The BCS-BEC crossover

Literature: Chapter 9. ”The BCS-BEC Crossover” by Meera M. Parish, and Chap-ter 10. ”Spectroscopies - Theory” by Paivi Torma in ”Quantum Gas Experiments –Exploring Many-Body States”, P. Torma and K. Sengstock (Eds.), Ref. [1]. MeeraParish is thanked for providing the pictures for this lecture.

Learning goals

• To learn the connection between BCS-type superfluids of fermions and Bose-Einstein condensation of composite bosons.

• To get some understanding of what is the concept of contact.• To be aware of some of the key quantum gas experiments that have proventhe existence of the BCS-BEC crossover.

9 Quantum phase transition vs. crossover

A crossover usually refers in physics to a situation where the system parameterschange from one regime to another. For instance, the interactions in a system maychange from weak to strong. Or, there can be a dimensional crossover from e.g.one to two-dimensional configuration. There can be also crossovers between twoqualitatively di↵erent many-body states. In general, the word crossover refers tosmooth and continuous evolution between the regimes or states. This is in contrastto quantum phase transitions (or usual thermal phase transitions). There the changefrom one system state to another is abrupt: thermodynamic quantities diverge.Remind yourself also about the di↵erence between a usual thermal phase transitionand a quantum phase transition: in the former, entropy and other system energiescompete and the transition occurs at finite temperature (e.g. Ising model) while inthe latter two di↵erent energies of the system (for instance kinetic and interactionenergies) compete and the transition can happen also at zero temperature when asystem parameter is changed (e.g. superfluid - Mott insulator transition).

Important• (Quantum) phase transition: abrupt change in system state, thermodynamicquantities diverge

• Crossover: smooth evolution of quantities from one qualitatively di↵erentstate/regime to another

10 The BCS-BEC crossover

Previously, you have learned about Bose-Einstein condensation (BEC) and, in theprevious two lectures, about the BCS theory of superconductivity. The BCS theorypresents superconductivity (superfluidity) as condensation of Cooper pairs. Thetheory describing the condensate of these pairs is, however, quite di↵erent from thetheory of BEC of bosons: the many-body wavefunction and the excitation spectrumlook quite di↵erent. However, one can actually argue that these two are connected.The Cooper pairs are actually just correlations in momentum space, so one can saythat the radius of the pair is very large: so large that several Cooper pairs overlapspatially. Now think about tuning the interparticle interaction between the particles

22

so that the pair size becomes smaller and smaller, eventually so small that a paircan be considered a point-like particle (that is, the pair size is much much smallerthan the average interparticle distance in the system). That is, we have formedcomposite bosons. Now, are these bosons still condensed forming a BEC, and is the

evolution from Cooper pairs to BEC of composite bosons completely smooth or are

there perhaps quantum phase transitions? This is a question that the Nobel laureateAnthony Leggett and other researchers such as Eagles, Nozieres and Schmitt-Rinkasked already in the 1980’s and even earlier [2, 3, 4]. The actual answer to thisquestion was obtained by ultracold gas systems only during the last decade. Let usnow approach the question first from a simple mean-field many-body descriptionpoint of view, following the original arguments of in Refs. [2, 3, 4], and a recentreview by Meera Parish [5] (apart from the book, a (not fully up-to-date) versionof the article is available in the arXiv, arXiv:1402.5171).

Let us consider a Fermi gas with two components (", #) in three dimensions(3D), with the Hamiltonian that is the same as in the previous lecture:

H � µN =X

k�

(✏k � µ) c†k� ck� +V0

V

X

k,k0,q

c†k"c†q�k#cq�k0#ck0", (10.1)

where the spin � = {", #}, the momentum dispersion ✏k = ~2k2

2m , V is the systemvolume, and µ the chemical potential. An attractive contact interaction with thestrength V

0

< 0 is considered. Note that here the mean-field approximation wasnot done, this is just the initial Hamiltonian (8.4), expressed with field operators,transformed to the momentum basis using the transformation of field operatorsto the momentum basis via a Fourier type transform, Eq.(8.17). Then performingone spatial integration in the Hamiltonian (8.4) imposes a restriction to the fourmomenta in the Fourier transforms and summation over three moemnta are left.Now approaching a mean-field approximation from this transformed Hamiltonian,one notices that Hamiltonian implies also pairs with a finite momentum q. In theBCS theory, it is assumed that a macroscopic number of pairs starts to accumulatewith q = 0 and the order parameter is chosen accordingly. This Hamiltonian canthus support both loosely bound Cooper pairs, as we have learned in previous lec-tures, and tightly bound molecule-type pairs which can be considered as compositebosons. We will come back to this point later, but for the moment, assume thatthese di↵erent types of pairs exist.

Let us now consider the composite boson regime. One can write the operatorof a composite boson as

b†q =X

k

'kc†k"c

†q�k# (10.2)

where 'k is the relative two-body wave function in momentum space. The many-body wave-function of a BEC of non-interacting bosons can be expressed as a co-herent state:

| iBEC = N e�ˆb†0 |0i (10.3)

where N is a normalization constant and � = h |b0

| i is the condensate orderparameter. In other words, |�|2/V corresponds to the condensate density. Notethat this assumes a weakly interacting boson gas with negligible quantum depletion

23

of the condensate due to interactions. This is a good approximation deep in thebosonic limit.

Using the definition of the composite boson operator (10.2) the state is

| iBEC = N e�P

k 'kc†k"c

†�k# |0i. (10.4)

On the other hand we know from previous lectures that the BCS state can beexpressed by the many-body wavefunction

| iBCS =Y

k

(uk + vkc†k"c

†�k#)|0i, (10.5)

where |uk|2 + |vk|2 = 1 due to normalization. The beautiful discovery of Eagles [2],Leggett [3] as well as Nozieres and Schmitt-Rink [4] was that the two simple mean-field many-body wavefunctions are actually the same, | iBEC = | iBCS ⌘ | i,that is,

| i = N e�P

k 'kc†k"c

†�k# |0i =

Y

k

(uk + vkc†k"c

†�k#)|0i (10.6)

if one requires vk/uk = �'k and N =Q

k uk. It is a nice exercise to prove theequality (10.6). This means that the BEC regime where composite bosons con-dense and the BCS regime of condensation of Cooper pairs are described by onewavefunction that evolves smoothly from one regime to the other. That is, thereis a BCS-BEC crossover — instead of, for instance, a quantum phase transitionbetween the regimes. Of course, this was just a prediction based on approximatemean-field theory which may be inadequate especially in the regime between theBCS and BEC sides. Therefore, it was a fundmentally important achiement to showby quantum gas experiments that it is indeed a crossover. The simple theory givenabove actually describes the crossover qualitatively quite well at low temperatures.However, it is not su�cient for quantitative predictions, especially at elevated tem-peratures; other methods such as quantum Monte Carlo (QMC) calculations haveto be applied and there are still many open questions related to the theory of thecrossover regime at high temperatures.

10.1 Connection to scattering length and the Feshbach resonance

Now, let us relate the above discussion more closely to ultracold gas experiments.Interactions between atoms in ultracold gas systems are often characterized by the s-wave scattering length aS . The s-wave is for most atoms the only relevant scatteringchannel at the ultralow temperatures. Furthermore, since the gases are dilute, therange of the potential is usually very small compared to the interparticle distanceand therefore the (in principle distance-dependent and complicated) potential canbe approximated by a contact potential characterized by a single quantity, thescattering length.

The bare interaction V0

in the Hamiltonian (10.1) is related to the scatteringlength aS via the renormalization relation we discussed in last lecture (renormal-ization is needed due to the unphysical assumption of a contact interaction)

m

4⇡~2aS=

1

V0

+1

V

⇤X

k

1

2✏k. (10.7)

24

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

k/kF

|vk|2

1/kFaS = −2

1/kFaS = 0

1/kFaS = 0.55

1/kFaS = 1.5

Figure 1: The momentum distribution v2k for interactions 1/kFaS varied across theBCS–BEC crossover.

Here we have introduced an ultraviolet cut-o↵ ⇤ which will be the upper limit of thesummation (integral). Its physical meaning can be understood as the inverse of therange of the interaction potential: in e↵ect, we now put back in the lengthscale thatwas neglected when a contact interaction with zero range was done. The dilutenessand low energies of the ultracold collisions mean that ⇤� 1/aS , ⇤� kF . Formally,one can take the limits V

0

! 0, ⇤! 1 while keeping the left hand side of Eq. (10.7)fixed and finite. The value of the bare interaction is thus irrelevant as well as thecuto↵, and by scaling the Hamiltonian by the Fermi energy (a nice exercise) one seesthat the ground state of the system depends on a single dimensionless parameterkFaS . The phenomenon of Feshbach resonance can be used to tune aS . A two-bodybound state appears around 1/aS = 0. The BCS and BEC regimes are reached,respectively, in the limits 1/kFaS � �1 and 1/kFaS � 1. The crossover, or socalled unitarity, region is defined as kF |aS | > 1. Thus by tuning the scatteringlength by magnetic field, one can experimentally explore the BCS-BEC crossover.

10.2 Momentum distribution

Let us now go back to the BCS-BEC wavefunction (10.6). One can do a mean-field description of the problem, corresponding to this wavefunction (c.f. earlierlectures) and use the scattering length as in the Hamiltonian instead of the bareinteraction. The momentum distribution nk = |vk|2 then evolves throughout thecrossover as depicted in Figure 1. In the BCS regime, the Fermi surface is onlyslightly smoothened due to pairing for momenta around the Fermi momentum.Towards the BEC regime, the distributions become flatter and flatter; more andmore momentum states are needed to build up tightly bound pairs (in the Fouriersense, when the two fermions localize very close to each other, a wide range ofmomenta of the individual particles is needed to create such a localized entity).

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10.3 Ground state properties

In the ground state, the free energy ⌦ = h |H � µN | i is minimized. The freeenergy is

⌦ = 2X

k

(✏k � µ)|vk|2 +V0

V

X

kk0

v⇤kukvk0u⇤k0 +

V0

V

X

kk0

|vk|2|vk0 |2. (10.8)

We take u, v to be real as before. The last term in the free energy corresponds tothe mean-field Hartree energy V

0

n2 which can be neglected since for short rangeinteractions the bare interaction V

0

can be considered small (remember in the previ-ous lecture we combined the Hartree term with the chemical potential which couldobvious be done also here). Minimizing the free energy at fixed chemical potential(take a derivative with respect to vk and use u2

k + v2k = 1 to get duk/(dvk)) gives

2(✏k � µ)ukvk + (u2

k � v2k)V0

V

X

k0

uk0vk0 = 0. (10.9)

Remarkably, in the limit vk ! 0, where the e↵ects of Pauli exclusion should benegligible, this reduces to the Schrodinger equation (just replace uk and u2

k by one)for the two-body bound state with wave function vk/

pN and binding energy �2µ:

2(✏k � µ)vk +V0

V

X

k0

vk0 = 0. (10.10)

This is another way of seeing how the wave function familiar from the BCS contextalso produces (approximately) something that describes a bound boson (or a con-densate of them). In the BEC regime thus the chemical potential becomes directlyrelated to the binding energy of the bound pair, µ ! �"B/2.

When not taking the limit vk ! 0, one has to solve the following equationsoriginating from Eq.(10.9):

� ⌘ U

V

X

k

ukvk = �U

V

X

k

�

2Ek(10.11)

n =1

V

X

k

v2k =1

2V

X

k

✓1� ✏k � µ

Ek

◆(10.12)

Ek =p(✏k � µ)2 +�2 (10.13)

Here Eqs. (10.11) and (10.12) correspond, respectively, to the usual gap and num-ber equations which we have seen in the previous lecture, and � is the BCS orderparameter. Note that sometimes in context of BCS theory especially for low temper-ature superconductors, only gap equation is solved. However, for large interactions,this starts to produce unphysically large order parameters (gap energies) �. To keepthese in control when approching strong interactions, it is necessary to solve alsothe number equation self-consistently together with the gap equation. This is some-times called the BCS-Leggett theory, and it can describe the BCS-BEC crossoverqualitatively well near T = 0.

It is of interest to consider the single particle excitation energy Ek in thetwo regimes of the crossover (note that we could have obtained Ek also by doinga Bogoliubov transformation to the mean-field Hamiltonian, as done in the previ-ous lecture). In the BCS regime, chemical potential is positive and the excitation

26

−2 −1 0 1 2−2

−1

0

1

2

1/kFaS

Δ/εF

µ/εF

Figure 2: The chemical potential µ/"F and the order parameter �/"F througoutthe crossover.

energy becomes essentially the BCS gap energy � for momenta such that ✏k ⇠ µi.e. close to the Fermi level. This is the single particle energy gap known from theBCS theory. On the BEC side the chemical potential becomes negative and large inabsolute value, since it corresponds to the binding energy. The order parameter �still remains finite, but the chemical potential dominates the single particle excita-tion energy Ek so that it becomes essentially µB/2 i.e. half of the binding energy.The single particle excitations in the BEC regime thus involve breaking a tightlybound pair. The development of the order parameter and the chemical potentialthroughout the crossover are depicted in Figure 2.

We have thus seen that both the momentum distribution (Figure 1) and thechemical potential and order parameter (Figure 2) evolve smoothly from the BCSto the BEC regimes.

Apart from single particle excitations, there is a low energy bosonic collectivemode (gapless Goldstone mode) associated with fluctuations of �. This mode couldbe calculated by introducing interactions between the quasiparticles. This collectivemode also evolves smoothly from one regime to another. Since the binding energyand thus the single particle excitation energy is very large in the BEC regime, thecollective mode is then the only low-energy excitation.

10.4 Unitarity regime

We have now seen how the two-component Fermi gas can evolve from Cooper pairs,which are correlated mainly in the momentum space and thus overlap physically,to tightly bound pairs in the BEC regime. Figure 3 shows this schematically; inthe crossover regime |kFaS | > 1, the pair size becomes of the order of the inter-particle spacing. One may anticipate that this can lead to some interesting newphysics. The limit 1/kFaS = 0 is called the unitarity limit and it gives rise to auniversal strongly interacting Fermi gas which is independent of any interactionlength scale. That is, formally the scattering length goes to infinity, but that actu-ally does not make any of the physical quantities singular. Instead of the scatteringlength, the system is characterized by a universal constant ⇠. At zero temperature,all thermodynamic quantities only depend on density via ⇠; for instance, the chem-

27

Figure 3: Change in the pair size in the BCS-BEC crossover in a two-componentFermi gas.

ical potential µ = ⇠"F and the total energy E = ⇠ 3

5

"FN . Ultracold gases were thefirst realization of the concept of a unitary Fermi gas. For those who wish to studythe topic in depth, Ref. [6] is recommended.

In addition to 1/kFaS = 0, there is another special point in the unitarityregime, namely the point where µ = 0. This is a turning point since the quasiparticleenergy Ek =

p(✏k � µ)2 +�2 has a minimum at a finite k for positive µ and at

k = 0 for negative chemical potential, the former corresponding to the BCS sideand the latter to the BEC of bound pairs. In fact, µ = 0 can be understood as thepoint where the Fermi surface disappears; the boundary between the BCS and BECregimes. It does not occur exactly at the Feshbach resonance point 1/kFaS = 0 buta bit on the repulsive interaction (positive scattering length) side of the resonance,namely 1/kFaS ' 0.55 as is seen from Figure 2, c.f. here the results of the task inExercise sheet 10 where a simple calculation on two-particle scattering restrictedby a Fermi sea also shows that the bound state appears for a finite value of thepositive scattering length.

10.5 Finite temperature

The transition temperature is determined by the low-energy excitations of the con-densate. In the BCS regime, the pairing gap is small and the single particle excita-tions that break Cooper pairs are the relevant low-energy excitations. The criticaltemperature of pair condensation thus coincides with the one of pair formation. Byminimizing the mean-field free energy at finite temperature

⌦(T ) = �V�2

V0

+X

k

(✏k � µ� Ek)� 2kBTX

k

ln⇣1 + e�Ek/kB

T⌘

(10.14)

one can show that Tc ⇠ �(0) (here kB = 1) and thus Tc/"F goes to zero exponen-tially when 1/kFaS ! �1 as known from BCS theory.

In the BEC regime, the energy needed for pair breaking and thus singleparticle excitations is very large, while the collective modes provide the low-energyexcitations that determine the transition temperature. Thus the critical temper-ature for pairing T ⇤ given by mean-field theory no longer coincides with Tc. Inthe limit 1/kFaS ! 1, Tc/"F saturates to the transition temperature for a non-interacting BEC, where Tc/"F ' 0.218. Describing the gas at finite temperatureand in the strongly interacting unitarity regime is a major open theoretical chal-lenge. Especially the nature of the normal state just above the critical temperature

28

-2 -1 0 1 21/kFaS

0

0.1

0.2T/�F

Figure 4: Condensation transition temperature Tc throughout the BCS–BECcrossover, calculated using the Nozieres–Schmitt-Rink approach [4] which includesGaussian fluctuations. The dashed line marks the temperature T ⇤ around whichpairs start to form. The filled circle marks the latest experimentally measured Tc

at unitarity [7], which is consistent with quantum Monte Carlo predictions [8, 9].Figure from [5].

is not known. It might be, for instance, a Fermi liquid, or a so called pseudo-gapstate which has gap-formation without superfluidity even above Tc. Ultracold gasexperiments have not yet been able to unanimously decide between these options.The question is of fundamental imporatance due to the universal nature of thegas, as well as because some high-temperature superfluids are believed to be in thestrong interaction regime between the BCS and BEC ends. Indeed the transitiontemperature seems to be high at the unitarity regime, see Figure 4.

10.6 The concept of Contact

One reason ultracold quantum gases are interesting is that they allow to studystrongly interacting, strongly correlated systems. Theoretical description of suchsystems is often highly di�cult: if the interparticle interaction is strong, it does noto↵er a small parameter for perturbation theory. Moreover, the quantum correlationsmay become so significant that even non-perturbative mean-field theories (such asthe BCS theory) and other non-perturbative methods fail. This is often the situationin quantum gas systems with large scattering lengths.

However, even in systems with large scattering length, it is possible to deriveuniversal relations. In the heart of these derivations is the notion that evenwhen the interactions are strong, there is still something that is very small in thesesystems: the range of the interaction. In the universal relations, the short scale andlarge scale physics are e↵ectively separated, yielding simple relations that can betested. The universal relations are indeed universal, i.e. valid for many-body, few-body, non-interacting, interacting, superfluid etc. systems. A central quantity in theuniversal relations is the so-called contact. The universal relations have been mostextensively studied so far in the context of the two-component interacting Fermigas. Many of the universal relations were first derived by Shina Tan [10, 11, 12]

29

and are therefore sometimes called Tan’s relations. The relations can be derivedin several ways; see e.g. Refs. [13] and [14] for other types of early derivations. Anexcellent review and pedagogical presentation of the topic is given in Ref. [15]. Ifyou want a brief (only four pages) summary of the concept and the most importantrelated experiments, it can be found as the Chapter 10.8. in Ref. [1].

The physical meaning of the universal relations can be described by variousapproaches, here I present my favourite which is the viewpoint given by the deriva-tion in Ref. [13]. It is based on the concept of operator product expansion (OPE)which expresses a product of two operators at di↵erent locations in the followingway

OA

⇣R+

r

2

⌘OB

⇣R� r

2

⌘=

X

C

fCA,B(r)OC (R) . (10.15)

Here fCA,B(r) are called Wilson coe�cients. The OPE is an asymptotic expansion

under certain conditions. Such an expansion is in general well justified in manyultracold quantum gas systems. Note that such an operator product could describe,for instance, density, spin or pairing correlations, that is, characterize the importantphysics of the system. Interestingly, the OPE means that one can express the origi-nal, non-local product in terms of a local operator and a function that only dependson the relative coordinate. If we now think that r approaches zero, it means that theWilson coe�cients depend on the short-range (meaning well below the scatteringlength) physics only. E↵ective separation of the short-range physics from the restis at the heart of deriving the universal relations. The ”rest” is what is describedby the contact C; note that this is often the interesting part of the physics, e.g. theproperties of the many-body state. The short-distance, short-time dynamics thenlead to the universal relations, with the contact appearing as a parameter in them.

In Ref. [13], the contact is defined as

C =

Zd3Rh�†�(R)i, (10.16)

where �(R) = g0

2

1

(R) and g0

is a cut-o↵ dependent coupling constant. Thecontact is thus related to the expectation value of the interaction term in the Hamil-tonian. The contact can also be understood as a measure of the increased likelihoodof finding two particles close (i.e. closer than the scale of the scattering length)to each other, caused by the strong interactions. Consequently, in the momentumdomain, it is a measure of atoms with large momentum. The contact itself is oftenhighly di�cult to calculate. However, it appears in several universal relations whichcan each be experimentally verified and mutually compared.

Perhaps the most important relation is the adiabatic relation. It is a ther-modynamic relation connecting the contact and the scattering length:

✓dE

da�1

S

◆

S

=

✓dF

da�1

◆

T

= � ~4⇡m

C, (10.17)

where E is the energy and F = E � TS the free energy, S the entropy and aS thescattering length. The contact thus tells how the system energy changes with thescattering length: if we know how the contact depends on the scattering length, thethermodynamics of the system is known.

Another example is the tail of the RF spectrum which is proportional tothe contact in the following way, in the case of a two-component gas with the RF

30

transfer to a third state that does not interact significantly with the two components(no final state interactions):

I(!) ! ⌦2

p~

4⇡pm!3/2

C. (10.18)

Here C is the contact between the two species.As third example, we mention the tail of the static

S(k) ! 1

8

✓1

k� 4

⇡aSk2

◆C, (10.19)

where C is the contact density (related to the contact by C =Rd3rC(r)).

10.7 Experiments on the BCS-BEC crossover

It is impossible to review exhaustively the ultracold gas experiments related to theBCS-BEC crossover in this lecture; I refer to Refs. [5, 1, 16] for more references.Note that the problem has been approached also in other systems than ultracoldgases although they o↵er so far the only clean and controllable platform to studythe crossover. Here I mention only some examples.

The first important crossover studies appeared in the year 2004 when two-component strongly interacting Fermi gases were cooled down and manipulatedwith the Feshbach resonance in such a way that condensates of molecules formed ofthe two components (bound papers) were achieved. This was done simultaneouslyby the groups of Rudi Grimm (Innsbruck) and Debbie Jin (JILA) [17, 18] and verysoon after by the group of Nobel laureate Wolfgang Ketterle [19]. These experimentsshowed that a condensate is possible on the BEC side of the Feshbach resonance. In2004, the groups of Jin [18] and Ketterle [20] showed the existence of Fermion pairsat the unitarity regime between the BCS and BEC regimes, and the experiments ofGrimm’s group using radio-frequency spectroscopy revealed the many-body natureof the pairing at the unitarity regime by showing that the pairing energy dependedon the Fermi energy[21, 22]. Studies of the whole crossover then followed fromthe groups of John Thomas [23], Rudi Grimm [24], Christophe Salomon [25] andRandy Hulet[26] focusing e.g. on characterizing collective modes and pairing. Theproof that the system is indeed a superfluid came in 2005 from Wolfgang Ketterle’sgroup with the observation of quantized vortices in the gas [27]. All the experimentsshowed that the evolution from the BCS to BEC side is smooth and continuous,and thus indeed it is a crossover. The was a remarkable achievement proving correctthe predictions of Eagles and Leggett.

The early experiments opened an avenue for a a large number of beautifulexperiments studying the crossover physics in depth – it would take too much timeto describe all of them in these lectures (one by M. Zwierlein’s group where forinstance Tc at unitarity was measured [7] was mentioned already in the context ofFigure 4). However, since I briefly introduced the concept of contact, I mention heresome of the experiments that have actually measured it.

In Refs. [28] and [29] Bragg spectroscopy was used for verifying the universalrelation for the tail of the static structure factor, Eq. (10.19). The contact wasobtained in Ref. [30] from several di↵erent universal relations: by measuring the tailof the momentum distribution from ballistic expansion and by using the momentum-resolved RF spectroscopy (photoemission spectroscopy), and the tail of the RFspectrum Eq. (10.18). The values of contact obtained from these three independentmeasurements were in good agreement with each other.

31

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