Mottness

Annals of Physics 321 (2006) 1634–1650

www.elsevier.com/locate/aop

Mottness

Philip Phillips *

Department of Physics, University of Illinois, Urbana-Champaign, IL 61801-3080, USA

Received 23 February 2006; accepted 3 April 2006

Abstract

We review several of the normal state properties of the cuprates in an attempt to establish an orga-nizing principle from which pseudogap phenomena, broad spectral features, T-linear resistivity, andspectral weight transfer emerge. We first show that standard field theories with a single critical lengthscale cannot capture the T-linear resistivity as long as the charge carriers are critical. What seems to bemissing is an additional length scale, which may or may not be critical. Second, we prove a generalisedversion of Luttinger’s theorem for a Mott insulator. In particular, we show that for Mott insulators,regardless of the spatial dimension, the Fermi surface of the non-interacting system is converted into asurface of zeros of the single-particle Green function when the underlying band structure has particle-hole symmetry. Only in the presence of particle-hole symmetry does the volume of the surface of zerosequal the particle density. The surface of zeros persists at finite doping and is shown to provide aframework from which pseudogaps, broad spectral features, spectral weight transfer on the Mottgap scale, and the additional length scale required to capture T-linear resistivity can be understood.� 2006 Elsevier Inc. All rights reserved.

1. Introduction

The normal state of the high-temperature copper-oxide superconductors (cuprates forshort) is anything but normal. First, all parent cuprates are Mott insulators [1]. Such insu-lators do not insulate because the band is full, but rather because strong local electron cor-relations dynamically generate a charge gap by splintering a half-filled band into lower andupper Hubbard bands. Second, once doped, Mott insulators still insulate. In fact, numerousexperiments [2–5] in which a large magnetic field is applied to kill superconductivity reveal

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doi:10.1016/j.aop.2006.04.003

* Fax: +1 2172447704.E-mail address: [email protected].

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P. Phillips / Annals of Physics 321 (2006) 1634–1650 1635

that the resistivity diverges as the temperature tends to zero below a critical doping level, xc.Above xc, traditional Fermi liquid behaviour is observed in the overdoped regime. Third,electron quasiparticles are nowhere to be found in the underdoped regime where the ubiq-uitous pseudogap [7–9] phenomenon obtains. Fourth, near optimal doping, the resistivityscales as a linear [10] function of temperature as opposed to the quadratic behaviour indic-ative of a Fermi liquid. Such oddities did not, of course, beset a theoretical understanding ofsuperconductivity in metals such as aluminum because the normal state is accurately rootedin Fermi liquid theory. In the cuprates, the central problem of the normal state is under-standing how the disparate phenomena just described arise from strong electron correla-tions intrinsic to the Mott state. Simply stated, the question is: is there a generalorganizing principle that captures Mottness? Mottness refers to those features of dopedMott insulators that are not deducible from ordering. While it might be that ordering isdeducible from Mottness, Mottness is distinct nomologically from ordering. Indeed, it isMottness that makes the cuprate problem largely intractable. It is for this reason thatnumerous theories of the cuprates get Mottness clearly wrong, ignore it, or deny its exis-tence. In this paper, we prove two theorems which bare directly on the nature of the low-energy theory for a doped Mott insulator. Both theorems point to physics beyond thatwhich is captured by standard field theories based on one-parameter correlation length scal-ing. In the first theorem, we show explicitly that within single parameter scaling, quantumcriticality cannot account for the linear T resistivity. Second, we prove a generalised versionof Luttinger’s theorem for a Mott insulator. Specifically, we show that for Mott insulators,the Fermi surface of the non-interacting system is converted into a surface of zeros of thesingle-particle Green function when the underlying band structure has particle-hole symme-try. While the surface of zeros is shown to persist in the absence of particle-hole symmetry,only in the presence of such a symmetry does the volume of the surface of zeros equal theparticle density. The surface of zeros provides a framework from which pseudogaps, broadspectral features, and spectral weight transfer on the Mott gap scale can be understood.

2. T-linear resistivity

One of the leading explanations for the T-linear resistivity is quantum criticality [11,12].We do not analyse here specific proposals such as the highly successful marginal-Fermiliquid theory [13], which is in principle consistent with a quantum critical point, but ratherdissect the assumptions of quantum criticality which lead to T-linear resistivity. At thequantum critical coupling, or in the quantum critical regime, kBT is the only energy scalegoverning collisions between quasiparticle excitations of the order parameter. Conse-quently, the transport relaxation rate,

1

str

/ kBT�h; ð1Þ

scales as a linear function of temperature, thereby implying a T-linear resistivity if (naive-ly) the scattering rate is what solely dictates the transport coefficients. That temperature isthe only scale in the quantum critical regime regardless of the nature of the quasiparticleinteractions is a consequence of universality. Eq. (1) holds as long as the inequalitiesT > |D| and t < 1/|D| are maintained, D the energy scale measuring the distance from thecritical point and t the observation time. The energy scale D / dzm varies as a functionof the tuning parameter d = g � gc, where m is the correlation length exponent and z is

1636 P. Phillips / Annals of Physics 321 (2006) 1634–1650

the dynamical exponent. At the critical coupling d = 0 or g = gc, D vanishes. Ultimately,the observation time constraint, t < 1/|D| implies that only at the quantum critical pointdoes the T-linear scattering rate obtain for all times. That the quantum critical regimeis funnel-shaped follows from the inequality T > |D|. The funnel-shaped critical regionshould be bounded by a temperature Tupper above which the system is controlled byhigh-energy processes. Whether or not quantum criticality is operative up to temperaturesof order T = 1000 K in the cuprates is questionable.

We have recently formulated [14] a general scaling argument for the conductivity tounderstand if T-linear resistivity is consistent with the general claims of quantum critical-ity. The argument is based on three general assumptions: (1) the critical degrees of freedomcarry the current, (2) only a single length scale is relevant near the critical point, and (3)charge is conserved. Granted true, these assumptions rule out quantum criticality as thecause of the T-linear resistivity.

Before we present the proof, consider as a precursor to the Drude formula for theconductivity,

rDrude ¼1

4p

x2plstr

1þ x2s2tr;

ð2Þ

with xpl the plasma frequency and str the transport relaxation time. If we substitute Eq. (1)into Eq. (2), T-linear resistivity follows in the zero-frequency limit. However, explicitlyappearing in the formula for the resultant conductivity is the xpl. While the plasma fre-quency is commonplace in typical transport theories of metals, we show below that sucha degree of freedom does not exist in quantum critical theories. Consequently, the predic-tions of Drude theory are not compatible with those of quantum criticality. That inconsis-tencies lie with the Drude formula and quantum criticality can be seen immediately fromthe work of van der Marel [15] and co-workers who used Eq. (2) to collapse their opticalconductivity data as a function of x/T. True quantum critical behaviour would be de-scribed by an optical conductivity of the form, T�lf (x/T). Contrastly, they find [31] thatl = 1 for x/T < 1.5 and l � 0.5 for x/T > 3 if f (x/T) is described by the Drude formula.

Consider a general action S whose microscopic details are unimportant as long as thecurrent is critical. An externally applied electromagnetic vector potential Al,l = 0,1, . . . ,d, couples to the electrical current, jl, such that

S ! S þZ

dsddxAljl: ð3Þ

Under the one-parameter scaling hypothesis for quantum systems, the spatial correlations ina volume smaller than the correlation volume, nd, and temporal correlations on a time scaleshorter than nt / nz are small, and space-time regions of size ndnt behave as independentblocks. Using this hypothesis, we write the scaling form for the singular part of the logarithmof the partition function by counting the number of correlated volumes in the whole system

ln Z ¼ Ldb

ndnt

F ðdndd ; fAikn

dAgÞ: ð4Þ

In this expression, L is the system size, b = 1/kBT the inverse temperature, d the distancefrom the critical point, and dd and dA the scaling dimensions of the critical coupling andvector potential, respectively. The variables Ai

k ¼ Aiðx ¼ kn�1t Þ correspond to the (uni-

form, k = 0) electromagnetic vector potential at the scaled frequency k = xnt, and


i = 1, . . . ,d labels the spatial components. Two derivatives of the logarithm of the parti-tion function with respect to the electromagnetic gauge Ai(x),

rijðx; T Þ ¼1

Ldb

1

xd2 ln Z

dAið�xÞdAjðxÞ

¼ n�d n�1t

xn2dA

d2

dAi��kdAj

�k

F ðd ¼ 0Þ��k¼xntfAi

k¼0g

¼ Q2

�hn2dA�dRijðxntÞ

ð5Þ

determine the conductivity for carriers with charge Q. We have explicitly set d = 0 as ourfocus is the quantum critical regime. At finite temperature, the time correlation length iscutoff by the temperature as nt / 1/T, and nt / nz. The engineering dimension of the elec-tromagnetic gauge is unity (dA = 1). From charge conservation, the current operators can-not acquire an anomalous dimension; hence, dA = 1 is exact [16]. We then arrive at thegeneral scaling form

rðx; T Þ ¼ Q2

�hT ðd�2Þ=zR

�hxkBT

� �ð6Þ

for the conductivity where R is an explicit function only of the ratio, x/T. (The ij tensorindices have been dropped for simplicity.) This scaling form generalizes to finite T and xthe T = 0 frequency dependent critical conductivity originally obtained by Wen [16]. Thegeneric scaling form, Eq. (6), is also in agreement with that proposed by Damle and Sach-dev [17] in their extensive study of collision-dominated transport near a quantum criticalpoint (see also the scaling analysis in Ref. [18]). What the current derivation lays plain isthat regardless of the underlying statistics or microscopic details of the Hamiltonian, be itbosonic (as in the work of Damle and Sachdev [17]) or otherwise, be it disordered or not,the general scaling form of the conductivity is unchanged. The Anderson metal-insulatortransition in d = 2 + �, which can be thought of as a quantum phase transition where thedimensionless disorder strength is the control parameter [20,19], obeys the scaling functionderived here for the conductivity.

At zero frequency, the dc limit, we obtain

rðx ¼ 0Þ ¼ Q2

�hRð0Þ kBT

�hc

� �ðd�2Þ=z

: ð7Þ

In general R (0) „ 0. Else, the conductivity is determined entirely by the non-singular andhence non-critical part of the free energy. The cuprates are anisotropic 3-dimensional sys-tems. Hence, the relevant dimension for the critical modes is d = 3 not d = 2. In the lattercase, the temperature prefactor is constant. For d = 3, we find that T-linear resistivityobtains only if z = �1. Such a negative value of z is unphysical in standard commutative[24] field theories as it implies that energy scales diverge for long wavelength fluctuations atthe critical point.

While it is certainly reasonable to assume that R (0) is finite at zero temperature, it iscertainly a possibility that R (0) might in fact diverge. A possible scenario of how this stateof affairs might obtain is that a dangerously irrelevant operator could govern the conduc-

T

FL

Quantum critical?ρ∼Τ

(strange metal)

SCPG

QCPdoping

AF

SG

Fig. 1. Heuristic phase diagram of the cuprate superconductors as a function of temperature and hole dopinglevel. AF, antiferromagnet; SG, spin glass; SC, superconductor; PG, pseudogap regime in which the single-particle specrum develops a dip; FL, Fermi liquid. The spin-glass phase ceases at a critical doping level (quantumcritical point, QCP) inside the dome. The dashed lines represent crossovers not critical behaviour. The strange-metal behaviour, T-linear resistivity, in the funnel-shaped regime has been attributed to quantum criticalbehaviour. A scaling analysis of the conductivity at the quantum critical point rules out this scenario, however.


tivity. In this case, R (0) � 1/T p, and T-linear behaviour obtains if p = (d � 2)/z + 1. Thiscan only occur above the upper critical dimension. In this regime, all criticality is mean-field-like. Hence, a possibility that cannot be eliminated at this time is that all criticalityin the cuprates is inherently mean-field and dangerously irrelevant operators control theconductivity. For strongly correlated systems, however, it is unlikely that criticality isinherently mean-field.

Consequently, three options are available at this point: (1) quantum criticality has noth-ing to do with the problem, (2) the current is carried by non-critical degrees of freedom, or(3) new quantum critical scenarios in which additional length scales describe the physics.In a scenario involving non-critical degrees of freedom, fermionic charge carriers in thenormal state of the cuprates could couple to a critical bosonic mode. Such an accountis similar to that in magnetic systems [11] in which fermions scatter off massless bosonicdensity or spin fluctuations and lead to an array of algebraic forms for the resistivity[22,23] ranging from T 4/3 to T 3/2 in antiferromagnetic and ferromagnetic systems, respec-tively. While disorder can alter the exponent [23], T-linear resistivity results only in arestricted parameter space. The robustness of T-linear resistivity in the cuprates makes thistype of scenario unlikely. What about new scenarios? Let us entertain the possibility thatan additional length ~n is relevant and diverges as ~n / na, with a > 1. In the calculation ofthe correlation volume in Eq. (4), one must replace nd with nd fi ‘d = nd. Consequently,hð~n=nÞ, with h (y) = y�k a general scaling function. In essence, one is reducing the effectivedimensionality such that d fi d * = d � k (a � 1). T-linear resistivity obtains if z = 2 � d *.The reduction in the effective dimensionality, k (a � 1), can now be fine-tuned so thatd * 6 1, thereby resulting in physically permissible values of the dynamical exponent,z P 1. Such fine scripting of two length scales is also without basis at this time (Fig. 1).

3. Spectral weight transfer

It is worth pointing out that non-commutative [24] field theories do permit z < 0. Cen-tral to z < 0 is UV-IR mixing and as a consequence a breakdown of the standard Wilso-nian renormalization scheme. While a field theory based on non-commuting coordinates isundoubtedly not applicable to the cuprates, UV-IR mixing is certainly present as it has


been documented experimentally both in the normal state [25–29] as well as the supercon-ducting state [30–32]. Hence, the relevant question is: can spectral weight transfer give riseto an additional length scale?

To answer this question, we recount the well known argument [33,34] on doping-depen-dent spectral-weight transfer across the Mott gap. Consider the half-filled one-dimensionalchain of one-electron atoms shown in Fig. 2 described by the Hubbard Hamiltonian,

H ¼ �Xi;j;r

tijcyircjr þ U

Xi

ni"ni# � lX

ir

nir; ð8Þ

in which electrons hopping on a lattice between neighbouring sites with amplitude tij = taij

and chemical potential l pay an energy cost U anytime they doubly occupy the same site.The operator cirðcyirÞ annihilates (creates) an electron on site i with spin r and nir is theoccupancy on site i with spin r. When the hopping term vanishes, we can treat the half-filled system as having one electron per site. For a chain containing N sites, there are N

ways to remove an electron and N ways to add an electron with an energy cost U. Theoperators that describe such excitations are nir = cir (1 � ni � r) which adds an electronto an unoccupied site and gyir ¼ cyirni�r which creates a doubly occupied site. Such opera-tors create excitations in the lower and upper Hubbard bands, respectively, noted also inFig. 2 as the photo-electron and inverse photo-electron spectra. When a single hole is cre-ated, the number of ways to remove an electron is now N � 1. Surprisingly, the number ofways of adding an electron to the system so that the energy cost is U also decreases toN � 1. This means that there are two less states at high and low energies. Where arethe remaining two states? They correspond to the two ways of occupying the empty sitewith spin up and spin down electrons. Such states correspond to the addition part ofthe low-energy spectral weight and hence lie immediately above the chemical potential

N1 2 N1U

EF

Doped Mott Insulator

N N

PES IPES

U

EF

Fig. 2. Spectral weight transfer in a doped Mott insulator. The Photoelectron spectrum (PES) denotes theelectron removal states while the electron-addition states are located in the inverse photo-electron spectrum(IPES). The on-site charging energy is U. Removal of a single electron results in the creation of two single particlestates at the top of the lower Hubbard band. By state conservation, one state comes from the lower and the otherfrom the upper Hubbard band and hence spectral weight transfer across the Mott gap.


as illustrated in Fig. 2. One of the states must come from the upper Hubbard band (UHB)as the high energy part now has a spectral weight of N � 1, the other from the lower Hub-bard band. Hence, for a single hole, there is a net transfer of one state from high to lowenergy. In general, simple state counting yields 2x for the growth of addition part of thelow-energy spectral weight, K (x) and 1 � x for the depletion of the high energy sector. In aFermi liquid, adding holes simply creates quasiparticles near the chemical potential andhence cannot involve the high energy part. In actuality, the dynamical contribution[33,34] to the LESW results in K (x) > 2x. The dynamical LESW corresponds to virtualexcitations to the UHB. Hence, in a strongly correlated system, some of the low-energydegrees of freedom arise from the high energy scale. A bonus that can be explained fromspectral weight transfer is the sign change of the Hall coefficient seen widely [35–37] in thecuprates. Particle-hole symmetry is restored in the lower Hubbard band when the additionand removal parts of the low-energy spectral weights are equal. Equating the atomic(U =1) values for these weights, namely 1 � x for the addition spectrum and 2x forthe removal part, we obtain that the maximum doping level at which the Hall coefficientchanges sign is x = 1/3. This upper bound correlates with the experimentally observed val-ue for the vanishing of the Hall coefficient [35–37] in the cuprates and is precisely the valueobtained [38,39] from strong-coupling calculations at large U �1.

Spectral weight transfer described above can be reformulated in terms of the compositeoperators for the lower and upper Hubbard bands. Write the electron operator ascir = nir + gir and the corresponding spectral function,

Aðk;xÞ ¼ �1

pImF T ðhðt � t0ÞhfcirðtÞ; cyjrðt0ÞgiÞ

¼ Agg þ Ann þ 2Agn

ð9Þ

explicitly the imaginary part of the single particle Green function, contains diagonal termsas well as a mixing term which carries the information regarding the interconnectednessbetween the high and low energy scales. It is from the mixing term that spectral weighttransfer arises. Shown in Fig. 3 is the mixing term computed using a self-consistent clustermethod [40]. Regardless of filling, the cross term has both negative and positive contribu-tions. This structure arises necessarily because the integral of Agn over all frequencyZ 1

�1Agnðk;xÞdx ¼ 0 ð10Þ

yields the equal time correlator hfnır; gyirgi ¼ 0, whose vanishing maintains the Pauli

principle. This implies that Agn is either zero, which it is not, or it must have both po-sitive and negative parts, representing constructive and destructive interference, respec-tively, between different regions in energy. At half-filling, upper panel, if particle-holesymmetry is present, the contributions of the cross term below and above the chemicalpotential sum to zero independently, indicating that all energy scales need not be re-tained to ensure the Pauli principle. Such is not the case, however, at finite doping.The lower panel in Fig. 3 indicates that a pseudogap develops at the chemical potential,indicating an orthogonality catastrophe. Because a pseudogap subtracts spectral weightat low energies and transfers it to intermediate to high energy, the sum rule which en-sures the Pauli principle is satisfied only when Agn is integrated over all energy scalesnot simply up to the chemical potential (or some intermediate energy cutoff) as wouldbe the case in projected models.

-10 0 10

ω/t

-0.02

-0.01

0

0.01

0.02

0.03

cros

s co

rrel

atio

nU = 8tU = 12t

-10 0 10

ω/t

-0.02

-0.01

0

0.01

0.02

cros

s co

rrel

atio

n U = 8t

0 0.2 0.4T/t

0

0.02

0.04

0.06

0.08

D(ε

F)

n = 1

n = 0.95

Fig. 3. Cross correlation or quantum interference Agn between the upper and lower Hubbard bands at half-filling,n = 1 and at n = 0.95 at T = 0.1t. The dip at the chemical potential in the lower panel represents the pseudogap.The inset shows that this dip leads to a vanishing density of states at zero temperature and hence an orthogonalitycatastrophe.


That projection changes fundamentally the statistics of the particles arises from thetruncation of the Hilbert space. Consider the standard way of formulating the t � J modelfrom the Hubbard model: block diagonalize the Hubbard model into sectors with a fixednumber of doubly occupied sites via a similarity transformation [41], S, which connectssectors that differ by a single double occupancy. In such a scheme, the electron operatorin the low-energy sector is transformed to

cir ! Pe�ScireSP � air; ð11Þwhere P removes double occupancy. To lowest order in t/U, air = nir and nyi"n

yi# ¼ 0. That

is, projected operators in Eq. (11) block two states from being occupied rather one aswould be the case for fermions. Simply, projection of the electron onto the lower Hubbardband does not result in a fermion. In fact, such projected objects are exclusions obeyinggeneralized exclusion statistics [42,43].

Since, truncating the Hilbert space from four to three states per site changes the statis-tics, the limits U fi1 and L fi1 do not commute. Whether or not this changes the phys-ics fundamentally is open for debate. However, if transport is governed by a finite lengthscale for double occupancy, ndo, then the correct order of limits is L fi1 followed byU fi1. Projected models such as the t � J model correspond to the opposite regime,U fi1 and then L fi1. In this case, ndo =1. A finite value of ndo would certainly func-tion as the additional length scale that is required to describe T-linear resistivity. In recentcluster calculations [44] on the Hubbard model, we observed an insulating state (see Fig. 4)

0.1 1

T/t

0

10

20

30

40

50

ρ/ρ

0

n = 0.97n = 0.95n = 0.9n = 0.85n = 0.8

0 0.02 0.04 0.06 0.08 0.1doping x

0

0.2

0.4

0.6

0.8

1

T*/

J

x % of D.O.0.03 3.60.05 3.40.06 3.20.08 2.70.10 2.0

Fig. 4. Resistivity as a function of temperature for the Hubbard model (with U = 10t) using the spectral functioncomputed previously by Stanescu and Phillips [47] for fillings n = 0.97, 0.95, 0.9, 0.85, and 0.80. q0 = h/e2. Theinset shows the doping dependence of the pseudogap energy scale, T*. The dependence obeys the functional form,Jð1� xn2

do=4a2Þ, where ndo is the average distance between doubly occupied sites, the percentage of which isindicated in the table.


below a critical doping level as is seen experimentally [2–6]. The pseudogap arises [44] fromthe energy gap between local cluster states with total spin S and S � 1. This energy differ-ence is essentially the triplet-singlet splitting and hence scales as t2/U. We argued that theinsulating state persists as long as nhn

2do < L2, nh = x (L/a)2 the number of holes. nhn

2do ¼ L2

defines the percolation limit. By calculating the percentage of doubly occupied sites, weobtained ndo numerically and plotted the T*-line, J (1 � cx (ndo/a)2), in the inset ofFig. 4. The agreement of this phenomenological fit with the resistivity data in which ametallic state obtains at x = 0.1 lends credence to our assertion that ndo is the relevantlength scale for the pseudogap. Similar calculations on the t � J model find a metallic state[45,46] consistent with the lack of commutativity of the limits U fi1 and L fi1. Butthis question is far from settled.

4. Unifying principle: zeros

To help resolve the potential non-commutativity of U fi1 and L fi1, we seek a uni-fying principle from which spectral weight transfer, pseudogap phenomena, and broadspectral features emerge. Such a principle does exist and stems from the analytical struc-ture of the single-particle Green function whenever a gap is dynamically generated. In Fer-mi liquids, the single-particle Green function has poles at the quasiparticle excitations. Thedivergence of the single-particle Green function defines the Fermi surface. In insulators, nosuch divergence obtains because quasiparticles are absent. However, Luttinger’s [48]theorem

NV � nLutt

¼ 2

ZGð0;pÞ>0

d3p

ð2pÞ3ð12Þ


has been argued recently [49–51] to still hold. The essence is that the integral definingthe particle density is performed over a slice in momentum space where the Green func-tion is positive. Green functions change sign and become positive either at poles or atzeros. While numerous calculations [50,51] have found zeros in 1D or quasi-1D sys-tems, it is unclear if the zeros are an artifact of the approximation. Further, shouldit not be the case that a sum rule exists on the Luttinger surface, then a new formof Eq. (12) must be derived to describe the particle density. In fact, the breakdownof the Luttinger sum rule has been noted [52] for a specific model for a Kondo insu-lator. Should the volume of the surface of zeros not equal the particle density, then anew form of Eq. (12) must be derived [51] which do in fact preserve the Luttingervolume.

Recently, we have found a general proof [53] that allows us to identify precisely wherethe surface of zeros occurs regardless of spatial dimension provided certain symmetriesobtain. We expand on that proof here. To proceed, we note that the causal nature ofthe Green function permits it to be constructed entirely from the spectral function (thatis, the imaginary part)

Grðk;xÞ ¼Z 1

�1dx0

Arðk; �0Þ�� 0 þ ig

ð13Þ

through the standard Hilbert representation. For a Mott insulator, a gap of order U oc-curs in the spectral function. We will take the gap to have a width 2D centered about 0.Within the gap, Aðk; �Þ ¼ 0. This is a necessary condition for any gap. Consequently, inthe presence of a gap, the real part of the Green function evaluated at the Fermi energyreduces to

Rrðk; 0Þ ¼ �Z �D�

�1d�0

Arðk; �0Þ�0

�Z 1

Dþ

d�0Arðk; �0Þ

�0: ð14Þ

We now prove that when particle-hole symmetry is present, the retarded Green function isan even function of frequency at the non-interacting Fermi surface. As a consequence, Eq.(14) is identically zero along that momentum surface. To proceed, we consider the mo-ments

MrnðkÞ �

Zdx2p

xndxGrðk;xÞ ð15Þ

of the Green function. For simplicity, we have set �h = 1. Using the Heisenberg equationsof motion, we reduce [54] the moments in real space

Mrnði; jÞ ¼ 1

2hf½H ; ½H � � � ½H ; cir� � � � �n times; c

yjrgi þ hfcir; ½� � � ½cyjr;H � � � �H �;H �n timesgi

� �ð16Þ

to a string of commutators of the electron creation or annihilation operators withthe Hubbard Hamiltonian. The right-hand side of this expression is evaluated atequal times. To evaluate the string of commutators, it suffices to focus on the prop-erties of

KðnÞir ¼ ½� � � ½cir;H �; � � �H �n times; ð17Þ


where by construction, Kð0Þir ¼ cir. We write the Hubbard Hamiltonian as H = Ht + HU,where HU includes the interaction as well as the chemical potential terms. The form ofthe first commutator,

Kð1Þir ¼X

j

tijcjr þ Ucirni�r � lcir ð18Þ

suggests that we seek a solution of the form

KðnÞir ¼X

j

tijKðnÞjr þ QðnÞir ; ð19Þ

where QðnÞir ¼ ½� � � ½cir;H U �; � � �H U �n times involves a string containing HU n times and in Kjr,Ht appears at least once. Our proof hinges on the form of QðnÞir which we write in general asQðnÞir ¼ ancirni�r þ bncir. The solution for the coefficients

anþ1 ¼ ðU � lÞan þ Uð�lÞn

bn ¼ ð�lÞnð20Þ

is determined from the recursion relationship Qðnþ1Þir ¼ ½QðnÞir ;HU �. In the moments, the

quantity which appears is

QðnÞir ; cyjr

n oD E¼ dij anhni�ri þ bn½ � � dijcn: ð21Þ

Consequently, the moments simplify to

Mrnði; jÞ ¼ dijcn þ

1

2

Xl

til KðnÞlr ; cyjr

n oD Eþ h:c:

� �: ð22Þ

The criterion for the zeros of the Green function now reduces to a condition on the parityof the right-hand side of Eq. (22). Consider the case of half-filling and nearest-neighbourhopping. Under these conditions, Æniræ = 1/2 and by particle-hole symmetry, l = U/2. Theexpressions for an and bn lays plain that the resultant coefficients

cn ¼U2

� �n1þ ð�1Þn

2ð23Þ

vanish for n odd. Consequently, Gr (k,x) is an even function if the second term in Eq. (22)vanishes. In Fourier space, the second term is proportional to the non-interacting bandstructure, which in the case of nearest-neighbour hopping is tðkÞ ¼ �2t

Pdi¼1 cos ki. As a

result, the Green function only has even moments at the momenta for whichPdi¼1 cos ki ¼ 0. The vanishing of t (k) defines the non-interacting Fermi surface. Conse-

quently, the surface of zeros is pinned at the Fermi surface of the non-interacting systemat half-filling whenever a Mott gap opens in the presence of particle-hole symmetry. Thisconstitutes one of the few exact results for Mott insulators that is independent of spatialdimension. The only condition for the applicability of our proof is that the form of the gapleads to finite integrals in Eq. (14). Hence, the minimal condition is that the density ofstates vanishes at zero frequency as xa, a > 0. Note, this encompasses even the Luttingerliquid case in which 0 < a < 1.

To prove that this observation applies regardless of the range of the Coulomb interac-tions, we consider an argument directly from particle-hole symmetry. Under a general par-ticle-hole transformation,


cir ! eiQ�ri cyir; ð24Þwith the chemical potential fixed at l = U/2 in the Hubbard model, the spectral functionbecomes

Arðk;xÞ ¼ Arð�k�Qþ 2np;�xÞ; ð25Þ

Hence, the spectral function is an even function of frequency for k ¼ Q=2þ np. Considerone dimension and nearest-neighbour hopping. In this case, the symmetry points are ±p/2,the Fermi points for the half-filled non-interacting band. In two dimensions, this proof issufficient to establish the existence of only two points, not a surface of zeros. To determinethe surface, we take advantage of an added symmetry in higher dimensions. For example,in two dimensions, we can interchange the canonical x and y axes leaving the Hamiltonianunchanged. This invariance allows us to interchange kx and ky on the left-hand side of Eq.(25) resulting in the conditions

ky ¼ �kx � qþ 2np ð26Þand by reflection symmetry

�ky ¼ �kx � qþ 2np; ð27Þ

where Q ¼ ðq; qÞ. For nearest neighbour hopping, the resultant condition,kx ± ky = � p + 2np, is the solution to coskx + cosky = 0, which defines the Fermi sur-face for the non-interacting system. When nearest–neighbour Coulomb interactions arepresent, the chemical potential changes to (U + 2V)/2 at the particle-hole symmetricpoint and the proof follows as before. Arbitrary density–density interactions and henceall Coulomb interactions are independent of Q under a particle-hole transformation.Hence, our proof applies regardless of the range of the Coulomb interactions. A pro-posed proof [55] for the existence of zeros in 1D relies explicitly on rotational invarianceof the Green function. Such an invariance is not applicable to the Hubbard model as thecharge and spin velocities are necessarily different as long as U „ 0. Finally, existing cal-culations based on the random phase approximation in which zeros were found at thenon-interacting Fermi surface in 1D [50] and quasi-1D [51,52] systems are consistentwith our proof.

While this proof applies strictly at half-filing, it places constraints on how the Fermisurface can evolve at finite doping in Mott systems. The consequences of this result areas follows. First, in the presence of the Mott gap, the self-energy diverges at zero frequencyalong the surface of zeros. To prove this, observe that the single-particle Green functioncan be written as Grðk;xÞ ¼ 1=ðx� �ðkÞ �RRrðk;xÞ � iIRrðk;xÞÞ, where R is the self-energy. Near the Luttinger surface, G / k � kL. IRðk; 0Þ / dðk � kLÞ. Note, however, thatIGðk;xÞ ¼ 0 for all energies within the gap. Now simply invert the Green function toobtain that RRrðk;xÞ / ðk � kLÞ�1, thereby proving the assertion. Such a divergence isa clear indication that perturbation theory breaks down. Zeros are a concrete manifesta-tion of this breakdown. The divergence of RR prevents the renormalized energy band,�ðkÞ þRRrðk; 0Þ from crossing the chemical potential, thereby resulting in an insulatingstate. This reinterpretation of the Mott insulating state provides a general way of under-stand how insulating states arise through dynamical generation of a gap in the absence ofsymmetry breaking. In dynamical cluster calculations, the divergence of R (p, 0) [56] hasbeen observed but attributed to antiferromagnetism. What our proof makes clear is that


the divergence of the self-energy is a result of Mottness itself; that is, it is not concomitantto T = 0 ordering. Second, the volume of the surface of zeros equals the particle densityonly when particle-hole symmetry is present. In general, the particle density in Eq. (12)contains an additional term [48] given by the frequency and momentum integral ofGdxR. Under Eq. (25), Gðk;xÞ ! �Gð�k � Q;�xÞ and Rðk;xÞ ! �Rð�k � Q;�xÞ,the Hubbard model remains invariant with the chemical potential fixed at l ¼ U=2and the additional contribution to the electron density integrates to zero by symmetry.In the absence of particle-hole symmetry, the GdxR contains a singular contribution whichalways integrates to a finite value. The explicit details of this calculation are givenelsewhere [53].

Third, Zeros are absent from projected models at half-filling. Under projection, the realpart of the Green function reduces to the first integral in Eq. (14) because projection doesnot preserve the contribution above the gap. This integral is always positive and hencezeros are absent. Transforming the operators in the t � J model to respect [57] the no dou-ble occupancy condition is of no help as the problem stems from the loss of spectral weightabove the gap once projection occurs. This result points to asymptotic slavery [40] of theHubbard model or equivalently an example of the non-commutativity discussed earlier inthat a low-energy reduction changes the physics. Stated another way, the t � J model vio-lates Luttinger’s theorem at half-filling. The essence of this breakdown is that there is nokinetic energy in the t � J model at half-filling, simply interacting spins. If there is nokinetic energy, there can be no surface of zeros. Such is not the case in the Hubbard modelsince no eigenstate of the Hubbard model has a fixed number of doubly occupied sites.

Fourth, even at infinitesimal doping, the t � J and Hubbard models are not equivalent.Strictly speaking, at half-filling, it is not entirely appropriate to compare the t � J andHubbard models because the former has no spectral weight above the chemical potential.Perhaps the proper way is to compare the two models in the limit of infinitesimal doping(see Figs. 5a and c). For one hole (n = 1�), numerical and analytical studies [58–60] find aquasiparticle in the t � J model with weight J/t at (p/2,p/2) whereas in the Hubbard model[61], the quasiparticle weight vanishes as Z / L�h, h > 0, L the system size. This result isconsistent because the one-hole system should closely mirror the half-filled system, whichfor the Hubbard model must have a surface of zeros for n = 1 while no such constraintexists in the t � J model. This has profound consequences for the minimal model neededto describe the cuprates.

Fifth, band structure cannot affect the existence of zeros in the presence of the Mottgap. At present, our proof applies to any kind of band structure that is generated fromhopping processes which remain unchanged after the application of Eq. (24). In general,the two kinds of hopping processes transform as � (p � kx,p � ky) = � � (kx,ky) andt 0 (p � kx,p � ky) = t 0 (kx,ky). The latter is relevant to the cuprates. If only such hoppingis present, the surface of zeros is no longer the diagonal (p, 0) to (0,p) (or the point ±p/2 in 1D), but rather the ‘‘cross’’ (0,p/2) to (p,p/2) and (p/2,0) to (p/2,p) (or, in 1D, thepoints ±p/4 and ±3p/4). When both types of hopping are present, no symmetry argu-ments can be made. However, at this point it is imperative that we distinguish betweenthe 1D and 2D cases. In 1D, if only two Fermi points exist, the Luttinger theorem requiresthat the zeros be at the symmetric Fermi points, ±p/2. But in principle there can be 2nFermi points in 1D. Consider the case of t and t 0 hopping. Four Fermi points exist fort 0 > t/2. In such cases, the zeros need not be at the Fermi surface of the non-interactingsystem to preserve the Luttinger volume. Such 1D cases share the complexity of the

Fig. 5. Evolution of the surface of zeros in the first quadrant of the FBZ. Yellow indicates RG > 0 while blueRG < 0. The Hubbard model is constrained to have a surface of zeros as n fi 1 whereas the t � J model is not.The two options upon doping represent weak-coupling (1a or c–d) and strong-coupling (1a–b). The transitionfrom (a) to (b) requires a critical point at nc whereas (d) does not. Experiments [2,6] indicating an insulating statefor n > nc are consistent with an abrupt transition from (a) to (b). (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this paper.)


general case of an asymmetric band structure in 2D. When this state of affairs obtains, wecan establish the existence of zeros by a key assumption: the Green function is a contin-uous function of the hopping parameters t and t 0. When only t is present, Rr (k, 0) hasone sign (plus) near k = (0,0) (or, in 1D, k = 0), and the opposite (minus) neark = (p,p) (k = p in 1D) and will vanish on the zero line. Alternatively, if we have t 0 hop-ping, Rr (k, 0) will have a certain sign near k = (0, 0) and k = (p,p), and the opposite signnear k = (0,p) and k = (p, 0) and will vanish on the ‘‘cross’’. From continuity, for t 0 � t,Rr (k;t, t 0) will have the same sign structure as Rr (k; t, t 0 = 0). That is, it will change signwhen going from (0, 0) to (p,p) regardless of the path taken. Therefore, the line of zerosexists for small enough t’, the relevant limit for the cuprates. In the opposite limit, t 0 t,a similar argument holds.

Finally, the surface of zeros defines the pseudogap at finite non-zero doping. At finitehole-doping (x = 1 � n), the chemical potential jumps into the lower Hubbard band. For


x � 1, Rr (k, 0) must vanish on some momentum surface because most of the spectralweight at (p,p) still resides above the chemical potential, whereas at (0,0) it resides below.Consequently, a surface of zeros should still be present so long as a gap, commonly knownas the pseudogap, opens up to some doping xc. Satisfying the zero condition, Eq. (14),requires spectral weight to lie immediately above the chemical potential. Spectral weighttransfer across the Mott gap depicted in Fig. 2 and discussed extensively previously medi-ates the zeros. The pseudogap that results is necessarily asymmetrical for small x becausethe 2x peak must lie closer to the chemical potential than the 1 � x contribution in orderfor Eq. (14) to vanish. Hence, an asymmetric pseudogap is a direct indication that Mott-ness rather than weak-coupling/ordering physics is the operative cause. In fact, such anasymmetry has been observed in the pseudogap regime in the cuprates [62]. The real ques-tion is how do the zeros evolve upon doping. As shown in Fig. 5, either the surface of zerosvanishes abruptly above a critical doping or it shrinks smoothly. In either case, the pseudo-gap must vanish at a finite number of k-points. A hard gap would involve all momenta.Hence, the pseudogap must vanish as xa, with a > 0. Because poles cannot exist withinan � neighbourhood of a zero if the pseudogap arises from Mottness, quasiparticles cannotsurvive at the nodes. Weak-coupling or Fermi liquid-type scenarios permit nodal quasipar-ticles. Numerics [63–65] point to such a non-symmetry breaking pseudogap in the Hub-bard model at finite doping whose cause is Mottness. Also, Stanescu and Kotliar [66]have shown convincingly using a cellular dynamical cluster method on the Hubbard modelwith nearest-neighbour hopping that the spectral features of a momentum-dependentpseudogap are directly related to the surface of zeros. This is the first extensive numericalstudy that has established the hard connection between zeros and the pseudogap.

5. Concluding remarks

Because of the divergence of the self energy, the surface of zeros plays a significant rolein doped Mott insulators. However, the only when particle-hole symmetry is present doesthe volume of the surface of zeros equal the particle density. In Fermi liquids, no spectralweight transfer occurs, no pseudogap phenomena exists and quasiparticles are welldefined. Just the opposite obtains in doped Mott systems. That broad spectral featuresnaturally follows from the surface of zeros is clear because sharp quasiparticles requirethe renormalized energy band to cross the chemical potential. This is not possible as longas RRrðkL; 0Þ diverges as has been demonstrated. Hence, one of the most nettling problemswith the cuprates, namely broad spectral features [8] has a natural resolution on the sur-face of zeros. The zeros also offer a concrete realisation of the non-commutativity ofU fi1 and L fi1 as they are present in the Hubbard model at half-filling but not inthe projected t � J model. A vanishing of the surface of zeros above a critical doping levelwill define a length scale associated with a gap. Such a length scale will undoubtedly play arole in the transport properties in the strange metallic regime. A complete treatment of thetransport properties and the nature of the excitations on the surface of zeros will necessi-tate a much sought after field theory of Mottness.

Acknowledgments

I wish to acknowledge several of my students who have worked assiduously on theideas presented in this review: Tudor D. Stanescu, whose thesis was entitled ‘‘Mottness,’’


Ting-Pong Choy, and Dimitrios Galanakis. I also acknowledge the collaboration withClaudio Chamon which led to the proof on T-linear resistivity. I also thank E. Fradkinwhose insistence that zeros did not exist forced us to construct the proof on zeros in Sec-tion 4 and A. Tsvelik for an e-mail exchange on the 1D zero problem. This work was fund-ed by the NSF Grant No. DMR-0305864. The Workshop on Mottness and QuantumCriticality was made possible by a Grant from NSF and the DOE.

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