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Applying BCS-BEC Crossover Theory To High Temperature Superconductors and UltracoldAtomic Fermi Gases

Qijin Chen1, Jelena Stajic2, and K. Levin11James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637 and

2Los Alamos National Laboratory, Los Alamos, New Mexico, 87545(Dated: January 1, 2018)

This review is written at the time of the twentieth anniversary of the discovery of high temperature super-conductors, which, nearly coincides with the important discovery of the superfluid phases of ultracold trappedfermionic atoms. We show how these two subjects have much in common. Both have been addressed from theperspective of the BCS-Bose Einstein condensation (BEC) crossover scenario, which is designed to treat shortcoherence length superfluids with transition temperatureswhich are “high”, with respect to the Fermi energy. Ageneralized mean field treatment of BCS-BEC crossover at general temperaturesT , based on the BCS-Leggettground state, has met with remarkable success in the fermionic atomic systems. Here we summarize this suc-cess in the context of four different cold atom experiments,all of which provide indications, direct or indirect,for the existence of a pseudogap. This scenario also provides a physical picture of the pseudogap phase in theunderdoped cuprates which is a central focus of highTc research. We summarize successful applications ofBCS-BEC crossover to key experiments in highTc systems including the phase diagram, specific heat, andvortex core STM data, along with the Nernst effect, and exciting recent data on the superfluid density in veryunderdoped samples,

PACS numbers: 03.75.-b, 74.20.-zKeywords: Bose-Einstein condensation, BCS-BEC crossover, fermionic superfluidity, highTc superconductivity

Contents

I. Introduction 1A. Historical Background 1B. Fermionic Pseudogaps and Meta-stable Pairs: Two

Sides of the Same Coin 2C. Introduction to highTc Superconductivity:

Pseudogap Effects 4D. Summary of Cold Atom Experiments: Crossover in

the Presence of Feshbach Resonances 6

II. Theoretical Formalism for BCS-BEC crossover 7A. Many-body Hamiltonian and Two-body Scattering

Theory 7B. T- Matrix-Based Approaches to BCS-BEC

Crossover in the Absence of Feshbach Effects 8C. Extending conventional Crossover Ground State to

T 6= 0: T-matrix scheme in the presence ofclosed-channel molecules 9

III. Physical Implications: Ultracold AtomSuperfluidity 10A. Tc Calculations and Trap Effects 10B. Thermodynamical Experiments 11C. Temperature Dependent Particle Density Profiles 12D. RF Pairing gap Spectroscopy 13E. Collective Breathing Modes atT ≈ 0 14

IV. Physical Implications: High Tc Superconductivity 15A. Phase Diagram and Superconducting Coherence 15B. Electrodynamics in the superconducting phase 15C. Bosonic Power Laws and Pairbreaking Effects 16D. Anomalous Normal State Transport: Nernst

Coefficient 18

V. Conclusions 18

Acknowledgments 19

References 19

I. INTRODUCTION

A. Historical Background

Most workers in the field of highTc superconductivitywould agree that we have made enormous progress in thelast 20 years in characterizing these materials and in identi-fying key theoretical questions and constructs. Experimentalprogress, in large part, comes from transport studies [1, 2]in addition to three powerful spectroscopies: photoemission[3, 4], neutron [5, 6, 7, 8, 9, 10, 11, 12] and Josephson inter-ferometry [13, 14, 15]. Over the last two decades, theoristshave emphasized different aspects of the data, beginning withthe anomalous normal state associated with the highestTc sys-tems (“optimal doping”) and next, establishing the nature andimplications of the superconducting phase, which was ulti-mately revealed to have ad-wave symmetry. Now at the timeof this twenty year anniversary, one of the most exciting areasof research involves the normal state again, but in the lowTc

regime, where the system is “underdoped” and in proximityto the Mott insulating phase. We refer to this unusual phase asthe “pseudogap state”.

This pseudogap phase represents a highly anomalous formof superconductivity in the sense that there is an excitationgap present at the superfluid transition temperatureTc wherelong range order sets in. The community has struggled withtwo generic classes of scenarios for explaining the pseudogap

2

and its implications belowTc. Either the excitation gap is inti-mately connected to the superconducting order reflecting, forexample, the existence of “pre-formed pairs”, or it is extrin-sic and associated with a competing ordered state unrelatedtosuperconductivity.

The emphasis of this Review is on the pseudogap state asaddressed by a particular preformed pair scenario which hasits genesis in what is now referred to as “BEC-Bose Einsteincondensation (BEC) crossover theory”. Here one contem-plates that the attraction (of unspecified origin) which leadsto superconductivity is stronger than in conventional super-conductivity. In this way fermion pairs form before they Bosecondense, much as in a Bose superfluid. In support of thisviewpoint for the cuprates are the observations that: (i) the co-herence lengthξ for superconductivity is anomalously short,around10A as compared with1000A for a typical supercon-ductor. Moreover (ii) the transition temperatures are anoma-lously high, and (iii) the systems are close to two dimen-sional (2D) (where pre-formed pair or fluctuation effects areexpected to be important). Finally, (iv) the pseudogap has thesamed-wave symmetry [16] as the superconducting order pa-rameter [3, 4] and there seems to be a smooth evolution of theexcitation gap from above to belowTc.

To investigate this BCS-BEC crossover scenario we havethe particular good fortune today of having a new classof atomic physics experiments involving ultracold trappedfermions which, in the presence of an applied magnetic field,have been found to have a continuously tunable attractive in-teraction. At high fields the system exhibits BCS-like super-fluidity, whereas at low fields one sees BEC-like behavior.

This Review presents a consolidated study of both the pseu-dogap phase of the cuprates and recent developments in ultra-cold fermionic superfluids. The emphasis of these cold atomexperiments is on the so-called unitary or strong scatteringregime, which is between the BEC and BCS limits, but onthe fermionic side. The superfluid state in this intermediateregime is also referred to in the literature as a “resonant super-fluid” [17, 18]. Here we prefer to describe it as the “pseudogapphase”, since that is more descriptive of the physics and un-derlines the close analogy with highTc systems. Throughoutthis Review we will use these three descriptive phrases inter-changeably.

B. Fermionic Pseudogaps and Meta-stable Pairs: Two Sides ofthe Same Coin

BCS-BEC crossover theory is based on the observations ofEagles [19] and Leggett [20] who independently noted thatthe BCS ground state wavefunction

Ψ0 = Πk(uk + vkc†kc

†−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 BEC. What is essential is that the

chemical potentialµ of the fermions be self consistently com-puted asU varies.

The variational parametersvk and uk are usually repre-sented by the two more directly accessible parameters∆sc(0)andµ, which characterize the fermionic system. Here∆sc(0)is the zero temperature superconducting order parameter.These fermionic parameters are uniquely determined in termsof U and the fermionic densityn. The variationally deter-mined self consistency conditions are given by two BCS-likeequations which we refer to as the “gap” and “number” equa-tions respectively.

∆sc(0) = −U∑

k

∆sc(0)1

2Ek

n = 2∑

k

[

1− ǫk − µ

Ek

]

(2)

where

Ek ≡√

(ǫk − µ)2 +∆2sc(0) (3)

andǫk = ~2k2/2m are the dispersion relations for the Bo-

goliubov quasiparticles and free fermions, respectively.Anadditional advantage of this formalism is that Bogoliubov deGennes theory, a real space implementation of this groundstate, can be used to address the effects of inhomogeneity andexternal fields atT = 0. This has been widely used in thecrossover literature.

Within this ground state there have been extensive studies[21] of collective modes [22, 23] and effects of two dimen-sionality [22]. Nozieres and Schmitt-Rink were the first [24]to address non-zeroT . We will outline some of their con-clusions later in this Review. Randeria and co-workers refor-mulated the approach of Nozieres and Schmitt-Rink (NSR)and moreover, raised the interesting possibility that crossoverphysics might be relevant to high temperature superconduc-tors [22]. Subsequently other workers have applied this pic-ture to the highTc cuprates [25, 26, 27] and ultracold fermions[17, 18, 28, 29] as well as formulated alternative schemes[30, 31] for addressingT 6= 0. Importantly, a number of ex-perimentalists, most notably Uemura [32], have claimed evi-dence in support [33, 34, 35] of the BCS-BEC crossover pic-ture for highTc materials.

Compared to work on the ground state, considerably lesshas been written on crossover effects at non-zero temperaturebased on Eq. (1). Because our understanding has increasedsubstantially since the pioneering work of NSR, and becausethey are the most interesting, this review is focused on thesefinite T effects.

The importance of obtaining a generalization of BCS the-ory which addresses the crossover from BCS to BEC groundstate at general temperaturesT ≤ Tc cannot be overesti-mated. BCS theory as originally postulated can be viewedas a paradigm among theories of condensed matter systems; itis complete, in many ways generic and model independent,and well verified experimentally. The observation that thewavefunction of Eq. (1) goes beyond strict BCS theory, sug-gests that there is a larger mean field theory to be addressed.

3

0 1 2U/Uc

-1

0

1

µ/E

FBCS PG BEC

FIG. 1: Behavior of theT = 0 chemical potentialµ in the threeregimes.µ is essentially pinned at the Fermi temperatureEF in theBCS regime, whereas it becomes negative in the BEC regime. ThePG (pseudogap) case corresponds to non-Fermi liquid based super-conductivity in the intermediate regime.

Equally exciting is the possibility that this mean field the-ory can be discovered and simultaneously tested in a verycontrolled fashion using ultracold fermionic atoms [17, 18].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. 1, 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 of a Fermi surface,and we say that it is “fermionic”. For negativeµ, the Fermisurface is gone and the material is “bosonic”.

The new and largely unexplored physics of this problem liesin the fact that once outside the BCS regime, but before BEC,superconductivity or superfluidity emerge out of a very exotic,non-Fermi liquid normal state. Emphasized in Fig. 1 is thisintermediate (i.e., pseudogap or PG) regime having positiveµ which we associate with non-Fermi liquid based supercon-ductivity [25, 36, 37]. Here, the onset of superconductivityoccurs in the presence of fermion pairs. Unlike their counter-parts in the BEC limit, these pairs are not infinitely long lived.Their presence is apparent even in the normal state where anenergy must be applied to create fermionic excitations. Thisenergy cost derives from the breaking of the metastable pairs.Thus we say that there is a “pseudogap” (PG) at and aboveTc. It will be stressed throughout this Review that gaps in thefermionic spectrum and bosonic degrees of freedom are two

Τc

sc∆

∆∆(Τ)

T*

FIG. 2: Contrasting behavior of the excitation gap∆(T ) and or-der parameter∆sc(T ) versus temperature in the pseudogap regime.The height of the shaded region reflects the number of noncondensedpairs, at each temperature.

sides of the same coin. A particularly important observationto make is that the starting point for crossover physics is basedon the fermionic degrees of freedom. A non-zero value of theexcitation gap∆ is equivalent to the presence of metastableor stable fermion pairs. And it is only in this indirect fashionthat we can probe the presence of these “bosons”, within theframework 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. More generally, in the presence ofa moderately strong attractive interaction it pays energeticallyto take some advantage and to form pairs (say roughly at tem-peratureT ∗) within the normal state. Then, for statistical rea-sons these bosonic degrees of freedom ultimately are drivento condense atTc < T ∗, as in BEC.

Just as there is a distinction betweenTc andT ∗, there mustbe a distinction between the superconducting order param-eter ∆sc and the excitation gap∆. In Fig. 2 we present aschematic plot of these two energy parameters. It may be seenthat the order parameter vanishes atTc, as in a second orderphase transition, while the excitation gap turns on smoothlybelow T ∗. It should also be stressed that there is only onegap energy scale in the ground state [20] of Eq. (1). Thus∆sc(0) = ∆(0).

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 fermionic pairs are pre-formed,∆is essentially 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 behavioris illustrated in Fig. 3.

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 4 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.

4

∆

Tc T TTc

∆

FIG. 3: Comparison of temperature dependence of excitationgapsin BCS (left) and BEC (right) limits. The gap vanishes atTc for theformer while it is essentiallyT -independent for the latter.

BCS Pseudogap (PG) BECc c c

FIG. 4: 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. For atomic Fermi gases,the“virtual molecules” in the PG case consist primarily of “Cooper”pairs of fermionic atoms.

C. 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 stateat T = 0 to T = Tc. We begin with a brief overview [1, 2]of pseudogap effects in 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 [26, 27, 38, 39, 40] that a BCS-BEC crossover-induced pseudogap is the origin of the mys-terious normal state gap observed in high temperature super-conductors. While this is a highly contentious subject someofthe arguments in favor of this viewpoint (beyond those listedin Section I A) rest on the following observations: (i) To agood approximation the pseudogap onset temperature [41, 42]T ∗ ≈ 2∆(0)/4.3 which satisfies the BCS scaling relation.(ii) There is widespread evidence for pseudogap effects bothabove [1, 2] as well as (iii) below [43, 44]Tc. (iv) In addi-tion, it has also been argued that short coherence length su-perconductors may quite generally exhibit a distinctive formof superconductivity [32] which sets them apart from conven-tional superconductors. One might want, then, to concentrateon this more generic feature, (rather than on more exotic as-

Hole concencentration x

Tem

pera

ture

T

AFM

Pseudo Gap

Tc

T*

SC

FIG. 5: 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.

0

5

10

15

20

25

0 50 100 150 200 250 300

T (K)

Spe

ctra

l Gap

FIG. 6: Temperature dependence of the excitation gap at the antin-odal point (π,0) in Bi2Sr2CaCu2O8+δ (BSCCO) for three differentdoping concentrations from near-optimal (discs) to heavy underdop-ing (inverted triangles), as measured by angle-resolved photoemis-sion spectroscopy (from Ref. 3).

pects), which they have in common with other superconduc-tors in their distinctive class.

In Fig. 5 we show a sketch of the phase diagram for thehole-doped copper oxide superconductors. Herex representsthe concentration of holes which can be controlled by, say,adding Sr substitutionally to La1−xSrxCuO4. At zero andsmallx the system is an antiferromagnetic (AFM) insulator.Precisely at half filling (x = 0) we understand this insulatorto derive from Mott effects. These Mott effects may or maynot be the source of the other exotic phases indicated in thediagram, i.e., the superconducting (SC) and the “pseudogap”(PG) phases. Once AFM order disappears the system remainsinsulating until a critical hole concentration (typicallyarounda few percent) when an insulator-superconductor transition isencountered. Here photoemission studies [3, 4] suggest thatonce this line is crossed,µ appears to be positive. Forx ≤ 0.2,the superconducting phase has a non-Fermi liquid (or pseudo-gapped) normal state [2]. We note an important aspect of thisphase diagram at lowx. As the pseudogap becomes stronger(T ∗ increases), superconductivity as reflected in the magni-

5

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. 7: Temperature dependence of fermionic excitation gaps∆ and superfluid densityρs for various doping concentrations (from Ref. 44).When∆(Tc) 6= 0, there is little correlation between∆(T ) andρs(T ); this figure suggests that something other than fermionic quasi-particles(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 ).

tude ofTc becomes weaker.Figure 6 indicates the temperature dependence of the ex-

citation gap for three different hole stoichiometries. Thesedata [3] were taken from angle resolved photoemission spec-troscopy (ARPES) measurement. For one sample shown ascircles, (corresponding roughly to “optimal” doping) the gapvanishes roughly atTc as might be expected for a BCS su-perconductor. At the other extreme are the data indicated byinverted 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 under-doped sample (smallx), which from the phase diagram canbe seen to have a rather lowTc. Moreover,Tc is not evidentin these data on underdoped samples.

While the highTc community has focused on pseudogapeffects aboveTc, there is a good case to be made that these ef-fects also persist below. STM data [33] taken belowTc withina vortex core indicate that there is a clear depletion of the den-sity of states around the Fermi energy in the normal phasewithin the core. These data underline the fact that the exis-tence of an energy gap has little or nothing to do with the ex-istence of phase coherent superconductivity. It also underlinesthe fact that pseudogap effects effectively persist belowTc; thenormal phase underlying superconductivity forT ≤ Tc is nota Fermi liquid.

Analysis of thermodynamical data [2, 43] has led to a sim-ilar inference. For the PG case, the entropy extrapolated intothe superfluid phase, based on Fermi liquid theory, becomesnegative. In this way Loram and co-workers [43] deducedthat the normal phase underlying the superconducting stateis not a Fermi liquid. Rather, they claimed to obtain properthermodynamics, it must be assumed that this state containsa persistent pseudogap. In this way they argued for a dis-tinction between the excitation gap∆ and the superconduct-ing order parameter, within the superconducting phase. Tofit their data they presume a modified fermionic dispersionEk =

√

(ǫk − µ)2 +∆2(T ) where

∆2(T ) = ∆2sc(T ) + ∆2

pg (4)

Here∆pg is taken on phenomenological grounds to beT -independent. While Eq. (4) is also found in BCS-BEC

crossover theory, there are important differences. In the lat-ter approach∆pg → 0 asT → 0.

Finally, Fig. 7 makes the claim for a persistent pseudogapbelowTc in an even more suggestive way. Figure 7a repre-sents a schematic plot of excitation gap data such as are shownin Fig. 6. Here the focus is on temperatures belowTc. Mostimportantly, this figure indicates that theT dependence in∆varies dramatically as the stoichiometry changes. Thus, intheextreme underdoped regime, where PG effects are most in-tense, 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 ). Figure 7 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.

The phase diagram also suggests that pseudogap effects be-come stronger with underdoping. How does one accommo-date this in the BCS-BEC crossover scenario? At the simplestlevel one may argue that as the system approaches the Mottinsulating limit, fermions are less mobile and the effective-ness of the attraction increases. In making the connection be-tween the strength of the attraction and the variablex in thecuprate phase diagram we will argue that it is appropriate tosimply fit T ∗(x). In this Review we do not emphasize Mottphysics because it is not particularly relevant to the atomicphysics problem. It also seems to be complementary to theBCS-BEC crossover scenario. It is understood that both com-ponents are important in highTc superconductivity. It shouldbe stressed that hole concentrationx in the cuprates plays therole of applied magnetic field in the cold atom system. Theseare the external parameters which serve to tune the BCS-BECcrossover.

Is there any evidence for bosonic degrees of freedom in

6

the normal state of highTc superconductors? The answeris unequivocally yes:meta-stable bosons are observable assuperconducting fluctuations. These effects are enhanced inthe presence of the quasi-two dimensional lattice structure ofthese materials. In the underdoped case, one can think ofT ∗

as marking the onset of preformed pairs which are closelyrelated to fluctuations of conventional superconductivitythe-ory, but which are made more robust as a result of BCS-BECcrossover effects, that is, stronger pairing attraction. Anum-ber of people have argued [45, 46] that fluctuating normalstate vortices are responsible for the anomalous transportbe-havior of the pseudogap regime. It has been proposed [47] thatthese data may alternatively be interpreted as suggesting thatbosonic degrees of freedom are present in the normal state.

D. Summary of Cold Atom Experiments: Crossover in thePresence 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 and 40K, which have been trapped and cooledvia magnetic and optical means. Typically these traps con-tain 105 atoms at very low densities≈ 1013 cm−3. Herethe Fermi temperature in a trap can be estimated to be ofthe order of a micro-Kelvin. It was argued on the basis ofBCS theory alone [48], and rather early on (1997), that thetemperatures associated with the superfluid phases may beattainable in these trapped gases. This set off a search forthe “holy grail” of fermionic superfluidity. That a Fermi de-generate state could be reached at all is itself quite remark-able; this was was first reported [49] by Jin and deMarco in1999. By late 2002 reports of unusual hydrodynamics in adegenerate Fermi gas indicated that strong interactions werepresent [50]. 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 [51, 52] and in quark-gluon plasmas [53, 54]. More-over, there has been a fairly extensive body of analytic workon the ground state properties of this regime [55, 56], whichgoes beyond the simple mean field wave 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 [58, 59, 60] as reported in mid-2003either via three body recombination [61] or by sweeping themagnetic field across a Feshbach resonance. Moreover, theyare extremely long lived [59]. From this work it was rel-atively straightforward to anticipate that a Bose condensatewould also be achieved. Credit goes to theorists such as Hol-land [17] and to Griffin [29] and their co-workers for recog-nizing that the superfluidity need not be only associated withcondensation of long lived bosons, but in fact could also de-rive, as in BCS, from fermion pairs. In this way, it was arguedthat a suitable tuning of the attractive interaction via Feshbachresonance effects, would lead to a realization of a BCS-BECcrossover.

FIG. 8: (Color) Spatial density profiles of a molecular cloudoftrapped40K atoms in the BEC regime in the transverse directionsafter 20 ms of free expansion (from Ref. 57), showing thermalmolec-ular cloud aboveTc (left) and a molecular condensate (right) belowTc. (a) shows the surface plots, and (b) shows the cross-sectionsthrough images (dots) with bimodal fits (lines).

By late 2003 to early 2004, four groups [57, 62, 63, 64] hadobserved the “condensation of weakly bound molecules” (thatis, on theas > 0 side of resonance), and shortly thereafter anumber had also reported evidence for superfluidity on theBCS side [65, 66, 67, 68]. The BEC side is the more straight-forward since the presence of the superfluid is reflected in abi-modal distribution in the density profile. This is shown inFig. 8 from Ref. 57, and is conceptually similar to the behav-ior for condensed Bose atoms [69]. On the BEC side but nearresonance, the estimatedTc is about0.3TF , with condensatefractions varying from 20% or so to nearly 100%. The con-densate lifetimes are relatively long in the vicinity of reso-nance, 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 particu-larly exciting. The atomic community, for the most part, feltthe previous counterpart observations on the BEC side wereexpected and not significantly different from condensationinBose atoms. The evidence for this new form of “fermionic su-perfluidity” rests on studies [65, 66] that perform fast sweepsfrom negativeas to positiveas across the resonance. The fieldsweeps allow, in principle, a pairwise projection of fermionicatoms (on the BCS side) onto molecules (on the BEC side). Itis presumed that in this way one measures the momentum dis-tribution of fermion pairs. The existence of a condensate wasthus inferred. Other experiments which sweep across the Fes-hbach resonance adiabatically, measure the size of the cloudafter release [64] or within a trap [70].

Evidence for superfluidity on the BCS side, which does notrely on the sweep experiments, has also been deduced fromcollective excitations of a fermionic gas [67, 71]. Pairinggapmeasurements with radio frequency (RF) spectroscopy probes

7

[68] have similarly been interpreted [72] as providing supportfor the existence of superfluidity, although more directly theseexperiments establish the existence of fermion pairs. Quiterecently, evidence for a phase transition has been presentedvia thermodynamic measurements and accompanying theory[73]. The latter, like the theory [72] of RF experiments [68],is based on the formalism presented in this Review. A mostexciting and even more recent development has been the ob-servation of vortices [74] which appears to provide a smokinggun for the existence of the superfluid phase.

II. THEORETICAL FORMALISM FOR BCS-BECCROSSOVER

A. Many-body Hamiltonian and Two-body Scattering Theory

We introduce the Hamiltonian [17, 29, 75] used in thecold atom and highTc crossover studies. The most generalform for this Hamiltonian consists of two types of interac-tion effects: those associated with the direct interactionbe-tween fermions parametrized byU , and those associated with“fermion-boson” interactions, whose strength is governedbyg.

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.)

(5)

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

q = q2/2M , andν is an important parameterwhich represents the magnetic field-induced “detuning”. Herewe use the convention~ = kB = c = 1. In this two channelproblem the ground state wavefunction is slightly modifiedand given by

Ψ0 = Ψ0 ⊗ΨB0 (6)

where the molecular or Feshbach boson contributionΨB0 is as

given in Ref. 76.Whether both forms of interactions are needed in either sys-

tem is still under debate. The bosons (b†k) of the cold atomproblem [17, 18] are referred to as belonging to the “closedchannel”. These spin-singlet molecules represent a separatespecies, not to be confused with the (“open channel”) fermionpairs (a†ka

†−k), which are associated with spin triplet. As a

result of virtual occupation of the bound state of the closedchannel the interaction between open channel fermions canbe tuned (through applied magnetic field) to vary from weakto very strong.

In this review we will discuss the behavior of crossoverphysics both with and without the closed-channel. Previousstudies of highTc superconductors have invoked a similarbosonic term [27, 77, 78, 79] as well, although less is knownabout its microscopic origin. This fermion-boson couplingis

not to be confused with the coupling between fermions anda “pairing-mechanism”-related boson ([b + b†]a†a) such asphonons in a metal superconductor. The couplingb†aa andits Hermitian conjugate represent a form of sink and sourcefor creating fermion pairs, in this way inducing superconduc-tivity in some ways, as a by-product of Bose condensation.

It is useful at this stage to introduce thes-wave scatteringlength,a, defined by the low energy limit of two body scatter-ing in vacuum. We begin with the effects ofU only, presum-ing thatU is always an attractive interaction (U < 0) whichcan be arbitrarily varied,

m

4πa≡ 1

U+∑

k

1

2ǫk(7)

We may define a critical valueUc of the potential as that asso-ciated with the binding of a two particle state in vacuum. Wecan write down an equation forUc given by

U−1c = −

∑

k

1

2ǫk(8)

although specific evaluation ofUc requires that there be acut-off imposed on the above summation, associated withthe range of the potential.The fundamental postulate ofcrossover theory is that even though the two-body scatter-ing length changes abruptly at the unitary scattering condi-tion (|a| = ∞), in the N-body problem the superconductivityvaries smoothly through this point.

Provided we redefine the appropriate “two body” scatteringlength, Equation (7) holds even in the presence of Feshbacheffects [28, 29]. It has been shown thatU in the above equa-tions is replaced by

U → Ueff ≡ U +g2

2µ− ν(9)

and we writea → a∗. Experimentally, the two body scatteringlengtha∗ varies with magnetic fieldB. Thus we have

m

4πa∗≡ 1

Ueff+∑

k

1

2ǫk(10)

More precisely the effective interaction between two fermionsis momentum and energy dependent. It arises from a secondorder process involving emission and absorption of a closed-channel molecular boson. The net effect of the direct plusindirect interactions is given byUeff (Q) ≡ U + g2D0(Q),whereD0(Q) ≡ 1/[iΩn−ǫmb

q −ν+2µ] is the non-interactingmolecular boson propagator. Here and throughout we usea four-momentum notation,Q ≡ (q, iΩn), and its analyt-ical continuation,Q → (q,Ω + i0+), and write

∑

Q ≡T∑

Ωn

∑

q, whereΩn is a Matsubara frequency. What ap-pears in the gap equation, however, isUeff (Q = 0) whichwe define to beUeff . When the open-channel attractionUis weak, clearly,2µ ≤ ν is required so that the Feshbach-induced interaction is attractive. In the extreme BEC limitν = 2µ. However, when a deep bound state exists in theopen channel, such as in40K, the system may evolve into a

8

600 700 800 900 1000B [G]

-3

-2

-1

0

1

2

3

k Fa s

BEC PG BCS

834 G

FIG. 9: Characteristic behavior of the scattering length for 6Li in thethree regimes.

metastable state such that2µ > ν in the BEC regime andthere is a point on the BCS side whereUeff = 0 precisely.

Figure 9 presents a plot of this scattering lengthkF as ≡kFa

∗ for the case of6Li. It follows that as is negative whenthere is no bound state, it tends to−∞ at the onset of thebound state and to+∞ just as the bound state stabilizes. Itremains positive but decreases in value as the interaction be-comes increasingly strong. The magnitude ofas is arbitrarilysmall in both the extreme BEC and BCS limits, but with op-posite sign.

B. T- Matrix-Based Approaches to BCS-BEC Crossover in theAbsence of Feshbach Effects

To address finite temperature in a way which is consistentwith Eq. (1), or with alternative ground states, one introducesa T -matrix approach. Here one solves self consistently forthe single 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 ina (generalized) mean field averaging procedure; they arise ex-clusively from the fermions and are sufficiently weak so as notto lead to any incomplete condensation in the ground state, asis compatible with Eq. (1).

In this section we demonstrate that at theT -matrix levelthere are three distinct schemes which can be implemented toaddress BCS-BEC crossover physics. AboveTc, quite gener-ally one writes for thet-matrix

t(Q) =U

1 + Uχ(Q)(11)

and theories differ only on what is the nature of the pairsusceptibilityχ(Q), and the associated self energy of thefermions. BelowTc one can also consider aT -matrix ap-proach 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 C.

In analogy with Gaussian fluctuations, Nozieres andSchmitt-Rink considered [24]

χ0(Q) =∑

K

G0(K)G0(Q−K) (12)

with self energy

Σ0(K) =∑

Q

t(Q)G0(Q −K) , (13)

whereG0(K) is the noninteracting fermion Green’s function.The number equation of the Nozieres Schmitt-Rink scheme[22, 24] is then deduced in an approximate fashion [80] byusing a leading order series forG with

G = G0 + G0Σ0G0 . (14)

It is straightforward, however, to avoid this approximation inDyson’s equation, and a number of groups [31, 37] have ex-tended NSR in this way.

Similarly one can consider

χ(Q) =∑

K

G(K)G(Q −K) (15)

with self energy

Σ(K) =∑

Q

t(Q)G(Q −K) . (16)

This latter scheme has been also extensively discussed in theliterature, by among others, Haussmann [81], Tchernyshyov[82] and Yamada and Yanatse [40].

Finally, we can contemplate the asymmetric form [25] forthe T -matrix, so that the coupled equations fort(Q) andG(K) are based on

χ(Q) =∑

K

G(K)G0(Q−K) (17)

with self energy

Σ(K) =∑

Q

t(Q)G0(Q−K) . (18)

It should be noted, however, that this asymmetric form can bederived from the equations of motion by truncating the infi-nite series at the three particle level,G3, and then factorizingtheG3 into one- and two-particle Green’s functions [83]. Theother two schemes are constructed diagrammatically or fromagenerating functional, (as apposed to derived from the Hamil-tonian).

It will be made clear in what follows that, if one’s goal isto extend the usual crossover ground state of Eq. (1) to finitetemperatures, then one must choose the asymmetric form forthe pair susceptibility, as shown in Eq. (17). Other approachessuch as the NSR approach toTc, or that of Haussmann leadto different ground states which should, however, be very in-teresting in their own right. These will need to be charac-terized in future. Indeed, the work of Strinati group has also

9

emphasized that the ground state associated with theTc calcu-lations based on NSR is distinct from that in the simple meanfield theory of Eq. (1), and they presented some aspects of thiscomparison in Ref. 84.

Other support for thisGG0-basedT -matrix scheme comesfrom its equivalence to self consistent Hartree-approximatedGinzburg-Landau theory [85]. Moreover, there have been de-tailed studies [86] to demonstrate how the superfluid densityρs can be computed in a fully gauge invariant (Ward Identityconsistent) fashion, so that it vanishes at the self consistentlydeterminedTc. Such studies are currently missing for the caseof the other twoT -matrix schemes.

C. Extending conventional Crossover Ground State toT 6= 0:T-matrix scheme in the presence of closed-channel molecules

In the T -matrix scheme we employ, the pairs are de-scribed by the pair susceptibilityχ(Q) =

∑

K G0(Q −K)G(K)ϕ2

k−q/2 whereG depends on a BCS-like self en-

ergy Σ(K) ≈ −∆2G0(−K)ϕ2k. Throughout this section

ϕk ≡ exp−k2/2k20 introduces a momentum cutoff, wherek0 represents the inverse range of interaction, which is as-sumed infinite for a contact interaction.

The noncondensed pairs [87] have propagatortpg(Q) =Ueff (Q)/[1 + Ueff (Q)χ(Q)], whereUeff is the effectivepairing interaction which involves the direct two-body interac-tion U as well as virtual excitation processes associated withthe Feshbach resonance [29, 87]. At smallQ, tpg can be ex-panded, after analytical continuation (iΩn → Ω + i0+), as

tpg(Q) ≈ Z−1

Ω− Ωq + µpair + iΓQ. (19)

The parameters appearing in Eq. (19) are discussed in moredetail in Ref. 75. HereZ−1 is a residue andΩq = q2/2M∗

the pair dispersion, whereM∗ is the effective pair mass. Thelatter parameter as well as the pair chemical potentialµpair

depends on the important, but unknown, gap parameter∆through the fermion self energyΣ. The decay widthΓQ isnegligibly small for smallQ belowTc.

While there are alternative ways of deriving the self con-sistent equations which we use, (such as a decoupling of theGreen’s function equations of motion [83]), here we presentan approach which shows how thisGG0-basedT -matrixscheme has strong analogies with the standard theory of BEC.But, importantly this BEC is embedded in a self consistenttreatment of the fermions. Physically, one should focus on∆ as reflecting the presence of bosonic degrees of freedom.In the fermionic regime (µ > 0), it represents the energy re-quired to break the pairs, so that∆ is clearly associated withthe presence of “bosons”. In the bosonic regime,∆2 directlymeasures the density of pairs.

In analogy with the standard theory of BEC, it is expected[87] that∆ contains contributions from both noncondensedand condensed pairs. The associated densities are propor-tional to ∆2

pg(T ) and ∆2sc(T ), respectively. We may write

the first of several constraints needed to close the set of equa-tions. (i) One has a constraint on thetotal number of pairs[75]which can be viewed as analogous to the usual BEC numberconstraint

∆2(T ) = ∆2sc(T ) + ∆2

pg(T ) . (20)

To determine∆, (ii) one imposes the BEC-like constraintthat the pair chemical potential vanishes in the superfluidphase:

µpair = 0 T ≤ Tc . (21)

This yields

t−1pg (Q → 0) = 0 = U−1

eff (0) + χ(0) (22)

so that

U−1

eff (0) +∑

k

1− 2f(Ek)

2Ek

ϕ2k = 0 , (23)

Importantly, belowTc, ∆ satisfies the usual BCS gap equa-tion. Here we introduce the quasiparticle dispersionEk =√

(ǫk − µ)2 +∆2ϕ2k, and f(x) is the Fermi distribution

function.(iii) In analogy with the standard derivation of BEC, the

total contribution ofnoncondensedpairs is readily computedby simply adding up their number, based on the associatedpropagator

∆2pg ≡ −

∑

Q

tpg(Q) . (24)

One can rewrite Eq. (24) so that it looks more directly likea number equation, by introducing the Bose distribution func-tion b(x) for noncondensed pairs as

∆2pg = Z−1

∑

b(Ωq, T ) , (25)

so that the noncondensed pair density is given byZ∆2pg. Note

that the right hand sides of the previous two equations dependon the unknown∆ through the self energy appearing inGwhich, in turn enterstpg or Ωq. Also note that atT = 0,∆pg = 0 so that all pairs are condensed as is consistent withthe mean-field BCS-Leggett ground state.

Finally, in analogy with the standard derivation of BEC,(iv) one can then compute the number ofcondensedpairs as-sociated with∆sc, given that one knows the total∆ and thenoncondensed component.

Despite this analogy with BEC, fermions are the fundamen-tal particles in the system. It is their chemical potentialµ thatis determined from the number conservation constraint

n = nf + 2nb0 + 2nb ≡ nf + 2ntotb . (26)

Herenb0 andnb represent the density of condensed and non-condensed closed-channel molecules, respectively,ntot

b is thesum, andnf = 2

∑

K G(K) is the atomic density associ-ated with the open-channel fermions. These closed channel

10

-2 0 2 41/kFa

0

0.1

0.2

0.3

Tc/T

F

BECPGBCS

FIG. 10: Typical behavior ofTc as a function of1/kF a in a ho-mogeneous system.Tc follows the BCS predictions and approachesthe BEC asymptote0.218TF in the BEC limit. In the intermediateregime, it reaches a maximum around1/kF a = 0 and a minimumaround whereµ = 0.

fermions have a propagatorD(Q), which we do not discussin much detail in order to make the presentation simpler. Here

nb = −∑

Q

D(Q) ≈ Zb

∑

q

b(Ωq) , (27)

where b(x) is the Bose distribution function. The renor-malized propagatorD(Q) is given by the same equation asEq. (19) with a different residueZ−1 → Zb.

In this way the system of equations is complete [87]. Thenumerical scheme is straightforward in principle. We compute∆ (andµ) via Eqs. (23) and (26), to determine the contributionfrom the condensate∆sc via Eqs. (20) and (24). AboveTc,this theory must be generalized to solve self consistently forµpair which no longer vanishes [88].

III. PHYSICAL IMPLICATIONS: ULTRACOLD ATOMSUPERFLUIDITY

In this section we compare four distinct classes of exper-iments on ultracold trapped fermions with theory. These arethermodynamics [73, 88], temperature dependent density pro-files [89], RF pairing gap spectroscopy [68, 90, 91], and col-lective mode measurements [67, 71]. We address all fourexperiments in the context of the mean field ground state ofEq. (1), and its finite temperature extension discussed in Sec-tion II C. That there appears to be good agreement betweentheory and experiment lends rather strong support to the sim-ple mean field theory, which is at the center of this Review.Interestingly, pseudogap effects are evident in various ways inthese experiments and this serves to tie the ultracold fermionsto the highTc superconductors.

A. Tc Calculations and Trap Effects

Before turning to experiment, it is important to discuss thebehavior of the transition temperature which is plotted as a

-2 0 2 41/kFa

0

0.1

0.2

0.3

0.4

0.5

Tc/T

F

BECPGBCS

FIG. 11: Typical behavior ofTc of a Fermi gas in a trap as a functionof 1/kF a. It follows BCS prediction in the weak coupling limit,1/kF a << −1, and approaches the BEC asymptote0.518TF in thelimit 1/kF a → +∞. In contrast to the homogeneous case in Fig. 10,the BEC asymptote is much higher due to a compressed profile fortrapped bosons.

function of scattering length in Fig. 10 for the homogeneouscase, presumings-wave pairing. We discuss the effects ofd-wave pairing in Section IV in the context of application tothe cuprates. Starting from the BCS regime this figure showsthatTc initially increases as the interaction strength increases.However, this increase competes with the opening of a pseu-dogap or excitation gap∆(Tc). Technically, the pairs becomeeffectively heavier before they form true bound states. Even-tually Tc reaches a maximum (very near unitarity) and thendecreases slightly until field strengths corresponding to thepoint whereµ becomes zero. At this field value (essentiallywhereTc is minimum), the system becomes a “bosonic” su-perfluid, and beyond this pointTc increases slightly to reachthe asymptote corresponding to an ideal Bose gas. Trapeffects change these results only quantitatively as seen inFig. 11. To treat these trap effects one introduces the localdensity approximation (LDA) in whichTc is computed underthe presumption that the chemical potentialµ → µ − V (r). Here we consider a spherical trap withV (r) = 1

2mω2r2.

The Fermi energyEF is determined by the total atom num-berN via EF ≡ kBTF = ~ω(3N)1/3 ≡ ~

2k2F /2m, wherekF is the Fermi wavevector at the center of the trap. It canbe seen that the homogeneous curve is effectively multipliedby an “envelope” curve when a trap is present. This envelope,with a higher BEC asymptote, reflects the fact that the particledensity at the center of the trap is higher in the bosonic, rela-tive to the fermionic case. In this wayTc is relatively higherin the BEC regime, as compared to BCS, whenever a trap ispresent.

Figure 12 presents a plot of the position dependent excita-tion gap∆(r) and particle densityn(r) profile over the extentof the trap. An important point needs to be made: becausethe gap is largest at the center of the trap, bosonic excitationswill be dominant there. At the edge of the trap, by contrast,where fermions are only weakly bound (since∆(r) is small),the excitations will be primarily fermionic. We will see theimplications of these observations as we examine thermody-namic and RF spectra data in the ultracold gases.

11

0 0.5 1r/RTF

0

0.2

0.4

0.6

0.8

1

n(r)

(ar

b. u

nit)

, ∆/E

F

∆(r)

n(r)

At unitarity

FIG. 12: Typical spatial profile ofT = 0 densityn(r) and fermionicexcitation gap∆(r) of a Fermi gas in a trap. The curves are com-puted at unitarity, where1/kF a = 0. HereRTF is the Thomas-Fermi radius.

B. Thermodynamical Experiments

Figure 13 present a plot which compares experiment andtheory in the context of thermodynamic experiments [73, 88]on trapped fermions. Plotted on the vertical axis is the en-ergy which can be input in a controlled fashion experimen-tally. The horizontal axis is temperature which is calibratedtheoretically based on an effective temperatureT introducedphenomenologically, and discussed below. The experimen-tal data are shown for the (effectively) non-interacting case aswell as unitary. In this discussion we treat the non-interactingand BCS cases as essentially equivalent since∆ is so smallon the scale of the temperatures considered. The solid curvescorrespond to theory for the two cases. Although not shownhere, even without a temperature calibration, the data suggestsa phase transition is present in the unitary case. This can beseen as a result of the change in slope ofE(T ) as a functionof T .

The phenomenological temperatureT is relatively easy tounderstand. What was done experimentally to deduce thistemperature was to treat the unitary case as an essentially freeFermi gas to, thereby, infer the temperature from the widthof the density profiles, but with one important proviso: a nu-merical constant is introduced to account for the fact that thedensity profiles become progressively narrower as the systemvaries from BCS to BEC. This systematic variation in the pro-file widths reflects the fact that in the free Fermi gas case,Pauli principle repulsion leads to a larger spread in the particledensity than in the bosonic case. And the unitary regime hasa profile width which is somewhere in between, so that oneparametrizes this width by a simple function ofβ. We canthink of β as reflecting bosonic degrees of freedom, withinan otherwise fermionic system. Atβ ≡ 0 the system is afree Fermi gas. The principle underlying this rescaling of thenon-interacting gas is known as the “universality hypothesis”[92, 93]. At unitarity, the Fermi energy of the non-interactingsystem is the only energy scale in the problem (for the widelyused contact potential) since all other scales associated withthe two-body potential drop out whenas → ±∞. We re-fer to this phenomenological fitting temperature procedureas

0.1 1T/TF

0.01

0.1

1

10

E(t

heat

)/E

0 -1

0 0.2 0.8√1+β ~

T0

0.2

0.8

T/T

F

T /TF =0.27

Tc

FIG. 13: (color) EnergyE vs physical temperatureT . The uppercurve and data points correspond to the BCS or essentially free Fermigas case, and the lower curve and data correspond to unitarity. Thelatter provide indications for a phase transition. The inset shows howtemperature must be recalibrated belowTc. From Ref. [73].

Thomas-Fermi (TF) fits.An interesting challenge was to relate this phenomenolog-

ical temperatureT to the physical temperatureT ; more pre-cisely one compares

√1 + βT andT . This relationship is

demonstrated in the inset of Fig. 13. And it was in this waythat the theory and experiment could be plotted on the samefigure, as shown in the main body of Fig. 13. The inset wasobtained using theory only. The theoretically produced pro-files were analyzed just as the experimental ones to extract√1 + βT and compare it to the actualT . AboveTc no re-

calibration was needed as shown by the straight line goingthrough the diagonal. BelowTc the phenomenologically de-duced temperatures were consistently lower than the physicaltemperature. That the normal state temperatures needed noadjustment shows that the phenomenology captures importantphysics. It misses, however, an effect associated with the pres-ence of a condensate which we will discuss shortly.

We next turn to a more detailed comparison of theory andexperiment for the global and lowT thermodynamics. Fig-ure 14 presents a blow-up ofE at the lowestT comparing theunitary and non-interacting regimes. The agreement betweentheory and experiment is quite good. In the figure, the tem-perature dependence ofE reflects primarily fermionic excita-tions at the edge of the trap, although there is a small bosoniccontribution as well. It should be noted that the theoreticalplots were based on fittingβ to experiment by picking a mag-netic field very slightly off resonance. [In the simple meanfield theoryβ = −0.41, and in Monte Carlo simulations [56]β = −0.54. Both these theoretical numbers lie on either sideof experiment [73] whereβ = −0.49].

Figure 15 presents a wider temperature scale plot which,again, shows very good agreement. Importantly one can seethe effect of a pseudogap in the unitary case. The tempera-tureT ∗ can be picked out from the plots as that at which thenon-interacting and unitary curves intersect. This correspondsroughly toT ∗ ≈ 2Tc.

12

0 0.1 0.2 0.3 0.4 0.5T/TF

0

1

2

E/E

F

Tc=0.29

Theory, noninteractingTheory, unitarynoninteractingunitary

FIG. 14: (color) Low temperature comparison of theory (curves) andexperiments (symbols) in terms ofE/EF (EF = kBTF ) per atomas a function ofT/TF , for both unitary and noninteracting gases ina Gaussian trap. From Ref. 73.

0 0.5 1 1.5T/TF

0

2

4

E/E

F

Tc=0.29

Theory, noninteractingTheory, unitarynoninteractingunitary

FIG. 15: (color) Same as Fig. 14 but for a much larger range of tem-perature. The quantitative agreement between theory and experimentis very good.The fact that the two experimental (and the two theoret-ical) curves do not merge until higherT ∗ > Tc is consistent with thepresence of a pseudogap.

C. Temperature Dependent Particle Density Profiles

In order to understand more deeply the behavior of the ther-modynamics, we turn next to a comparison of finiteT densityprofiles. Experiments which measure these profiles [70, 94]all report that they are quite smooth at unitarity, without anysigns of the bimodality seen in the BEC regime. We discussthese profiles in terms of the four panels in Fig. 16. These fig-ures are a first step in understanding the previous temperaturecalibration procedure.

In this figure we compare theory and experiment for theunitary case. The experimental data were estimated to corre-spond to roughly this same temperature (T/TF = 0.19) basedon the calibration procedure discussed above. The profilesshown are well within the superfluid phase (Tc ≈ 0.3TF atunitarity). This figure presents Thomas-Fermi fits [94] to (a)the experimental and (b) theoretical profiles as well as (c) their

-1 0 1x (100µm)

0

0.5

1

1.5

-1 0 1x/RTF

0

0.5

1

1.5TheoryExperiment

0 0.2 0.4 0.6T/TF

0

0.005

0.01

0.015

Tc=0.272

-1 0 1x/RTF

0

0.5

1

1.5Comparison

n(x) n(x)

n(x)χ2

(a) (b)

(c)(d)

FIG. 16: (color) Temperature dependence of (a) experimental one-dimensional spatial profiles (circles) and TF fit (line) fromRef. 94,(b) TF fits (line) to theory both atT ≈ 0.7Tc ≈ 0.19TF (circles) and(c) overlay of experimental (circles) and theoretical (line) profiles, aswell as (d) relative rms deviations (χ2) associated with these fits totheory at unitarity. The circles in (b) are shown as the line in (c). Theprofiles have been normalized so thatN =

∫

n(x)dx = 1, and wesetRTF = 100 µm in order to overlay the two curves.χ2 reaches amaximum aroundT = 0.19TF .

comparison, for a chosenRTF = 100 µm, which makes itpossible to overlay the experimental data (circles) and theo-retical curve (line). Finally Fig. 16d indicates the relativeχ2

or root-mean-square (rms) deviations for these TF fits to the-ory. This figure was made in collaboration with the authorsof Ref. [94]. Two of the three dimensions of the theoreticaltrap profiles were integrated out to obtain a one-dimensionalrepresentation of the density distribution along the transversedirection:n(x) ≡

∫

dydz n(r).This figure is in contrast to earlier theoretical studies which

predict a significant kink at the condensate edge which ap-pears not to have been seen experimentally [70, 94]. More-over, the curves behave monotonically with both temperatureand radius. Indeed, in the unitary regime the generalized TFfitting procedure of Ref. 94 works surprisingly well. Andthese reasonable TF fits apply to essentially all temperaturesinvestigated experimentally [94], as well as theoretically, in-cluding in the normal state.

It is important to establish why the profiles are so smooth,and the condensate is, in some sense, rather invisible, exceptfor its effect on the TF-inferred-temperature. This apparentsmoothness can be traced to the presence of noncondensedpairs of fermions which need to be included in any consistenttreatment. Indeed, these pairs belowTc are a natural counter-part of the pairs aboveTc which give rise to pseudogap effects.

To see how the various contributions enter into the trap pro-file, in Fig. 17 we plot a decomposition of this profile for var-ious temperatures from below to aboveTc. The various colorcodes indicate the condensate along with the noncondensedpairs and the fermions. This decomposition is based on thesuperfluid density so that all atoms participate in the conden-

13

0

0.02

0.04

0.06nnpair

nsnQP

0

0.02

0.04

Den

sity

Pro

files

0 0.5 1 1.50

0.02

0.04

00.511.5r/RTF

0 0.5 1 1.50 0.5 1 1.5 00.511.5r/RTF

0 0.5 1 1.5

T/Tc = 1.5

1.0

0.75

0.5

0.25

0

Unitary

FIG. 17: (color) Decomposition of density profiles at various temper-atures at unitarity. Here green (light gray) refers to the condensate,red (dark gray) to the noncondensed pairs and blue (black) tothe ex-cited fermionic states.Tc = 0.27TF , andRTF is the Thomas-Fermiradius.

sation atT = 0. This, then, forms the basis for addressingboth thermodynamics and RF pairing-gap spectroscopy in thisReview.

The figure shows that byT = Tc/2 there is a reasonablenumber of excited fermions and bosons. As anticipated earlierin Section II C, the latter are at the trap edge and the formerin the center. ByT = Tc the condensate has disappeared andthe excitations are a mix of fermions (at the edge) and bosonstowards the center. Indeed, the noncondensed bosons are stillpresent byT = 1.5Tc, as a manifestation of a pseudogap ef-fect. Only for somewhat higherT ≈ 2Tc do they disappearaltogether, so that the system becomes a non-interacting Fermigas.

Two important points should be made. The noncondensedpairs clearly are responsible for smoothing out what otherwisewould be a discontinuity [92, 95] between the fermionic andcondensate contributions. Moreover, the condensate shrinksto the center of the trap asT is progressively raised. It is thisthermal effect which is responsible for the fact that the TFfitting procedure for extracting temperature leads to an under-estimate as shown in the inset to Fig. 13. The presence of thecondensate tends to make the atomic cloud smaller so that thetemperature appears to be lower in the TF fits.

D. RF Pairing gap Spectroscopy

Measurements [68] of the excitation gap∆ have been madeby using a third atomic level, called|3〉, which does not partic-ipate in the superfluid pairing. Under application of RF fields,one component of the Cooper pairs, called|2〉, is presumablyexcited to state|3〉. If there is no gap∆ then the energy it takesto excite|2〉 to |3〉 is the atomic level splittingω23. In the pres-ence of pairing (either above or belowTc) an extra energy∆must be input to excite the state|2〉, as a result of the breakingof the pairs. Figure 18 shows a plot of the spectra near unitar-ity for four different temperatures, which we discuss in moredetail below. In general for this case, as well as for the BCS

0.4

0.4

0.0

0.0

0.0

0.4

0.4

0.0

1.1

0.85

< 0.4

T/T = 1.2c

−20 0 20 40

Fra

ctio

nal l

oss

in |2

>

RF offset (kHz)

FIG. 18: Experimental RF Spectra at unitarity. The temperatures la-beled in the figure were computed theoretically at unitaritybased onadiabatic sweeps from BEC. The two top curves, thus, correspondto the normal phase, thereby, indicating pseudogap effects. HereEF = 2.5µK, or 52 kHz. From Ref. 68.

and BEC limits, there are two peak structures which appear inthe data: the sharp peak atω23 ≡ 0 which is associated with“free” fermions at the trap edge and the broader peak whichreflects the presence of paired atoms; more directly this broadpeak derives from the distribution of∆ in the trap. At highT (compared to∆), only the sharp feature is present, whereasat low T only the broad feature remains. The sharpness ofthe free atom peak can be understood as coming from a largephase space contribution associated with the2 → 3 excita-tions [91]. Clearly, these data alone do not directly indicatethe presence of superfluidity, but rather they provide strongevidence for pairing.

As pointed out in Ref. [90] these experiments serve as acounterpart to superconducting tunneling in providing infor-mation about the excitation gap. A theoretical understandingof these data was first presented in Ref. 72 using the frame-work of Section II C. Subsequent work [91] addressed thesedata in a more quantitative fashion as plotted in Fig. 19. Herethe upper and lower panels correspond respectively to inter-mediate and low temperatures. For the latter one sees thatthe sharp “free atom” peak has disappeared, so that fermionsat the edge of the trap are effectively bound at these lowT . Agreement between theory and experiment is quite sat-isfactory, although the total number of particles was adjustedsomewhat relative to the experimental estimates.

It is interesting to return to the previous figure (Fig. 18)and to discuss the temperatures in the various panels. Whatis measured experimentally are temperaturesT ′ which cor-

14

0

0.1

0.2

0.3

0.4

-20 0 20 40RF detuning (kHz)

0

0.1

0.2

0.3

0.4

T/TF = 0.25

T/TF = 0.09

Spe

ctra

l int

ensi

ty (

arb.

uni

ts)

FIG. 19: Comparison of calculated RF spectra (solid curve,Tc ≈0.29TF ) with experiment (symbols) in a harmonic trap calculated at822 G for the two lower temperatures. The temperatures were chosenbased on Ref. [68]. The particle number was reduced by a factor of 2,as found to be necessary in addressing another class of experiments[84]. The dashed lines are a guide to the eye. From Ref. 91.

respond to the temperature at the start of a sweep from theBEC limit to unitarity. Here fits to the BEC-like profiles areused to deduceT ′ from the shape of the Gaussian tails in thetrap. Based on knowledge about thermodynamics (entropyS),and givenT ′, one can then compute the final temperature inthe unitary regime, assumingS is constant. Indeed, this adi-abaticity has been confirmed experimentally in related work[70]. We find that the four temperatures are as indicated in thefigures. Importantly, one can conclude that the first two casescorrespond to a normal state, albeit close toTc. Importantly,these figures suggest that a pseudogap is present as reflectedby the broad shoulder above the narrow free atom peak.

E. Collective Breathing Modes atT ≈ 0

We turn, finally, to a comparison between theory [96, 97]and experiment [67, 71, 93, 98] for the collective breathingmodes within a trap atT ≈ 0. The very good agreementhas provided some of the earliest and strongest support forthe simple mean field theory of Eq. (1). Interestingly, MonteCarlo simulations which initially were viewed as a superiorapproach, lead to significant disagreement between theory andexperiment [99]. Shown in Fig. 20 is this comparison for theaxial mode in the inset and the radial mode in the main bodyof the figure as a function of magnetic field. The experimentaldata are from Ref. 67. The original data on the radial modesfrom Ref. 71, was in disagreement with that of Ref. 67, butthis has since been corrected [99], and there is now a consis-tent experimental picture from both the Duke and the Inns-bruck groups for the radial mode frequencies.

At T = 0, calculations of the mode frequencies can be re-

FIG. 20: Breathing mode frequencies as a function ofκ ≈1.695(kF a)

−1, from Tosiet al. [96]. The main figure and inset plotthe transverse and axial frequencies, respectively. The solid curvesare calculations [96] based on BCS-BEC crossover theory atT = 0,and the symbols plot the experimental data from Kinastet al. [67].

duced to a calculation of an equation of state forµ as a func-tion of n. One of the most important conclusions from thisfigure is that the behavior in the near-BEC limit (which is stillfar from the BEC asymptote) shows that the mode frequenciesdecreasewith increasing magnetic field. This is opposite toearlier predictions [100] based on the behavior of true bosonswhere a Lee-Yang term would lead to an increase. Indeed, thepair operators do not obey the commutation relations of truebosons except in the zero density orkF a → 0+ limit [101].Figure 20, thus, underlines the fact that fermionic degreesoffreedom (or compositeness) are still playing a role at thesemagnetic fields. There are predictions in the literature [102]that one needs to achievekFa somewhat less than0.3 (ex-perimentally, the smallest values for these experiments are 0.3and 0.7 for the various groups) in order to approach the truebosonic limit. At this point, then, the simple mean field theorywill no longer be adequate. Indeed, there are other indications[103] of the breakdown of this mean field in the extreme BEClimit which are, physically, reflected in the width of the parti-cle density profiles. This originates from an overestimate (byroughly a factor of 3) of the size of the effective “inter-boson”scattering length.

Overall the mean field theory presented here looks verypromising. Indeed, the agreement between theory and exper-iment is better than one might have anticipated. For the col-lective mode frequencies, it appears to be better than MonteCarlo calculations [99]. Nevertheless, uncertainties remain.Theories which posit a different ground state will need to becompared with the four experiments discussed here. It is, fi-nally, quite possible that incompleteT = 0 condensation willbecome evident in future experiments. If so, an alternativewavefunction will have to be contemplated [102, 104]. Whatappears to be clear from the current experiments is that, justas in highTc superconductors, the ultracold fermionic super-fluids exhibit pseudogap effects. These are seen in thermo-dynamics, in RF spectra and in the temperature dependenceof the profiles (through the noncondensed pair contributions).

15

Moreover, while not discussed here, at finiteT , damping ofthe collective mode frequencies seems to change qualitatively[93] at a temperature which is close to the estimatedT ∗.

Looking to the future, at an experimental level, new pairinggap spectroscopies appear to be emerging at a fairly rapid pace[105, 106]. These will further test the present and subsequenttheories. Indeed, recently, a probe of the closed channel frac-tion [106] has been analyzed [107] within the present frame-work and has led to good quantitative agreement between the-ory and experiment.

IV. PHYSICAL IMPLICATIONS: HIGH Tc

SUPERCONDUCTIVITY

A. Phase Diagram and Superconducting Coherence

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 21 shows a plot of the two impor-tant temperaturesTc andT ∗ as a function of increasing attrac-tive coupling. On the left is BCS and the right is PG. The BECregime is not visible. This is becauseTc disappears before itcan be accessed. This disappearance ofTc is relatively easyto understand. Because thed-wave pairs are more extended(than theirs-wave counterparts) they experience Pauli princi-ple repulsion more intensely. Consequently the pairs localize(their mass is infinite) well before the fermionic chemical po-tential is negative [38].

The competition betweenT ∗ andTc, in which asT ∗ in-creases,Tc decreases, is also apparent in Fig. 21. This isa consequence of pseudogap effects. More specifically, thepairs become heavier as the gap increases in the fermionicspectrum, competing with the increase ofTc due to the in-creasing pairing strength. It is interesting to compare Fig. 21with the experimental phase diagram plotted as a function ofthe doping concentrationx in Fig. 5. If one inverts the hori-zontal axis (and ignores the unimportant AFM region) the twoare very similar. To make an association from couplingU tothe variablex, it is reasonable to fitT ∗. It is not particularlyuseful to implement this last step here, since we wish to em-phasize crossover effects which are not complicated by “Mottphysics”.

Because of quasi-two dimensionality, the energy scales ofthe vertical axis in Fig. 21 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-culations show that in a strictly 2D 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 in 2D.

The presence of pseudogap effects raises an interesting setof issues surrounding the signatures of the transition which

0 1 2Attractive coupling constant

0

0.1

0.2

T/E

F

Pseudogap

Superconductor

NormalT*Tc

FIG. 21: Typical phase diagram for a quasi-two dimensionald-wavesuperconductor on a tight-binding lattice at high fillingn ≈ 0.85 perunit cell; here the horizontal axis corresponds to−U/4t, wheret isthe in-plane hopping matrix element.

the 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, given that a gap is already present at theonset of superconductivity. One understands how phase co-herence shows up in BCS theory, since the ordered state isalways accompanied by the appearance of an excitation gap.

To address these coherence effects one has to introduce adistinction [110] between the self energy associated with non-condensed and condensed pairs. This distinction is blurredby the approximations made in Section II C. Within this im-proved scheme [110] superconducting coherence effects canbe probed as, presented in Fig. 22, along with a compari-son to experiment. Shown are the results of specific heatand tunneling calculations and their experimental counter-parts [2, 33]. The latter measures, effectively, the density offermionic states. Here the label “PG” corresponds to an ex-trapolated normal state in which we set the superconductingorder parameter∆sc to zero, but maintain the the total excita-tion gap∆ to be the same as in a phase coherent, supercon-ducting state. Agreement between theory and experiment issatisfactory.

B. Electrodynamics in the superconducting phase

In some ways the subtleties of phase coherent pairing areeven more perplexing in the context of electrodynamics. Fig-ure 7 presents a paradox in which the excitation gap forfermions appears to have little to do with the behavior ofthe superfluid density. This superfluid density can be readilycomputed within the BCS-BEC crossover scenario [25, 44].Particularly important is to include all excitations of thecon-densate in a fully consistent fashion, compatible with ther-

16

-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. 22: Extrapolated normal state (PG) and superconducting state (SC) contributions to SIN tunneling and thermodynamics (left), as well ascomparison with experiments (right) on tunneling for BSCCO[33] and on specific heat for Y0.8Ca0.2Ba2Cu3O7−δ [108]. The theoretical SINcurve is calculated forT = Tc/2, while the experimental curves are measured outside (dashed line) and inside (solid line) a vortex core.

modynamics, and which is also manifestly gauge invariant.To make contact with electrodynamic experiments, one has tointroduce the variablex and this is done via a fit toT ∗(x)in the phase diagram. In addition it is also necessary to fitρs(T = 0, x) to experiment, and we do so here, noting that[32] the Uemura relationρs(0, x) ∝ Tc(x) no longer holdsfor very underdoped samples [109, 111]. By fitting thesex-dependent quantities we are, in effect accounting for at leastsome aspects of Mott physics. The paradox raised by Fig. 7is resolved by noting that there are bosonic excitations of thecondensate [25] and that these become more marked with un-derdoping, as pseudogap effects increase. In this wayρs doesnot exclusively reflect the fermionic gap, but rather vanishes“prematurely” before this gap is zero, as a result of pair exci-tations of the condensate.

This theory can be quantitatively compared with experi-ment. Figure 23 presents theoretical and experimental plotsof the lower critical field,Hc1(T ), for a group of severely un-derdoped YBCO crystals as considered in Ref. 109. There itwas argued thatHc1(T ) ∝ ρs(T ), so that the lower criticalfield effectively measures the in-plane superfluid density.Ex-perimentally what is directly measured is the magnetizationwith applied field parallel to thec-axis. The experimental re-sults are shown on the lower two panels and theory on theupper two. The left hand figures plotHc1(T ) vs T and theright hand figures correspond to a rescaling of this functioninthe formHc1(T )/Hc1(0) vsT/Tc. Theoretically, it is foundthat the fermionic contribution leads to a linearT dependenceat lowT , associated withd-wave pairing, whereas the bosonicterm introduces aT 3/2 term. Quite remarkably even when theUemura relation no longer holds, there is still a “universality”in the normalized plots as shown in both theory and exper-iment by the right hand figures. It should be noted that the

experimental plot contains (atTc = 55.5K) a slightly differ-ent cuprate phase known as the ortho-II phase, which does notlie on the universal curves. The universality found here canbeunderstood as associated with the fact thatTc(x), rather than∆(x), is the fundamental energy scale inρs(T, x). The reasonthat∆(x) is not the sole energy scale is that bosonic degreesof freedom are also present, and help to driveρs to zero atTc. By contrast, Fermi-liquid based approaches [109, 112] as-sume that the fermions are the only relevant excitations, andthey account for this data by introducing a phenomenologicalparameterα which corresponds to the effective charge of thefermionic quasi-particles.

As anticipated in earlier theoretical calculations [25] thebosonic contribution begins to dominate in severely under-doped systems so that the slopedHc1/dT (associated with thelowest temperatures reached experimentally) shoulddecreasewith underdoping. Although observed a number of years afterthis prediction, this is precisely what is seen experimentally,as shown in Fig. 24. Here the inset plots the experimentalcounterpart data. It can be seen that theory and experimentare in reasonably good quantitative agreement. This theoret-ical viewpoint is very different from a “Fermi-liquid” basedtreatment of the superconducting state, for which the strongdecrease in the slope ofHc1 was not expected. Within thepresent formalism, the optical conductivity [1]σ(ω) is simi-larly modified [113] to include bosonic as well as fermioniccontributions.

C. Bosonic Power Laws and Pairbreaking Effects

The existence of noncondensed pair states belowTc af-fects thermodynamics, in the same way that electrodynamics

17

0 5 10 15 20 25T (K)

0

10

20

30

40

50

Hc1

(O

e)Tc = 8.9 K13.7 K15.5 K17.6 K22.0 K

0 0.2 0.4 0.6 0.8 1T/Tc

0

0.2

0.4

0.6

0.8

1

Hc1

(T)/

Hc1

(0)

FIG. 23: (color) Comparison between calculated lower critical field,Hc1, as a function ofT (upper left panel), and experimental data (lowerleft) from Ref. 109, with variable doping concentrationx. The right column shows normalized plots,Hc1(T )/Hc1(0) versusT/Tc, for theoryand experiment, respectively, revealing a quasi-universal behavior with respect to doping, with the exception of theTc = 55.5K ortho-IIphase. Both theory plots share the same legends. The quantitative agreement between theory and experiment is quite good.

is affected, as discussed above. Moreover, one can predict[36] that theq2 dispersion will lead to ideal Bose gas powerlaws in thermodynamical and transport properties. These willbe present in addition to the usual power laws or (fors-wave) exponential temperature dependencies associated withthe fermionic quasi-particles. Note that theq2 dependenceis dictated by the ground state of Eq. (1). Clearly this meanfield like state is inapplicable in the extreme BEC limit, where,presumably inter-boson effects become important and leadto a linear dispersion. Presumably, in the PG or near-BECregimes, fermionic degrees of freedom are still dominant andit is reasonable to apply Eq. (1). Importantly, at present nei-ther the cuprates nor the cold atom systems access this trueBEC regime.

The consequences of these observations can now be listed[36]. For a quasi-two dimensional system,Cv/T will ap-pear roughly constant at the lowest temperatures, althoughitvanishes strictly atT = 0 asT 1/2. The superfluid densityρs(T ) will acquire aT 3/2 contribution in addition to the usualfermionic terms. By contrast, for spin singlet states, there is

no explicit pair contribution to the Knight shift. In this waythe lowT Knight shift reflects only the fermions and exhibitsa scaling withT/∆(0) at low temperatures. Experimentally,in the cuprates, one usually sees a finite lowT contribution toCv/T . A Knight shift scaling is seen. Finally, also observedis a deviation from the predictedd-wave linear inT powerlaw in ρs. The new power laws inCv andρs are convention-ally attributed to impurity effects. Experiments are not yetat a stage to clearly distinguish between these two alternativeexplanations.

Pairbreaking effects are viewed as providing important in-sight into the origin of the cuprate pseudogap. Indeed, the dif-ferent pairbreaking sensitivities ofT ∗ andTc are usually pro-posed to support the notion that the pseudogap has nothing todo with superconductivity. To counter this incorrect inference,a detailed set of studies was conducted, (based on the BEC-BCS scenario), of pairbreaking in the presence of impurities[114, 115] and of magnetic fields [116]. These studies makeit clear that the superconducting coherence temperatureTc isfar more sensitive to pairbreaking than is the pseudogap onset

18

0 20 40Tc (K)

1

1.5

2

2.5

dHc1

/dT (

Oe/

K)

FIG. 24: Comparison of theoretically calculated lowT slopedHc1/dT (main figure) for various doping concentrations (corre-sponding to differentTc) in the underdoped regime with experimen-tal data (inset) from Ref. 109. The theoretical slopes are estimatedusing the low temperature data points accessed experimentally. Thequantitative agreement is very good.

temperatureT ∗. Indeed, the phase diagram of Fig. 21 whichmirrors its experimental counterpart, shows the very different,even competing nature ofT ∗ andTc, despite the fact that botharise from the same pairing correlations.

D. Anomalous Normal State Transport: Nernst Coefficient

Much attention is given to the anomalous behavior of theNernst coefficient in the cuprates [45]. This coefficient israther simply related to the transverse thermoelectric coeffi-cientαxy which is plotted in Fig. 25. In large part, the ori-gin of the excitement in the literature stems from the fact thatthe Nernst coefficient behaves smoothly through the supercon-ducting transition. BelowTc it is understood to be associatedwith superconducting vortices. AboveTc if the system werea Fermi liquid, there are arguments to prove that the Nernstcoefficient should be essentially zero. Hence the observationof a non-negligible Nernst contribution has led to the pictureof fluctuating “normal state vortices”.

The formalism of Ref. 47 can be used to address these datawithin the framework of BCS-BEC crossover. The results areplotted in Fig. 25 with a subset of the data plotted in the upperright inset. It can be seen that the agreement is reasonable.Inthis way a “pre-formed pair” picture appears to be a viable al-ternative to “normal state vortices”. It will, ultimately,be nec-essary to take these transport calculations belowTc. This is aproject for future research and in this context it will be impor-tant to establish in this picture how superconducting statevor-tices are affected by the noncondensed pairs and conversely.

V. CONCLUSIONS

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

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. 25: Calculated transverse thermoelectric response, which ap-pears in the Nernst coefficient, as a function of temperaturefor theunderdoped cuprates.

context, we explored the intersection of two very differentfields: highTc superconductivity and cold atom superfluid-ity. 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 can beassociated with meta-stable pairs of fermions; a (pseudogap)energy must be supplied to break these pairs apart into theirseparate components. The pseudogap also persists belowTc

in the sense that there are noncondensed fermion pair excita-tions of the condensate.

The recent discovery of superfluidity in cold fermion gasesopens the door to a set of fascinating problems in condensedmatter physics. Unlike the bosonic system, there is no coun-terpart of Gross-Pitaevskii theory. A new theory which goesbeyond BCS and encompasses BEC in some form or anotherwill have to be developed in concert with experiment.As ofthis writing, there are four experiments where the simple meanfield theory discussed in this review is in reasonable agree-ment with the data. These include the collective mode studiesover the entire range of accessible magnetic fields [96, 97].Inaddition in the unitary regime, RF spectroscopy-based pair-ing gap studies [72, 91], as well as density profile [89] andthermodynamic studies [73, 88] all appear to be compatiblewith this theory. Interestingly, all of these provide indicationsfor a pseudogap either directly through the observation of thenormal state energy scales,T ∗ and∆, or indirectly, throughthe observation of noncondensed pairs. The material in thisReview is viewed as the first of many steps in a long pro-cess. It is intended to provide continuity from one community(which has addressed the BCS-BEC crossover scenario, sincethe early 1990’s) to another.

19

Acknowledgments

We gratefully acknowledge the help of our many closecollaborators over the years: Jiri Maly, Boldizsar Janko,Ioan Kosztin, Ying-Jer Kao, Andrew Iyengar, Shina Tan andYan He. We also thank our co-authors John Thomas, An-

drey Turlapov and Joe Kinast, as well as Thomas Lem-berger, Brent Boyce, Joshua Milstein, Maria Luisa Chiofaloand Murray Holland. This work was supported by NSF-MRSEC Grant No. DMR-0213765 (JS,ST and KL), NSFGrant No. DMR0094981 and JHU-TIPAC (QC).

[1] T. Timusk and B. Statt, Rep. Prog. Phys.62, 61 (1999).[2] J. L. Tallon and J. W. Loram, Physica C349, 53 (2001).[3] H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Ran-

deria, M. R. Norman, T. Mochiku, K. Hadowaki, and J. Giap-intzakis, Nature382, 51 (1996).

[4] R. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys.75, 473 (2003).

[5] S. W. Cheong, G. Aeppli, T. E. Mason, H. A. Mook, S. M.Hayden, P. C. Canfield, Z. Fisk, K. N. Clausen, and J. L. Mar-tinez, Phys. Rev. Lett.67, 1791 (1991).

[6] H. F. Fong, B. Keimer, P. W. Anderson, D. Reznik, F. Dogan,and I. A. Aksay, Phys. Rev. Lett.75, 316 (1995).

[7] M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. Endoh, Rev.Mod. Phys.70, 897 (1998).

[8] G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, andJ. Kulda, Science278, 1432 (1997).

[9] G. Aeppli, T. E. Mason, H. A. Mook, A. Schroeder, and S. M.Hayden, Physica C282-287, 231 (1997).

[10] H. A. Mook, P. Dai, S. M. Hayden, G. Aeppli, T. G. Perring,and F. Dogan, Nature395, 580 (1998).

[11] J. Rossat-Mignod, L. P. Regnault, C. Vettier, P. Bourges,P. Burlet, J. Bossy, J. Y. Henry, and G. Lapertot, Physica C185, 86 (1991).

[12] J. M. Tranquada, P. M. Gehring, G. Shirane, S. Shamoto, andM. Sato, Phys. Rev. B46, 5561 (1992).

[13] D. A. Wollman, D. J. Van, Harlingen, J. Giapintzakis, andD. M. Ginsberg, Phys. Rev. Lett.74, 797 (1995).

[14] C. C. Tsuei, J. R. Kirtley, C. C. Chi, L. S. Yu-Jahnes, A. Gupta,T. Shaw, J. Z. Sun, and M. B. Ketchen, Phys. Rev. Lett.73, 593(1994).

[15] A. Mathai, Y. Gim, R. C. Black, A. Amar, and F. C. Wellstood,Phys. Rev. Lett.74, 4523 (1995).

[16] To be precise, the gap and pseudogap do not contain a phaseand thus have|dx2

−y2 | symmetry, while the order parameterhasdx2

−y2 symmetry.[17] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and

R. Walser, Phys. Rev. Lett.87, 120406 (2001).[18] E. Timmermans, K. Furuya, P. W. Milonni, and A. K. Kerman,

Phys. Lett. A285, 228 (2001).[19] D. M. Eagles, Phys. Rev.186, 456 (1969).[20] A. J. Leggett, inModern Trends in the Theory of Condensed

Matter (Springer-Verlag, Berlin, 1980), pp. 13–27.[21] R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod.

Phys.62, 113 (1990).[22] M. Randeria, in Bose Einstein Condensation, edited by

A. Griffin, D. Snoke, and S. Stringari (Cambridge Univ. Press,Cambridge, 1995), pp. 355–92.

[23] R. Cote and A. Griffin, Phys. Rev. B48, 10404 (1993).[24] P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys.59, 195

(1985).[25] Q. J. Chen, I. Kosztin, B. Janko, and K. Levin, Phys. Rev. Lett.

81, 4708 (1998).[26] R. Micnas and S. Robaszkiewicz, Cond. Matt. Phys.1, 89

(1998).[27] J. Ranninger and J. M. Robin, Phys. Rev. B53, R11961

(1996).[28] J. N. Milstein, S. J. J. M. F. Kokkelmans, and M. J. Holland,

Phys. Rev. A66, 043604 (2002).[29] Y. Ohashi and A. Griffin, Phys. Rev. Lett.89, 130402 (2002).[30] A. Griffin and Y. Ohashi, Phys. Rev. A67, 063612 (2003).[31] P. Pieri, L. Pisani, and G. C. Strinati, Phys. Rev. Lett.92,

110401 (2004).[32] Y. J. Uemura, Physica C282-287, 194 (1997).[33] C. Renner, B. Revaz, K. Kadowaki, I. Maggio-Aprile, and

O. Fischer, Phys. Rev. Lett.80, 3606 (1998).[34] G. Deutscher, Nature397, 410 (1999).[35] A. Junod, A. Erb, and C. Renner, Physica C317-318, 333

(1999).[36] Q. J. Chen, I. Kosztin, and K. Levin, Phys. Rev. Lett.85, 2801

(2000).[37] J. Maly, B. Janko, and K. Levin, Physica C321, 113 (1999).[38] Q. J. Chen, I. Kosztin, B. Janko, and K. Levin, Phys. Rev. B

59, 7083 (1999).[39] P. Pieri and G. C. Strinati, Phys. Rev. B61, 15370 (2000).[40] Y. Yanase, J. Takanobu, T. Nomura, H. Ikeda, T. Hotta, and

K. Yamada, Phys. Rep.387, 1 (2003).[41] M. Kugler et al., Phys. Rev. Lett.86, 4911 (2001).[42] M. Oda, K. Hoya, R. Kubota, C. Manabe, N. Momono,

T. Nakano, and M. Ido, Physica C281, 135 (1997).[43] J. W. Loram, K. Mirza, J. Cooper, W. Liang, and J. Wade, J.

Superconductivity7, 243 (1994).[44] J. Stajic, A. Iyengar, K. Levin, B. R. Boyce, and T. R. Lem-

berger, Phys. Rev. B68, 024520 (2003).[45] Z. Xu, N. Ong, Y. Want, T. Kakeshita, and S. Uchida, Nature

406, 486 (2000).[46] J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bo-

zovic, Nature398, 221 (1999).[47] S. Tan and K. Levin, Phys. Rev. B69, 064510(1) (2004).[48] M. Houbiers, R. Ferwerda, H. T. C. Stoof, W. McAlexander,

C. A. Sackett, and R. G. Hulet, Phys. Rev. A56, 4864 (1997).[49] B. DeMarco and D. S. Jin, Science285, 1703 (1999).[50] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade,

and J. E. Thomas, Science298, 2179 (2002).[51] J. G. A. Baker, Phys. Rev. C60, 054311 (1999).[52] H. Heiselberg, J. Phys. B37, 141 (2004).[53] K. Itakura, Nucl. Phys A715, 859 (2003).[54] P. F. Kolb and U. Heinz (2003), preprint nucl-th/0305084.[55] H. Heiselberg, Phys. Rev. A63, 043606 (2001).[56] J. Carlson, S. Chang, V. Pandharipande, and K. Schmidt,Phys.

Rev. Lett.91, 050401 (2003).[57] M. Greiner, C. A. Regal, and D. S. Jin, Nature426, 537 (2003).[58] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature424,

47 (2003).[59] K. E. Strecker, G. B. Partridge, and R. Hulet, Phys. Rev.Lett.

91, 080406 (2003).[60] J. Cubizolles, T. Bourdel, S. Kokkelmans, G. Shlyapnikov, and

20

C. Salomon, Phys. Rev. Lett.91, 240401 (2003).[61] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin,

J. H. Denschlag, and R. Grimm, Phys. Rev. Lett.91, 240402(2003).

[62] S. Jochim et al., Science302, 2101 (2003).[63] M. W. Zwierlein et al., Phys. Rev. Lett.91, 250401 (2003).[64] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy,

M. Teichmann, L. Tarruell, S. J. Kokkelmans, and C. Salomon,Phys. Rev. Lett.93, 050401 (2004).

[65] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett.92,040403 (2004).

[66] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach,A. J. Kerman, and W. Ketterle, Phys. Rev. Lett.92, 120403(2004).

[67] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E.Thomas, Phys. Rev. Lett.92, 150402 (2004).

[68] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim,J. Hecker-Denschlag, and R. Grimm, Science305, 1128(2004).

[69] F. Dalfovo et al., Rev. Mod. Phys.71, 463 (1999).[70] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin,

D. Hecke, and G. R, Phys. Rev. Lett.92, 120401 (2004).[71] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin,

J. Denschlag, and R. Grimm, Phys. Rev. Lett.92, 203201(2004).

[72] J. Kinnunen, M. Rodriguez, and P. Torma, Science305, 1131(2004).

[73] J. Kinast, A. Turlapov, J. E. Thomas, Q. J. Chen, J. Stajic,and K. Levin, Science307, 1296 (2005), published online 27January 2005; doi:10.1126/science.1109220.

[74] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, and W.Ket-terle, Nature435, 170404 (2005).

[75] Q. J. Chen, J. Stajic, S. N. Tan, and K. Levin, Phys. Rep.412,1 (2005).

[76] T. Kostyrko and J. Ranninger, Phys. Rev. B54, 13105 (1996).[77] V. Geshkenbein, L. Ioffe, and A. Larkin, Phys. Rev. B55, 3173

(1997).[78] R. Friedberg and T. D. Lee, Phys. Lett. A138, 423 (1989).[79] T. Friedberg and T. D. Lee, Phys. Rev. B40, 6745 (1989).[80] J. W. Serene, Phys. Rev. B40, 10873 (1989).[81] R. Haussmann, Z. Phys. B91, 291 (1993).[82] O. Tchernyshyov, Phys. Rev. B56, 3372 (1997).[83] L. P. Kadanoff and P. C. Martin, Phys. Rev.124, 670 (1961).[84] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett.93,

100404 (2004).[85] J. Stajic, A. Iyengar, Q. J. Chen, and K. Levin, Phys. Rev. B

68, 174517 (2003).[86] I. Kosztin, Q. J. Chen, Y.-J. Kao, and K. Levin, Phys. Rev. B

61, 11662 (2000).[87] J. Stajic, Q. J. Chen, and K. Levin, Phys. Rev. A71, 033601

(2005).[88] Q. J. Chen, J. Stajic, and K. Levin (2004), arXiv:cond-

mat/0411090.[89] J. Stajic, Q. J. Chen, and K. Levin, Phys. Rev. Lett.94, 060401

(2005).[90] J. Kinnunen, M. Rodriguez, and P. Torma, Phys. Rev. Lett. 92,

230403 (2004).[91] Y. He, Q. J. Chen, and K. Levin, Phys. Rev. A72, 011602(R)

(2005).[92] T.-L. Ho, Phys. Rev. Lett.92, 090402 (2004).[93] J. Kinast, A. Turlapov, and T. J. E, Phys. Rev. Lett.94, 170404

(2005).[94] J. Kinast, A. Turlapov, and J. E. Thomas (2004), preprint

cond-mat/0409283.[95] M. L. Chiofalo, S. J. J. M. F. Kokkelmans, J. N. Milstein,and

M. J. Holland, Phys. Rev. Lett.88, 090402 (2002).[96] H. Hu, A. Minguzzi, X.-J. Liu, and M. P. Tosi, Phys. Rev.

Lett. 93, 190403 (2004), see also the preprint version, cond-mat/0404012v1. The transverse breathing mode data fromBartenstein et al shown in Fig. 2 of the published version hasrecently been corrected so that they agree with both theory andthe data from Kinast et al.

[97] H. Heiselberg, Phys. Rev. Lett.93, 040402 (2004).[98] J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. A70,

051401(R) (2004).[99] C. Chin and R. Grimm, private communication.

[100] S. Stringari, Europhys. Lett.65, 749 (2004).[101] J. R. Schrieffer,Theory of Superconductivity(Perseus Books,

Reading, MA, 1983), 3rd ed.[102] S. Tan and K. Levin (2005), arXiv:cond-mat/0506293.[103] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev.

Lett. 93, 090404 (2004).[104] M. J. Holland, C. Menotti, and L. Viverit (2004), preprint,

cond-mat/0404234.[105] M. Greiner, C. A. Regal, and D. S. Jin, Phys. Rev. Lett.94,

070403 (2005).[106] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack,and

R. G. Hulet, Phys. Rev. Lett.95, 020404 (2005).[107] Q. J. Chen and K. Levin (2005), arXiv:cond-mat/0505689.[108] J. W. Loram, K. A. Mirza, J. R. Cooper, and J. L. Tallon, J.

Phys. Chem. Solids59, 2091 (1998).[109] R. X. Liang, D. A. Bonn, W. N. Hardy, and B. D, Phys. Rev.

Lett. 94, 117001 (2005).[110] Q. J. Chen, K. Levin, and I. Kosztin, Phys. Rev. B63, 184519

(2001).[111] Y. Zuev, M. S. Kim, and T. Lemberger (2004), arXiv:cond-

mat/0410135.[112] P. A. Lee and X.-G. Wen, Phys. Rev. Lett.78, 4111 (1997).[113] A. Iyengar, J. Stajic, Y.-J. Kao, and K. Levin, Phys. Rev. Lett.

90, 187003 (2003).[114] Q. J. Chen and J. R. Schrieffer, Phys. Rev. B66, 014512

(2002).[115] Y.-J. Kao, A. Iyengar, J. Stajic, and K. Levin, Phys. Rev. B 68,

214519 (2002).[116] Y.-J. Kao, A. Iyengar, Q. J. Chen, and K. Levin, Phys. Rev. B

64, 140505 (2001).