecular bosons andfermion pairs should be stressed. Indeed, there is just onebranchΩq for bosonic-like excitations, so that at smallQ

D(Q) =Z−1

b

Ω − Ωq + µboson + iΓQ, (95)

where belowTc, µpair = µboson = 0. HereΓQ containsall the imaginary part of the inverseT -matrix and it vanishesrapidly asQ → 0. As the coupling is decreased from theBEC limit towards BCS, the character of the pairs changesfrom pre-dominantly molecular-boson like to (noncondensed)Cooper pairs, composed only of the fermionic atoms.

D. Nature of the Pair Dispersion: Size and Lifetime ofnoncondensed Pairs Below Tc

We may rewrite the pair susceptibility (Chenet al., 1998,1999) of Eq. (77) (after performing the Matsubara sum andanalytically continuing to the real axis) in a relatively simpleform as

χ(Q) =∑

k

[ 1 − f(Ek) − f(ξk−q)

Ek + ξk−q − Ω − i0+u2k

− f(Ek) − f(ξk−q)

Ek − ξk−q + Ω + i0+v2k

]

, (96)

whereu2k andv2

k are given by their usual BCS expressions interms of∆ andξk ≡ ǫk − µ. In the long wavelength, lowfrequency limit, the inverse oftpg can be written as

a1Ω2 + Z0(Ω − q2

2M∗+ µpair + iΓQ). (97)

We are interested in the moderate and strong couplingcases, where we can drop thea1Ω

2 term in Eq. (97), and hencewe have Eq. (93) with

Ωq ≡ q2

2M∗=q2ξ2pg

Z0. (98)

This establishes a quadratic dispersion and defines the effec-tive pair mass,M∗. Analytical expressions for this mass arepossible via a smallq expansion ofχ, in Eq. (96). It is impor-tant to note that the pair mass reflects the effectivesizeξpg ofnoncondensed pairs. This serves to emphasize the fact that theq2 dispersion derives from the compositeness of the “bosons”,in the sense of their finite spatial extent. A description of thesystem away from the BEC limit must accommodate the factthat the pairs have an underlying fermionic character. Thispair mass has a different origin from the mass renormaliza-tion associated with real interacting bosons. There one finds amass shift which comes from a Hartree approximate treatmentof their q-dependent interactions. Finally, we note thatξpg iscomparable to the sizeξ of pairs in the condensate. In theweak coupling BCS limit, a smallQ expansion ofχ0 showsthat, because the leading order term inΩ is purely imaginary,theΩ2 contribution cannot be neglected. TheT -matrix, then,does not have the propagatingq2 dispersion, discussed above.

24

-300 -200 -100 0 100 200ν0/EF

0

0.1

0.2

0.3

Tc/T

F

BEC PG BCS

FIG. 21 Typical behavior ofTc in the homogeneous case. For thepurposes of illustration, this calculation includes Feshbach effects.Tc follows BCS for large magnetic detuning parameterν0 and ap-proaches the BEC asymptote0.218 for large negativeν0. It reaches amaximum at unitarity, and has a minimum when the fermionic chem-ical potentialµ changes sign.

It follows from Eq. (96) that the pair lifetime

ΓQ =π

Z0

∑

k

[1 − f(Ek) − f(ξk−q)] u2kδ(Ek + ξk−q − Ω)

+ [f(Ek) − f(ξk−q)] v2kδ(Ek − ξk−q + Ω) . (99)

HereΓQ reflects the rate of decay of noncondensed bosonsinto a bare and dressed fermion. Note that the excitation gapin Ek significantly restricts the contribution from theδ func-tions. Thus,the decay rate of pair excitations is greatly sup-pressed, due to the excitation gap (for fermions) in the super-conducting phase. Even in the normal phase pairs live longerthan one might have anticipated from “Pauli blocking” argu-ments which are based on evaluatingΓQ in a Fermi liquidstate. However, the same equations are not strictly valid aboveTc, because Eq. (89) no longer holds. To compute the pair life-time in the normal state requires a more extensive calculationinvolving the fullT -matrix self consistent equations and theirnumerical solution (Jankoet al., 1997; Malyet al., 1999a,b).

E. Tc Calculations: Analytics and Numerics

Calculations ofTc can be performed using Eqs. (69) and(70) along with Eq. (91). TheT -matrix is written in the ex-panded form of Eq. (93), which is based on the pair dispersionas derived in the previous section. For the most part the cal-culations proceed numerically.

A typical curve is plotted in Fig. 21, where the threeregimes BCS, BEC and PG are indicated. For positive, but de-creasing the magnetic detuningν0, Tc follows the BCS curveuntil the “pseudogap” or∆(Tc) becomes appreciable. Afterits maximum (near the unitary limit, at slightly negativeν0),Tc decreases, (as doesµ), to reach a minimum atµ ≈ 0. Thisdecrease inTc reflects the decreasing number of low energyfermions due to the opening of a pseudogap. Beyond thispoint, towards negativeν0, the system is effectively bosonic,and the superconductivity is no longer hampered by pseudo-gap effects. In the presence of FB, the condensate consists

of two contributions, although the weight of the fermion paircomponent rapidly disappears. SimilarlyTc rises, althoughslowly, towards the ideal BEC asymptote, following the in-verse effective boson mass. The corresponding curve basedon the NSR approach (Nozieres and Schmitt-Rink, 1985) hasonly one extremum, but nevertheless the overall magnitudesare not so different (Milsteinet al., 2002; Ohashi and Griffin,2002).

Analytic results are obtainable in the near-BEC limit only.The general expression for the effective mass1/M∗

0 in thislimit is given by

1

M∗0

=1

Z0∆2

∑

k

[

1

mv2k − 4Ekk

2

3m2∆2v4k

]

, (100)

where here FB effects have been dropped for simplicity. Inwhat follows, we expand Eq. (100) in powers ofna3

s and ob-tain after some algebra

M∗0 ≈ 2m

(

1 +πa3

sn

2

)

. (101)

We now invoke an important constraint, derived earlier inEq. (66) which corresponds to the fact that atTc all fermionsare constituents of uncondensed pairs

n

2=∑

q

b(Ωq, Tc) . (102)

From the above equation it follows that(M∗Tc)3/2 ∝ n =

const. which, in conjunction with Eq. (101) implies

Tc − T 0c

T 0c

= −πa3sn

2. (103)

HereT 0c is the transition temperature of the ideal Bose gas

with M0 = 2m. This downward shift ofTc follows the ef-fective mass renormalization, much as expected in a Hartreetreatment of GP theory atTc. Here, however, in contrast toGP theory for a homogeneous system with a contact potential(Dalfovo et al., 1999), there is a non-vanishing renormaliza-tion of the effective mass. This is a key point which under-lines the importance in this approach of the fermionic degreesof freedom, even at very strong coupling.

III. SELF CONSISTENCY TESTS

All T -matrix approaches to the BCS-BEC crossover prob-lem must be subject to self consistency tests. Among these,one should demonstrate thatnormal stateself energy effectswithin a T -matrix scheme are not associated with superflu-idity or superconductivity (Kosztinet al., 1998). While thisseems at first sight straightforward, all theories should beputto this test. Thus for a charged system, a meaningful resultfor the superfluid densityρs requires that there be an exactcancellation between diamagnetic and paramagnetic currentcontributions atTc. In this way self energy effects in the num-ber equation and gap equation must be treated in a consistent

25

fashion. In the following we address some of these self con-sistency issues in the context of the asymmetric (χ ≈ GG0)BCS-BEC crossover scheme. Other crossover theories withdifferent forms for the pair susceptibility should also be sub-jected to these tests. We begin with a summary of calculations(Kosztinet al., 2000; Stajicet al., 2003a) of the Meissner ef-fect, followed by a treatment of the collective mode spectrum.Our discussion in this section is presented for the one channelproblem, in the absence of Feshbach effects.

A. Important check: behavior of ρs

The superfluid density may be expressed in terms of thelocal (static) electromagnetic response kernelK(0)

ns =m

e2K(0) = n− m

3 e2Pαα(0) , (104)

with the current-current correlation function given by

Pαβ(Q) = −2 e2∑

K

λα(K,K +Q)G(K +Q)

×ΛEMβ (K +Q,K)G(K) . (105)

Here the bare vertexλ(K,K + Q) = 1m (k + q/2) and we

considerQ = (q, 0), with q → 0. We writeΛEM = λ +δΛpg + δΛsc, where the pseudogap contributionδΛpg to thevertex correction will be shown in Appendix B to satisfy aWard identity belowTc

δΛpg(K,K) =∂Σpg(K)

∂k. (106)

By contrast for the superconducting contributions, one has

δΛsc(K,K) = −∂Σsc(K)

∂k. (107)

This important difference in sign is responsible for the factthat the Meissner effect is associated with superconductivity,and not with a normal state self energy.

The particle densityn, after partial integration can berewritten asn = −(2/3)

∑

K k · ∂kG(K). Then, as a re-sult of Dyson’s equation, one arrives at the following generalexpression which relates to the diamagnetic contribution

n = −2

3

∑

K

G2(K)

[

k2

m+ k · ∂Σpg(K)

∂k+ k · ∂Σsc(K)

∂k

]

.

(108)Now, inserting Eqs. (108) and (105) into Eq. (104) one cansee that the pseudogap contribution tons drops out by virtueof Eq. (106); we find

ns =2

3

∑

K

G2(K)k ·(

δΛsc −∂Σsc

∂k

)

. (109)

We emphasize that the cancellation of this pseudogap con-tribution to the Meissner effect is the central physics of this

analysis, and it depends on treating self energy effects in aWard-identity- consistent fashion.

We also have that

δΛsc(K+Q,K) = ∆2scG0(−K−Q)G0(−K)λ(K+Q,K) .

(110)Inserting Eqs. (87), (89), and (110) into Eq. (109), after cal-culating the Matsubara sum, one arrives at

ns =4

3

∑

k

∆2sc

E2k

ǫk

[

1 − 2 f(Ek)

2Ek

+ f ′(Ek)

]

. (111)

This expression can be simply rewritten in terms of the BCSresult for the superfluid density

(ns

m

)

=∆2

sc

∆2

(ns

m

)BCS

. (112)

Here (ns/m)BCS is just (ns/m) with the overall prefactor∆2

sc replaced with∆2.Finally we can rewrite Eq. (112) using Eq. (90) as

(ns

m

)

=

[

1 −∆2

pg

∆2

]

(ns

m

)BCS

. (113)

In this form it is evident that (via∆2pg) pair excitations out of

the condensate are responsible for a suppression of the super-fluid density relative to that obtained from fermionic excita-tions only (Stajicet al., 2003b).

B. Collective Modes and Gauge Invariance

The presence of pseudogap self energy effects greatly com-plicates the computation of collective modes (Kosztinet al.,2000). This is particularly apparent at nonzero temperature.Once dressed Green’s functionsG enter into the calculationalschemes, the collective mode polarizabilities and the elec-tromagnetic (EM) response tensor must necessarily includevertex corrections dictated by the form of the self-energyΣ,which depends on theT-matrix which, in turn depends on theform of the pair susceptibilityχ. These necessary vertex cor-rections are associated with gauge invariance in the same way,as was seen forρs, and discussed in Appendix B. Collectivemodes are important in their own right, particularly in neutralsuperfluids, where they can be directly detected as signaturesof long range order. They also must be invoked to arrive ata gauge invariant formulation of electrodynamics. It is rela-tively straightforward to introduce these collective modeef-fects into the electromagnetic response in a completely gen-eral fashion that is required by gauge invariance. The diffi-culty is in the implementation.

In the presence of a weak externally applied EM field, withfour-vector potentialAµ = (φ,A), the four-current densityJµ = (ρ,J) is given by

Jµ(Q) = Kµν(Q)Aν(Q) , (114)

whereKµν(Q) is the EM response kernel.

26

The incorporation of gauge invariance into a general mi-croscopic theory may be implemented in several ways. Herewe do so via a generalized matrix Kubo formula (Kuliket al.,1981) in which the perturbation of the condensate is includedas additional contributions∆1 + i∆2 to the applied externalfield. These contributions are self consistently obtained (byusing the gap equation) and then eliminated from the final ex-pression forKµν . We now implement this procedure. Letη1,2 denote the change in the expectation value of the pair-ing field η1,2 corresponding to∆1,2. For the case of ans-wave pairing interactionU < 0, the self-consistency condi-tion ∆1,2 = Uη1,2/2 leads to the following equations:

Jµ = KµνAν = Kµν0 Aν + Rµ1∆1 +Rµ2∆2 ,(115a)

η1 = −2∆1

|U | = R1νAν +Q11∆1 +Q12∆2 , (115b)

η2 = −2∆2

|U | = R2νAν +Q21∆1 +Q22∆2 , (115c)

where

Kµν0 (ω,q) = Pµν(ω,q) +

ne2

mgµν(1 − gµ0) (116)

is the usual Kubo expression for the electromagnetic re-sponse. We define the current-current correlation functionPµν(τ,q) = −iθ(τ)〈[jµ(τ,q), jν(0,−q)]〉. In the aboveequation,gµν is a (diagonal) metric tensor with elements(1,−1,−1,−1). We define

Rµi(τ,q) = −iθ(τ)〈[jµ(τ,q), ηi(0,−q)]〉 (117)

with µ = 0, . . . , 3, andi, j = 1, 2; and

Qij(τ,q) = −iθ(τ)〈[ηi(τ,q), ηj(0,−q)]〉 (118)

Finally, it is convenient to define

Qii = 2/U +Qii . (119)

In order to demonstrate gauge invariance and reduce thenumber of component polarizabilities, we first rewriteKµν ina way which incorporates the effects of the amplitude contri-butions via a renormalization of the relevant generalized po-larizabilities,

K ′µν0 = Kµν

0 − Rµ1R1ν

Q11

, (120)

It can be shown, after some analysis, that the gauge invari-ant form for the response tensor is given by

Kµν = K ′µν0 −

(

K ′µν′

0 qν′

)(

qν′′K ′ν′′ν

0

)

qµ′K ′µ′ν′

0 qν′

. (121)

The above equation satisfies two important requirements: itis manifestly gauge invariant and, moreover, it has been re-duced to a form that depends principally on the four-current-current correlation functions. (The word “principally” appears

because in the absence of particle-hole symmetry, there areeffects associated with the order parameter amplitude contri-butions that add to the complexity of the calculations).

The EM response kernel of a superconductor contains apole structure that is related to the underlying Goldstone bo-son of the system. Unlike the phase mode component of thecollective mode spectrum, this Anderson-Bogoliubov (AB)mode is independent of Coulomb effects. The dispersion ofthis amplitude renormalized AB mode is given by

qµK′µν0 qν = 0 . (122)

For an isotropic systemK ′αβ0 = K ′11

0 δαβ , and Eq. (122) canbe rewritten as

ω2K ′000 + q2K ′11

0 − 2ωqαK′0α0 = 0 , (123)

with α = 1, 2, 3, and in the last term on the left hand sideof Eq. (123) a summation over repeated Greek indices is as-sumed.

It might seem surprising that from an analysis which incor-porates a complicated matrix linearized response approach,the dispersion of the AB mode ultimately involves onlythe amplitude renormalized four-current correlation func-tions, namely the density-density, current-current and density-current correlation functions. The simplicity of this result is,nevertheless, a consequence of gauge invariance.

At zero temperatureK ′0α0 vanishes, and the sound-like AB

mode has the usual linear dispersionω = ωq = c|q| with the“sound velocity” given by

c2 = K ′110 /K

′000 . (124)

We may, thus, interpret the AB mode as a special type of col-lective mode which is associated withAν = 0. This mode cor-responds to free oscillations of∆1,2 with a dispersionω = cqgiven by the solution to the equation

det|Qij | = Q11Q22 −Q12Q21 = 0 . (125)

The other collective modes of this system are derived by in-cluding the coupling to electromagnetic fields. Within self-consistent linear response theory the fieldδφ must be treatedon an equal footing with∆1,2 and formally can be incorpo-rated into the linear response of the system by adding an extratermKµ0

0 δφ to the right hand side of Eqs. (115a), (115b), and(115c). Note that, quite generally, the effect of the “externalfield” δφ amounts to replacing the scalar potentialA0 = φ byA0 = φ = φ+ δφ. In this way one arrives at the following setof three linear, homogeneous equations for the unknownsδφ,∆1, and∆2

0 = R10δφ+ Q11∆1 +Q12∆2 , (126a)

0 = R20δφ+Q21∆1 + Q22∆2 , (126b)

δρ =δφ

V= K00

0 δφ+R01∆1 +R02∆2 , (126c)

whereV is an effective particle-hole interaction which mayderive from the pairing channel or, in a charged superconduc-tor, from the Coulomb interaction. The dispersion of the col-lective modes of the system is given by the condition that the

27

0 0.2 0.4 0.6 0.8 1T/Tc

0

0.5

1R

e, Im

c(T)

/c(0

) Rec

Im c

U/Uc = 0.9BCS (U/Uc = 0.5)

FIG. 22 Temperature dependence of the real and imaginary parts ofthe AB collective mode velocity for moderate (solid lines) and weak(dashed lines) coupling. The velocity vanishes atTc for both cases.

above equations have a nontrivial solution

∣

∣

∣

∣

∣

∣

Q11 + 1/U Q12 R10

Q21 Q22 + 1/U R20

R01 R02 K000 − 1/V

∣

∣

∣

∣

∣

∣

= 0 . (127)

In the BCS limit where there is particle-hole symmetryQ12 =Q21 = R10 = R01 = 0 and, the amplitude mode decouplesfrom the phase and density modes; the latter two are, however,in general coupled.

The above formalism (or its equivalent RPA variations(Belkhir and Randeria, 1992; Cote and Griffin, 1993)) hasbeen applied to address collective modes in the crossover sce-nario. The most extensive studies have been atT = 0 basedon the ground state of Eq. (1). There one finds a smoothchange in the character of the Anderson-Bogoliubov (AB)mode. At weak coupling one obtains the usual BCS valuec = vF /

√3. By contrast at strong coupling, the collective

mode spectrum reflects (Belkhir and Randeria, 1992) an effec-tive boson-boson interaction deriving from the Pauli statisticsof the constituent fermions. This is most clearly seen in jel-lium models (Griffin and Ohashi, 2003; Kosztinet al., 2000)where the AB sound velocity is equivalent to that predictedfor a 3D interacting Bose gasc = [4πnaB/m

2B]1/2. Here,

as earlier, the inter-boson scattering length is twice thatof theinter-fermionic counterpart, at strong coupling.6

The effects of finite temperature on the AB mode have beenstudied in (Kosztinet al., 2000), by making the approxima-tion that the temperature dependence of the order parameteramplitude contribution is negligible. Then the calculation ofthis dispersion reduces to calculations of the electromagnetic

6 In the neutral case, for the full collective modes of Eq. (127), there are nu-merical differences (of order unity) in the prefactors of the mode frequency,so that the collective modes of the crossover theory are not strictly the sameas in GP theory. In these calculationsV in Eq. (127) should be associ-ated with the pairing interaction, which enters in the context of fermionicdensity-density correlation effects within the particle-hole channel.

response, as discussed, for example inρs. See also AppendixB. At finite T , the AB mode becomes damped and the real andimaginary parts of the sound velocity have to be calculatednumerically. Here one finds, as expected, that the complexc → 0 asT → Tc. The results are plotted in Fig. 22 for bothmoderate and weak coupling. Because the real and imaginaryparts are comparable there, the mode is highly damped nearTc.

C. Investigating the Applicability of a Nambu MatrixGreen’s Function Formulation

Diagrammatic schemes appropriate to BCS superconduc-tors are based on a Nambu matrix Green’s function approach.The off-diagonal or anomalous Green’s functions in this ma-trix are given by

F (K) ≡ ∆scG(K)G0(−K) (128)

as well as its Hermitian conjugate. This Nambu formalismwas designed to allow study of perturbations to the BCSstate, due to, for example, external fields or impurities. Forthese perturbations the central assumption is that they actonboth the “normal” (with BCS Green’s functionG(K) fromEq. (30)) and anomalous channels in a symmetrical way.

Understanding as we now do that BCS theory is a veryspecial case of superconductivity, this raises the cautionthatonce one goes beyond BCS, some care should be taken to jus-tify this Nambu approach. At the very least the distinctionbetween the order parameter∆sc and the excitation gap∆raises ambiguity about applying a diagrammatic theory basedon Nambu-Gor’kov Green’s functions.

More importantly, in treating the effects of the pseudogapself energyΣpg as we do here, it should be clear that this selfenergy doesnot play a symmetric role in the anomalous andnormal channels. It is viewed here as an entirely normal stateeffect. However this theory accommodates the equivalent(Kosztin et al., 2000) of the Nambu-Gor’kov “F”- functionin general response functions such as in the Maki-Thompsondiagrams of Appendix B through the asymmetric combina-tion GG0 which always arises in pairs (for example, as theFF combinations of BCS theory). In this regard theGG0

formalism appears to differ from all otherT -matrix schemeswhich are designed to go belowTc, in that the Nambu schemeis not assumed at the start. Nevertheless, many features ofthis formalism seem to naturally arise in large part becauseof Eq. (128), which demonstrates an intimate connection be-tweenGG0 and the conventional diagrams of BCS theory.

IV. OTHER THEORETICAL APPROACHES TO THECROSSOVER PROBLEM

A. T 6= 0 Theories

We have noted in Section I.G that there are three main theo-retical approaches to the crossover problem based onT -matrixtheories. Their differences are associated with differentforms

28

for the pair susceptibilityχ. The resulting calculations ofTc

show similar variations. When two full Green’s functions arepresent inχ (as in the FLEX approach),Tc varies monotoni-cally (Haussmann, 1993) with increasing attractive coupling,approaching the ideal gas Bose-Einstein asymptote from be-low. When two bare Green’s functions are present inχ, asin the work of Nozieres and Schmitt-Rink, and of Rande-ria and co-workers, thenTc overshoots (Sa de Meloet al.,1993; Nozieres and Schmitt-Rink, 1985; Randeria, 1995) theBEC asymptote and ultimately approaches it from above. Fi-nally when there is one bare and one dressed Green’s function,Tc first overshoots and then decreases to a minimum aroundµ = 0, eventually approaching the asymptote from below(Maly et al., 1999b). This last appears to be a combinationof the other two approaches. Overall the magnitudes are rela-tively similar and the quantitative differences are small.Thereis, finally, another body of work (Giovannini and Berthod,2001) based on theGG0 scheme which should be noted here.The approach by the Swiss group seeks to make contact withthe phase fluctuation scenario for highTc superconductors andaddresses STM and ARPES data.

Bigger differences appear when theseT -matrix theories areextended belowTc. Detailed studies are most extensive forthe NSR approach. BelowTc one presumes that theT -matrix(or χ) contains only bare Green’s functions, but these func-tions are now taken to correspond to their Nambu matrix form,with the order parameter∆sc appearing in the dispersion rela-tion for excited fermions (Pieriet al., 2004; Randeria, 1995).A motivation for generalizing the below-Tc T -matrix in thisway is that one wants to connect to the collective mode spec-trum of the superconductor, so that the dispersion relationforpair excitations isΩq ≈ cq. In this way the system would bemore directly analogous to a true Bose system.

Self energy effects are also incorporated belowTc but onlyin the number equation, either in the approximate manner ofNSR (Griffin and Ohashi, 2003; Randeria, 1995) or throughuse of the full Dyson resummation (Pieriet al., 2004) ofthe diagonal Nambu-matrix componentG(Σ0). For the lat-ter scheme, Strinati’s group has addressed pseudogap effectsin some detail with emphasis on the experimentally observedfermionic spectral function. Some concern can be raised thatthe fermionic excitations in the gap equation do not incor-porate this pseudogap, although these pairs are presumed toemerge out of a normal state which has a pseudogap. Indeed,this issue goes back to the original formulation of NSR, whichincludes self energy effects in the number equation, and notinthe gap equation.

Tchernyshyov (Tchernyshyov, 1997) presented one of thefirst discussions belowTc for the FLEX-basedT -matrixscheme. He also addressed pseudogap effects and found asuppression of the fermion density of states at low energywhich allows for long-lived pair excitations inside this gap.At low momenta and frequencies, their dispersion is that of aBogoliubov-sound-like mode with a nonzero mass. Some ofthese ideas have been recently extended (Domanski and Ran-ninger, 2003) to apply to the boson-fermion model.

The work of Micnas and collaborators (Micnaset al., 1995)which is also based on the FLEX scheme predates the work

of Tchernyshyov, although their initial focus was aboveTc intwo dimensional Hubbard models. Here the fermionic spec-tral functions were studied in detail, using a numerically gen-erated solution of theT -matrix equations. A nice review oftheir large body of work on “real space pairing” in the cuprateswas presented based on a variety of different model Hamilto-nians (Micnas and Robaszkiewicz, 1998).

An extensive body of work on the FLEX scheme has beencontributed by Yamada and Yanatse (Yanaseet al., 2003) bothabove and belowTc. They point out important distinctionsbetween their approach and that of NSR. The effects of thebroken symmetry are treated in a generalized Nambu formal-ism, much as assumed by Pieriet al. ((Pieriet al., 2004)) andby Griffin and Ohashi ((Griffin and Ohashi, 2003)), but herethe calculations involve self energy effects in both the numberand gap equations. Their work has emphasized the effects ofthe pseudogap on magnetic properties, but they have discusseda wide variety of experiments in highTc and other exotic su-perconductors.

The nature of the ground states which result from these twoalternatives (χ ≈ GG andχ ≈ G0G0 ) has yet to be clearlyestablished. It should be noted that, despite the extensivebodyof work based on the NSR orG0G0 scheme, no detailed pic-ture has been presented in the literature of the nature of theground state which results when their calculations atTc areextrapolated (Pieriet al., 2004) down toT = 0. In largepart, the differences between other work in the literature andthat summarized in Sections II.C are due to the whether (as inNSR-based papers) or not (as here) the superconducting orderparameter∆sc alone characterizes the fermionic dispersionbelowTc.

One might well ask the question: because the underlyingHamiltonian [Eq. (11)] is associated with inter-fermionic, notinter-bosonic interactions, will this be reflected in the near-BEC limit of the crossover problem? The precise BEC limitis, of course, a non-interacting, or ideal Bose gas, but awayfrom this limit, fermionic degrees of freedom would seemto be relevant in ways that may not be accounted for by theanalogue treatment of the weakly interacting Bose gas. Themost extensive study (albeit, aboveTc) of this issue is due toPieri and Strinati (Pieri and Strinati, 2000). Their work, im-portantly, points out the inadequacies ofT -matrix schemes,particularly at strong coupling. While the ground state in theircalculations is unknown, it is necessarily different from theconventional crossover state of Eq. (1). There is much in-tuition to be gained by studying this simplest of all groundstates, as outlined in this Review, but it will clearly be of greatvalue in future to consider states, in which, for example, thereis less than full condensation.

Finally, we note that studies of non-s wave orbital pairingstates in the context of the BCS-BEC crossover problem haverapidly proliferated in the literature. This is, in large part be-cause of the observedd-wave pairing in the cuprates. Indeed,Leggett’s initial work (Leggett, 1980), addressedp-wave pair-ing in 3He. Randeria and co-workers (Randeria, 1995) alsoconsidered non-s wave pairing. Studies (Andrenacciet al.,1999; den Hertog, 1999) of this crossover for ad-wave lat-tice case are plentiful atT = 0. Even earlier work was done

29

(Chenet al., 1999) on extensions of this ground state to in-clude nonzero temperature. Moreover, ad-wave continuummodel has also been considered (Quintanillaet al., 2002). Itis expected that in the near future cold atom experiments willbegin to addressl 6= 0 pairing via appropriate choices of theFeshbach resonances (Ho and Zahariev, 2004).

B. Boson-Fermion Models for High Tc Superconductors

The notion that it might be appropriate to include bosonicdegrees of freedom in highTc superconductors goes back toT.D. Lee and R. Friedberg who introduced (Friedberg and Lee,1989a,b) the “boson-fermion” model within three years of thediscovery of cuprate superconductivity. The work of Lee andcolleagues is based on the Hamiltonian of Eq. (11) withoutthe direct fermion-fermion interactionU . To our knowledgethis is the first study which argues for something akin to “realspace pairing” in these materials. Interestingly, this work wasmotivated only by the observed short coherence length. Pseu-dogap effects, which make this case much stronger, had notyet been properly characterized. The analogies with He4 anda more BEC-like picture of superconductivity were discussed,as well as the superfluid density, and energy gap structure, al-beit fors-wave pairing.

Subsequent work on this model Hamiltonian in the con-text of the highTc copper oxides was pursued by Ranningerand Micnas and their collaborators. Indeed, Ranninger andRobaszkiewicz (Ranninger and Robaszkiewicz, 1985) werethe first to write down the boson-fermion model, althoughinitially, in an entirely different context, the bosons were as-sociated with phonons rather than electron pairs, and theywere presumed to be localized. In this context they addressedmean field theoretic solutions belowTc as well as more re-cently (Domanski and Ranninger, 2003) an approach based onrenormalization group techniques. The physical picture whichemerges leads to two gap structures belowTc, one associatedwith a pseudogap and another with the superconductivity. Inthis way it appears to be very different from approaches basedon Eq. (1).

V. PHYSICAL IMPLICATIONS: ULTRACOLD ATOMSUPERFLUIDITY

A. Homogeneous case

In this section we summarize the key characteristics offermionic superfluidity in ultracold gases, in the homogeneoussituation, without introducing the trap potential. Our resultsare based on numerical solution of the coupled Eqs. (69), (70)and (91). The upper left panel of Fig. 23 plots the fermionicchemical potential, the Feshbach boson condensate ratio andthe inverse scattering length as a function ofν0. Here we havechosen what we believe is the physically appropriate valuefor the Feshbach couplingg0 = −40EF/k

3F for 40K, as de-

termined from the experimentally measured scattering length

0

0.5

1

0

1

2

3

-600 -300 0ν0

0

0.1

0.2

-600 -300 0ν0

-2

0

2

-600 -300 0

0

0.4 m/4πξ

∆(0)∆(Tc)

m/4πas

µ2nb0/ntot

ξ -- pair sizeas -- scattering length

m/4πas

Tc

kFas

kFξ

µ

FIG. 23 (Color) Characteristics of the ground state andTc asa function of the magnetic detuningν0 in a homogeneous case.Shown are the chemical potentialµ, fraction of molecular conden-sate2nb0/ntot, inverse scattering lengthm/4πas, gap∆(0), inversepair sizem/4πξ, dimensionless relative pair sizekF ξ at T = 0 aswell asTc and∆(Tc). The pair size is roughly the same as inter-particle spacing,kF ξ = 1, at unitarity. Asν0 decreases, the gap∆increases and reaches a maximum, and then decreases slowly and ap-proaches its BEC asymptote (not shown) from above. Here we takeg0 = −40EF /k

3/2

F andU0 = −3EF /k3F , and we use the conven-

tion for unitsEF = kF = ~ = kB = 2m = 1. A large exponentialmomentum cutoff,kc/kF = 80 is assumed in the calculations.

data and resonance width.7 This parameter is roughly an orderof magnitude larger for6Li. Hereν0, the gaps andµ are all inunits ofEF , and the plots, unless indicated otherwise, are atzero temperature.

The upper right panel plots the excitation gap atTc as wellas the gap atT = 0. The lower left hand panel indicatesTc

along with the inverse pair sizeξ−1 in the condensate. Finallythe lower right panel plotsξ itself, along withas.

One can glean from the figure that the Feshbach boson frac-tion decreases, becoming negligible when the chemical po-tential passes through zero. This latter point marks the on-set of the PG regime, and in this regime the condensate con-sists almost entirely of fermionic pairs. The upper limit ofthePG regime, that is, the boundary line with the BCS phase, isreached once the pseudogap,∆(Tc) is essentially zero. Thishappens whenµ is close to its saturation value atEF . At thispoint the pair size rapidly increases.

These results can be compared with those derived froma smaller value of the Feshbach coupling constantg0 =

−10EF/k3/2F shown in Fig. 24. Now the resonance is effec-

7 We note that the precise values ofg0 andU0 are not critical for either6Li or40K as long as they give roughly the right (large) resonance width ∆B andbackground scatter lengthabg . In fact, these parameters usually changeduring experiment since the total particle numberN (and henceEF andkF ) decreases in the process of evaporative cooling. In order to comparewith experiment, one needs to compare at the correct value ofkF as.

30

0

0.5

1

0

0.5

1

-120 -60 0ν0

0

0.1

0.2

-120 -60 0ν0

-2

0

2

-120 -60 0

0

0.4m/4πξ

∆(0)∆(Tc)

m/4πas

µ2nb0/ntot

m/4πas

Tc

kFas

kFξ

µ

FIG. 24 (Color) Characteristics of the ground state andTc, as inFig. 23 except for a smallerg0. In comparison with Fig. 23, the Fes-hbach resonance becomes much narrower. Hereg0 = −10EF /k

3/2

F

andU0 = −3EF /k3F .

0

1

0

2

4

0.50.6|U|

0

0.1

0.2

0.50.6|U|

-20

0

20

0.50.6

0

0.4m/4πξ

∆(0)

∆(Tc)

m/4πas

µ

m/4πas

Tc kFas kFξ

µ

FIG. 25 (Color) Characteristics of the ground state andTc in theone-channel model, as in Fig. 23 but as a function of the attractionstrengthU . Hereg0 = 0. Different from the two-channel shown inFigs. 23-24 case, the gap∆ increases indefinitely with|U |. The unitsfor U areEF /k3

F . Note that|U | increases to the left.

tively narrower. Other qualitative features remain the same asin the previous figure.

One can plot the analogous figures in the absence of Fes-hbach effects. Here the horizontal axis is the inter-fermionicinteraction strength|U |, as is shown in Fig. 25. Three essentialdifferences can be observed. Note, first the obvious absenceofthe Feshbach boson or molecular condensatenb0. There is, ofcourse, a condensate associated with pairs of fermions (∆sc)and these pairs will become bound into “fermionic molecules”for sufficiently strong attraction. Secondly, note that theexci-tation gap∆(0) is monotonically increasing as|U | increasestowards the BEC limit. By contrast, from the upper rightpanels of Fig. 24 one can see that when Feshbach effects are

0

0.5

1

0

0.5

1

-120 -60 0ν0

0

0.1

0.2

-120 -60 0ν0

-2

0

2

-120 -60 0

0

0.4m/4πξ

∆(0)

∆(Tc)m/4πas

µ2nb0/ntot

m/4πas

Tc

kFas

kFξ

µ

FIG. 26 (Color) Characteristics of the ground state andTc, as inFig. 24, except nowU0 = 0. As a consequence of the vanishing ofU0, the gap∆ is monotonic and approaches its BEC asymptote frombelow asν0 decreases. Hereg0 = −10EF /K

3/2

F .

-400 0 400ν0 /EF

0

0.1

0.2

0.3

Tc

/EF

U0 = -100

-80

-40

-20

-10

U0 = -3-800 0 4000

1

2

∆(T

c)/E

F

U0 = -3

-10

-20

-40 -100-80

FIG. 27 (Color) Behavior ofTc (main figure) and∆(Tc) (inset) atfixedg0 = −40EF /k

3/2

F with variableU0. AsU0 → −∞, unitarityis approached from the BCS side at large positiveν0. At the sametime,∆ ∝ |g| also decreases.

present∆(0) ≈ ∆(Tc) decreases towards zero in the extremeBEC limit. It can also be seen that the shape of the scatteringlength curvevs U is different from the plots in the previoustwo figures. Hereas more rapidly increases in magnitude oneither side of the unitary limit. Nevertheless it is important tostress that,except for the nature of the Bose condensate in theBEC limit, the physics of the Feshbach resonance-induced su-perfluidity is not so qualitatively different from that associatedwith a direct inter-fermionic attraction.

Figure 26 presents the analogous plots for the case whenthe direct attraction between fermionsU vanishes. Here thecoupling constantg0 is the same as in Fig. 24, and pseudo-gap effects are present here just as in all other cases. Themost notable difference in the physics is that the formation

31

-20 0 200

0.01N

(ω)

T=Tc+

T=0.75Tc

T=0.5Tc

-2 0 2ω/EF

0

0.3

N(ω

)

-0.05 0 0.05ω/EF

0

1

2BEC

PG

BCS(a)

(b)

(c)

FIG. 28 Typical density of states (DOS)N(ω) vs energyω in thethree regimes at indicated temperatures. There is no depletion of theDOS atTc in the BCS case. By contrast, in the BEC case, there isa large gap in DOS atTc, roughly determined by|µ|. In the inter-mediate, i.e., pseudogap, regime, there is a strong depletion of theDOS already atTc. OnceT decreases belowTc, the DOS decreasesrapidly, signalling the superfluid phase transition.

of the Cooper condensate is driven exclusively by couplingto the molecular bosons. A contrast between the two fig-ures is in the behavior of∆(Tc) and∆(0). For theU = 0case and in the BEC limit,∆(0) approaches the asymptoticvalue (|g|

√

n0b) from below. When the system is the fermionic

regime (µ > 0), aside from quantitative details, there is verylittle to distinguish the effects ofU from those ofg, exceptwhen these parameters are taken to very extreme limits.

Finally, in Fig. 27 we plotTc vs ν0 in the presence of Fes-hbach effects with variableU0, from weak to strong back-ground coupling. The lower inset shows the behavior of theexcitation gap atTc. With very strong direct fermion attrac-tionU , we see thatTc has a very different dependence onν0.In this limit there is a molecular BEC to PG crossover (whichmay be inaccessible in actual experiments, sinceU is not suf-ficiently high). Nevertheless, it is useful for completeness toillustrate the entire range of theoretical behavior.

Figures 28(a)-28(c) show the fermionic density of states forthe BEC, PG and BCS limits. These figures are important inestablishing a precise visual picture of a “pseudogap”. Thetemperatures shown are just aboveTc, and forT = 0.75Tc

andT = 0.5Tc. The methodology for arriving at these plotswill be discussed in the following section, in the context ofhigh Tc superconductors. Only in the BCS case is there aclear signature ofTc in the density of states, but the gap is sosmall andTc is so low, that this is unlikely to be experimen-tally detectable. Since the fermionic gap is well establishedin the BEC case, very little temperature dependence is seenas the system goes from the normal to the superfluid states.Only the PG case, whereTc is maximal, indicates the pres-ence of superfluidity, not so much atTc but once superfluidorder is well established atT = 0.5Tc, through the presenceof sharper coherence features, much as seen in the cuprates.

0 1 2 3k/kF

0

0.5

1

n k

BEC,T=0,T=TcPG, T=0PG, T=TcBCS, T=0BCS, T=Tc

0 1T/Tc

0

1

∆(T

)/∆(0

)

BECPGBCS

FIG. 29 Typical behavior of the fermionic momentum distributionfunctionnk atT = 0 andTc in the three regimes for a homogeneoussystem. The inset indicates the corresponding normalized gaps as afunction ofT/Tc. In all three regimes,nk barely changes betweenT = Tc andT = 0.

The inset of Fig. 29 plots the temperature dependence of∆(T ). It should be stressed thatTc is only apparent in∆(T )in the BCS case. To underline this point, in the main body ofFig. 29 we plot the fermionic momentum distribution functionnk, which is the summand in the number equation, Eq. (70),atT = 0 andT = Tc. The fact that there is very little changefrom T = 0 to T = Tc makes the important point that thismomentum distribution function in a homogeneous system isnot a good indicator of phase coherent pairing. For the PGcase, this, in turn, derives from the fact that∆(T ) is nearlyconstant. For the BEC limit the excitation gap, which is dom-inated byµ, similarly, does not vary throughTc. In the BCSregime,∆(T ) is sufficiently small so as to be barely percepti-ble on the scale of the figure.

1. Unitary Limit and Universality

There has been considerable emphasis on the behavior offermionic gases near the unitary limit whereas = ±∞. At-tention (Ho, 2004; Viveritet al., 2004) has concentrated onthis limit in the absence of Feshbach bosons. We now wantto explore the more general two-channel problem here. Atunitarity we require that

4πa∗sm

= U0 +g20

2µ− ν0(129)

diverges. This will be consistent, provided

ν0 = 2µ . (130)

It should be clear that the parameterU0 is of no particular rel-evance in the unitary limit. This is to be contrasted with theone channel problem, where Feshbach bosons are absent andunitarity is achieved by tuning the direct inter-fermion inter-action,U .

32

0

0.4

0.8

0

0.4

0.8

0 20 40|g0|

0

1

2

3

0 20 40|g0|

0

0.4

0.8

µ /EF = ν0/2EF

∆ /EF

EF’ = EF (1-2nb0/n)2/3

2nb0/n

kF’ = kF (1-2nb0/n)1/3

kFξ

µ /EF’∆ /EF’

kF’ ξ

FIG. 30 (Color) Characteristics of the unitary limit atT = 0 as afunction of |g0| in the two-channel problem. Herenb0 is the den-sity of condensed Feshbach bosons.E′

F = k′2F /2m defines an

effective Fermi energy of the fermionic component with a densityn = ntot − 2nb0. After rescaling,µ/E′

F , ∆/E′F , andk′

F ξ are es-sentially a constant ing0. This indicates that at unitarity, the systemcan be treated effectively as a two-fluid model atT = 0. The unitsfor g0 areEF /k

3/2

F .

In the unitary limit, theT = 0 gap and number equationsbecome

1 + Uc

∑

k

1

2Ek

= 0, (131)

and

ntot = 2n0b + n =

2∆20

g20

+∑

k

(

1 − ǫk − µ

Ek

)

, (132)

where we have used the fact that at unitarity

n0b =

g2∆20

[(2µ− ν)U + g2]2=

∆20

g20

, (133)

and we define∆0 ≡ ∆sc(T = 0). HereUc is the criticalcoupling, given in Eq. (17). From these equations it followsthat in this limit the values ofµ and∆0 in the ground state areentirely determined by the Feshbach couplingg0.

Figure 30 presents a plot of zero temperature propertiesshowing their dependence ong0. The two endpoints of thecurves are of interest. For arbitrarily small but finiteg0 theunitary limit corresponds roughly toµ ≈ 0. Moreover, herethe condensate consists exclusively of Feshbach bosons. Atlarge g0, (which appear to be appropriate to current exper-iments on6Li and 40K), the behavior at unitarity is closer tothat of the one channel problem (Ho, 2004). Here one finds theso-called universal behavior with parametersµ/EF = 0.59,and ∆/EF = 0.69 associated with the simple crossoverground state and a contact potential interaction8.

8 In our plots we find small numerical corrections associated with our non-

0

0.4

0.8

0

0.4

0.8

0 20 40|g0|

0.1

0.2

0 20 40|g0|

0

0.4

0.8

0 500

3

µ/EF = ν0/2EF ∆(Tc)/EF

2nb/ntot

kFξ

Tc/EF

2np/ntot

FIG. 31 (Color) Characteristics of the unitary limit atT = Tc asa function of |g0| in the two channel problem. In contrast to theprevious figure forT = 0, there is no rescaling which would lead touniversal behavior ofµ, ∆ or ξ with respect tog0. More than20% ofthe system is composed of fermionic quasiparticles when|g0| → 0.Herenb andnp denote the density of noncondensed Feshbach bosonsand of the sum of FB and fermion pairs, respectively. The units forg0 areEF /k

3/2

F .

To what extent is the unitary limit of the two channel prob-lem universal? It clearly is not in an absolute sense. However,there is a kind of universality implicit in theseT = 0 calcula-tions, as can be seen from Fig. 30 if we take into account thefact that there are a reduced number of fermions, as a resultof the Bose condensate. We, thus, divide the two zero temper-ature energy scales by an effective Fermi energy defined byE′

F = EF (n/ntot)2/3 with n = ntot − 2n0b . Figure 30 shows

that asg0 decreases, the effective or rescaled Fermi energyE′

F decreases while the Bose condensate fraction increases.The rescaled quantitiesµ/E′

F and∆/E′F are universal, that

is, they are essentially independent ofg0, as seen in the Figure.A similar scaling can be done for the Cooper pair sizeξ. Witha reduced effective Fermi momentumk′F = kF (n/ntot)1/3,the dimensionless productk′F ξ is also independent ofg0. Wecan arrive at some understanding of this modified universality,by referring back to Sec. V.A. In summary, the breakdown ofuniversality seen in Fig. 30 is to be associated with a reducednumber of fermions, which, in turn derives from the Bose con-densaten0

b . Oncen0b is negligible, the distinction between one

and two channel models disappears.Finite temperature properties, such asTc and ∆(Tc) can

also be calculated in the unitary limit, as a function ofg0.These are plotted in Fig. 31. Just as for the zero temperatureenergy scales, these parameters [along withµ(Tc)] increasemonotonically withg0. However, at finite temperatures thereare noncondensed Feshbach bosons as well as noncondensedfermion pairs, which interact strongly with each other. These

infinite cutoff.

33

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

-8 -4 0 4ν0

0

0.1

0.2

-8 -4 0 4ν0

-2

-1

0

1

2

-8 -4 0 4-0.2

0

0.2m/4πξ

∆(0)

∆(Tc)

m/4πas

µ2nb0/ntot

m/4πas

Tc

kFas

kFξ

µ

FIG. 32 (Color) Characteristics of the smallg0 limit as a functionof ν0 in the two-channel model. Unless indicated otherwise the plotscorrespond toT = 0. The molecular condensate fraction is veryhigh and there is no maximum at unitarity. The gap∆ is very small(∝ g0). Here we takeg0 = −1EF /k

3/2

F andU0 = −3EF /k3F .

Note the very different scales for the abscissa when comparing withFigs. 23 and 24.

noncondensed states are represented in the lower right inset.Plotted there is the fraction of noncondensed bosons [nb(Tc)]and sum of both noncondensed fermion pairs and noncon-densed bosons [np(Tc)] at Tc. These quantities correspondto the expressions shown in Eqs. (80) and (92), respectively.

These noncondensed particles are responsible for the factthat atTc a simple rescaling will not yield the universalityfound atT = 0. Moreover, these noncondensed bosons andfermion pairs influence calculations ofTc, itself, via pseudo-gap effects which enter as∆(Tc). These pseudogap effects,which are generally ignored in the literature, compete withsuperconductivity, so thatTc is bounded with its a maximumgiven by≈ 0.25TF , just as seen in Fig. 25, for the one channelproblem. Despite the fact that there is no universality, even inthe rescaled sense, atTc one can associate the largeg0 limitswith the behavior of the one channel system which has beenrather extensively studied in the unitary limit (Ho, 2004).Thislargeg0 limit seems to be reasonably appropriate for the cur-rently studied Feshbach resonances in6Li and 40K. In thissense, these particular atomic resonances are well treatedby aone channel problem, near the unitary limit.

In summary, comparison between Fig. 30 and Figure 31 re-veals that (i) atT = 0, all Feshbach bosons and pairs arecondensed and the two (fermionic and bosonic) condensatesare relatively independent of each other. (ii) By contrast,at T = Tc, fermionic quasiparticles, noncondensed fermionpairs and Feshbach bosons all coexist and interact stronglywith each other. This destroys the simple scaling found atT = 0.

2. Weak Feshbach Resonance Regime: |g0| = 0+

The effects of very small yet finiteg0 lead to a nearlybi-modal description of crossover phenomena. Figure 32presents the behavior for very narrow Feshbach resonances.It shows that there is a rather abrupt change as a function ofmagnetic field (viaν0) from a regime in which the conden-sate is composed only of Feshbach bosons, to one in whichit contains only fermion pairs. Depending on the size ofg0, this transition point (taken for concreteness to be where2n0

b/ntot ≈ 0.5, for example) varies somewhat, but it seems

to occur close to the unitary limit and, importantly, for nega-tive scattering lengthsi.e., on the BCS side. This should becontrasted with the largeg0 case). This smallg0 regime ofparameter space was explored by Ohashi and Griffin (Ohashiand Griffin, 2002). Indeed the behavior shown in the figure issimilar to the physical picture which Falco and Stoof (Falcoand Stoof, 2004) presented.

This figure represents a natural evolution in the behavioralready indicated by Figs. 23 and 24. Here, however, in con-trast to these previous figures the behavior ofTc is essentiallymonotonic, and the weak maximum and minimum found ear-lier has disappeared. For this regime, the small pseudogapwhich can be seen in the upper right panel arises predomi-nantly from the contribution of Feshbach bosons via Eqs. (84)and (91). At the same time it should be noted that the transi-tion from mostly molecular bosons to mostly fermion pairs inthe condensate is continuous. While it may look sharp par-ticularly for a plot with a more extended range ofν0 thanshown here, this closeup view gives a more precise pictureof a smooth crossover transition.

We end by noting that in this smallg0 regime, the system isfar from the one channel limit obtained wheng0 is identically0, as shown in Fig. 25. It should be clear from the abovediscussion thatFeshbach bosons play a more important rolein the regime of unitary scattering the smaller isg0, providedthat it does not strictly vanish.

B. Superfluidity in Traps

Most experiments on fermionic superfluids are done in thepresence of a trap potentialV (r). For the most part, this ismost readily included at the level of the local density approx-imation (LDA). In this section, we summarize the self con-sistent equations for spherical trapsV (r) = 1

2mω2r2. Tc is

defined as the highest temperature at which the self consistentequations are satisfied precisely at the center. At a tempera-tureT lower thanTc the superfluid region extends to a finiteradiusRsc. The particles outside this radius are in a normalstate. Also important is the radiusR∆ beyond which the ex-citation gap is effectively zero, to a chosen level of numericalaccuracy.

The self consistent equations (with simplified units~ = 1,fermion massm = 1

2 , and Fermi energyEF = ~ω(3N)1

3 =1, whereω is the trap frequency) are given in terms of theFeshbach coupling constantg and inter-fermion attractive in-

34

0 0.5 10

0.1

0.2

0.3

n(r)

T = 00.25Tc

0.5Tc

0.75Tc

T = Tc

0 0.5 1r/RTF

0

0.03

0.06

0 0.5 10

0.02

0.04

BEC Unitary BCS

FIG. 33 (Color) Representative spatial profiles of fermionsin a harmonic trap potential at various temperatures fromTc to T = 0 in the threeindicated regimes. As the system evolves from the BCS to the deep BEC regime, the fermion cloud shrinks, and a bimodal distribution of thedensity develops belowTc. Note the different scales for the vertical axes.

teractionU by a gap equation

1 +

[

U +g2

2µ− 2V (r) − ν

]

∑

k

1 − 2f(Ek)

2Ek

= 0 . (134)

which characterizes the excitation gap∆(r) at positionsrsuch thatµpair(r) = 0. Moreover, the pseudogap contribu-tion to∆2(T ) = ∆2

sc(T ) + ∆2pg(T ) is given by

∆2pg =

1

Z

∑

q

b(Ωq − µpair) . (135)

Hereb represents the usual Bose function. The density of par-ticles at radiusr:

n(r) = 2n0b +

2

Zb

∑

q

b(Ωq − µboson)

+ 2∑

k

[

v2k(1 − f(Ek)) + u2

kf(Ek)]

. (136)

The important quantityµpair = µboson is identically zeroin the superfluid regionr < Rsc, and must be solved for selfconsistently at larger radii. HereM is the bare boson massequal to2m, and the various residues,Z andZb and pair dis-persionΩq are described in Section II.C.

The transition temperatureTc is numerically determined asfollows: (i) A chemical potential is assigned to the center ofthe trapµ(0). The chemical potential at radiusr will then beµ(r) = µ(0) − V (r). (ii) The gap equation [Eq. (134)] andpseudogap equation [Eq. (135)] are solved at the center (set-ting ∆pg = ∆) to find Tc and∆(0, Tc) (iii) Next the radiusR∆ is determined. (It can be defined as the point where thegap has a sufficiently small value, such as∆(0, Tc)/1000).(iv) Following this, the pseudogap equation (Eq. (135)) issolved for∆(r, Tc) up to the radiusR∆. Thenn(r) is deter-mined using Eq. (136). (v) Finally,n(r) is integrated over allspace (

∫

d3rn(r)) and compared with the given total numberof fermionsN . The procedure is repeated until the prescribedaccuracy has been reached. To extend this algorithm belowTc

a similar approach is used which now includes an additionalstep in whichRsc is computed.

Figure 33 shows a plot of the density profiles obtained fromsolving the self consistent equations. Here the Thomas-Fermiradius, calledRTF , is calculated for the non-interacting case.In the BCS regime there is virtually no temperature depen-dence in the density profile as expected from the small val-ues for∆. In the unitary and BEC regimes as the system ap-proachesTc, n(r) can be reasonably well fit to a Gaussian. Abi-modality in the distribution is evident in the BEC regime,thus providing a clear signature of the superfluid phase. How-ever, in the PG or unitary it is more difficult to establish super-fluidity, since the profiles do not show a clear break at the po-sition where the condensate first appears (Stajicet al., 2005a).

These observations differ somewhat from other theoreti-cal predictions in the literature (Chiofaloet al., 2002; Ho,2004). The differences between the present theory and thisother work can be associated with the inclusion of noncon-densed pairs or pseudogap effects. These noncondensed pairssmooth out the otherwise abrupt transition between the con-densate and the fermionic excitations. In contrast to earlierwork (Peraliet al., 2004b) these studies, which are based onthe standard crossover ground state (Leggett, 1980), yieldonlymonotonic density distributions. Future experiments willbeneeded to look for the secondary maxima predicted by Strinatiand co-workers (Peraliet al., 2004a) in the BEC and unitaryregimes.

Finally, Fig. 34 shows the behavior ofTc (inset) in a trap,together with the superfluid density (main figure). The latteris computed following the discussion in Sec. III. From theinset one can read off thatTc/EF ≈ 0.3 at resonance, whichis very close to (Peraliet al., 2004a), or slightly smaller thanother estimates (Ohashi and Griffin, 2003) in the literature.

The structure seen in the plot ofTc for the trapped gas re-flects the minimum and maximum found for the homogeneoussolution (Fig. 23); these are related, respectively, to whereµ ≈ 0 and where unitary scattering occurs. TheTc curve fol-lows generally the decrease in the density at the center of thetrap from BEC to BCS. The temperature dependences of the(integrated overr) superfluid density reflect the nature of ex-citations of the condensate. The BEC limit has only bosonic,while the BCS has only fermionic excitations. The unitarycase is somewhere in between. The fermionic contributionsare apparent in the BCS regime, whereNs decreases almost

35

0 0.5 1T/Tc

0

0.5

1N

s(T

)/N

s(0)

BCSUnitaryBEC

-200 0-400ν0/EF

0

0.2

0.4

Tc/E

F

FIG. 34 Behavior of superfluid densityNs as a function ofT in thethree different regimes. Note that the negative curvaturesat low Tfor the unitary and BEC cases are different from the linear behaviorin the BCS case. In the BEC limit,N − Ns ∝ T 3/2. In the insetwe showTc as a function of magnetic detuningν0 in a trap. Wehave takenU0 = −0.9EF /k3

F andg0 = −35EF /k3/2

F , based onexperimental data for40K.

linearly at lowT . For ans-wave superfluid this reflects thecontributions from low∆(r) ≤ T regions, near the edge ofthe trap. These are present to a lesser degree in the unitarycase. The bosonic contribution toNs and to thermodynamics(in both the BEC and unitary regimes) is governed by the im-portant constraint thatµpair(r) vanishes at positionsr wherethere is a co-existent superfluid. A similar constraint is im-posed at the level of Hartree-Fock-Bogoliubov theory for truebosons (Shi and Griffin, 1998). These gapless pair excitationsare responsible for aT 3/2 temperature dependence inNs. Inthis way the lowT power law dependences are similar to theiranalogues in the homogeneous case of Section VI.D, much asfound for true Bose gases (Carret al., 2004) in a trap.

As for the BEC regime there is an interesting observationto make about the contribution to the density profile associ-ated with the condensate. Experimentalists generally assumea Gaussian form throughout the trap for the noncondensedpairs. In this way they estimate the condensate fraction. Herewe find (Stajicet al., 2005a) that this Gaussian form is notcorrect in the superfluid region of the trap, and that this Gaus-sian approximation will underestimate (by≈ a factor of 1.5)the condensate fraction in the BEC regime.

C. Comparison Between Theory and Experiment: ColdAtoms

As time progresses it will be important to assess the as-sumed ground state wavefunction (Leggett, 1980) and its twochannel variant (Milsteinet al., 2002; Ohashi and Griffin,2002; Stajicet al., 2005b) in more detail. A first genera-tion set of measurements and theories have indicated an ini-tial level of success in addressing four experiments: RF pair-ing gap spectroscopy and collective mode measurements, aswell as more recently, the particle density profiles (Stajic

et al., 2005a) and thermodynamics (Kinastet al., 2005) ofFermi gases in the unitary regime. The first of these ex-periments (Chinet al., 2004) seem to provide some supportfor the fact that the Bogoliubov excitations with dispersion,Ek ≡

√

(ǫk − µ)2 + ∆2(T ) have a “gap parameter” or pseu-dogap,∆(T ) which is in general different from the supercon-ducting order parameter, as a consequence of noncondensedpairs at and aboveTc. Torma and colleagues (Kinnunenet al.,2004b) have used the formalism outlined in Section II.B, toaddress RF spectroscopy. Subsequently (Kinnunenet al.,2004a) they have combined this analysis with an approximateLDA treatment of the trap to make contact with RF experi-ments (Chinet al., 2004). Important to their work is the factthat the edges of a trap have a free atom character, where thereis a vanishing∆(r).

Additional support for the wavefunction comes from thecollective mode behavior nearT = 0. The mode frequen-cies can be connected to an equation of state (Stringari, 2004).Numerical studies (Heiselberg, 2004b; Huet al., 2004) of thisequation for the standard crossover ground state demonstrate agood fit to the measured breathing mode frequencies (Barten-steinet al., 2004a; Kinastet al., 2004). Importantly, in thenear-BEC limit the frequencies decrease with increasing mag-netic field; this is opposite to the behavior predicted for truebosons (Stringari, 2004), and appears to reinforce the notionthat for the magnetic fields studied thus far, the “BEC” regimeshould be associated with composite rather than real bosons.

At T = 0, calculations of the collective mode frequenciesare based on an equation of state. Moreover, the zero temper-ature equation of state associated with the mean field groundstate can be computed analytically in the near-BEC limit

µ = − 1

2ma2s

+πasn

m− 3(πna3

s)2

4ma2s

+O(n3),

where the first two terms on the right hand side were pre-viously discussed following Eq. (55). Importantly, the thirdterm is responsible for the observed agreement between the-ory and collective mode experiments, in the near-BEC regime.

A serious shortcoming of the wavefunction ansatz has beenemphasized in the literature. This is associated with the ra-tio of the inter-boson to the fermionic scattering length nearthe unitary regime, but on the BEC side of the Feshbach reso-nance. Experimentally the numbers are considerably smaller(Bartensteinet al., 2004b; Zwierleinet al., 2004) (0.6) thanthe factor of 2 predicted for the one channel model discussedbelow Eq. (55). They are consistent, however, with more exactfew body calculations (Petrovet al., 2004). These few bodycalculations apply to a regime of magnetic fields wherekF as

is smaller than about0.3 -0.4. This seems to be close to theboundary for the “quasi-BEC” regime which can be accessedexperimentally (Bartensteinet al., 2004b) in6Li.

It is interesting to repeat theT = 0 many body calcula-tions now with the inclusion of FB, since these two channeleffects begin to be important atkF as smaller than around0.2-0.25. Here, one sees (Stajicet al., 2005b), that the factor of 2found in the one channel calculations is no longer applicable,and the values of the ratio (of bosonic to fermionic scattering

36

0 1 2Attractive coupling constant

0

0.1

0.2

T/E

F

Pseudogap

Superconductor

NormalT*Tc

FIG. 35 Phase diagram for a quasi-two-dimensional (2D)d-wavesuperconductor on a lattice. Here the horizontal axis corresponds tothe strength of attractive interaction−U/4t, wheret is the in-planehopping matrix element.

lengths) are considerably smaller, and, thus, tending in the di-rection which is more in line with experiment. This reflectsthe decrease in the number of fermionic states which mediatethe interaction between molecular bosons.

In this way the mean field theory presented here lookspromising. Nevertheless, uncertainties remain. Current theo-retical work (Kinnunenet al., 2004a) on the RF experiments islimited to the intermediate coupling phase, while experimentsare also available for the near-BEC and near-BCS regimes,and should be addressed theoretically. In addition, the pre-dictions of alternative approaches (Peraliet al., 2004b) whichposit a different ground state will need to be compared withthese RF data. The collective mode calculations (Heiselberg,2004b; Huet al., 2004) which have been performed have notyet considered the effects of Feshbach bosons, which maychange the behavior in the near-BEC regime.

It is, finally, quite possible that incompleteT = 0 conden-sation will become evident in future experiments. If so, an al-ternative wavefunction will have to be contemplated. Indeed,it has been argued (Hollandet al., 2004) that the failure of theone channel model to match the calculations of Petrov and co-workers (Petrovet al., 2004) is evidence that there are seriousweaknesses. At an experimental level, new pairing gap spec-troscopies appear to be emerging at a fairly rapid pace (Bruunand Baym, 2004; Greineret al., 2005) which will further testthese and subsequent theories.

VI. PHYSICAL IMPLICATIONS: HIGH Tc

SUPERCONDUCTIVITY

We begin with an important caveat that many of the exper-iments outlined in this section can alternatively be addressedfrom a Mott physics point of view (Leeet al., 2004). Herewe summarize the interpretation of these experiments withinBCS-BEC crossover theory with the hope that this discussionwill be of value to the cold atom community, and that ulti-mately, it will help to establish precisely how relevant is the

0 1 2 3-U/4t

0

1

2

m/M

*

20Tc /4t

m/M*

D-wave, quasi-2D square lattice

n=0.85

FIG. 36 Inverse pair mass (solid line) andTc (dashed line) as a func-tion of coupling strength−U/4t for a d-wave superconductor on aquasi-2D square lattice, wheret is the in-plane hopping matrix ele-ment. Tc vanishes when the pair mass diverges, before the bosonicregime is reached. The electron density is taken to ben = 0.85 perunit cell, corresponding to optimal hole doping in the cuprate super-conductors.

BCS-BEC crossover scenario for understanding high temper-ature superconductivity.

A. Phase Diagram and Superconductor-InsulatorTransition: Boson Localization Effects

The highTc superconductors are different from the ultra-cold fermionic superfluids in one key respect; they ared-wavesuperconductors and their electronic dispersion is associatedwith a quasi-two dimensional tight binding lattice. In manyways this is not a profound difference from the perspective ofBCS-BEC crossover. Figure 35 shows a plot of the two im-portant temperaturesTc andT ∗ as a function of increasing at-tractive coupling. On the left is BCS and the right is PG. TheBEC regime is not apparent. This is becauseTc disappearsbefore it can be accessed. At the point whereTc vanishes,µ/EF ≈ 0.8.

A competition between increasingT ∗ andTc is also appar-ent in Fig. 35. This is a consequence of pseudogap effects.There are fewer low energy fermions around to pair, asT ∗

increases. It is interesting to compare Fig. 35 with the exper-imental phase diagram plotted as a function ofx in Fig. 7. Ifone inverts the horizontal axis (and ignores the AFM region,which is not addressed here) the two are very similar. To makean association between the couplingU and the variablex, itis reasonable to simply fitT ∗(x). We do not dwell on this laststep here, since we wish to emphasize crossover effects whichare not complicated by “Mott physics”.

Because of quasi-two dimensionality, the energy scales ofthe vertical axis in Fig. 35 are considerably smaller than theirthree dimensional analogues. Thus, pseudogap effects areintensified, just as conventional fluctuation effects are moreapparent in low dimensional systems. This may be one ofthe reasons why the cuprates are among the first materials toclearly reveal pseudogap physics. Moreover, the present cal-

37

0

40

80

-0.2 -0.1 0 0.1 0.20

40

0

40

80

-0.2 -0.1 0 0.1 0.2ω

0

40

0

40

80

120

-0.2 -0.1 0 0.1 0.20

100

200

A(ω

)

T ≥Tc

0.99Tc

0.95Tc

0.9Tc 0.7Tc

0.8Tc(a)

(b)

(c)

(d)

(e)

(f )

FIG. 37 Evolution of spectral functions of a pseudogapped superconductor with temperature. As soon asT drops belowTc, the spectral peakssplit up rapidly, signalling the onset of phase coherence.

culations show that in a strictly2D material,Tc is driven tozero, by bosonic or fluctuation effects. This is a direct reflec-tion of the fact that there is no Bose condensation in2D.

Figure 36 presents a plot (Chenet al., 1999) of boson lo-calization effects associated withd-wave pairing. A diver-gence in the effective massm/M∗ occurs when the cou-pling strength, or equivalentlyT ∗ becomes sufficiently large.Once the bosons localize the system can no longer supportsuperconductivity andTc vanishes. It is generally believedthat a system of bosons with short range repulsion is eithera superfluid or a Mott insulator in its ground state (Fisheret al., 1989). Moreover, the insulating phase is presumed tocontain some type of long range order. Interestingly, thereare recent reports of ordered phases (Fuet al., 2004; Ver-shinin et al., 2004) in the very underdoped cuprates. Theseappear on the insulating side and possibly penetrate acrossthe superconductor-insulator boundary; they have been inter-preted by some as Cooper pair density waves (Chenet al.,2004; Melikyan and Tesanovic, 2004; Tesanovic, 2004).Theobservation of boson localization appears to be a natural con-sequence of BCS-BEC theory; it should be stressed that it isassociated with thed-wave lattice case. Becaused-wave pairsare more spatially extended (than theirs-wave counterparts)they experience stronger Pauli repulsion (Chenet al., 1999)and this inhibits their hopping.

B. Superconducting Coherence Effects

The presence of pseudogap effects raises an interesting setof issues surrounding the signatures of the transition whichthe highTc community has wrestled with, much as the coldatom community is doing today. For a charged superconduc-tor there is no difficulty in measuring the superfluid density,through the electrodynamic response. Thus one knows withcertainty whereTc is. Nevertheless, people have been con-cerned about precisely how the onset of phase coherence isreflected in thermodynamics, such asCV or in the fermionicspectral function. One understands how phase coherenceshows up in BCS theory, since the ordered state is always ac-companied by the appearance of an excitation gap. When agap is already well developed atTc, how do these signatures

emerge?To address these coherence effects one has to introduce a

distinction between the self energy (Chenet al., 2001) associ-ated with noncondensed and condensed pairs. This distinctionis blurred by the approximation of Eq. (89). Above, but nearTc, or at any temperature below we now use an improved ap-proximation (Malyet al., 1999a,b)

Σpg ≈ ∆2

ω + ξk + iγ(137)

This is to be distinguished fromΣsc where the condensedpairs are infinitely long-lived and there is no counterpart forγ. The value of this parameter, and even itsT -dependence isnot particularly important, as long as it is nonzero.

Figure 37 plots the fermionic spectral function atξk = 0,calledA(ω), as the system passes from above to belowTc.One can see in this figure that just belowTc, A(ω) is zeroat a pointω = 0, and that as temperature further decreasesthe spectral function evolves smoothly into approximatelytwoslightly broadened delta functions, which are just like theircounterparts in BCS. In this way there is a clear signature as-sociated with superconducting coherence. To compare withexperimental data is somewhat complicated, since measure-ments of the spectral function (Campuzanoet al., 2004; Dam-ascelli et al., 2003) in the cuprates also reveal other higherenergy features (“dip, hump and kink”), not specifically asso-ciated with the effects of phase coherence. Nevertheless, thisFigure, like its experimental counterpart, illustrates that sharpgap features can be seen in the spectral function, but only be-low Tc.

Analogous plots of superconducting coherence effects arepresented in Fig. 38 in the context of more direct compari-son with experiment. Shown are the results of specific heatand tunneling calculations and their experimental counterparts(Renneret al., 1998; Tallon and Loram, 2001). The lattermeasures, effectively, the density of fermionic states. Herethe label “PG” corresponds to an extrapolated normal statein which we set the superconducting order parameter∆sc tozero, but maintain the the total excitation gap∆ to be the sameas in a phase coherent, superconducting state. Agreement be-tween theory and experiment is satisfactory. We present this

38

-0.4 -0.2 0 0.2 0.4V

0

5

10

dI/d

V

SCPG

0 1 2 3T/Tc

0

0.1

0.2S

/(T/T

c), C

v/(T

/Tc)

SCPG

S/T

Cv /T

x = 0.125

FIG. 38 Superconducting state (SC) and extrapolated normalstate (PG) contributions to SIN tunneling characteristicsdI/dV (upper panels)and thermodynamicsS/T andCv/T (lower panels). We compare theoretical calculations (left) with experiments (right) on tunneling forBSCCO from (Renneret al., 1998) and on specific heat for Y0.8Ca0.2Ba2Cu3O7−δ (YBCO) from (Tallon and Loram, 2001). The theoreticalSIN curve is calculated forT = Tc/2, while the experimental curves are measured outside (dashed line) and inside (solid line) a vortex core.Thedashed and solidcurves in the upper left panel are for SC and PG state, respectively. In both lower panels, thesolid and dashedcurvesindicate the SC and PG state, respectively.

0 0.5 1T/Tc

0

0.5

1

ρ s(T

)/ρ s

(0)

0 10

1

x=0.18x=0.15x=0.10x=0.055

FIG. 39 Comparison of normalized superfluid densityρs/ρs(0) ver-susT/Tc for experiment (main figure) and theory (inset) in highTc

superconductors with variable doping concentrationx. Both exper-iment and theory show a quasi-universal behavior with respect todoping.

artificial normal state extrapolation in discussing thermody-namics in order to make contact with its experimental coun-terpart. However, it should be stressed that in zero magneticfield, there is no coexistent non-superconducting phase. BCStheory is a rather special case in which there are two possiblephases belowTc, and one can, thereby, use this coexistence tomake a reasonable estimate of the condensation energy. WhenBose condensation needs to be accommodated, there seems tobe no alternative “normal” phase belowTc.

0 50 100T (K)

-30

-20

-10

0

ρ s(T

)-ρ s

(0)

(meV

)

0 40 80

-20

-10

0

x = 0.055x = 0.10x = 0.15x = 0.18

FIG. 40 Offset plot of superfluid densityρs comparing experiment(main figure) with theory (inset). Both experiment and theory showthe same quasi-universal slopedρs/dT for different hole concentra-tionsx.

C. Electrodynamics and Thermal Conductivity

In some ways the subtleties of phase coherent pairing areeven more perplexing in the context of electrodynamics. Fig-ure 12 presents a paradox in which the excitation gap forfermions appears to have little to do with the behavior of thesuperfluid density. To address these data (Stajicet al., 2003b)one may use the formalism of Sec. III.A. One has to introducethe variablex (which accounts for Mott physics) and this isdone via a fit toT ∗(x) in the phase diagram. Here for Fig. 40

39

it is also necessary to fitρs(T = 0, x) to experiment, althoughthis is not important in Fig. 39. The figures show a reasonablecorrespondence (Stajicet al., 2003b) with experiment. Theparadox raised by Fig. 12 is resolved by noting that there arebosonic excitations of the condensate, as in Eq. (113) and thatthese become more marked with underdoping, as pseudogapeffects increase. In this wayρs does not exclusively reflect thefermionic gap, but rather vanishes “prematurely” before thisgap is zero, as a result of pair excitations.

The optical conductivityσ(ω) is similarly modified (Iyen-garet al., 2003; Timusk and Statt, 1999). Indeed, there is anintimate relation betweenρs andσ(ω) known as the f-sumrule. Optical conductivity studies in the literature, boththeoryand experiment, have concentrated on the lowω, T regimeand the interplay between impurity scattering andd wave su-perconductivity (Berlinskyet al., 2000). Also of interest areunusually highω tails (Molegraafet al., 2002; Santander-Syroet al., 2003) in the real part ofσ(ω) which can be inferredfrom sum-rule arguments and experiment. Figure 12 raises athird set of questions which pertain to the more global behav-ior of σ. In the strong PG regime, where∆ has virtually noTdependence belowTc, the BCS-computedσ(ω) will be sim-ilarly T -independent. This is in contrast to what is observedexperimentally whereσ(ω) reflects the sameT dependence asin ρs(T ), as dictated by the f-sum rule.

One may deduce a consequence of this sum rule, based onEq. (113). If we associate the fermionic contributions withthefirst term in square brackets in this equation, and the bosoniccontributions with the second. Note that the fermionic trans-port terms reflect the full∆ just as do the single particle prop-erties. (This may be seen by combining the superconductingand Maki Thompson or PG diagrams in Appendix B). Wemay infer a sum rule constraint on the bosonic contributions,which vanish as expected in the BCS regime where∆pg van-ishes. We write

2

π

∫ ∞

0

dΩ σbosons(Ω, T ) =∆2

pg

∆2

(ns

m

)BCS

(T ) . (138)

The bosonic contributions can be determined most readilyfrom a framework such as time dependent Ginzburg Landautheory, which represents rather well the contributions from theAslamazov-Larkin diagrams, discussed in Appendix B. Thebosons make a maximum contribution atTc.

The resulting optical conductivity (Iyengaret al., 2003) inthe superconducting state is plotted in Fig. 41 below. The keyfeature to note is the decrease in width of the so-called Drudepeak with decreasing temperature. In the present picture thispseudogap effect can be understood as coming from the con-tribution of bosonic excitations of the condensate, which arepresent atTc, but which disappear at lowT . This behavioris reasonably consistent with what is reported experimentally(Timusk and Statt, 1999), where “the effect of the opening ofa pseudogap is a narrowing of the coherent Drude peak at lowfrequency”. The calculations leading to Fig. 41, indicate thatby 20 K most of the conductivity is fermionic except for ahigh frequency tail associated with the bosons. This tail maybe responsible in part for the anomalously highω contribu-tions toσ required to satisfy the sum rule (Molegraafet al.,

0 100 200 300ω [meV]

0

4000

Re

σ [

Ω-1

cm

-1]

20 K88.7 K

150 2000 50 100

10K

300K65K

Re

σ

FIG. 41 Real part of ac conductivity at low and high T belowTc.Here pseudogap effects, which insure that∆(T ) ≈ constant belowTc, require that one go beyond BCS theory to explain the data shownin the inset (Puchkovet al., 1996). In the theory, inelastic and othereffects which presumably lead to the large constant background termobserved in the data have been ignored.

2002; Santander-Syroet al., 2003).Although the normal state calculations have not been done

in detail, once the temperature is less thanT ∗, pseudogap ef-fects associated with the fermionic contribution will alsoleadto a progressive narrowing in the Drude peak with decreasingT , down toTc, due to the opening of an excitation gap. Thismechanism (which is widely ascribed to (Timusk and Statt,1999) in the literature) reflects longer lived electronic states,associated with the opening of this gap. However, it cannot beapplied to explain the progressive narrowing of the conductiv-ity peaks, once∆(T ) becomes temperature independent, as isthe case at lowT . In this way, one can see that pseudogap ef-fects show up in the optical conductivity of underdoped sam-ples, both from the fermionic contributions (near and belowT ∗) and the bosonic contributions (belowTc) as a narrowingof the Drude peak.

The data shown in the inset of Fig. 41 reveal that the con-densate is made from frequency contributions primarily up to≈ 0.15 eV. These data pose a challenge for applying strictBCS theory to this problem, since one consequence of a tem-perature independent∆ in the underdoped phase is the ab-sence ofT dependence in the optical conductivity. This is thesame challenge as posed by Fig. 12. Strikingly there is no ex-plicit “gap feature” in the optical conductivity in the normalstate. Such a gap feature, that is a near vanishing ofσ foran extended range of frequencies, will occur in models wherethe pseudogap is unrelated to superconductivity; it is not seenexperimentally, thus far

By contrast with the electrical conductivity, the thermalconductivity is dominated by the fermionic contributions atessentially allT . This is because the bosonic contributionto the heat current (as in standard TDGL theory (Larkin andVarlamov, 2001)), is negligibly small, reflecting the low en-ergy scales of the bosons. Thermal conductivity, experiments(Sutherlandet al., 2003) in the highTc superconductors pro-

40

vide some of the best evidence for the presence of fermionicd-wave quasiparticles belowTc. In contrast to the ac conduc-tivity, here one sees a universal lowT limit (Lee, 1993), andthere is little to suggest that something other than conventionald-wave BCS physics is going on here. This cannot quite be thecase however, since in the pseudogap regime, the temperaturedependence of the fermionic excitation gap is highly anoma-lous, as shown in Fig. 3, compared to the BCS analogue.

D. Thermodynamics and Pair-breaking Effects

The existence of noncondensed pair states belowTc affectsthermodynamics, in the same way that electrodynamics is af-fected, as discussed above. Moreover, one can predict (Chenet al., 2000) that theq2 dispersion will lead to ideal Bosegas power laws in thermodynamical and transport properties.These will be present in addition to the usual power laws or(for s-wave) exponential temperature dependences associatedwith the fermionic quasiparticles. Note that theq2 dependencereflects the spatial extentξpg, of the composite pairs , and thissize effect has no natural counterpart in true Bose systems.Itshould be stressed that numerical calculations show that thesepair masses, as well as the residueZ are roughlyT indepen-dent constants at lowT . As a result, Eq. (94) implies that∆2

pg = ∆2(T ) − ∆2sc(T ) ∝ T 3/2.

The consequences of these observations can now be listed(Chen et al., 2000). For a quasi-two dimensional system,Cv/T will appear roughly constant at the lowest temperatures,although it vanishes strictly atT = 0 asT 1/2. The superfluiddensityρs(T ) will acquire aT 3/2 contribution in addition tothe usual fermionic terms. By contrast, for spin singlet states,there is no explicit pair contribution to the Knight shift. Inthis way the lowT Knight shift reflects only the fermions andexhibits a scaling withT/∆(0) at low temperatures. Experi-mentally, in the cuprates, one usually sees a finite lowT con-tribution toCv/T . A Knight shift scaling is seen. Finally, alsoobserved is a deviation from the predictedd-wave linear inTpower law inρs. The new power laws inCv andρs are con-ventionally attributed to impurity effects, whereρs ∝ T 2, andCv/T ∝ const. Experiments are not yet at a stage to clearlydistinguish between these two alternative explanations.

Pair-breaking effects are viewed as providing important in-sight into the origin of the cuprate pseudogap. Indeed, thedifferent pair-breaking sensitivities ofT ∗ andTc are usuallyproposed to support the notion that the pseudogap has nothingto do with superconductivity. To counter this inference, a de-tailed set of studies was conducted, (based on the BEC-BCSscenario), of pair-breaking in the presence of impurities (Chenand Schrieffer, 2002; Kaoet al., 2002) and of magnetic fields(Kao et al., 2001). These studies make it clear that the su-perconducting coherence temperatureTc is far more sensitiveto pair-breaking than is the pseudogap onset temperatureT ∗.Indeed, the phase diagram of Fig. 35 which mirrors its experi-mental counterpart, shows the very different, even competingnature ofT ∗ andTc, despite the fact that both arise from thesame pairing correlations.

E. Anomalous Normal State Transport: Nernst and OtherPrecursor Transport Contributions

Much attention is given to the anomalous behavior of theNernst coefficient in the cuprates (Xuet al., 2000). This coef-ficient is rather simply related to the transverse thermoelectriccoefficientαxy which is plotted in Fig. 13. In large part, theorigin of the excitement in the literature stems from the factthat the Nernst coefficient behaves smoothly through the su-perconducting transition. BelowTc it is understood to be as-sociated with superconducting vortices. AboveTc if the sys-tem were a Fermi liquid, there are arguments to prove thatthe Nernst coefficient should be essentially zero. Hence theobservation of a non-negligible Nernst contribution has led tothe picture of normal state vortices.

Another way to view these data is that there are bosonicdegrees of freedom present in the normal state, as in the BCS-BEC crossover approach. To describe this dynamics we beginwith the TDGL scheme outlined in Sec. I.H.1. This formalismmust be modified somewhat however. When “pre-formed”pairs are present at temperaturesT ∗ high compared toTc,the classical bosonic fields of conventional TDGL must be re-placed by their quantum counterparts (Tan and Levin, 2004).We investigate this fairly straightforward extension of TDGLin the next sub-section, and subsequently explore the resultingimplications for transport.

1. Quantum Extension of TDGL

To make progress on the quantum analogue of TDGL westudy a Hamiltonian describing bosons coupled to a parti-cle reservoir of fermion pairs. The latter are addressed ina quantum mechanical fashion. Our treatment of the reser-voir has strong similarities to the approach of Caldeira andLeggett (Caldeira and Leggett, 1983). These bosons are in thepresence of an electromagnetic field, which interacts with thefermion pairs of the reservoir as well, since they have chargee∗ = 2e. For simplicity we assume that the fermions are dis-persionless. This Hamiltonian is given by

H =∑

lm

εlmψ†l (t) exp

(

−ie∗Clm(t))

ψm(t)

+∑

l

e∗φψ†ψ+∑

il

(ai + e∗φ)w†iwi + ηiψ

†wi + η∗iw†iψ

+∑

il

(bi − e∗φ)v†i vi + ζiψ†v†i + ζ∗i viψ

. (139)

Here εlm is the hopping matrix element for the bosons.Clm(t) is the usual phase factor associated with the vectorpotential. Annihilation operators for the reservoir,wi andvi (with infinitesimal coupling constantsηi andζi), representpositive and negative frequencies respectively and need tobetreated separately. The energiesai andbi of the reservoir areboth positive.

The bosonic propagatorT (k, ω) can be exactly computedfrom the equations of motion. HereA(k, ω) = Re2iT (k, ω)

41

0 0.5 1.0 1.5 2.0 2.5

0

1

2

3

4

T (100K)

α xy

T ∗ /Tc=3.1

T ∗ /Tc=7.4

T ∗ /Tc=∞

0 1 20

1

2

3

4

T (100K)

α xy

FIG. 42 Calculated transverse thermoelectric response, which ap-pears in the Nernst coefficient. Inset plots experimental (Xu et al.,2000) data. Both the theoretical and experimental values ofαxy arenormalized by the value ofαxy for an optimally doped sample at2Tc, 0.0232(B/T) V/(KΩm).

is the boson spectral function, and

T (k, ω) ≡(

ω − ε(k) − Σ1(ω) +i

2Σ2(ω)

)−1

(140)

We have

Σ2(ω) ≡ 2π∑

i

|ηi|2δ(ω−ai)−2π∑

i

|ζi|2δ(ω+bi), (141)

with Σ1(ω) defined by a Kramers Kronig transform.The reservoir parametersai, bi, ηi andζi which are of no

particular interest, are all subsumed into the boson self energyΣ2(ω). From this point forward we may ignore these quan-tities in favor of the boson self energy. We reiterate that thistheory makes an important simplification, that the reservoirpairs have no dispersion.For this case one can solve for theexact transport coefficients.

For small, but constant magnetic and electric fields andthermal gradients we obtain the linearized response. For no-tational convenience, we definevab(k) ≡ ∂2

∂ka∂kbε(k). Then

we can write a few of the in-plane transport coefficients whichappear in Eqs. (8) and (9) as

σ = −e∗2

2

∫

d3k

(2π)3dω

2πvxvxA

2(k, ω)∂b(ω)

∂ω(142)

and

αxy = −e∗2Bz

6T

∫

d3k

(2π)3dω

2πvxvxvyyωA

3(k, ω)∂b(ω)

∂ω.

(143)

These equations naturally correspond to their TDGL coun-terparts (except for different phenomenological coefficients)whenT ≈ Tc.

2. Comparison with Transport Data

The formalism of the previous sub-section and in particu-lar, Eqs. (142) and (143), can be used to address transport data

0 0.5 1.0 1.5 2.0 2.5 3.00

0.2

0.4

0.6

0.8

1.0

T (100K)

ρ

2D-UD

1 20

1

2

T (100K)

τ−1(1

00K

)

2D-UD

1 20

1

T (100K)

α xy

2D-UD

FIG. 43 dc resistivityρ (normalized byρQ ≡ s/e2) vs. T , forquasi-2D underdoped (UD) samples with two different forms forµpair. These are shown plotted in the left inset asτ−1 ≡ −µpair/Imγ.The right inset shows correspondingαxy , and arrows indicateTc.

within the framework of BCS-BEC crossover. The results forαxy, which relates to the important Nernst effect, are plottedin Fig. 42 with a subset of the data plotted in the upper rightinset. It can be seen that the agreement is reasonable. In thisway a “pre-formed pair” picture is, at present, a viable alter-native to “normal state vortices”.

Within the crossover scenario, just as in TDGL, there arestrong constraints on other precursor transport effects: para-conductivity, diamagnetism and optical conductivity are all in-directly or directly connected to the Nernst coefficient, inthesense that they all derive from the same dynamical equation ofmotion for the bosons. Analogous inferences have been made(Geshkenbeinet al., 1997), based on similar bosonic general-izations of TDGL.

The calculated behavior of a number of these other proper-ties is summarized (Tan and Levin, 2004) in Figs. 43 and 44below, which indicate both quasi-two dimensional and morethree dimensional behavior. In the plots for the resistivity,ρ, a fairly generic (Leridonet al., 2001) linear-in-T contri-bution 9 from the fermions has been added, as a backgroundterm. A central theme of concern in the literature is whetherone can understand the behavior ofρ simultaneously with thatof αxy. Experiments (Leridonet al., 2001; Watanabeet al.,1997) indicate that there is a small feature corresponding toT ∗ at whichρ deviates from the background linear fermionicterm, but that the deviation is not noticeably large except inthe immediate vicinity ofTc. This is in contrast to the behav-ior of αxy which shows a significant high temperature onset.A key observation from the theoretical calculations shown inthe figures is that the onset temperatures for the Nernst coef-ficient and the resistivity are rather different with the formerbeing considerably higher than the latter. Moreover all onsetsappear to be significantly less thanT ∗. Qualitative agreement

9 In reality, one should include an additional temperature dependence frompseudogapped fermions which presumably has a small upturn,relative to alinear curve, asT decreases.

42

1 20

1

T (100K)

ρ

3D-UD

1 20

1

2

3

T (100K)α x

y

3D-UD

0 1 2−1

0

T (100K)

M 2D,3D

FIG. 44 dc resistivity andαxy, for an underdoped, more three-dimensional, system like YBCO. The arrow indicatesTc. The mag-netizationM is plotted as the inset, and is independent of dimension-ality.

with experiment is not unreasonable. However, a more quan-titative comparison with experiment has not been established,since it depends in detail on a phenomenological input: thetemperature dependence of the pair chemical potentialµpair.

Another important feature of these plots is the fact that thereis virtually no diamagnetic precursor, as was also found to bethe case in simple TDGL. To see this, note that the magne-tization can be written (Schmid, 1969; Tan and Levin, 2004)

µ0M

B= −4

3

αkBT

~c

ξ2ab

s√

1 + (2ξc/s)2,

whereα is the fine structure constant,c is the speed of light,sis the interlayer spacing, andξab (ξc) are coherence lengths inthe direction parallel (perpendicular) to the planes. Fromthisresult we can estimate the order of magnitude of the diamag-netic susceptibility due to preformed pairs. For a typical highTc superconductor,T ∼ 50K, so thatλ ≡ hc

kBT ∼ 3×10−4m,andλ/α ∼ 4 × 10−2m. Because the coherence lengths andthe interlayer spacing are on the nanometer length scale, pre-cursor effects in the diamagnetic susceptibility are of theorder10−6, and thus extremely small, as is consistent with experi-ment (Wanget al., 2001).

In order to make the BCS-BEC scenario more convincingit will be necessary to take these transport calculations belowTc. This is a project for future research and in this context itwill ultimately be important to establish in this picture howsuperconducting state vortices are affected by the persistentpseudogap. Magnetic field induced magnetic order which hasbeen associated with vortices (Lakeet al., 2002) will, more-over, need to be addressed. The analogous interplay of vor-tices and pseudogap will also be of interest in the neutral su-perfluids.

VII. CONCLUSIONS

In this Review we have summarized a large body of workon the subject of the BCS-BEC crossover scenario. In this

context, we have explored the intersection of two very differ-ent fields: highTc superconductivity and cold atom superflu-idity. Theories of cuprate superconductivity can be crudelyclassified as focusing on “Mott physics” which reflects theanomalously small zero temperature superfluid density and“crossover physics”, which reflects the anomalously short co-herence length. Both schools are currently very interestedin explaining the origin of the mysterious pseudogap phase.In this Review we have presented a case for its origin incrossover physics. The pseudogap in the normal state canbe associated with meta-stable pairs of fermions; a (pseudo-gap) energy must be supplied to break these pairs apart intotheir separate components. The pseudogap also persists be-low Tc in the sense that there are non condensed fermion pairexcitations of the condensate. These concepts have a naturalanalogue in self consistent theories of superconducting fluctu-ations, but for the crossover problem the width of the “criticalregion” is extremely large. This reflects the much stronger-than-BCS attractive interaction.

It was not our intent to shortchange the role of Mott physicswhich will obviously be of importance in our ultimate under-standing of the superconducting cuprates. There is, however,much in this regard which is still uncertain associated withestablishing the simultaneous relevance and existence of spin-charge separation (Anderson, 1997), stripes (Kivelsonet al.,2003), and hidden order parameters (Chakravartyet al., 2001).What we do have in hand, though, is a very clear experimen-tal picture of an extremely unusual superconductor in whichsuperconductivity seems to evolve gradually from aboveTc tobelow. We have in this Review tried to emphasize the commonground between highTc superconductors and ultracold super-fluids. These Mott issues may nevertheless, set the agenda forfuture cold atom studies of fermions in optical lattices (Hof-stetteret al., 2002).

The recent discovery of superfluidity in cold fermion gasesopens the door to a new set of fascinating problems in con-densed matter physics. Unlike the bosonic system, there isno ready-made counterpart of Gross-Pitaevskii theory. A newmean field theory which goes beyond BCS and encompassesBEC in some form or another will have to be developed inconcert with experiment.As of this writing, there are fourexperiments where the simple mean field theory discussed inthis review is in reasonable agreement with the data. Theseare the collective mode studies over the entire range of acces-sible magnetic fields (Heiselberg, 2004b; Huet al., 2004). Inaddition in the unitary regime, RF spectroscopy-based pairinggap studies (Kinnunenet al., 2004a), as well as density pro-file (Stajicet al., 2005a) and thermodynamic studies (Kinastet al., 2005) all appear to be compatible with this theory.

The material in this Review is viewed as the first of manysteps in a long process. It was felt, however, that some con-tinuity should be provided from another community whichhas addressed and helped to develop the BCS-BEC crossover,since the early 1990’s when the early signs of the cupratepseudogap were beginning to appear. As of this writing, itappears likely that the latest experiments on cold atoms probea counterpart to this pseudogap regime. That is, on both sides,but near the resonance, fermions form in long lived metastable

43

pair states at higher temperatures than those at which theyBose condense.

Acknowledgments

We gratefully acknowledge the help of our many closecollaborators over the years: J. Maly, B. Janko, I. Kosztin,Y.-J. Kao and A. Iyengar. We also thank our colleaguesT. R. Lemberger, B. R. Boyce, J. N. Milstein, M. L. Chio-falo and M.J. Holland, and most recently, J. E. Thomas, J.Kinast, and A. Turlapov. This work was supported by NSF-MRSEC Grant No. DMR-0213765 (JS,ST and KL), NSFGrant No. DMR0094981 and JHU-TIPAC (QC).

APPENDIX A: Derivation of the T = 0 variationalconditions

The ground state wavefunction is given by

Ψ0 = e−λ2/2+λb†0 |0〉 ⊗ Πk(uk + vka

†ka

†−k)|0〉 (A1)

Thus one has three variational parametersλ, uk, vk with nor-malization conditionu2

k + v2k = 1. We can defineθk such

thatuk = sin θk andvk = cos θk, so that we have only twovariational parameters,λ andθk.

We then have

〈Ψ0|H − µN |Ψ0〉 = (ν − 2µ)λ2 + 2∑

k

(ǫk − µ)v2k

+ 2U∑

k,k′

ukvkuk′vk′ + 2gλ∑

k

ukvk ,

where we have chosenϕk = 1 in Eq. (11) for simplicity.Let us define

∆sc = −2U∑

k

ukvk. (A2)

The variational conditions are

δ〈H − µN〉δλ

= 0 (A3)

and

δ〈H − µN〉δθk

= 0 (A4)

These conditions yield two equations: a relationship betweenthe two condensates

gλ

∆sc=

1

U

g2

ν − 2µ(A5)

and theT = 0 gap equation:

1 + (U +g2

2µ− ν)∑

k

1

2Ek

= 0 . (A6)

AL1 AL2pgΛδ

t pg sct

Σpg Σsc

MT

+ +=

b)Λ

Σ =

a) +

FIG. 45 Self energy contributions (a) and response diagramsforthe vertex correction corresponding toΣpg (b). Heavy lines are fordressed, while light lines are for bare Green’s functions. Wavy linesindicatetpg. See text for details.

Similarly, we can write an equation for the total number offermions

ntot = 〈Ψ0|2∑

k

b†kbk + 2∑

k

a†kak|Ψ0〉 = 2λ2

+∑

k

[

1 − ǫk − µ

Ek

]

where

Ek =

√

(ǫk − µ)2 + ∆2sc, (A7)

with ∆sc = ∆sc − gλ. Thus we have obtained the three equa-tions which characterize the system atT = 0. It should beclear that the variational parameterλ is the same as the con-densate weight so thatλ = φm ≡ 〈bq=0〉.

APPENDIX B: Proof of Ward Identity Below Tc

We now want to establish that the generalized Ward identityin Eq. (106) is correct for the superconducting state as well.The vertexδΛpg may be decomposed into Maki-Thompson(MT ) and two types of Aslamazov-Larkin (AL1, AL2) dia-grams, whose contribution to the response is shown here inFig. 45b. We write

δΛpg ≡ δΛMT + δΛ1AL + δΛ2

AL(Λ) . (B1)

Using conventional diagrammatic rules one can see that theMT term has the same sign reversal as the anomalous super-conducting self energy diagram. This provides insight intoEq. (107). Here, however, the pairs in question are noncon-densed and their internal dynamics (viatpg as distinguishedfrom tsc) requires additionalAL1 andAL2 terms as well,which will ultimately be responsible for the absence of aMeissner contribution from this normal state self energy ef-fect.

44

Note that theAL2 diagram is specific to ourGG0 scheme,in which the field couples to the fullG appearing in theT -matrix through a vertexΛ. It is important to distinguish thevertexΛ which appears in theAL2 diagram from the full EMvertexΛEM .

In particular, we have

Λ = λ + δΛpg − δΛsc , (B2)

where the sign change of the superconducting term (relativeto ΛEM ) is a direct reflection of the sign in Eq. (107).

We now show that there is a precise cancellation betweentheMT andAL1 pseudogap diagrams atQ = 0. This can-cellation follows directly from the normal state Ward identity

Q · λ(K,K +Q) = G−10 (K) −G−1

0 (K +Q) , (B3)

which implies

Q · [δΛ1AL(K,K +Q) + δΛMT (K,K +Q)] = 0 (B4)

so thatδΛ1AL(K,K) = −δΛMT (K,K) is obtained exactly

from theQ→ 0 limit.Similarly, it can be shown that

Q · Λ(K,K +Q) = G−1(K) −G−1(K +Q) (B5)

The above result can be used to infer a relation analogous toEq. (B4) for theAL2 diagram, leading to

δΛpg(K,K) = −δΛMT (K,K) , (B6)

which expresses this pseudogap contribution to the vertexentirely in terms of the Maki-Thompson diagram shown inFig. 45b. It is evident thatδΛMT is simply the pseudogapcounterpart ofδΛsc, satisfying

−δΛMT (K,K) =∂Σpg(K)

∂k. (B7)

Therefore, one observes that forT ≤ Tc

δΛpg(K,K) =∂Σpg(K)

∂k, (B8)

which establishes that Eq. (106) applies to the superconduct-ing phase as well. As expected, there is no direct Meissnercontribution associated with the pseudogap self-energy.

APPENDIX C: Quantitative Relation between BCS-BECcrossover and Hartree-TDGL

In this appendix we make more precise the relation betweenHartree-approximated TDGL theory and theT -matrix of ourGG0 theory. The Ginzburg-Landau (GL) free energy func-tional in momentum space is given by (Hassing and Wilkins,1973)

F [Ψ] = N(0)∑

q

|Ψq|2(ǫ+ ξ21q2)

+b02

∑

qi

Ψ∗q1

Ψ∗q2

Ψq3Ψq1+q2−q3

, (C1)

whereΨq are the Fourier components of the order parame-ter Ψ(r, t), ǫ = T−T∗

T∗ , ξ1 is the GL coherence length,b0 =

[1/T 2π2]78ζ(3), T ∗ is the critical temperature when feedbackeffects from the quartic term are neglected, andN(0) is thedensity of states at the Fermi level in the normal state. Thequantity〈|Ψq|2〉 is determined self-consistently via

〈|Ψq|2〉 =

∫

DΨe−βF [Ψ]|Ψq|2∫

DΨe−βF [Ψ], (C2)

whereF [Ψ] is evaluated in the Hartree approximation (Sta-jic et al., 2003b). Our self consistency condition can be thenwritten as

〈|Ψq|2〉 =T

N(0

[

ǫ+ b0∑

q′

〈|Ψq′0|2〉 + ξ21q2

]−1

. (C3)

If we sum Eq. (C3) overq and identify∑

q〈|Ψq0|2〉 with ∆2,we obtain a self-consistency equation for the energy ”gap” (orpseudogap)∆ aboveTc

∆2 =∑

q

T

N(0)

[

ǫ+ b0∆2 + ξ21q

2

]−1

, (C4)

∆2 =∑

q

T

N(0)

1

−µpair(T ) + ξ21q2, (C5)

where

µpair(T ) = −ǫ− b0∆2 . (C6)

Note that the critical temperature is renormalized downwardwith respect toT ∗ and satisfies

µpair(Tc) = 0 . (C7)

To compare with GL theory we expand ourT -matrix equa-tions to first order in the self energy correction. TheT -matrixcan be written in terms of the attractive coupling constantUas

t(Q) =U

1 + Uχ0(Q) + Uδχ(Q), (C8)

where

χ0(Q) =∑

K

G0(Q−K)G0(K) . (C9)

Defining

∆2 = −∑

Q

t(Q) (C10)

we arrive at the same equation as was derived in Section II.C

Σ(K) ≈ −G0(−K)∆2 (C11)

45

where one can derive a self consistency condition on∆2 interms of the quantityδχ(0) (which is first order inΣ), whichsatisfies

δχ(0) = −b0N(0)∆2, (C12)

implying that

δχ(0) = −b0T∑

q

1

ǫ+ ξ21q2 − δχ(0)/N(0)

, (C13)

which coincides with the condition derived earlier in Eq. (C4).

APPENDIX D: Convention and Notation

1. Notation

We follow standard notations as much as possible. They aresummarized below.EF — Fermi energykF — Fermi momentum~ — Planck constantkB — Boltzmann constantc — Speed of lighte — Electron chargeTc — Critical temperature for (superfluid/superconducting)

phase transitionT ∗ — Pair formation or pseudogap onset temperature.T — Temperatureµ, µpair , µB — Fermionic, pair and bosonic chemical po-

tential, respectively.∆ — Fermionic excitation gap∆sc — Superconducting/superfluid order parameter∆pg — PseudogapFour vectorK ≡ (iωn,k),

∑

K ≡ T∑

n

∑

k, whereωn =(2n+ 1)πT is the odd (fermionic) Matsubara frequency.

Four vectorQ ≡ (iΩn,q),∑

Q ≡ T∑

n

∑

q, whereΩn =2nπT is the even (bosonic) Matsubara frequency.f(x) = 1/(ex/kBT + 1) — Fermi distribution functionb(x) = 1/(ex/kBT − 1) — Bose distribution functionG(K), G0(K) — Full and bare Green’s functions for

fermionsD(Q), D0(Q) — Full and bare Green’s functions for Fes-

hbach bosons.Σ(K), ΣB(Q) — Self energy of fermions and bosons, re-

spectively.χ(Q) — Pair susceptibilityt(Q) — T -matrixλ, Λ — Bare and full vertex, respectively.ν, ν0 — Magnetic detuning parameter.as, a∗s — S-wave inter-fermion scattering lengthU , U0 — Renormalized and bare strength of the attrac-

tive interaction between fermions in the open-channel.U0 ∝as(ν = +∞).Uc — Critical coupling strength at which the two-body

scattering length diverges.A = (φ,A) — Four vector potentialJ = (ρ,J) — Four current

Kµν(Q) — Electromagnetic response kernelǫk = k2/2m — Bare fermion dispersion in free spaceξk = ǫk − µ — Bare fermion dispersion measured from

chemical potential.Ek — Bogoliubov quasiparticle dispersionu2k = 1

2 (1 + ξk/Ek), v2k = 1

2 (1 − ξk/Ek) — Coherencefactor as given in BCS theory.ntot = n+2(nb+nb0) — Total, fermion, finite momentum

boson, condensed boson density, respectively.V (r) = 1

2mω2r2 — Harmonic trap potential

We use alphabetic ordering when multiple authors appeartogether.

We always refer to the absolute value when we refer to theinteraction parametersU , U0, g, g0 as increasing or decreas-ing.

2. Convention for units

Throughout this Review, we use the convention for unitswhere it is not explicitly spelled out:

~ = kB = c = 1.EF = TF = kF = 2m = 1.In a harmonic trap,EF , TF andkF are defined by the non-

interacting Fermi gas with the same total particle numberN .EF = ~ω(3N)1/3. We further take the Thomas-Fermi radiusRTF = 1.

In this convention, the fermion density in a homogeneousgas in three dimensions isn = 1/3π2, the harmonic trapfrequencyω = 2, and the total fermion number in a trap isN = 1/24.

In the two-channel model, the units areEF /k3F for U and

U0, andEF /k3/2F for g andg0.

Our fermionic chemical potentialµ is measured with re-spect to the bottom of the energy band, which leads to (i)µ = EF in the non-interacting limit atT = 0, and (ii) µchanges sign when the system crosses the boundary betweenfermionic and bosonic regimes.

3. Abbreviations

BCS — Bardeen-Cooper-Schrieffer (theory)BEC — Bose-Einstein condensationGL — Ginzburg-Landau (theory)TDGL — Time-dependent Ginzburg-Landau (theory)RPA — Random phase approximationLDA — Local density approximationGP — Gross-PitaevskiiTF — Thomas-Fermi (approximation)NSR — Nozieres and Schmitt-RinkFLEX — FLuctuation EXchangeAB — Anderson-Bogoliubov (mode)PG — PseudogapSC — SuperconductorAFM — AntiferromagnetFB — Feshbach bosonLSCO — La1−xSrxCuO4

46

BSCCO — Bi2Sr2CaCu2O8+δ

YBCO — YBa2Cu3O7−δ

ARPES — Angle-resolved photoemission spectroscopySTM — Scan tunneling microscopyRF – Radio frequency (spectroscopy)SIN — Superconductor-insulator-normal metal (tunneling

junction)SI — Superconductor-insulator (transition)3D — Three dimensions

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