applying bcs–bec crossover theory to high-temperature...

Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5, p. 538–560

Applying BCS–BEC crossover theory to

high-temperature superconductors and ultracold atomic

Fermi gases

(Review Article)

Qijin Chen1, Jelena Stajic2, and K. Levin1

1James Franck Institute and Department of Physics, University of ChicagoChicago, Illinois, 60637 USA

E-mail: [email protected]

2Los Alamos National Laboratory, Los Alamos, New Mexico, 87545 USA

Received September 13, 2005

This review is written at the time of the twentieth anniversary of the discovery of high-temper-ature superconductors, which, nearly coincides with the important discovery of the superfluidphases of ultracold trapped fermionic atoms. We show how these two subjects have much in com-mon. Both have been addressed from the perspective of the BCS–Bose–Einstein condensation(BEC) crossover scenario, which is designed to treat short coherence length superfluids with tran-sition temperatures which are «high», with respect to the Fermi energy. A generalized mean fieldtreatment of BCS–BEC crossover at general temperatures T, based on the BCS–Leggett groundstate, has met with remarkable success in the fermionic atomic systems. Here we summarize thissuccess in the context of four different cold atom experiments, all of which provide indications, di-rect or indirect, for the existence of a pseudogap. This scenario also provides a physical picture ofthe pseudogap phase in the underdoped cuprates which is a central focus of high Tc research. Wesummarize successful applications of BCS–BEC crossover to key experiments in high Tc systems in-cluding the phase diagram, specific heat, and vortex core STM data, along with the Nernst effect,and exciting recent data on the superfluid density in very underdoped samples.

PACS: 74.20.Fg, 71.10.Ca

Keywords: high-temperature Tc superconductivity, Bose–Einstein condensation, fermionic atomic systems.

1. Introduction

1.1. Historical background

Most workers in the field of high Tc superconduc-tivity would agree that we have made enormous prog-ress in the last 20 years in characterizing these materi-als and in identifying key theoretical questions andconstructs. Experimental progress, in large part, co-mes from transport studies [1,2] in addition to threepowerful spectroscopies: photoemission [3,4], neutron[5–12] and Josephson interferometry [13–15]. Overthe last two decades, theorists have emphasized differ-ent aspects of the data, beginning with the anomalousnormal state associated with the highest Tc systems

(«optimal doping») and next, establishing the natureand implications of the superconducting phase, whichwas ultimately revealed to have a d-wave symmetry.Now at the time of this twenty year anniversary, oneof the most exciting areas of research involves the nor-mal state again, but in the low Tc regime, where thesystem is «underdoped» and in proximity to the Mottinsulating phase. We refer to this unusual phase as the«pseudogap state».

This pseudogap phase represents a highly anoma-lous form of superconductivity in the sense that thereis an excitation gap present at the superfluid transi-tion temperature Tc where long range order sets in.The community has struggled with two generic classes

© Qijin Chen, Jelena Stajic, and K. Levin, 2006

of scenarios for explaining the pseudogap and its im-plications below Tc. Either the excitation gap is inti-mately connected to the superconducting order re-flecting, for example, the existence of «pre-formedpairs», or it is extrinsic and associated with a compet-ing ordered state unrelated to superconductivity.

The emphasis of this Review is on the pseudogapstate as addressed by a particular preformed pair sce-nario which has its genesis in what is now referred toas «BEC–Bose–Einstein condensation (BEC) cross-over theory». Here one contemplates that the attrac-tion (of unspecified origin) which leads to super-conductivity is stronger than in conventional supercon-ductivity. In this way fermion pairs form before theyBose condense, much as in a Bose superfluid. In sup-port of this viewpoint for the cuprates are the observa-tions that: (i) the coherence length � for superconduc-tivity is anomalously short, around 10 � as comparedwith 1000 � for a typical superconductor. Moreover(ii) the transition temperatures are anomalously high,and (iii) the systems are close to two-dimensional(2D) (where pre-formed pair or fluctuation effects areexpected to be important). Finally, (iv) the pseudogaphas the same d-wave symmetry [16] as the supercon-ducting order parameter [3,4] and there seems to be asmooth evolution of the excitation gap from above tobelow Tc.

To investigate this BCS–BEC crossover scenariowe have the particular good fortune today of having anew class of atomic physics experiments involvingultracold trapped fermions which, in the presence ofan applied magnetic field, have been found to have acontinuously tunable attractive interaction. At highfields the system exhibits BCS-like superfluidity,whereas at low fields one sees BEC-like behavior.

This Review presents a consolidated study of boththe pseudogap phase of the cuprates and recent devel-opments in ultracold fermionic superfluids. The em-phasis of these cold atom experiments is on theso-called unitary or strong scattering regime, which isbetween the BEC and BCS limits, but on the fermio-nic side. The superfluid state in this intermediate re-gime is also referred to in the literature as a «resonantsuperfluid» [17,18]. Here we prefer to describe it asthe «pseudogap phase», since that is more descriptiveof the physics and underlines the close analogy withhigh Tc systems. Throughout this Review we will usethese three descriptive phrases interchangeably.

1.2. Fermionic pseudogaps and metastable pairs:two sides of the same coin

BCS–BEC crossover theory is based on the obser-vations of Eagles [19] and Leggett [20] who independ-ently noted that the BCS ground state wavefunction

� �0 0� � ��k k( )|† †u v c ck k k (1)

had a greater applicability than had been appreciatedat the time of its original proposal by Bardeen, Coo-per and Schrieffer (BCS). As the strength of the at-tractive pairing interaction U (< 0) between fermionsis increased, this wavefunction is also capable of de-scribing a continuous evolution from BCS like behav-ior to a form of BEC. What is essential is that thechemical potential � of the fermions be self consis-tently computed asU varies.

The variational parameters vk and uk are usuallyrepresented by the two more directly accessible pa-rameters � sc( )0 and �, which characterize the fer-mionic system. Here � sc( )0 is the zero temperaturesuperconducting order parameter. These fermionic pa-rameters are uniquely determined in terms of U andthe fermionic density n. The variationally determinedself consistency conditions are given by two BCS-likeequations which we refer to as the «gap» and «num-ber» equations respectively:

� �sc scUE

( ) ( )0 01

2� �

k k;

nE

�

��

��2 1

�k

kk

�(2)

where

E sck k� �( ) ( )� � 2 2 0� (3)

and �k � �2 2 2k / m are the dispersion relations for the

Bogoliubov quasiparticles and free fermions, respec-tively. An additional advantage of this formalism is thatBogoliubov–de Gennes theory, a real space implementa-tion of this ground state, can be used to address the ef-fects of inhomogeneity and external fields at T � 0. Thishas been widely used in the crossover literature.

Within this ground state there have been extensivestudies [21] of collective modes [22,23] and effects oftwo dimensionality [22]. Nozieres and Schmitt–Rinkwere the first [24] to address non-zero T. We will out-line some of their conclusions later in this Review.Randeria and co-workers reformulated the approach ofNozieres and Schmitt–Rink (NSR) and moreover,raised the interesting possibility that crossover physicsmight be relevant to high-temperature superconduc-tors [22]. Subsequently other workers have appliedthis picture to the high Tc cuprates [25–27] andultracold fermions [17,18,28,29] as well as formulatedalternative schemes [30,31] for addressing T � 0.Importantly, a number of experimentalists, most nota-bly Uemura [32], have claimed evidence in support[33–35] of the BCS–BEC crossover picture for highTc materials.

Applying BCS–BEC crossover theory to high temperature superconductors and ultracold atomic Fermi gases

Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 539

Compared to work on the ground state, considerablyless has been written on crossover effects at non-zerotemperature based on Eq. (1). Because our understand-ing has increased substantially since the pioneeringwork of NSR, and because they are the most interest-ing, this review is focused on these finite T effects.

The importance of obtaining a generalization ofBCS theory which addresses the crossover from BCSto BEC ground state at temperatures T Tc� cannot beoverestimated. BCS theory as originally postulatedcan be viewed as a paradigm among theories of con-densed matter systems; it is complete, in many waysgeneric and model independent, and well verified ex-perimentally. The observation that the wavefunctionof Eq. (1) goes beyond strict BCS theory, suggeststhat there is a larger mean field theory to be ad-dressed. Equally exciting is the possibility that thismean field theory can be discovered and simulta-neously tested in a very controlled fashion usingultracold fermionic atoms [17,18]. Mean field ap-proaches are always approximate. We can ascribe thesimplicity and precision of BCS theory to the fact thatin conventional superconductors the coherence length� is extremely long. As a result, the kind of averagingprocedure implicit in mean field theory becomesnearly exact. Once � becomes small BCS is not ex-pected to work at the same level of precision. Never-theless even when they are not exact, mean field ap-proaches are excellent ways of building up intuition.And further progress is not likely to be made withoutinvestigating first the simplest of mean field ap-proaches, associated with Eq. (1).

The effects of BEC–BCS crossover are most di-rectly reflected in the behavior of the fermionic chemi-cal potential �. We plot the behavior of � in Fig. 1,which indicates the BCS and BEC regimes. In the

weak coupling regime � � EF and ordinary BCS the-ory results. However at sufficiently strong coupling, �begins to decrease, eventually crossing zero and thenultimately becoming negative in the BEC regime,with increasing | |U . We generally view � � 0 as a cross-ing point. For positive � the system has a remnant of aFermi surface, and we say that it is «fermionic». Fornegative �, the Fermi surface is gone and the materialis «bosonic».

The new and largely unexplored physics of thisproblem lies in the fact that once outside the BCSregime, but before BEC, superconductivity or super-fluidity emerge out of a very exotic, non-Fermi liquidnormal state. Emphasized in Fig. 1 is this intermedi-ate (i.e., pseudogap or PG) regime having positive �which we associate with non-Fermi liquid based super-conductivity [25,36,37]. Here, the onset of supercon-ductivity occurs in the presence of fermion pairs. Un-like their counterparts in the BEC limit, these pairsare not infinitely long lived. Their presence is appar-ent even in the normal state where an energy must beapplied to create fermionic excitations. This energycost derives from the breaking of the metastable pairs.Thus we say that there is a «pseudogap» (PG) at andabove Tc. It will be stressed throughout this Reviewthat gaps in the fermionic spectrum and bosonic de-grees of freedom are two sides of the same coin. A par-ticularly important observation to make is that thestarting point for crossover physics is based on thefermionic degrees of freedom. A non-zero value of theexcitation gap � is equivalent to the presence ofmetastable or stable fermion pairs. And it is only inthis indirect fashion that we can probe the presence ofthese «bosons», within the framework of Eq. (1).

In many ways this crossover theory appears to rep-resent a more generic form of superfluidity. Withoutdoing any calculations we can anticipate some of theeffects of finite temperature. Except for very weakcoupling, pairs form and condense at different tem-peratures. More generally, in the presence of a moder-ately strong attractive interaction it pays energeti-cally to take some advantage and to form pairs (sayroughly at temperature T�) within the normal state.Then, for statistical reasons these bosonic degrees offreedom ultimately are driven to condense at T Tc �

� ,as in BEC.

Just as there is a distinction between Tc and T� ,there must be a distinction between the superconduct-ing order parameter � sc and the excitation gap �. InFig. 2 we present a schematic plot of these two energyparameters. It may be seen that the order parametervanishes at Tc, as in a second order phase transition,while the excitation gap turns on smoothly below T� .It should also be stressed that there is only one gap

540 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5

Qijin Chen, Jelena Stajic, and K. Levin

0 1 2U/Uc

–1

0

1

�/E

F

BCS PG BEC

Fig. 1. Behavior of the T � 0 chemical potential � in thethree regimes. � is essentially pinned at the Fermi temper-ature EF in the BCS regime, whereas it becomes negativein the BEC regime. The PG (pseudogap) case correspondsto non-Fermi liquid based superconductivity in the inter-mediate regime.

energy scale in the ground state [20] of Eq. (1). Thus� �sc( ) ( )0 0� .

In addition to the distinction between � and � sc,another important way in which bosonic degrees offreedom are revealed is indirectly through the tem-perature dependence of �. In the BEC regime wherefermionic pairs are pre-formed, � is essentially con-stant for all T Tc� (as is �). By contrast in the BCSregime it exhibits the well known temperature de-pendence of the superconducting order parameter.This behavior is illustrated in Fig. 3.

Again, without doing any calculations we can makeone more inference about the nature of crossover phys-ics at finite T. The excitations of the system mustsmoothly evolve from fermionic in the BCS regime tobosonic in the BEC regime. In the intermediate case,the excitations are a mix of fermions and metastablepairs. Figure 4 characterizes the excitations out of thecondensate as well as in the normal phase. This sche-matic figure will play an important role in our think-ing throughout this Review.

1.3. Introduction to high Tc superconductivity:pseudogap effects

This Review deals with the intersection of twofields and two important problems: high-temperaturesuperconductors and ultracold fermionic atoms inwhich, through Feshbach resonance effects, the at-tractive interaction may be arbitrarily tuned by a mag-netic field. Our focus is on the broken symmetry phaseand 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 super-conductors. There is an extensive body of theoreticalliterature which aims to understand these pseudogapeffects from a variety of different viewpoints. We listsome of these in Refs. 38–40 for the interested reader.A study of concrete data in these systems providesa rather natural way of building intuition aboutnon-Fermi liquid based superfluidity, and this should,in turn, be useful for the cold atom community.

It has been argued by some [26,27,42–44] that aBCS–BEC crossover-induced pseudogap is the originof the mysterious normal state gap observed in high-tem-perature superconductors. While this is a highly con-tentious subject some of the arguments in favor of thisviewpoint (beyond those listed in Section 1.1) rest onthe following observations: (i) To a good approxi-mation the pseudogap onset temperature [45,46]T /* ( ) .� 2 0 4 3� which satisfies the BCS scaling rela-tion. (ii) There is widespread evidence for pseudogapeffects both above [1,2] as well as (iii) below [47,48]Tc. (iv) In addition, it has also been argued that shortcoherence length superconductors may quite generallyexhibit a distinctive form of superconductivity [32]which sets them apart from conventional superconduc-tors. One might want, then, to concentrate on thismore generic feature (rather than on more exotic as-pects), which they have in common with other super-conductors in their distinctive class.



SC

(T)

Tc T

Fig. 2. Contrasting behavior of the excitation gap �( )Tand order parameter � sc T( ) versus temperature in thepseudogap regime. The height of the shaded region reflectsthe number of noncondensed pairs, at each temperature.

Tc T

TTc

a

b

Fig. 3. Comparison of temperature dependence of excita-tion gaps in BCS (a) and BEC (b) limits. The gap van-ishes at Tc for the former while it is essentially T-inde-pendent for the latter.

BCS Pseudogap BEC

Fig. 4. The character of the excitations in the BCS–BECcrossover both above and below Tc. The excitations areprimarily fermionic Bogoliubov quasiparticles in the BCSlimit and bosonic pairs (or «Feshbach bosons») in theBEC limit. For atomic Fermi gases, the «virtual mole-cules» in the PG case consist primarily of «Cooper» pairsof fermionic atoms.

In Fig. 5 we show a sketch of the phase diagram forthe hole-doped copper oxide superconductors. Here xrepresents the concentration of holes which can becontrolled by, say, adding Sr substitutionally toL1�xSr xCuO4. At zero and small x the system is anantiferromagnetic (AFM) insulator. Precisely at halffilling (x � 0) we understand this insulator to derivefrom Mott effects. These Mott effects may or may notbe the source of the other exotic phases indicated inthe diagram, i.e., the superconducting (SC) and the«pseudogap» phases. Once AFM order disappears thesystem remains insulating until a critical hole concen-tration (typically around a few percent) when an insu-lator-superconductor transition is encountered. Herephotoemission studies [3,4] suggest that once this lineis crossed, � appears to be positive. For x � 0 2. , thesuperconducting phase has a non-Fermi liquid (orpseudogapped) normal state [2]. We note an importantaspect of this phase diagram at low x. As the pseudogapbecomes stronger (T� increases), superconductivity asreflected in the magnitude of Tc becomes weaker.

Figure 6 indicates the temperature dependence ofthe excitation gap for three different hole stoichio-metries. These data [3] were taken from angle resolvedphotoemission spectroscopy (ARPES) measurement.For one sample shown as circles, (correspondingroughly to «optimal» doping) the gap vanishesroughly at Tc as might be expected for a BCS super-conductor. At the other extreme are the data indicatedby inverted triangles in which an excitation gap ap-pears to be present up to room temperature, with verylittle temperature dependence. This is what is referredto as a highly underdoped sample (small x), whichfrom the phase diagram can be seen to have a rather

low Tc. Moreover, Tc is not evident in these data onunderdoped samples.

While the high Tc community has focused onpseudogap effects above Tc, there is a good case to bemade that these effects also persist below. STM data[33] taken below Tc within a vortex core indicate thatthere is a clear depletion of the density of statesaround the Fermi energy in the normal phase withinthe core. These data underline the fact that the exis-tence of an energy gap has little or nothing to do withthe existence of phase coherent superconductivity. Italso underlines the fact that pseudogap effects effec-tively persist below Tc; the normal phase underlyingsuperconductivity for T Tc� is not a Fermi liquid.

Analysis of thermodynamical data [2,47] has led toa similar inference. For the PG case, the entropy ex-trapolated into the superfluid phase, based on Fermiliquid theory, becomes negative. In this way Loramand co-workers [47] deduced that the normal phase un-derlying the superconducting state is not a Fermi liq-uid. Rather, they claimed to obtain proper thermody-namics, it must be assumed that this state contains apersistent pseudogap. In this way they argued for a dis-tinction between the excitation gap � and the super-conducting order parameter, within the superconduct-ing phase. To fit their data they presume a modified

fermionic dispersion E Tk k� �( ) ( )� � 2 2� where

� � �2 2 2( ) ( )T Tsc pg� � (4)

Here � pg is taken on phenomenological grounds to beT-independent. While Eq. (4) is also found inBCS–BEC crossover theory, there are important dif-ferences. In the latter approach � pg � 0 as T � 0.



Hole concentration x

Te

mp

era

ture

AFM

Pseudogap

Tc

T

SC

Fig. 5. Typical phase diagram of hole-doped high Tc super-conductors. The horizontal axis is hole doping concentration.There exists a pseudogap phase above Tc in the underdopedregime. Here SC denotes superconductor, and T� is the tem-perature at which the pseudogap smoothly turns on.

0

5

10

15

20

25

50 100 150 200 250 300T, K

Sp

ect

ralg

ap

Fig. 6. Temperature dependence of the excitation gap atthe antinodal point (�,0) in Bi2Sr2CaCu2O8� � (BSCCO)for three different doping concentrations from near-opti-mal (discs) to heavy underdoping (inverted triangles), asmeasured by angle-resolved photoemission spectroscopy(from Ref. 3).

Finally, Fig. 7 makes the claim for a persistentpseudogap below Tc in an even more suggestive way.Figure 7,a represents a schematic plot of excitationgap data such as are shown in Fig. 6. Here the focus ison temperatures below Tc. Most importantly, this fig-ure indicates that the T dependence in � varies dra-matically as the stoichiometry changes. Thus, in theextreme underdoped regime, where PG effects aremost intense, there is very little T dependence in � be-low Tc. By contrast at high x, when PG effects are lessimportant, the behavior of � follows that of BCS the-ory. What is most impressive however, is that thesewide variations in �( )T are not reflected in thesuperfluid density � s T( ). Figure 7 then indicates 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 un-derstand if the fermionic degrees of freedom through�( )T are dominating at all x. Rather this figure sug-

gests that something other than fermionic excitationsis responsible for the disappearance of superconductiv-ity, particularly in the regime where �( )T is relativelyconstant in T. At the very least pseudogap effects mustpersist below Tc.

The phase diagram also suggests that pseudogap ef-fects become stronger with underdoping. How doesone accommodate this in the BCS–BEC crossover sce-nario? At the simplest level one may argue that as thesystem approaches the Mott insulating limit, fermionsare less mobile and the effectiveness of the attractionincreases. In making the connection between thestrength of the attraction and the variable x in thecuprate phase diagram we will argue that it is appro-priate to simply fit T x*( ). In this Review we do notemphasize Mott physics because it is not particularlyrelevant to the atomic physics problem. It also seemsto be complementary to the BCS–BEC crossover sce-nario. It is understood that both components are im-portant in high Tc superconductivity. It should bestressed that hole concentration x in the cupratesplays the role of applied magnetic field in the coldatom system. These are the external parameters whichserve to tune the BCS–BEC crossover.

Is there any evidence for bosonic degrees of freedomin the normal state of high Tc superconductors? Theanswer is unequivocally yes: metastable bosons are ob-servable as superconducting fluctuations. These ef-fects are enhanced in the presence of the quasi-two-di-mensional lattice structure of these materials. In theunderdoped case, one can think of T* as marking theonset of preformed pairs which are closely related tofluctuations of conventional superconductivity the-ory, but which are made more robust as a result ofBCS–BEC crossover effects, that is, stronger pairingattraction. A number of people have argued [49,50]that fluctuating normal state vortices are responsiblefor the anomalous transport behavior of the pseudogapregime. It has been proposed [51] that these data mayalternatively be interpreted as suggesting that bosonicdegrees of freedom are present in the normal state.

1.4. Summary of cold atom experiments: crossoverin the presence of Feshbach resonances

There has been an exciting string of developmentsover the past few years in studies of ultracoldfermionic atoms, in particular, 6Li and 40K, whichhave been trapped and cooled via magnetic and opticalmeans. Typically these traps contain105 atoms at verylow densities � 1013 cm�3. Here the Fermi tempera-ture in a trap can be estimated to be of the order of amicrokelvin. It was argued on the basis of BCS theoryalone [52], and rather early on (1997), that the tem-peratures associated with the superfluid phases may be



0

1a

T = 48 Kc

T = 90 Kc

1T/Tc

(T)/

(0)

T = 90 KCT = 90 KC

T = 88 KCT = 65 KCT = 55 KC

T = 50 KCT = 48 KC

0

–15

–300 40 80

T, K

ss

(T)–

(0)(

me

V)

b

Fig. 7. Temperature dependence of fermionic excitationgaps � and superfluid density � s for various doping con-centrations (from Ref. 48). When �( )Tc � 0, there is littlecorrelation between �( )T and � s T( ); this figure suggeststhat something other than fermionic quasiparticles (e.g.,bosonic excitations) may be responsible for the disappear-ance of superconductivity with increasing T. Figure (b)shows a quasi-universal behavior for the slope d /dTs� atdifferent doping concentrations, despite the highlynon-universal behavior for �( )T .

attainable in these trapped gases. This set off a searchfor the «holy grail» of fermionic superfluidity. That aFermi degenerate state could be reached at all is itselfquite remarkable; this was first reported [53] by Jinand deMarco in 1999. By late 2002 reports of unusualhydrodynamics in a degenerate Fermi gas indicatedthat strong interactions were present [54]. Thisstrongly interacting Fermi gas (associated with theunitary scattering regime) has attracted widespreadattention independent of the search for superfluidity,because it appears to be a prototype for analogous sys-tems in nuclear physics [55,56] and in quark-gluonplasmas [57,58]. Moreover, there has been a fairly ex-tensive body of analytic work on the ground stateproperties of this regime [59,60], which goes beyondthe simple mean field wave function ansatz.

As a consequence of attractive s-wave interactionsbetween fermionic atoms in different hyperfine states,it was anticipated that dimers could also be made.Indeed, these molecules formed rather efficiently[62–64] as reported in mid-2003 either via three bodyrecombination [65] or by sweeping the magnetic fieldacross a Feshbach resonance. Moreover, they are ex-tremely long lived [63]. From this work it was rela-tively straightforward to anticipate that a Bose con-densate would also be achieved. Credit goes totheorists such as Holland [17] and to Griffin [29] andtheir co-workers for recognizing that the superfluidityneed not be only associated with condensation of longlived bosons, but in fact could also derive, as in BCS,from fermion pairs. In this way, it was argued that asuitable tuning of the attractive interaction viaFeshbach resonance effects, would lead to a realiza-tion of a BCS–BEC crossover.

By late 2003 to early 2004, four groups [61,66–68]had observed the «condensation of weakly bound mol-ecules» (that is, on the as � 0 side of resonance), andshortly thereafter a number had also reported evidencefor superfluidity on the BCS side [69–72]. The BECside is the more straightforward since the presence ofthe superfluid is reflected in a bimodal distribution inthe density profile. This is shown in Fig. 8 fromRef. 61, and is conceptually similar to the behavior forcondensed Bose atoms [73]. On the BEC side but nearresonance, the estimated Tc is about 0 3. TF , with con-densate overtions varying from 20 or so to nearly 100 .The condensate lifetimes are relatively long in the vi-cinity of resonance, and fall off rapidly as one goesdeeper into the BEC. However, for as � 0 there is noclear expectation that the density profile will providea signature of the superfluid phase.

These claims that superfluidity may have beenachieved on the BCS side (as � 0) of resonance wereviewed as particularly exciting. The atomic commu-

nity, for the most part, felt the previous counterpartobservations on the BEC side were expected and notsignificantly different from condensation in Bose at-oms. The evidence for this new form of «fermionicsuperfluidity» rests on studies [69,70] that performfast sweeps from negative as to positive as across theresonance. The field sweeps allow, in principle, apairwise projection of fermionic atoms (on the BCSside) onto molecules (on the BEC side). It is pre-sumed that in this way one measures the momentumdistribution of fermion pairs. The existence of a con-densate was thus inferred. Other experiments whichsweep across the Feshbach resonance adiabatically,measure the size of the cloud after release [68] orwithin a trap [74].

Evidence for superfluidity on the BCS side, whichdoes not rely on the sweep experiments, has also beendeduced from collective excitations of a fermionic gas[71,75]. Pairing gap measurements with radio fre-quency (RF) spectroscopy probes [72] have similarlybeen interpreted [76] as providing support for the ex-istence of superfluidity, although more directly theseexperiments establish the existence of fermion pairs.Quite recently, evidence for a phase transition hasbeen presented via thermodynamic measurements andaccompanying theory [77]. The latter, like the theory[76] of RF experiments [72], is based on the formal-ism presented in this Review. A most exciting andeven more recent development has been the observa-tion of vortices [78] which appears to provide a smok-ing gun for the existence of the superfluid phase.



0

1.0

2.0

–200 0 –2000

1.0

2.0

–200 0 –200position, m position, m

op

tical

de

nsi

ty

a

b

Fig. 8. Spatial density profiles of a molecular cloud oftrapped 40K atoms in the BEC regime in the transversedirections after 20 ms of free expansion (from Ref. 61),showing thermal molecular cloud above Tc (left) and amolecular condensate (right) below Tc. (a) shows the sur-face plots, and (b) shows the cross-sections through im-ages (dots) with bimodal fits (lines).

2. Theoretical formalism for BCS–BECcrossover

2.1. Many-body Hamiltonian and two-bodyscattering theory

We introduce the Hamiltonian [17,29,79] used inthe cold atom and high Tc crossover studies. The mostgeneral form for this Hamiltonian consists of twotypes of interaction effects: those associated with thedirect interaction between fermions parametrized byU, and those associated with «fermion–boson» inter-actions, whose strength is governed by g.

H N ��

� � � �� ( ) ( ),

, ,k

k k†

kq

q q†

q�

� �� a a b bmb� 2

� ��

� � � � � � � � ��U a a a a/ / / /

( , ), ,

,†

,†

, ,q k k

q k q k q k q kk k2 2 2 2 �

�

� �� ( ( ) ),

†, ,

q kq q k q kkg b a a/ /2 2 h. c. .

(5)

Here the fermion and boson kinetic energies are givenby �k � k / m2 2 , and �q

mb q / M� 2 2 , and � is an impor-tant parameter which represents the magneticfield-induced «detuning». Here we use the conven-tion � � � �k cB 1. In this two channel problem theground state wavefunction is slightly modified andgiven by

� � �0 0 0� � B(6)

where the molecular or Feshbach boson contribution�0

B is as given in Ref. 80.Whether both forms of interactions are needed in

either system is still under debate. The bosons (bk†) of

the cold atom problem [17,18] are referred to as be-longing to the «closed channel». These spin-singletmolecules represent a separate species, not to be con-fused with the («open channel») fermion pairs(a ak –k

† † ), which are associated with spin triplet. As aresult of virtual occupation of the bound state of theclosed channel the interaction between open channelfermions can be tuned (through applied magneticfield) to vary from weak to very strong.

In this review we will discuss the behavior of cross-over physics both with and without the closed-chan-nel. Previous studies of high Tc superconductors haveinvoked a similar bosonic term [27, 81–83] as well, al-though less is known about its microscopic origin.This fermion–boson coupling is not to be confusedwith the coupling between fermions and a «pair-ing-mechanism»-related boson ([ ]† †b b a a� ) such asphonons in a metal superconductor. The coupling b aa†

and its Hermitian conjugate represent a form of sink

and source for creating fermion pairs, in this way in-ducing superconductivity in some ways, as aby-product of Bose condensation.

It is useful at this stage to introduce the s-wave scat-tering length, a, defined by the low energy limit oftwo-body scattering in vacuum. We begin with the ef-fects ofU only, presuming thatU is always an attractiveinteraction (U � 0) which can be arbitrarily varied,

ma U4

1 12�

� � ��kk

. (7)

We may define a critical valueUc of the potential asthat associated with the binding of a two particlestate in vacuum. We can write down an equation forUc given by

Uc� � �1 1

2�kk(8)

although specific evaluation ofUc requires that therebe a cut-off imposed on the above summation, associ-ated with the range of the potential. The fundamen-tal postulate of crossover theory is that even thoughthe two-body scattering length changes abruptly atthe unitary scattering condition (| | )a � � , in theN-body problem the superconductivity variessmoothly through this point.

Provided we redefine the appropriate «two body»scattering length, Equation (7) holds even in the pres-ence of Feshbach effects [28,29]. It has been shownthatU in the above equations is replaced by

U U Ug

� � �eff

2

2� �(9)

and we write a a� � . Experimentally, the two-bodyscattering length a� varies with magnetic field B.Thus we have

m

a U4

1 12� �

� � �eff �kk

. (10)

More precisely the effective interaction between twofermions is momentum and energy dependent. Itarises from a second order process involving emissionand absorption of a closed-channel molecular boson.The net effect of the direct plus indirect interactionsis given by

~ ( , , ) ( )g Q K K g Qeff eff� � �� k k

U Q U g D Qeff ( ) ( ),� � 20

where D Q / i nmb

0 1 2( ) [ ]� �� q � � is the non-in-teracting molecular boson propagator. Here andthroughout we use a four-momentum notation,Q i n� ( , )q � , and its analytical continuation,



Q i� � �( , )q � 0 , and write � Q QTn q , where

� n is a Matsubara frequency. What appears in thegap equation, however, is U Qeff ( )� 0 which we de-fine to be Ueff . When the open-channel attraction Uis weak, clearly, 2� �� is required so that theFeshbach-induced interaction is attractive. In the ex-treme BEC limit � �� 2 . However, when a deepbound state exists in the open channel, such as in40K, the system may evolve into a metastable statesuch that 2� �� in the BEC regime and there is apoint on the BCS side whereUeff � 0 precisely.

Figure 9 presents a plot of this scattering lengthk a k aF s F� * for the case of 6Li. It follows that as isnegative when there is no bound state, it tends to � atthe onset of the bound state and to �� just as the boundstate stabilizes. It remains positive but decreases invalue as the interaction becomes increasingly strong.The magnitude of as is arbitrarily small in both the ex-treme BEC and BCS limits, but with opposite sign.

2.2. T-matrix-based approaches to BCS–BEC cross-over in the absence of Feshbach effects

To address finite temperature in a way which isconsistent with Eq. (1), or with alternative groundstates, one introduces a T-matrix approach. Here onesolves self consistently for the single fermion propaga-tor G and the pair propagator t. That one stops at thislevel without introducing higher order Green’s func-tions (involving three, and four particles, etc) is be-lieved to be adequate for addressing a leading ordermean field theory such as that represented by Eq. (1).One can see that pair–pair (boson–boson) interactionsare only treated in a (generalized) mean field averag-ing procedure; they arise exclusively from the fer-mions and are sufficiently weak so as not to lead toany incomplete condensation in the ground state, as iscompatible with Eq. (1).

In this section we demonstrate that at the T-matrixlevel there are three distinct schemes which can be im-

plemented to address BCS–BEC crossover physics.Above Tc, quite generally one writes for the t-matrix

t QUU Q

( )( )

��1 !

(11)

and theories differ only on what is the nature of thepair susceptibility !( )Q , and the associated self en-ergy of the fermions. Below Tc one can also consider aT-matrix approach to describe the particles and pairsin the condensate. For the most part we will defer ex-tensions to the broken symmetry phase to Section 2.3.

In analogy with Gaussian fluctuations, Nozieresand Schmitt–Rink considered [24]

!0 0 0( ) ( ) ( )Q G K G Q KK

� � (12)

with self energy

"0 0( ) ( ) ( )K t Q G Q KQ

� � , (13)

where G K0( ) is the noninteracting fermion Green’sfunction. The number equation of the Nozie-res–Schmitt–Rink scheme [22,24] is then deduced inan approximate fashion [84] by using a leading orderseries for G with

G G G G� �0 0 0 0" . (14)

It is straightforward, however, to avoid this approxi-mation in Dyson’s equation, and a number of groups[31,37] have extended NSR in this way.

Similarly one can consider

!( ) ( ) ( )Q G K G Q KK

� � (15)

with self energy

"( ) ( ) ( )K t Q G Q KQ

� � . (16)

This latter scheme has been also extensively discussedin the literature, by among others, Haussmann [85],Tchernyshyov [86] and Yamada and Yanatse [44].

Finally, we can contemplate the asymmetric form[25] for the T-matrix, so that the coupled equationsfor t Q( ) and G K( ) are based on

!( ) ( ) ( )Q G K G Q KK

� � 0 (17)

with self energy

"( ) ( ) ( )K t Q G Q KQ

� � 0 . (18)

It should be noted, however, that this asymmetricform can be derived from the equations of motion bytruncating the infinite series at the three-particle



600 700 800 900 1000B, G

–3

–2

–1

0

1

2

3BEC PG BCS

834 G

ka F

s

Fig. 9. Characteristic behavior of the scattering length for6Li in the three regimes.

level, G3, and then factorizing the G3 into one- andtwo-particle Green’s functions [87]. The other twoschemes are constructed diagrammatically or from agenerating functional, (as apposed to derived from theHamiltonian). It will be made clear in what followsthat, if one’s goal is to extend the usual crossoverground state of Eq. (1) to finite temperatures, thenone must choose the asymmetric form for the pair sus-ceptibility, as shown in Eq. (17). Other approachessuch as the NSR approach to Tc, or that of Haussmannlead to different ground states which should, how-ever, be very interesting in their own right . Thesewill need to be characterized in future. Indeed, thework of Strinati group has also emphasized that theground state associated with the Tc calculations basedon NSR is distinct from that in the simple mean fieldtheory of Eq. (1), and they presented some aspects ofthis comparison in Ref. 88.

Other support for this GG0-based T-matrix schemecomes from its equivalence to self consistentHartree-approximated Ginzburg–Landau theory [89].Moreover, there have been detailed studies [90] todemonstrate how the superfluid density � s can becomputed in a fully gauge invariant (Ward Identityconsistent) fashion, so that it vanishes at the self con-sistently determined Tc. Such studies are currentlymissing for the case of the other two T-matrixschemes.

Some concerns about the other two T-matrixschemes can be raised. In Ref. 24 pairing fluctuationeffects are not self consistently included so that selfenergy corrections appear in the number equation butnot in the gap equation. Similarly, the GG-basedT-matrix scheme of Ref. 85 fails to recover BCS theory[87] in the weak attraction limit where pairing fluctu-ations are negligible.

2.3. Extending conventional crossover ground stateto T � 0: T-matrix scheme in the presence of

closed-channel molecules

In the T-matrix scheme we employ, the pairs aredescribed by the pair susceptibility !( )Q ��

�K /G Q K G K0 22( ) ( )� k q where G depends on a

BCS-like self energy " �( ) ( )K G K� 20

2� k . Through-

out this section � k � exp( )k / k2022 introduces a mo-

mentum cutoff, where k0 represents the inverse rangeof interaction, which is assumed infinite for a contactinteraction.

The noncondensed pairs [91] have propagatort Q U Q U Q Qpg( ) ( ) / [ ( ) ( )]� �eff eff1 ! , where Ueff isthe effective pairing interaction which involves the di-rect two-body interactionU as well as virtual excita-tion processes associated with the Feshbach resonance

[29,91]. At smallQ, tpg can be expanded, after analyt-ical continuation (i in� �� 0 ), as

t QZ

ipgq Q

( ) � � �

�1

� � #�pair. (19)

The parameters appearing in Eq. (19) are discussed inmore detail in Ref. 79. Here Z �1 is a residue and� q q / M� 2 2 * the pair dispersion, where M* is theeffective pair mass. The latter parameter as well asthe pair chemical potential �pair depends on the im-portant, but unknown, gap parameter � through thefermion self energy ". The decay width #Q is negligi-bly small for small Q below Tc.

While there are alternative ways of deriving theself consistent equations which we use (such as a de-coupling of the Green’s function equations of motion[87]), here we present an approach which shows howthis GG0-based T-matrix scheme has strong analogieswith the standard theory of BEC. But, importantlythis BEC is embedded in a self consistent treatment ofthe fermions. Physically, one should focus on � as re-flecting the presence of bosonic degrees of freedom. Inthe fermionic regime (� � 0), it represents the energyrequired to break the pairs, so that � is clearly associ-ated with the presence of «bosons». In the bosonic re-gime, �2 directly measures the density of pairs.

In analogy with the standard theory of BEC, it isexpected [91] that � contains contributions from bothnoncondensed and condensed pairs. The associateddensities are proportional to � pg T2 ( ) and ~ ( )� sc T2 , re-spectively. We may write the first of several con-straints needed to close the set of equations. (i) Onehas a constraint on the total number of pairs [79]which can be viewed as analogous to the usual BECnumber constraint

� � �2 2 2( ) ~ ( ) ( )T T Tsc pg� � . (20)

To determine �, (ii) one imposes the BEC-like con-straint that the pair chemical potential vanishes in thesuperfluid phase:

�pair � �0 T Tc. (21)

This yields

t Q Upg� �� 1 10 0 0 0( ) ( ) ( )eff ! (22)

so that

Uf E

Ek

eff� �

��1 20

1 2

20( )

( )k

kk� . (23)

Importantly, below Tc, � satisfies the usual BCS gapequation. Here we introduce the quasiparticle disper-



sion Ek k k� �( )� � �2 2 2� , where �k � �2 2 2k / m is

the fermion kinetic energy, � is the fermionic chemi-cal potential, and f x( ) is the Fermi distribution func-tion.

(iii) In analogy with the standard derivation ofBEC, the total contribution of noncondensed pairs isreadily computed by simply adding up their number,based on the associated propagator

� pg pgQ

t Q2 � � ( ) . (24)

One can rewrite Eq. (24) so that it looks more di-rectly like a number equation, by introducing the Bosedistribution function b x( ) for noncondensed pairs as

� �pg qZ b T2 1� � � ( , ), (25)

so that the noncondensed pair density is given byZ pg�2 . Note that the right hand sides of the previoustwo equations depend on the unknown � through theself energy appearing inG which, in turn enters tpg or� q. Also note that at T � 0, � pg � 0 so that all pairsare condensed as is consistent with the mean-fieldBCS–Leggett ground state.

Finally, in analogy with the standard derivation ofBEC, (iv) one can then compute the number of con-densed pairs associated with ~� sc, given that oneknows the total � and the noncondensed component.

Despite this analogy with BEC, fermions are thefundamental particles in the system. It is their chemi-cal potential � that is determined from the numberconservation constraint

n n n n n nf b b f b� � � � �2 2 20tot . (26)

Here nb0 and nb represent the density of condensedand noncondensed closed-channel molecules, respec-tively, nb

tot is the sum, and n G Kf K� 2 ( ) is theatomic density associated with the open-channel fer-mions. These closed channel fermions have a propaga-tor D Q( ), which we do not discuss in much detail inorder to make the presentation simpler. Here

n D Q Z bbQ

bq

� �� ( ) ( )� , (27)

where b x( ) is the Bose distribution function. Therenormalized propagator D Q( ) is given by the sameequation as Eq. (19) with a different residueZ Zb� �1 .In this way the system of equations is complete

[91]. The numerical scheme is straightforward in prin-ciple. We compute � (and �) via Eqs. (23) and (24),to determine the contribution from the condensate ~� scvia Eqs. (20) and (24). Above Tc, this theory must be

generalized to solve self-consistently for �pair whichno longer vanishes [92].

3. Physical implications: ultracold atomsuperfluidity

In this section we compare four distinct classes ofexperiments on ultracold trapped fermions with the-ory. These are thermodynamics [77,92], temperaturedependent density profiles [93], RF pairing gap spec-troscopy [72,94,95], and collective mode measure-ments [71,75]. We address all four experiments in thecontext of the mean field ground state of Eq. (1), and itsfinite temperature extension discussed in Section 2.3.That there appears to be good agreement between the-ory and experiment lends rather strong support to thesimple mean field theory, which is at the center of thisReview. Interestingly, pseudogap effects are evident invarious ways in these experiments and this serves to tiethe ultracold fermions to the high Tc superconductors.

3.1. Tc calculations and trap effects

Before turning to experiment, it is important to dis-cuss the behavior of the transition temperature whichis plotted as a function of scattering length in Fig. 10for the homogeneous case, presuming s-wave pairing.We discuss the effects of d-wave pairing in Section 4in the context of application to the cuprates. Startingfrom the BCS regime this figure shows that Tc ini-tially increases as the interaction strength increases.However, this increase competes with the opening of apseudogap or excitation gap �( )Tc . Technically, thepairs become effectively heavier before they form truebound states. Eventually Tc reaches a maximum (verynear unitarity) and then decreases slightly until field



– 2 0 2 41/k aF

0

0.1

0.2

0.3

T/T

CF

BECPGBCS

Fig. 10. Typical behavior of Tc as a function of 1/k aF in ahomogeneous system. Tc follows the BCS predictions andapproaches the BEC asymptote 0218. TF in the BEC limit.In the intermediate regime, it reaches a maximum around1 0/k aF � and a minimum around where � � 0.

strengths corresponding to the point where � becomeszero. At this field value (essentially where Tc is mini-mum), the system becomes a «bosonic» superfluid,and beyond this point Tc increases slightly to reachthe asymptote corresponding to an ideal Bose gas.Somewhat different behavior in Tc appears in alterna-tive theories where Tc is found to have a maximumnear unitarity and to approach the BEC asympototefrom above [24] or to have no extremal points and toapproach the BEC asymptote from below [85].

Trap effects change these results only quantita-tively as seen in Fig. 11. However, the maximum in Tcmay no longer be visible. The calculated value of Tc( . )� 0 3TF at unitarity is in good agreement with ex-periment [77,96] and other theoretical estimates [97].To treat these trap effects one introduces the localdensity approximation (LDA) in which Tc is com-puted under the presumption that the chemical poten-tial � �� V r( ) . Here we consider a spherical trapwith V r / m r( ) ( )� 1 2 2 2$ . The Fermi energy EF is de-termined by the total atom number N viaE k N k / mF BT

/FF

� � �� $( )3 21 3 2 2 , where kF is theFermi wavevector at the center of the trap. It can beseen that the homogeneous curve is effectively multi-plied by an «envelope» curve when a trap is present.This envelope, with a higher BEC asymptote, reflectsthe fact that the particle density at the center of thetrap is higher in the bosonic, relative to the fermioniccase. In this way Tc is relatively higher in the BEC re-gime, as compared to BCS, whenever a trap is present.

Figure 12 presents a plot of the position dependentexcitation gap �( )r and particle density n r( ) profileover the extent of the trap. An important point needs tobe made: because the gap is largest at the center of thetrap, bosonic excitations will be dominant there. At the

edge of the trap, by contrast, where fermions are onlyweakly bound (since �( )r is small), the excitations willbe primarily fermionic. We will see the implications ofthese observations as we examine thermodynamic andRF spectra data in the ultracold gases.

3.2. Thermodynamical experiments

Figure 13 present a plot which compares experi-ment and theory in the context of thermodynamic ex-periments [77,92] on trapped fermions. Plotted on thevertical axis is the energy which can be input in a con-trolled fashion experimentally. The horizontal axis istemperature which is calibrated theoretically based onan effective temperature ~T introduced phenomeno-logically, and discussed below. The experimental dataare shown for the (effectively) non-interacting case aswell as unitary. In this discussion we treat the non-in-teracting and BCS cases as essentially equivalent since� is so small on the scale of the temperatures consid-ered. The solid curves correspond to theory for the twocases. Although not shown here, even without a tem-perature calibration, the data suggests a phase transi-tion is present in the unitary case. This can be seen as aresult of the change in slope of E T( )

~as a function of ~T.

The phenomenological temperature ~T is relativelyeasy to understand. What was done experimentally todeduce this temperature was to treat the unitary caseas an essentially free Fermi gas to, thereby, infer thetemperature from the width of the density profiles,but with one important proviso: a numerical constantis introduced to account for the fact that the densityprofiles become progressively narrower as the systemvaries from BCS to BEC. This systematic variation inthe profile widths reflects the fact that in the freeFermi gas case, Pauli principle repulsion leads to alarger spread in the particle density than in the



– 2 0 2 40

0.1

0.2

0.3

0.4

0.5BECPGBCS

1/k aF

T/T

CF

Fig. 11. Typical behavior of Tc of a Fermi gas in a trap as afunction of 1/k aF . It follows BCS prediction in the weakcoupling limit, 1 1/k aF �� , and approaches the BEC as-ymptote 0518. TF in the limit 1/k aF �. In contrast tothe homogeneous case in Fig. 10, the BEC asymptote ismuch higher due to a compressed profile for trapped bosons.

0.5 1r/RTF

0

0.2

0.4

0.6

0.8

1

n(r

,)ar

b.u

nit.

;/E

F

n(r)

At unitarity

(r)

Fig. 12. Typical spatial profile of T � 0 density n r( ) andfermionic excitation gap �( )r of a Fermi gas in a trap. Thecurves are computed at unitarity, where 1 0/k aF � . HereRTF is the Thomas–Fermi radius.

bosonic case. And the unitary regime has a profilewidth which is somewhere in between, so that oneparametrizes this width by a simple function of %. Wecan think of% as reflecting bosonic degrees of freedom,within an otherwise fermionic system. At% � 0 the sys-tem is a free Fermi gas. The principle underlying thisrescaling of the non-interacting gas is known as the«universality hypothesis» [96,98]. At unitarity, theFermi energy of the non-interacting system is the onlyenergy scale in the problem (for the widely used con-tact potential) since all other scales associated withthe two-body potential drop out when as � & �. Werefer to this phenomenological fitting temperatureprocedure as Thomas–Fermi (TF) fits.

An interesting challenge was to relate this pheno-menological temperature ~T to the physical temperatureT; more precisely one compares 1 � % ~T and T. This re-

lationship is demonstrated in the inset of Fig. 13. Andit was in this way that the theory and experimentcould be plotted on the same figure, as shown in themain body of Fig. 13. The inset was obtained usingtheory only. The theoretically produced profiles wereanalyzed just as the experimental ones to extract1 � % ~T and compare it to the actual T. Above Tc no

recalibration was needed as shown by the straight linegoing through the diagonal. Below Tc the phenome-nologically deduced temperatures were consistentlylower than the physical temperature. That the normalstate temperatures needed no adjustment shows thatthe phenomenology captures important physics. 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 the-ory and experiment for the global and low T thermo-dynamics. Figure 14 presents a blow-up of E at thelowest T comparing the unitary and non-interactingregimes. The agreement between theory and experi-ment is quite good. In the figure, the temperature de-pendence of E reflects primarily fermionic excitationsat the edge of the trap, although there is a smallbosonic contribution as well. It should be noted thatthe theoretical plots were based on fitting % to experi-ment by picking a magnetic field very slightly off res-onance. (In the simple mean field theory % � 0 41. ,and in Monte Carlo simulations [60] % � 0 54. . Boththese theoretical numbers lie on either side of experi-ment [77] where % � 0 49. ).

Figure 15 presents a wider temperature scale plotwhich, again, shows very good agreement. Impor-tantly one can see the effect of a pseudogap in the uni-tary case. The temperature T* can be picked out fromthe plots as that at which the non-interacting and uni-tary curves intersect. This corresponds roughly toT Tc� � 2 .

3.3. Temperature dependent particle densityprofiles

In order to understand more deeply the behavior ofthe thermodynamics, we turn next to a comparison offinite T density profiles. Experiments which measurethese profiles [74,99] all report that they are quitesmooth at unitarity, without any signs of the bimo-dality seen in the BEC regime. We discuss these pro-files in terms of the four panels in Fig. 16. These fig-



0.1 1T/TF

0.01

0.1

1

10E

(t)/

E–

1h

eat

0

0 0.2 0.81+ ~T0

0.2

0.8

T/T

F

T/T = 0.27F

Tc

Fig. 13. Energy E vs physical temperature T. The uppercurve and data points correspond to the BCS or essentiallyfree Fermi gas case, and the lower curve and data corre-spond to unitarity. The latter provide indications for aphase transition. The inset shows how temperature mustbe recalibrated below Tc. From Ref. 77.

0 0.1 0.2 0.3 0.4 0.5T/TF

1

2

E/E

F

Tc= 0.29

Theory, noninteractingTheory, unitarynoninteractingunitary

Fig. 14. Low temperature comparison of theory (curves)and experiments (symbols) in terms of E/EF (E k TF B F� )per atom as a function of T/TF, for both unitary andnoninteracting gases in a Gaussian trap. From Ref. 77.

ures are a first step in understanding the previoustemperature calibration procedure.

In this figure we compare theory and experimentfor the unitary case. The experimental data were esti-mated to correspond to roughly this same temperature(T/TF � 019. ) based on the calibration procedure dis-cussed above. The profiles shown are well within thesuperfluid phase (T Tc F� 0 3. at unitarity). This figurepresents Thomas–Fermi fits [99] to (a) the experi-mental and (b) theoretical profiles as well as (c) theircomparison, for a chosen RTF � 100 �m, which makesit possible to overlay the experimental data (circles)and theoretical curve (line). Finally Fig. 16,d indi-cates the relative !2 or root-mean-square (rms) devia-tions for these TF fits to theory. This figure was madein collaboration with the authors of Ref. 99. Two ofthe three-dimensions of the theoretical trap profileswere integrated out to obtain a one-dimensional repre-sentation of the density distribution along the trans-verse direction: n x dydz n r( ) ( )� � .

This figure is in contrast to earlier theoretical stud-ies which predict a significant kink at the condensateedge which appears not to have been seen experimen-tally [74,99]. Moreover, the curves behavemonotonically with both temperature and radius. In-deed, in the unitary regime the generalized TF fittingprocedure of Ref. 99 works surprisingly well. Andthese reasonable TF fits apply to essentially all tem-peratures investigated experimentally [99], as well astheoretically, including in the normal state.

It is important to establish why the profiles are sosmooth, and the condensate is, in some sense, rather in-visible, except for its effect on the TF-inferred-temper-ature. This apparent smoothness can be traced to thepresence of noncondensed pairs of fermions which need

to be included in any consistent treatment. Indeed,these pairs below Tc are a natural counterpart of thepairs above Tc which give rise to pseudogap effects.

To see how the various contributions enter into thetrap profile, in Fig. 17 we plot a decomposition of thisprofile for various temperatures from below to aboveTc. The various color codes indicate the condensatealong with the noncondensed pairs and the fermions.This decomposition is based on the superfluid densityso that all atoms participate in the condensation atT � 0. This, then, forms the basis for addressing boththermodynamics and RF pairing-gap spectroscopy inthis Review.

The figure shows that by T T /c� 2 there is a reason-able number of excited fermions and bosons. As antici-pated earlier in Section 2.3, the latter are at the trapedge and the former in the center. By T Tc� the con-densate has disappeared and the excitations are a mixof fermions (at the edge) and bosons towards the cen-ter. Indeed, the noncondensed bosons are still presentby T Tc� 15. , as a manifestation of a pseudogap effect.Only for somewhat higher T Tc� 2 do they disappear



0.5 1 1.5T/TF

0

2

4E

/EF

Tc=0.29

Theory, noninteractingTheory, unitarynoninteractingunitary

Fig. 15. Same as Fig. 14 but for a much larger range oftemperature. The quantitative agreement between theoryand experiment is very good. The fact that the two experi-mental (and the two theoretical) curves do not merge untilhigher T Tc

* � is consistent with the presence of apseudogap.

–1 0 1x (100 m)

0

0.5

1

1.5

–1 0 1x/RTF

0

0.5

1

1.5TheoryExperiment

0.2 0.4 0.6T/TF

0

0.005

0.01

0.015

Tc= 0.272

–1 0 10

0.5

1

1.5Comparison

n(x)

n(x)

n(x)

2

a b

cd

x/RTF

Fig. 16. Temperature dependence of (a) experimentalone-dimensional spatial profiles (circles) and TF fit (line)from Ref. 99, (b) TF fits (line) to theory both atT T Tc F 07 019. . (circles) and (c) overlay of experimen-tal (circles) and theoretical (line) profiles, as well as (d)

relative rms deviations (�2) associated with these fits to

theory at unitarity. The circles in (b) are shown as theline in (c). The profiles have been normalized so that

N n x dx� �� ( ) 1, and we set RTF � 100 �m in order to

overlay the two curves. �2 reaches a maximum around

T TF� 019. .

altogether, so that the system becomes a non-interact-ing Fermi gas.

Two important points should be made. The non-condensed pairs clearly are responsible for smoothingout what otherwise would be a discontinuity [98,100]between the fermionic and condensate contributions.Moreover, the condensate shrinks to the center of thetrap as T is progressively raised. It is this thermal ef-fect which is responsible for the fact that the TF fit-ting procedure for extracting temperature leads to anunderestimate as shown in the inset to Fig. 13. Thepresence of the condensate tends to make the atomiccloud smaller so that the temperature appears to belower in the TF fits.

3.4. RF pairing gap spectroscopy

Measurements [72] of the excitation gap � havebeen made by using a third atomic level, called |3�,which does not participate in the superfluid pairing.Under application of RF fields, one component of theCooper pairs, called |2�, is presumably excited to state|3�. If there is no gap � then the energy it takes to ex-cite |2� to |3� is the atomic level splitting $23. In thepresence of pairing (either above or below Tc) an extraenergy � must be input to excite the state |2�, as a re-sult of the breaking of the pairs. Figure 18 shows aplot of the spectra near unitarity for four differenttemperatures, which we discuss in more detail below.In general for this case, as well as for the BCS andBEC limits, there are two peak structures which ap-pear in the data: the sharp peak at $23 0� which is as-sociated with «free» fermions at the trap edge and thebroader peak which reflects the presence of paired at-oms; more directly this broad peak derives from thedistribution of � in the trap. At high T (compared to

�), only the sharp feature is present, whereas at low Tonly the broad feature remains. The sharpness of thefree atom peak can be understood as coming from alarge phase space contribution associated with the2 3� excitations [95]. Clearly, these data alone donot directly indicate the presence of superfluidity, butrather they provide strong evidence for pairing.

As pointed out in Ref. 94 these experiments serve asa counterpart to superconducting tunneling in provid-ing information about the excitation gap. A theoreti-cal understanding of these data was first presented inRef. 76 using the framework of Section 2.3. Subse-quent work [95] addressed these data in a more quan-titative fashion as plotted in Fig. 19. Here the upperand lower panels correspond respectively to interme-diate and low temperatures. For the latter one seesthat the sharp «free atom» peak has disappeared, sothat fermions at the edge of the trap are effectivelybound at these low T. Agreement between theory andexperiment is quite satisfactory, although the total



0

0.02

0.04

0.06 nnpairnsnQP

0

0.02

0.04

De

nsi

typ

rofil

es

0.5 1 1.50

0.02

0.04

00.51r/R TF

0 0.5 1 1.5 0 0.5 1 1.5 00.51r/RTF

0 0.5 1 1.5

T/Tc= 1.5

1.0

0.75

0.5

0.25

0

Unitary

Fig. 17. Decomposition of density profiles at various tem-peratures at unitarity. Here (light gray) refers to the con-densate, (dark gray) to the noncondensed pairs and(black) to the excited fermionic states. T Tc F� 027. , andRTF is the Thomas–Fermi radius.

0.4

0

0.4

0

0.4

0

0.4

0

–20 0 20 40

Frac

tion

allo

ssin

|2>

Therefore offset, kHz

T/T = 1.2c

1.1

0.85

<0.4

Fig. 18. Experimental RF spectra at unitarity. The tem-peratures labeled in the figure were computed theoreti-cally at unitarity based on adiabatic sweeps from BEC.The two top curves, thus, correspond to the normal phase,thereby, indicating pseudogap effects. Here EF � 25. �K,or 52 kHz. From Ref. 72.

number of particles was adjusted somewhat relative tothe experimental estimates.

It is interesting to return to the previous figure(Fig. 18) and to discuss the temperatures in the vari-ous panels. What is measured experimentally are tem-peratures T� which correspond to the temperature atthe start of a sweep from the BEC limit to unitarity.Here fits to the BEC-like profiles are used to deduce T�from the shape of the Gaussian tails in the trap. Basedon knowledge about thermodynamics (entropy S),and given T�, one can then compute the final tempera-ture in the unitary regime, assuming S is constant. In-deed, this adiabaticity has been confirmed experimen-tally in related work [74]. We find that the fourtemperatures are as indicated in the figures. Impor-tantly, one can conclude that the first two cases corre-spond to a normal state, albeit close to Tc. Impor-tantly, these figures suggest that a pseudogap ispresent as reflected by the broad shoulder above thenarrow free atom peak.

3.5. Collective breathing modes at T � 0

We turn, finally, to a comparison between theory[101,102] and experiment [71,75,96,103] for the col-lective breathing modes within a trap at T � 0. Thevery good agreement has provided some of the earliestand strongest support for the simple mean field theoryof Eq. (1). Interestingly, Monte Carlo simulationswhich initially were viewed as a superior approach,

lead to significant disagreement between theory andexperiment [104]. Shown in Fig. 20 is this comparisonfor the axial mode in the inset and the radial mode inthe main body of the figure as a function of magneticfield. The experimental data are from Ref. 71. Theoriginal data on the radial modes from Ref. 75, was indisagreement with that of Ref. 71, but this has sincebeen corrected [104], and there is now a consistent ex-perimental picture from both the Duke and theInnsbruck groups for the radial mode frequencies.

At T � 0, calculations of the mode frequencies canbe reduced to a calculation of an equation of state for� as a function of n. One of the most important conclu-sions from this figure is that the behavior in thenear-BEC limit (which is still far from the BEC as-ymptote) shows that the mode frequencies decreasewith increasing magnetic field. This is opposite to ear-lier predictions [105] based on the behavior of truebosons where a Lee-Yang term would lead to an in-crease. Indeed, the pair operators do not obey the com-mutation relations of true bosons except in the zerodensity or k aF � �0 limit [106]. Figure 20, thus, un-derlines the fact that fermionic degrees of freedom (orcompositeness) are still playing a role at these mag-netic fields. There are predictions in the literature[107] that one needs to achieve k aF somewhat lessthan 0 3. (experimentally, the smallest values for theseexperiments are 0.3 and 0.7 for the various groups) inorder to approach the true bosonic limit. At this point,then, the simple mean field theory will no longer be



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

Sp

ect

rali

nte

nsi

ty, a

rb. u

nits

Fig. 19. Comparison of calculated RF spectra (solid curve,T Tc F 029. ) with experiment (symbols) in a harmonic trapcalculated at 822 G for the two lower temperatures. Thetemperatures were chosen based on Ref. 72. The particlenumber was reduced by a factor of 2, as found to be neces-sary in addressing another class of experiments [88]. Thedashed lines are a guide to the eye. From Ref. 95.

–4 –2 0 2 41.52

1.54

1.56

1.58

1.60

–4 –2 0 2 4

x

��/

�

2.2

2.1

2.0

1.9

1.8

1.7–3 –1 1 3

x = (N a/a )1/6h0

–1

BEC limit

BCS limit

unitarity limit

Fig. 20. Breathing mode frequencies as a function of� �1695 1. ( )k aF , from Tosi et al. [10]. The main figureand inset plot the transverse and axial frequencies, respec-tively. The solid curves are calculations [101] based onBCS–BEC crossover theory at T � 0, and the symbols plotthe experimental data from Kinast et al. [71]. Here N istotal atom number, and aho the harmonical oscilator length.

adequate. Indeed, there are other indications [108] ofthe breakdown of this mean field in the extreme BEClimit which are, physically, reflected in the width ofthe particle density profiles. This originates from anoverestimate (by roughly a factor of 3) of the size ofthe effective «inter-boson» scattering length.

Overall the mean field theory presented here looksvery promising. Indeed, the agreement between theoryand experiment is better than one might have antici-pated. For the collective mode frequencies, it appearsto be better than Monte Carlo calculations [104]. Nev-ertheless, uncertainties remain. Theories which posit adifferent ground state will need to be compared withthe four experiments discussed here. It is, finally,quite possible that incomplete T � 0 condensation willbecome evident in future experiments. If so, an alter-native wavefunction will have to be contemplated[107,109]. What appears to be clear from the currentexperiments is that, just as in high Tc superconduc-tors, the ultracold fermionic superfluids exhibitpseudogap effects. These are seen in thermodynamics,in RF spectra and in the temperature dependence ofthe profiles (through the noncondensed pair contribu-tions). Moreover, while not discussed here, at finite T,damping of the collective mode frequencies seems tochange qualitatively [96] at a temperature which isclose to the estimated T� .

Looking to the future, at an experimental level,new pairing gap spectroscopies appear to be emergingat a fairly rapid pace [110,111]. These will further testthe present and subsequent theories. Indeed, recently,a probe of the closed channel overtion [111] has beenanalyzed [112] within the present framework and hasled to good quantitative agreement between theoryand experiment.

4. Physical implications: high Tcsuperconductivity

4.1. Phase diagram and superconducting coherence

The high Tc superconductors are different from theultracold fermionic superfluids in one key respect;they are d-wave superconductors and their electronicdispersion is associated with a quasi-two dimensionaltight binding lattice. In many ways this is not a pro-found difference from the perspective of BCS–BECcrossover. Figure 21 shows a plot of the two importanttemperatures Tc and T� as a function of increasing at-tractive coupling. On the left is BCS and the right isPG. The BEC regime is not visible. This is because Tcdisappears before it can be accessed. This disappear-ance of Tc is relatively easy to understand. Because thed-wave pairs are more extended (than their s-wavecounterparts) they experience Pauli principle repul-

sion more intensely. Consequently the pairs localize(their mass is infinite) well before the fermionic chem-ical potential is negative [42].

The competition between T� and Tc, in which as T�

increases, Tc decreases, is also apparent in Fig. 21.This is a consequence of pseudogap effects. More spe-cifically, the pairs become heavier as the gap increasesin the fermionic spectrum, competing with the in-crease of Tc due to the increasing pairing strength. Itis interesting to compare Fig. 21 with the experimen-tal phase diagram plotted as a function of the dopingconcentration x in Fig. 5. If one inverts the horizontalaxis (and ignores the unimportant AFM region) thetwo are very similar. To make an association from cou-plingU to the variable x, it is reasonable to fit T� . Itis not particularly useful to implement this last stephere, since we wish to emphasize crossover effectswhich are not complicated by «Mott physics».

Because of quasi-two dimensionality, the energyscales of the vertical axis in Fig. 21 are considerablysmaller than their three-dimensional analogues. Thus,pseudogap effects are intensified, just as conventionalfluctuation effects are more apparent in low-dimen-sional systems. This may be one of the reasons why thecuprates are among the first materials to clearly revealpseudogap physics. Moreover, the present calculationsshow that in a strictly 2D material, Tc is driven tozero, by bosonic or fluctuation effects. This is a directreflection of the fact that there is no Bose condensa-tion in 2D.

The presence of pseudogap effects raises an interest-ing set of issues surrounding the signatures of the tran-sition which the high Tc community has wrestledwith, much as the cold atom community is doing to-day. For a charged superconductor there is no diffi-culty in measuring the superfluid density, through the



0 1 2Attractive coupling constant

0.1

0.2

T/E

F Pseudogap

Superconductor

NormalTTc

Fig. 21. Typical phase diagram for a quasi-two dimen-sional d-wave superconductor on a tight-binding lattice athigh filling n 085. per unit cell; here the horizontal axiscorresponds to �U/ t4 , where t is the in-plane hopping ma-trix element.

electrodynamic response. Thus one knows with cer-tainty where Tc is. Nevertheless, people have beenconcerned about precisely how the onset of phase co-herence is reflected in thermodynamics, such as Cv orin the fermionic spectral function, given that a gap isalready present at the onset of superconductivity. Oneunderstands how phase coherence shows up in BCStheory, since the ordered state is always accompaniedby the appearance of an excitation gap.

To address these coherence effects one has to intro-duce a distinction [115] between the self energy asso-ciated with noncondensed and condensed pairs. Thisdistinction is blurred by the approximations made inSection 2.3. Within this improved scheme [115] super-conducting coherence effects can be probed as, pre-sented in Fig. 22, along with a comparison to experi-ment. Shown are the results of specific heat andtunneling calculations and their experimental coun-terparts [2,33]. The latter measures, effectively, thedensity of fermionic states. Here the label «PG» corre-sponds to an extrapolated normal state in which we setthe superconducting order parameter � sc to zero, butmaintain the the total excitation gap � to be the sameas in a phase coherent, superconducting state. Agree-ment between theory and experiment is satisfactory.

4.2. Electrodynamics in the superconducting phase

In some ways the subtleties of phase coherent pair-ing are even more perplexing in the context of electro-dynamics. Figure 7 presents a paradox in which theexcitation gap for fermions appears to have little to dowith the behavior of the superfluid density. Thissuperfluid density can be readily computed within theBCS–BEC crossover scenario [25,48]. Particularlyimportant is to include all excitations of the conden-sate in a fully consistent fashion, compatible withthermodynamics, and which is also manifestly gaugeinvariant. To make contact with electrodynamic ex-periments, one has to introduce the variable x and thisis done via a fit to T x*( ) in the phase diagram. In ad-dition it is also necessary to fit � s T x( , )� 0 to experi-ment, and we do so here, noting that [32] the Uemurarelation � s cx T x( , ) ( )0 ' no longer holds for veryunderdoped samples [114,116]. By fitting these x-de-pendent quantities we are, in effect accounting for atleast some aspects of Mott physics. The paradox raisedby Fig. 7 is resolved by noting that there are bosonicexcitations of the condensate [25] and that these be-come more marked with underdoping, as pseudogapeffects increase. In this way � s does not exclusively re-flect the fermionic gap, but rather vanishes «prema-turely» before this gap is zero, as a result of pair exci-tations of the condensate.

This theory can be quantitatively compared withexperiment. Figure 23 presents theoretical and experi-mental plots of the lower critical field, H Tc1( ), for agroup of severely underdoped YBCO crystals as con-sidered in Ref. 114. There it was argued thatH T Tc s1( ) ( )' � , so that the lower critical field effec-tively measures the in-plane superfluid density.Experimentally what is directly measured is the mag-netization with applied field parallel to the c-axis. Theexperimental results are shown on the lower two pan-els and theory on the upper two. The left hand figuresplot H Tc1( ) vs T and the right hand figures corre-spond to a rescaling of this function in the formH T /Hc c1 1 0( ) ( ) vs T/Tc. Theoretically, it is foundthat the fermionic contribution leads to a linear T de-pendence at low T, associated with d-wave pairing,whereas the bosonic term introduces a T /3 2 term.Quite remarkably even when the Uemura relation nolonger holds, there is still a «universality» in the nor-malized plots as shown in both theory and experimentby the right hand figures. It should be noted that theexperimental plot contains (at Tc � 55 5. K) a slightlydifferent cuprate phase known as the ortho-II phase,which does not lie on the universal curves. The univer-sality found here can be understood as associated withthe fact that T xc( ), rather than �( )x , is the fundamen-tal energy scale in � s T x( , ). The reason that �( )x isnot the sole energy scale is that bosonic degrees offreedom are also present, and help to drive � s to zeroat Tc. By contrast, Fermi-liquid based approaches[114,117] assume that the fermions are the only rele-



10

5

0

0.2

0.1

0

–4 –2 0 2 4

1 2 3 0

1

2

3

50 150 250

–300 –100 0 100 3000

1.0

2.0

dI/

dV

S/(

T/T

),C

/(T

/T)

cv

c

dI/

dV

,G–

1

V V , mVsample

SCPG

SCPG

4.2 KT = 83 Kc

(underdoped)

� = 0.439Y

S/T

E = 220 Kz

x = 0.125

S/T

C /TV

T/Tc T, K

Fig. 22. Extrapolated normal state (PG) and supercon-ducting state (SC) contributions to SIN tunneling andthermodynamics (left), as well as comparison with experi-ments (right) on tunneling for BSCCO [33] and on spe-cific heat for Y0 8. Ca02. Ba2Cu3O7�� [113]. The theoreticalSIN curve is calculated for T T /c� 2, while the experimen-tal curves are measured outside (dashed line) and inside(solid line) a vortex core.

vant excitations, and they account for this data by in-troducing a phenomenological parameter ( which cor-responds to the effective charge of the fermionicquasi-particles.

As anticipated in earlier theoretical calculations[25] the bosonic contribution begins to dominate in se-verely underdoped systems so that the slope dH /dTc1(associated with the lowest temperatures reached ex-perimentally) should decrease with underdoping. Al-though observed a number of years after this predic-tion, this is precisely what is seen experimentally, asshown in Fig. 24. Here the inset plots the experimen-tal counterpart data. It can be seen that theory and ex-periment are in reasonably good quantitative agree-ment. This theoretical viewpoint is very different froma «Fermi-liquid» based treatment of the supercon-ducting state, for which the strong decrease in theslope of Hc1 was not expected. Within the present for-malism, the optical conductivity [1] ) $( ) is similarlymodified [118] to include bosonic as well as fermioniccontributions.

4.3. Bosonic power laws and pairbreaking effects

The existence of noncondensed pair states below Tcaffects thermodynamics, in the same way that electro-

dynamics is affected, as discussed above. Moreover,one can predict [36] that the q2 dispersion will lead toideal Bose gas power laws in thermodynamical andtransport properties. These will be present in additionto the usual power laws or (for s-wave) exponentialtemperature dependencies associated with thefermionic quasiparticles. Note that the q2 dependenceis dictated by the ground state of Eq. (1). Clearly thismean field like state is inapplicable in the extremeBEC limit, where, presumably interboson effects be-come important and lead to a linear dispersion. Pre-sumably, in the PG or near-BEC regimes, fermionicdegrees of freedom are still dominant and it is reason-able to apply Eq. (1). Importantly, at present neitherthe cuprates nor the cold atom systems access this trueBEC regime.

The consequences of these observations can now belisted [36]. For a quasi-two dimensional system,C /Tvwill appear roughly constant at the lowest tempera-tures, although it vanishes strictly at T � 0 as T /1 2.The superfluid density � s T( ) will acquire a T /3 2 con-tribution in addition to the usual fermionic terms. Bycontrast, for spin singlet states, there is no explicitpair contribution to the Knight shift. In this way thelow T Knight shift reflects only the fermions and ex-hibits a scaling with T/�( )0 at low temperatures. Ex-perimentally, in the cuprates, one usually sees a finitelow T contribution to C /Tv . A Knight shift scaling isseen. Finally, also observed is a deviation from thepredicted d-wave linear in T power law in � s . The newpower laws inCv and � s are conventionally attributedto impurity effects. Experiments are not yet at a stageto clearly distinguish between these two alternativeexplanations.



10 15 20 25T, K

0

10

20

30

40

50

Tc

8.9 K13.7K15.5K17.6K22.0K

0.2 0.4 0.6 0.8 1T/Tc

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

5 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

T, K10 15 20 255

T/Tc

HO

ec1

,H

Oe

c1,

H/H

(0)

c1c1

H(T

)/H

(0)

c1c1

Tc

8.9 K13.7 K15.5 K17.6 K22.0 K55.5 K

Fig. 23. Comparison between calculated lower criticalfield, Hc1, as a function of T (upper left panel), and ex-perimental data (lower left ) from Ref. 114, with variabledoping concentration x. The right column shows normal-ized plots, H T /Hc c1 1 0( ) ( ) versus T/Tc, for theory and ex-periment, respectively, revealing a quasi-universal behav-ior with respect to doping, with the exception of theTc � 555. K ortho-II phase. Both theory plots share thesame legends. The quantitative agreement between theoryand experiment is quite good.

1

1.5

2

2.5

3

2

10 20 40 60

T , Kc

dH

/dT

(Oe

/K)

c1

T , Kc

dH

/dT

(Oe

/K)

c1

0 20 40

Fig. 24. Comparison of theoretically calculated low Tslope d dH / Tc1 (main figure) for various doping concentra-tions (corresponding to different Tc) in the underdoped re-gime with experimental data (inset) from Ref. 114. Thetheoretical slopes are estimated using the low temperaturedata points accessed experimentally. The quantitativeagreement is very good.

Pairbreaking effects are viewed as providing impor-tant insight into the origin of the cuprate pseudogap.Indeed, the different pairbreaking sensitivities of T�

and Tc are usually proposed to support the notion thatthe pseudogap has nothing to do with superconductiv-ity. To counter this incorrect inference, a detailed setof studies was conducted (based on the BEC–BCS sce-nario), of pairbreaking in the presence of impurities[119,120] and of magnetic fields [121]. These studiesmake it clear that the superconducting coherence tem-perature Tc is far more sensitive to pairbreaking thanis the pseudogap onset temperature T� . Indeed, thephase diagram of Fig. 21 which mirrors its experimen-tal counterpart, shows the very different, even com-peting nature of T� and Tc, despite the fact that botharise from the same pairing correlations.

4.4. Anomalous normal state transport:Nernst coefficient

Much attention is given to the anomalous behaviorof the Nernst coefficient in the cuprates [49]. This co-efficient is rather simply related to the transverse ther-moelectric coefficient ( xy which is plotted in Fig. 25.In large part, the origin of the excitement in the litera-ture stems from the fact that the Nernst coefficient be-haves smoothly through the superconducting transition.Below Tc it is understood to be associated with super-conducting vortices. Above Tc if the system were aFermi liquid, there are arguments to prove that theNernst coefficient should be essentially zero. Hence theobservation of a non-negligible Nernst contribution hasled to the picture of fluctuating «normal state vortices».

The formalism of Ref. 51 can be used to addressthese data within the framework of BCS–BEC cross-over. The results are plotted in Fig. 25 with a subsetof the data plotted in the upper right inset. It can beseen that the agreement is reasonable. In this way a«pre-formed pair» picture appears to be a viable alter-native to «normal state vortices». It will, ultimately,be necessary to take these transport calculations belowTc. This is a project for future research and in this con-text it will be important to establish in this picturehow superconducting state vortices are affected by thenoncondensed pairs and conversely.

5. Conclusions

In this Review we have summarized a large body ofwork on the subject of the BCS–BEC crossover sce-nario. In this context, we explored the intersection oftwo very different fields: high Tc superconductivityand cold atom superfluidity. Theories of cuprate su-perconductivity can be crudely classified as focusingon «Mott physics» which reflects the anomalously

small zero temperature superfluid density and «cross-over physics», which reflects the anomalously shortcoherence length. Both schools are currently very in-terested in explaining the origin of the mysteriouspseudogap phase. In this Review we have presented acase for its origin in crossover physics. The pseudogapin the normal state can be associated with metastablepairs of fermions; a (pseudogap) energy must be sup-plied to break these pairs apart into their separatecomponents. The pseudogap also persists below Tc inthe sense that there are noncondensed fermion pair ex-citations of the condensate.

The recent discovery of superfluidity in cold fer-mion gases opens the door to a set of fascinating prob-lems in condensed matter physics. Unlike the bosonicsystem, there is no counterpart of Gross–Pitaevskiitheory. A new theory which goes beyond BCS and en-compasses BEC in some form or another will have tobe developed in concert with experiment. As of thiswriting, there are four experiments where the simplemean field theory discussed in this review is in rea-sonable agreement with the data. These include thecollective mode studies over the entire range of acces-sible magnetic fields [101,102]. In addition in the uni-tary regime, RF spectroscopy-based pairing gap stud-ies [76,95], as well as density profile [93] andthermodynamic studies [77,92] all appear to be com-patible with this theory. Interestingly, all of theseprovide indications for a pseudogap either directlythrough the observation of the normal state energyscales, T* and �, or indirectly, through the observa-tion of noncondensed pairs. The material in this Re-view is viewed as the first of many steps in a long pro-cess. It is intended to provide continuity from onecommunity (which has addressed the BCS–BECcrossover scenario, since the early 1990’s) to another.



0 0.5 1.0 1.5 2.0 2.50

1

2

3

4

T (100 K)

�xy

T�/Tc =�

0 1 20

1

2

3

4

T (100 K)

xy

T�/Tc =7.4T�/Tc = 3.1

Fig. 25. Calculated transverse thermoelectric response,which appears in the Nernst coefficient, as a function oftemperature for the underdoped cuprates.

We gratefully acknowledge the help of our manyclose collaborators over the years: Jiri Maly, Boldiz�arJank�, Ioan Kosztin, Ying-Jer Kao, Andrew Iyengar,Shina Tan, and Yan He. We also thank our co-authorsJohn Thomas, Andrey Turlapov and Joe Kinast, aswell as Thomas Lemberger, Brent Boyce, JoshuaMilstein, Maria Luisa Chiofalo, and Murray Holland.This work was supported by NSF-MRSEC Grant No.DMR-0213765 (JS,ST and KL), NSF Grant No.DMR0094981 and JHU-TIPAC (QC).

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