the fermi-hubbard model - eth z · pdf filethe fermi-hubbard model ... they can be two...

Paivi Torma, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zurich

The Fermi-Hubbard model

Literature: Chapter 9. ”The BCS-BEC Crossover” by Meera M. Parish, and Chapter3. ”Quantum Gases in Optical Lattices” by Peter Barmettler and Corinna Kollathin ”Quantum Gas Experiments – Exploring Many-Body States”, P. Torma and K.Sengstock (Eds.), Ref. [1]. Corinna Kollath is thanked for providing a picture forthis lecture.

Learning goals

• To know the basics of the Fermi-Hubbard model.• To understand the basic features of the BCS-BEC crossover in a lattice withinthe attractive Hubbard model, and the di↵erences to the continuum case.

• To know the most important phases predicted for the repulsive Hubbardmodel: Mott insulator, antiferromagnet and d-wave superfluid.

• To be aware of the mapping between the repulsive and attractive Hubbardmodels.

11 The Hubbard model

In the ultracold gases context, the Bose-Hubbard model is often the one that istaught first in courses since it was experimentally realized by ultracold gases wellbefore the Fermi-Hubbard model. This is the case also in these series of quantum gaslectures at ETH: you have already learned about the Bose-Hubbard model and itsrealization in optical lattices in the fall semester. However, it is important to keep inmind that historically, it is the fermionic version that is the classic Hubbard model:the Hubbard model was designed in 1963 describe electrons (which are fermions) insolid state materials [31]. Only in eighties a bosonic version was considered[32, 33]and in the nineties it was suggested to be realized by optical lattices [34]. Now wewill learn about the ”original” Hubbard model, that is, the Fermi-Hubbard model.Since it indeed may describe electrons in materials and thereby various interestingstates of solid state materials, it is of great significance and has potential for instanceto shed light on the mechanism of high-temperature superconductivity which is stillnot fully understood. In recent years, the realization of the Fermi-Hubbard modelhas been achieved in optical lattices [35, 36]. This opens remarkable opportunitiesfor solving such questions with the help of quantum gas experiments.

The standard Fermi-Hubbard model considers two species of fermionic parti-cles which are confined in a lattice and move by hopping to the nearest neighboursites. The two species for electrons are the spin-up and spin-down electrons. Forultracold gases, they can be two di↵erent internal states of an atom or molecule(e.g. hyperfine states), or even two di↵erent species of atoms; in any case, we labelthem by a pseudospin � =", #. The particles interact only when at the same sitewithin the standard Hubbard model. There exist various extended Hubbard modelswith longer range interactions, but we do not discuss them here. In optical lattices,the nearest neighbour hopping and on-site interaction are realized by having a suf-ficiently strong lattice potential and short range of the interparticle interaction.The latter is the usual case for ultracold atoms where the interaction can be oftenapproximated even by contact interaction.

32


The Fermi-Hubbard Hamiltonian is

H = �JX

hn,mi

�c†n,� cn,� + h.c.

�+ U

X

n

nn,"nn,# ,

where c†n,� is the creation operator for a fermion with (pseudo-) spin � =", # and lat-tie site specified with discrete 3D (or 2D or 1D) set of integers marked with n. Herehn,mi denoted summation over nearest neighbours. The operator nn,� = c†n,� cn,� isthe density operator. The hopping coe�cient J and the interaction strength U aregiven by optical lattice parameters by an expansion of the annihilation and creationoperators of each species with the help of Wannier functions:

Jn,m = �Z

d3r w⇤(r� n)

✓� ~22m

r2 + VL(r)

◆w(r�m)

= � 1

V0

Z

k21st BZ

d3k eik·(n�m)Ek and (11.1)

Un,m,n0,m0 = g

Zd3r w⇤(r� n)w⇤(r�m)w(r� n0)w(r�m0).

The band energy Ek and the Wannier functions are obtained from the Bloch equa-tions for a single particle

✓~22m

(�ir+ k)2 + VL(r)

◆u(n)k (r) = E(n)

k u(n)k (r) . (11.2)

For approximate analytical formulas for J and U see e.g. [37]. If you have forgottenthese issues, please have a look at the autumn semester course.

The Fermi-Hubbard model leads to rich physics both for repulsive (U > 0)and attractive (U < 0) interactions. The fermionic statistics and the spin degree offreedom are behind many of the interesting phenomena found in the model. Here wechoose to discuss the attractive interactions first because this allows the describethe BCS-BEC crossover in a lattice and thus there is a direct connection to theprevious lecture. The repulsive interaction case is discussed after that. We will alsodiscuss a mapping between the attractive and repulsive Hubbard model.

11.1 BCS-BEC crossover in a lattice

In case of weak attractive interactions, the Hubbard model has a BCS-type super-fluid ground state. When the strength of the attractive interaction is increased, onecan observe the BCS–BEC crossover, but with some characteristics that are specificfor lattices. In particular, while the critical temperature is low in the BCS limit andhigher at unitarity both for the lattice and continuum cases, in the BEC limit thecritical temperature of a continuum gas stays nearly constant while the one in thelattice decreases dramatically. We will now learn what is the origin of this di↵erence.

Let us start from the Fermi-Hubbard Hamiltonian introduced above:

H = �JX

hn,mi

�c†n,� cn,� + h.c.

�+ U

X

n

nn,"nn,# , (11.3)

where again J is the hopping energy and U now corresponds to an attractive on-site interaction (U < 0). Mean-field approximation introducing pairing fields can be

33


done for this Hamiltonian in a similar way as done in the lecture where we discussedthe BCS theory. The superfluid order parameter now becomes

� = Uhc†"nc†#ni. (11.4)

The Hamiltonian, including the chemical potential as should be the case when usingthe grand canonical ensemble, is of the form (we again assume � real)

H � µN =�X

n

⇣µ"c

†"nc"n + µ#c

†#nc#n

⌘

+X

n

✓�c†"nc

†#n +�c#nc"n � �2

U

◆

� JX

hn,mi,�

c†�mc�n.

(11.5)

To diagonalize the Hamiltonian and to understand the nature of pairing, it is con-venient to write it in the (quasi-)momentum representation. This means doing aFourier transform by

c"n =1pM

X

k

eik·nc"k

c†"n =1pM

X

k

e�ik·nc†"k

c#n =1pM

X

k

eik·nc#k

c†#n =1pM

X

k

e�ik·nc†#k,

(11.6)

where M is the (finite) number of lattice sites and k runs through the reciprocallattice. The Hamiltonian becomes

bH =X

k

✓⇠"kc

†"kc"k + ⇠#kc

†#kc#k

+�c†"kc†#�k +�c#�kc"k

◆� �2

U,

(11.7)

where ⇠�k = ✏�k � µ� =P

↵ 2J(1 � cos(k↵)) � µ�. Here ↵ is x, y, z. We considerhere the case µ� ⌘ µ. Note that in order to get the dispersion correspond to thatof a free particle in the limit of small k, the following terms have been added to theHamiltonian:

2X

↵

X

k

⇣Jc†�kc�k

⌘=X

↵

2JN�. (11.8)

The momenta are restricted to the first Brillouin zone |kx|, |ky|, |kz| ⇡/a, wherea is the lattice spacing. Note, further, that U is finite in the lattice case, and a highmomentum cut-o↵ in the lattice is set by ⇡/a. Therefore renormalization proceduresthat were applied in the usual BCS theory in the previous lectures are not neededhere. For the same reasons, the Hartree term cannot be formally neglected – inpractice, it leads to a constant shift Un of the chemical potential and is assumed

34


to be included in the chemical potentials above. This means that the interactionenergy of the normal Fermi liquid phase is always included in the BCS mean-fieldtheory, unlike in the continuum case without the lattice. The Hamiltonian (11.7)can be diagonalized and gives solutions very similar to those in continuum BCStheory, for certain value range of the chemical potential µ; we will go through thedetails in the following lecture where the FFLO state in a lattice is discussed. Theextra length scale a given by the lattice means that the BCS-BEC crossover nowdepends separately on the density, defined by the dimensionless parameter "F /Jand the dimensionless interaction |U |/J . In the lattice, "F is defined as the chemicalpotential of the non-interacting Fermi gas with the same density n. Note that thereis a maximum density of n = 1 particle per site for each spin, corresponding to"F = 12J . The system is then simply a band insulator. For low densities "F ⌧ 12J ,the system behaves similarly to the continuum case in the BCS limit (note thatone should be in general cautious with the statements that low densities in a latticecorrespond to the continuum limit; this is strictly speaking not the case in allproblems). In the regime of small attractive interactions U , the Hubbard modeltherefore presents BCS-type pairing and superfluidity. Small interaction typicallymeans |U |/J < 1.

Increasing the interaction, the Hubbard model supports a two-body boundstate at |U |/J ' 7.9. The binding energy "B is given by

1

U=X

k

1

4J(3� cos(kxa)� cos(kya)� cos(kza)) + "B. (11.9)

When |U |/J � 12J , the size of the bound state is of order the lattice spacing a, with"B ' U . Then the size of the dimer is essentially constant, limited by a. The dimersmay then Bose condense. However, increasing U has another remarkable e↵ect. Onecan do second-order perturbation theory on Eq. (11.3) for small J/|U | and obtainan e↵ective model where there are only on-site dimers which hop from one site tothe neighbouring one with the hopping energy of approximately J2/|U |. Therefore,the hopping goes to zero as |U | ! 1; in e↵ect, the dimers have a very large e↵ectivemass. One can then anticipate that the BEC transition temperature goes down, asis predicted by large masses in the simplest order of magnitude estimates of BEC(i.e. that the thermal deBroglie wavelength of the particles should be similar to theaverage distance). This is indeed what happens in the BEC regime of the Hubbardmodel. While one still expects the system to approach a non-interacting BEC atzero temperature, the critical temperature Tc scales with the dimer hopping energy,i.e., Tc ⇠ J2/|U |, and it will thus zero instead of saturating like in Fig. 4, owing tothe localization of bosonic dimers in the lattice.

Figure 5 shows how Tc tends to zero in both the BCS and BEC limits, witha pronounced maximum in between. For further details, see e.g. [39]. Note that theHubbard model displays particle-hole symmetry at half-filling, "F = 6J . Thereforeabove half-filling, one observes the BCS-BEC crossover of holes instead of particles.Note also that on the BEC side we may again have two relevant temperatures: atemperature where on-site pairs form (basically of the order of U) and a much lowertemperature where these pairs may condense into a superfluid. In contrast, on theBCS side the pair formation and condensation happen at the same temperature.

In the context of quantum gases in optical lattices one has to bear in mindthat the one-band Hubbard model is unable to describe Feshbach-resonant gases atunitarity 1/aS = 0. There interactions scale with the lattice depth and thus can

35


Figure 5: BCS-BEC crossover in a lattice. Open symbols are Quantum Monte Carlocalculations for the critical temperature. The dashed line is not relevant here. Tem-perature is given in the units of Fermi energy of a free fermion system of the samedensity, V

0

is the lattice depth and ER is recoil energy. The main figure is for 6Liparameters and the inset for 40K. The scattering lengths are small (close to back-ground scattering length) and the crossover is driven by the strength of the latticepotential V

0

which e↵ectively determines the Hubbard interaction U . Figure from[38].

36


never be made small with respect to the band gap. Moreover, once a > aS > 0 i.e.deep in the BEC regime of the Feshbach resonances where tightly bound moleculescan exist, the inclusion of higher bands yields dimers that are smaller than thelattice spacing. In [40] the limits of validity of the single band approximation arecalculated when the scattering length is increased.

At the time of this teaching, although fermionic atoms have been loaded tooptical lattices, low enough temperatures for observing the BCS-BEC crossoverhave not been achieved yet.

11.2 Mott insulator, antiferromagnet, and d-wave superfluid

In case of repulsive interactions, the most prominent phases that the Fermi-Hubbardmodel presents are the Mott insulator and the emergence of the antiferromagnetic(Neel) order. The Mott insulator has alredy been observed with ultracold gas setups[41, 36] and there are also observations of short-range antiferromagnetic correlations[42, 43]. For a more detailed description of the phases in the context of ultracoldgases see Ref. [44].

At half filling, when the interaction energy U is large, charge fluctuations aresuppressed and a Mott-insulating state occurs in which charge degrees of freedomare localized on single lattice sites. It is intuitively understandable that the particlesof the two species try to avoid being at the same site since the large repulsiveinteraction dominates over kinetic energy (note that the particles of the same spinnaturally avoid being on the same site due to Pauli blocking). The lowest chargeexcitations are of particle hole-type and their energy is approximately �c ⇠ U .That is, a gap is opened in the charge section of the excitation spectrum. Notethat ”charge” is terminology from the Hubbard model for electrons, and in case ofultracold atoms means just the density (in contrast to the spin sector).

There exists a crossover between a liquid at low interaction strength and aMott-insulating state at strong interactions. At finite temperatures that are muchsmaller than the charge gap, characteristic suppression of charge fluctuations of theMott-insulating state still takes place.

The spin degrees of freedom also lead to interesting physics. The spin degreeslead to a highly degenerate ground state when U/J ! 1. One can think that deepin the Mott insulator state where the particles are almost perfectly localized, it doesnot matter whether the neighbour of a particle has the same or the opposite spin.Therefore, there are a large number of energetically equivalent ways of arranging theparticles in the lattice. This degeneracy is lifted for smaller U , where an e↵ectivemagnetic coupling between the spins emerges. This is because the system triesto lower energy by having at least some tunneling. At large interaction U � Jthis is done by the so-called superexchange process: neighboring fermions tunnel(hop) via an intermediate highly energetic doubly occupied state. This is a secondorder tunneling process. Due to Pauli blocking the doubly occupied state is onlypossible for fermions of di↵erent spin. Therefore the superexchange can occur onlyfor fermions of opposite spins, and the arising e↵ective coupling is antiferromagnetic.The coupling strength is given by Jex = 4J2

U [44] since it is a second order processin tunneling coupling 2J , with the energy o↵set U related to the intermediate state(c.f. the general formula for second order perturbation theory).

In three-dimensional cubic lattices, the superexchange coupling induces aphase transition to an antiferromagnet with long-range order at low temperature(Fig. 6a). The structure of the phase-boundary can be understood intuitively: at

37


large interaction the energy scale for the antiferromagnetic coupling diminishes asJ2/U , leading to a decreasing transition temperature with increasing interactions.In contrast, at low interaction, the charge gap becomes small and charge fluctuationscan destroy the magnetic ordering. In this regime of weak repulsive interaction, an-tiferromagnetic order is due to a spin-density-wave transition, in which the openingof the insulating gap and antiferromagnetic order occur simultaneously. See Figure6.

Due to the Mermin-Wagner theorem, Neel order at finite temperature is re-stricted to the 3D case. In 2D lattices, long-range antiferromagnetic order existsonly in the limit T = 0. In a 1D system, the ground state at half-filling in thepresence of repulsive interaction is a Mott insulator with algebraically decayingantiferromagnetic ordering. Ground state and finite temperature properties of theone-dimensional case can be obtained analytically from the Bethe ansatz solution[45].

The first experiments with non-interacting fermionic atoms loaded into a cu-bic optical lattice observed the change of the Fermi surface with increasing thenumber of atoms [46, 47]. The characteristic suppression of particle fluctuations inthe Mott-insulating phase has also been detected [41, 36]. Only recently have thefirst signs of short-range antiferromagnetic correlations been discovered by modu-lation spectroscopy [48, 49], and by using a superlattice [42] or Bragg spectroscopy[43]. However, the spontaneous formation of long-range antiferromagnetic order hasnot been realized so far. This is due to the relatively high temperatures, of the orderof the hopping amplitude [50]. An important challenge is therefore the design of ef-ficient cooling schemes [51]. Even lower temperatures than for the antiferromagnetare be required to address the long-standing question of unconventional supercon-ductivity in the doped 2D Hubbard model. There are predictions that a d-wavesuperfluid would exist in the doped case, but conclusive experimental as well asnumerical evidence is missing.

11.3 Mapping between the repulsive and attractive Hubbard models

The Hubbard model possesses certain symmetries with respect to the repulsive andattractive interactions. There exist a unitary transformation U

1

that connects thetwo at half filling, namely

U†1

ci"U = ✏(i)c†i" (11.10)

U †1

ci#U = c†i#, (11.11)

where ✏(i) = 1 for one sublattice of the bipartite lattice and ✏(i) = �1 for the other.With this transformation

U †1

Hhalf�filled(U)U1

= Hhalf�filled(�U) (11.12)

which means that the Hamiltonians for both signs of U must have the same energyspectrum and the same form of the ground state. Therefore, at half filling, the BCSsuperfluid maps to the Neel-ordered anti-ferromagnet and the BEC of on-site pairsto the Mott antiferromagnet at strong interactions. The existence of on-site pairsbefore one reaches the superfluid corresponds to having a Mott state before reachinga Mott antiferromagnet. This mapping explains the apparent similarity of Figures5 and 6. The mapping when being away from half filling will be discussed in thenext lecture. The identical behaviour at negative and positive U has been seen for

38


0 1 2 3 4 5 6 7

U/J

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

k

B

T

N

/JorEnergy/J

L

MI

SDW-Neel MI-Neel

(a)

T

N

J

ex

= 4J

2/U

�

c

⇠ U

Figure 6: Qualitative phase diagram of the Fermi-Hubbard model in three dimen-sions at half-filling as a function of interaction U and temperature [44]. The Neel-ordered antiferromagnet at low temperatures is understood as a spin-density wave(SDW-Neel) at weak interactions and a Mott antiferromagnet (MI-Neel) at stronginteractions. The latter is characterized by the superexchange constant Jex. Thecharge gap�c can be used to distinguish between the liquid (L) and Mott-insulating(MI) regimes in the normal phase at temperatures beyond the Neel temperatureTN . Figure from [37].

instance in expansion experiments of ultracold Fermi gases [52, 53]. Note that in anexperimental situation where the symmetry of the system may be broken by someunwanted or desired features, the mapping may not be exact.

39


Exotic superfluids: the Fulde-Ferrel-Larkin-Ovchinnikov(FFLO) state

Literature: T. Koponen, Fermionic superfluidity in optical lattices, Ph.D. Thesis,University of Jyvaskyla, Finland (2008),http://www.jyu.fi/static/fysiikka/vaitoskirjat/2008/timo-koponen-2008.pdfand the journal references mentioned in the lecture notes below. Timo Koponenis thanked for providing me the LaTex files of his Ph.D. thesis and, naturally, fordoing his Ph.D. in my group.

Learning goals

• To understand the question of whether fermionic superfluidity may coexistwith magnetization, and to know about the Chandrasekhar-Clogston limit.

• To learn in detail the theoretical description of the FFLO state in lattices.• To know that a similar treatment applies in continuum, and understand whythe FFLO state is more stable in a lattice than in continuum.

• To understand Luttinger’s theorem and (dis)appearance of Fermi surfaces.• To know why the FFLO state fulfills the Bloch theorem i.e. does not carrynet current in the ground state.

• To be aware of modern developments of the FFLO theory, for instance beyondmean-field calculations, 1D systems, inclusion of spin-orbit coupling.

12 Superconductivity and magnetization: theChandrasekhar-Clogston limit

Already in 1962 Chandrasekhar [54] and Clogston [55] asked the question whethersuperconductivity would survive in presence of magnetization. I follow here Clogston’sargument. What one usually expects for superconductors in presence of magneticfield is the Meissner e↵ect. A metal in a magnetic field may go from the supercon-ducting state to the normal due to free energy associated with the Meissner e↵ect(repelling of the magnetic field from the metal). The critical field H

0

for this isgiven by

FN = FS +H2

0

8⇡, (12.1)

where the F ’s are the free energies of the normal and the superconducting states.There is, however, a finite penetration depth for the magnetic field and therefore ifthe conductor has the right geometry, e.g. it is a thin cylinder, the Meissner e↵ectmay be very small and the critical field associated with it very high. One may thenask is there another limit for the field that penetrates into the superconductor? Alimit could come from the paramagnetic susceptibility �P of a normal metal, thatis, that the spins would like to align with the magnetic field (also referred to asPauli paramagnetism). Notably in the BCS theory, there is no such susceptibilitysince the spin-up and spin-down particles are correlated in pairs. From this, onegets another constraint for the field, namely

FN = FS +�PH2

0

2. (12.2)

The free energy di↵erence between a superfluid and a normal state is given byFN � FS = N(0)�2/2 where N(0) is the density of states at the Fermi level, while

40


�P = 2µ2

BN(0) where µB is the Bohr magneton. This gives the critical field

µBH0

=�p2. (12.3)

Often also a scaled version of the magnetic field is used:

h =�p2. (12.4)

Chandrasekhar’s argument is similar but produces only rough estimate of h ' �.The limit (12.3) is referred to as the Chandrasekhar-Clogston limit. It has beenobserved in ultracold Fermi gases [56] in trapped geometries which favour phaseseparation.

Now, the above argument is based on the simple BCS theory. People soonasked whether it would be modified if one allows a di↵erent type of paired su-perfluid state. One of such proposals is the Fulde-Ferrel-Larkin-Ovchinnikov state[57, 58, 58]. We will discuss it in detail in the following, and later mention also someother proposals. The FFLO state was originally proposed for a continuum system.However, is has been shown that it occurs in a rather small parameter window in3D continuum systems [59], and also in 3D traps as a small edge e↵ect [60]. In2007, we predicted [61] that the FFLO state appears in a much larger parameterarea in lattice geometries. Therefore we present the basic theory of the FFLO statebelow using the lattice context. As you have seen already from the BCS theory,the continuum and lattice descriptions are very similar, the di↵erence being justthe di↵erent dispersion in the energies, and issues related to the renormalization ofthe interaction and the treatment of the Hartree field. Once you know the FFLOformalism in the lattice, it will be easy to do the continuum calculation. A niceclassic reference about the continuum FFLO theory is [62].

13 FFLO state in a lattice

The simplest FFLO ansatz assumes that the Cooper pairs carry a finite momen-tum 2q and the order parameter is of the form (notation is otherwise the same asintroduced before in these lectures; now the Hubbard U is negative)

Uhc†"nc†#ni = �e2iq·n, (13.1)

where � � 0. This is equivalent with the standard BCS-theory in the limit of q = 0.This plane wave form of order parameter was first considered by Fulde and Ferrel[57]. Larkin and Ovchinnikov [58, 63] actually considered an ansatz which containsboth q and �q, thus the order parameter would be of the form

Uhc†"nc†#ni = 2� cos(2q · n). (13.2)

It seems to be the case in many known systems that the cosine wave LO ansatzgives a lower energy than the FF plane wave. This is actually fortunate becauseit is often simpler theoretically to use the FF ansatz. The argument then goesthat if FF minimizes the energy, LO would do it even better. So it is a reasonablefirst approach to try the FF ansatz. In general, the FFLO order parameter couldbe composed of any number of Fourier components with di↵erent q’s, leading tobasically arbitrarily complicated spatial dependence �(n) (or �(x) in general).

41


Remember here the lecture about the BCS-BEC crossover. There the initial(non-mean-field) Hamiltonian in the momentum space (10.1) clearly indicates thatthe pairs might have a finite momentum. Now in the FFLO state, the condensation,i.e. macroscopic population of Cooper pairs gathers to the pair momentum state 2qinstead of q = 0.

The Hamiltonian is now in the form (we allow here for spin-dependent andanisotropic hopping)

H � µ"N" � µ#N# =�X

n

⇣µ"c

†"nc"n + µ#c

†#nc#n

⌘

+X

n

✓�e2iq·nc†"nc

†#n +�e�2iq·nc#nc"n � �2

U

◆

�X

�

X

↵2{x,y,z}

J�↵X

hn,mi↵

c†�mc�n.

(13.3)

It is convenient to write the Hamiltonian in the (quasi-)momentum representation,i.e. represent the operators in the plane wave basis. This essentially means doing aFourier transform on (13.3) by

c"n =1pM

X

k

eik·nc"k

c†"n =1pM

X

k

e�ik·nc†"k

c#n =1pM

X

k

eik·nc#k

c†#n =1pM

X

k

e�ik·nc†#k,

(13.4)

where M is the (finite) number of lattice sites and k runs through the reciprocallattice. The density terms transform as

X

n

c†"nc"n =X

n

1pM

X

k

e�ik·nc†"k

! 1pM

X

k0

eik0·nc"k0

!

=1

M

X

n

X

k,k0

ei(k0�k)·nc†"kc"k0 =

X

k

c†"kc"k,

(13.5)

where the following identity is used:

1

M

X

n

ei(k0�k)·n = �k,k0 . (13.6)

This equation holds for all reciprocal lattice vectors k,k0.

42


Similarly, the interaction term becomes

X

n

⇣�e2iq·nc†"nc

†#n

⌘=

1

M

X

n,k,k0

⇣�e2iq·ne�ik·nc†"ke

�ik0·nc†#k0

⌘

=1

M

X

n,k,k0

⇣�ei(2q�k�k0

)·nc†"kc†#k0

⌘

=X

k,k0

1

M

X

n

ei(2q�k�k0)·n

!

| {z }=�k+k0

,2q

�c†"kc†#k0

=X

k

�c†"k+qc†#�k+q.

(13.7)

The nearest neighbour hopping term gives rise to a cosine dispersion, forexample the x-direction looks like

X

hn,mix

⇣c†"mc"n + c†#mc#n

⌘

=X

n

⇣(c†"n+(1,0,0) + c†"n�(1,0,0))c"n + (c†#n+(1,0,0) + c†#n�(1,0,0))c#n

⌘

=1

M

X

n,k,k0

e�i((k�k0)·n) �eikx + e�ik

x

�| {z }

=2 cos kx

⇣c†"kc"k0 + c†#kc#k0

⌘

=X

k

2 cos kx⇣c†"kc"k + c†#kc#k

⌘.

(13.8)

Finally, we have arrived at the mean field Hubbard Hamiltonian in momentumspace,

bH =X

k

✓⇠"kc

†"kc"k + ⇠#kc

†#kc#k

+�c†"k+qc†#�k+q +�c#�k+qc"k+q

◆� �2

U,

(13.9)

where ⇠�k = ✏�k � µ� =P

↵ 2J�↵(1� cos(k↵))� µ�. Note that in order to get thedispersion correspond to that of a free particle in the limit of small k, the followingterms have been added to the Hamiltonian:

2X

↵

X

k

⇣J�↵c

†�kc�k

⌘=X

↵

2J�↵N�. (13.10)

13.1 Bogoliubov transformation in the FFLO case

The Hamiltonian in (13.9) can be written in an equivalent form as

bH =X

k

✓⇠"k+qc

†"k+qc"k+q + ⇠#�k+q(1� c#�k+qc

†#�k+q)


◆� �2

U.

(13.11)

43


Now the operator part can be expressed as a matrix:

X

k

✓⇠"k+qc

†"k+qc"k+q � ⇠#�k+qc#�k+qc

†#�k+q


◆=

X

k

⇣c†"k+q c#�k+q

⌘✓⇠"k+q �� ⇠#�k+q

◆✓c"k+q

c†#�k+q

◆.

(13.12)

The essential point in this representation is that it is a sum of independent 2 ⇥ 2matrices that can be diagonalized separately. It is due to this feature that themean field approach under discussion is the most simple and straightforward wayto address superfluidity in a lattice theoretically.

Let us now derive a Bogoliubov transformation, B, for the terms of (13.12).The requirements are that B diagonalizes the matrix and that B is canonical, i.e. itpreserves the fermionic anticommutation relations. Without any loss of generalityB can be assumed real in this case. Denoting the new basis operators as �

+

and��, we have ✓

�+

�†�

◆= B

✓c"k+q

c†#�k+q

◆=

B

11

c" +B12

c†#B

21

c" +B22

c†#

!(13.13)

and we require

B

✓⇠"k+q �� ⇠#�k+q

◆B�1 =

✓E

+,k,q 00 �E�,k,q

◆. (13.14)

It should be pointed out that the sign in front of E� is a choice of notation at thispoint and does not a↵ect the results. From the anticommutation relations it followsthat

1 = {�+

, �†+

} = {B11

c" +B12

c†#, B11

c†" +B12

c#} = B2

11

+B2

12

1 = {��, �†�} = {B

21

c†" +B22

c#, B21

c" +B22

c†#} = B2

21

+B2

22

0 = {�+

, ��} = {B11

c" +B12

c†#, B21

c†" +B22

c#} = B11

B21

+B12

B22

.

(13.15)

These conditions imply B2

11

= B2

22

, B2

12

= B2

21

, and B11

B21

+ B12

B22

= 0, whichhas several physically equivalent solutions that di↵er only by the locations of minussigns. It is now possible to choose

B =

✓u �vv u

◆, (13.16)

where u, v 2 R and u2 + v2 = 1. This form satisfies all the requirements givenabove. Because B is a unitary matrix, it is enough to solve the eigenvalue problemof the Hamiltonian and the columns of B are the eigenvectors. The eigenvalues are

�± =⇠"k+q � ⇠#�k+q

2±

s✓⇠"k+q + ⇠#�k+q

2

◆2

+�2, (13.17)

and therefore

E±,k = Ek ± �⇠k2

:=

s✓⇠"k+q + ⇠#�k+q

2

◆2

+�2 ± ⇠"k+q � ⇠#�k+q

2. (13.18)

44


For comparison with the standard BCS theory it is good to note that if both speciesexperience the same potential, i.e. the single particle dispersions are identical, ✏"k =✏k = ✏#k, and in addition q = 0, this reduces to

E±,k =

s✓✏k � µ" + µ#

2

◆2

+�2 ± µ# � µ"2

. (13.19)

When µ" = µ#, the minimum value of both E+

and E� is �, which shows thatquasiparticle excitations have a minimum energy, i.e. energy gap, the magnitude ofwhich is �.

For further applications, it is relevant to know the values of u2, v2, and uv;they are

u2

k =1

2

✓1 +

⇠"k+q + ⇠#�k+q

2Ek

◆

v2k =1

2

✓1� ⇠"k+q + ⇠#�k+q

2Ek

◆

ukvk = � �

2Ek.

(13.20)

The Hamiltonian can now be written in the form

bH =X

k

✓⇣�†+,k ��,k

⌘✓E+,k 00 �E�,k

◆✓�+,k

��,k

◆+ ⇠#�k+q

◆� �2

U, (13.21)

with E± defined in (13.28). The operators �± are the quasiparticle operators. Theanticommutation relations are preserved by the canonical transformation. Thus thesystem is now described as an ideal Fermi gas with dispersions given by E

+

andE�:

bH =X

k

⇣E

+,k�†+,k�+,k + E�,k�

†�,k��,k + ⇠#�k+q � E�,k

⌘� �2

U, (13.22)

where the additional E� appears because of the normal ordering.

13.2 Self-consistent crossover equations

It is now possible to derive a set of equations from which �, µ", and µ# can besolved. These equations are called crossover equations. To start with the numberequations, note that the following holds for the total number of particles in either

one of the spin components: N� =P

k

Dc†�,kc�,k

E. It is straightforward to use the

inverse of the Bogoliubov transformation U to write the particle operators as linearcombinations of the quasiparticle operators as

✓c"c†#

◆= B�1

✓�+

�†�

◆= B†

✓�+

�†�

◆=

✓u v�v u

◆✓�+

�†�

◆=

✓u�

+

+ v�†�

u�†� � v�

+

◆. (13.23)

45


Thus we arrive at

N" =X

k

Dc†"kc"k

E=X

k

Dc†"k+qc"k+q

E

=X

k

D⇣u�†

+

+ v��⌘⇣

u�+

+ v�†�

⌘E

=X

k

Du2�†

+

�+

+ v2��†� + uv��+ + uv�†

+

�†�

E

=X

k

u2

D�†+

�+

E+ v2

D��

†�

E

=X

k

u2

knF

(E+,k) + v2knF

(�E�,k).

(13.24)

Note that expectation values of type h�+

��i are automatically zero. A similar equa-tion holds for the number of down particles:

N# =X

k

Dc†#kc#k

E

=X

k

D⇣u�†

� � v�+

⌘⇣u�� v�†

+

⌘E

=X

k

u2

D�†��

E+ v2

D�+

�†+

E

=X

k

u2

knF

(E�,k) + v2knF

(�E+,k).

(13.25)

Using the original definition of the order parameter, Uhc†"nc†#ni = �e2iq·n, it

is possible to derive the so called gap equation, which is derived here in the q = 0limit:

�

U=Dc†"nc

†#n

E=

*X

k,k0

e�i(k+k0)·nc†"kc

†#k0

+

=

*X

k,k0

e�i(k+k0)·n⇣uk�

†+,k + vk��,�k

⌘·

⇣�v�k0 �

+,�k0 + u�k0 �†�,k0

⌘+

=X

k,k0

e�i(k+k0)·n

� ukv�k0

D�†+,k�+,�k0

E

| {z }/�k0

,�k

+ u�k0vkD��,�k�

†�,k0

E

| {z }/�k0

,�k

!

=X

k

ukvk (1� nF

(E+,k)� n

F

(E�,k)) .

(13.26)

46


Finally, after substituting ��/2U for uv and dividing by �, the gap equation is

1 = �UX

k

1� nF

(E+,k)� n

F

(E�,k)

2Ek. (13.27)

Since U was defined as negative, the prefactor on the right hand side of the equationis positive.

The number equations (13.24) and (13.25), and the gap equation (13.27)together are equivalent with the standard BCS-Leggett theory. In the balancedcase, where N" = N#, it is possible to choose µ" = µ# by hand and eliminateone of the number equations. If the interactions are weak, i.e. |U | is small, thechemical potential can be approximated with the Fermi energy. However, when theinteraction strength increases, the chemical potential has to be solved from thenumber equation in order to get the correct results.

The calculation becomes more involved once non-zero values for q are allowed.In this case it is more feasible to solve � and q by minimizing the relevant freeenergy, as described in the next section.

13.3 The nature of the eigenenergies in the FFLO state

Now, let us have a closer look at the energy eigenvalues

E±,k = Ek ± �⇠k2

:=

s✓⇠"k+q + ⇠#�k+q

2

◆2

+�2 ± ⇠"k+q � ⇠#�k+q

2. (13.28)

It turns out that when solving for the FFLO state by minimizing the energy,the FFLO wave vector will be q ' kF"�kF#. The bigger the di↵erence in chemicalpotentials or particle numbers, the bigger is q. It is thus plausible that the �⇠k abovebecomes large when the spin-density imbalance grows. This means that some of theenergy eigenvalues become negative! In the BCS state, all of them are positive andthus there are no quasiparticles at zero temperature. In the FFLO state, however,there may exist single quasiparticles even in the ground state at zero temperature.These are particles that do not participate in pairing. In other words, a Fermisurface may exist in the system at the same time as it is a superfluid. This is calleda gapless superfluid. This means that even when most particles (momentum states)are associated with the pairing gap �, there are some momentum values wheregapless excitations are possible.

FFLO TO BE CONTINUED IN THE NEXT LECTURE...

References

[1] P. Torma. Spectroscopies – Theory. In eds. P. Torma and K. Sengstock,Quantum Gas Experiments: Exploring Many-Body States, pp. 199–250. Impe-rial College Press, London, (2015).

[2] D. M. Eagles, Possible pairing without superconductivity at low carrier con-centrations in bulk and thin-film superconducting semiconductors, Phys. Rev.186, 456–463, (1969).

[3] A. J. Leggett. Diatomic molecules and Cooper pairs. In eds. A. Pekalskiand J. Przystawa, Modern Trends in the Theory of Condensed Matter, p. 14.Springer-Verlag, Berlin, (1980).

47


[4] P. Nozieres and S. Schmitt-Rink, Bose condensation in an attractive fermiongas: From weak to strong coupling superconductivity, J. Low Temp. Phys. 59,195–211, (1985).

[5] M. M. Parish. The BCS-BEC Crossover. In eds. P. Torma and K. Sengstock,Quantum Gas Experiments: Exploring Many-Body States, pp. 179–197. Impe-rial College Press, London, (2015).

[6] W. Zwerger, Ed., The BCS–BEC Crossover and the Unitary Fermi Gas. vol.836, Lecture Notes in Physics, (Springer, 2012).

[7] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Revealingthe superfluid lambda transition in the universal thermodynamics of a unitaryFermi gas, Science. 335, 563–567, (2012).

[8] E. Burovski, E. Kozik, N. Prokof’ev, B. Svistunov, and M. Troyer, Critical tem-perature curve in BEC–BCS crossover, Phys. Rev. Lett. 101, 090402, (2008).

[9] O. Goulko and M. Wingate, Thermodynamics of balanced and slightly spin-imbalanced Fermi gases at unitarity, Phys. Rev. A. 82, 053621, (2010).

[10] S. Tan, Energetics of a strongly correlated Fermi gas, An-nals of Physics. 323(12), 2952 – 2970, (2008). ISSN 0003-4916. doi: http://dx.doi.org/10.1016/j.aop.2008.03.004. URLhttp://www.sciencedirect.com/science/article/pii/S0003491608000456.

[11] S. Tan, Large momentum part of a strongly correlated Fermigas, Annals of Physics. 323(12), 2971–2986, (2008). ISSN 0003-4916. doi: http://dx.doi.org/10.1016/j.aop.2008.03.005. URLhttp://www.sciencedirect.com/science/article/pii/S0003491608000432.

[12] S. Tan, Generalized virial theorem and pressure relation for a stronglycorrelated Fermi gas, Annals of Physics. 323(12), 2987–2990, (2008).ISSN 0003-4916. doi: http://dx.doi.org/10.1016/j.aop.2008.03.003. URLhttp://www.sciencedirect.com/science/article/pii/S0003491608000420.

[13] E. Braaten and L. Platter, Exact relations for a strongly interact-ing Fermi gas from the operator product expansion, Phys. Rev. Lett.100, 205301 (May, 2008). doi: 10.1103/PhysRevLett.100.205301. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.100.205301.

[14] S. Zhang and A. J. Leggett, Universal properties of the ultracold Fermi gas,Phys. Rev. A. 79, 023601 (Feb, 2009). doi: 10.1103/PhysRevA.79.023601. URLhttp://link.aps.org/doi/10.1103/PhysRevA.79.023601.

[15] E. Braaten. Universal relations for fermions with large scattering lengths. Ined. W. Zwerger, The BCS – BEC Crossover and the Unitary Gas, pp. 193–231.Springer, Heidelberg, (2012).

[16] W. E. Zwerger, The BCS – BEC Crossover and the Unitary Gas. (Springer,2012).

[17] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin,J. Hecker Denschlag, and R. Grimm, Bose-einstein condensation of molecules,Science. 302(5653), 2101–2103, (2003). doi: 10.1126/science.1093280. URLhttp://www.sciencemag.org/content/302/5653/2101.abstract.

[18] C. A. Regal, M. Greiner, and D. S. Jin, Observation of resonance condensationof fermionic atom pairs, Phys. Rev. Lett. 92, 040403, (2004).

[19] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Rau-pach, S. Gupta, Z. Hadzibabic, and W. Ketterle, Observation ofbose-einstein condensation of molecules, Phys. Rev. Lett. 91,250401 (Dec, 2003). doi: 10.1103/PhysRevLett.91.250401. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.91.250401.

48


[20] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman,and W. Ketterle, Condensation of pairs of fermionic atoms near a Feshbachresonance, Phys. Rev. Lett. 92, 120403, (2004).

[21] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag,and R. Grimm, Observation of the pairing gap in a strongly interacting Fermigas, Science. 305(5687), 1128–1130, (2004). doi: 10.1126/science.1100818. URLhttp://www.sciencemag.org/content/305/5687/1128.abstract.

[22] J. Kinnunen, M. Rodrıguez, and P. Torma, Pairing gap and in-gap excitations in trapped fermionic superfluids, Science. 305(5687), 1131–1133, (2004). doi: 10.1126/science.1100782. URLhttp://www.sciencemag.org/content/305/5687/1131.abstract.

[23] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas,Evidence for superfluidity in a resonantly interacting Fermi gas, Phys. Rev.Lett. 92, 150402 (Apr, 2004). doi: 10.1103/PhysRevLett.92.150402. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.92.150402.

[24] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H.Denschlag, and R. Grimm, Collective excitations of a degener-ate gas at the BEC – BCS crossover, Phys. Rev. Lett. 92,203201 (May, 2004). doi: 10.1103/PhysRevLett.92.203201. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.92.203201.

[25] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann,L. Tarruell, S. Kokkelmans, and C. Salomon, Experimental study of the BEC–BCS crossover region in lithium 6, Phys. Rev. Lett. 93, 050401, (2004).

[26] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G.Hulet, Molecular probe of pairing in the BEC – BCS crossover, Phys. Rev.Lett. 95, 020404 (Jul, 2005). doi: 10.1103/PhysRevLett.95.020404. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.95.020404.

[27] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ket-terle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature.435, 1047–1051, (2005).

[28] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drum-mond, P. Hannaford, and C. J. Vale, Universal behavior of pair cor-relations in a strongly interacting Fermi gas, Phys. Rev. Lett. 105,070402 (Aug, 2010). doi: 10.1103/PhysRevLett.105.070402. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.105.070402.

[29] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E.Drut, and S. Gandolfi, Precise determination of the structure fac-tor and contact in a unitary Fermi gas, Phys. Rev. Lett. 110,055305 (Jan, 2013). doi: 10.1103/PhysRevLett.110.055305. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.110.055305.

[30] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Verification ofuniversal relations in a strongly interacting Fermi gas, Phys. Rev. Lett.104, 235301 (Jun, 2010). doi: 10.1103/PhysRevLett.104.235301. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.104.235301.

[31] J. Hubbard, Electron correlations in narrow energy bands, Proceedings of theRoyal Society of London A: Mathematical, Physical and Engineering Sciences.276(1365), 238–257, (1963). ISSN 0080-4630. doi: 10.1098/rspa.1963.0204.

[32] R. M. Bradley, Quantum fluctuations in chains of Josephson junctions, Phys.Rev. B. 30(3), 1138–1147, (1984).

49


[33] M. P. A. Fisher, G. Grinstein, and D. S. Fisher, Boson localization and thesuperfluid-insulator transition, Phys. Rev. B. 40(1), 546–570, (1989).

[34] D. Jaksch, C. Bruder, J. Cirac, C. Gardiner, and P. Zoller, Cold bosonic atomsin optical lattices, Phys. Rev. Lett. 81(15), 3108–3111, (1998).

[35] R. Jordens, N. Strohmaier, K. Gunter, H. Moritz, and T. Esslinger,A Mott insulator of fermionic atoms in an optical lattice, Nature.455, 204 (Nov, 2008). doi: 10.1103/PhysRevLett.105.225301. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.105.225301.

[36] U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch, T. A. Costi, R. W.Helmes, D. Rasch, and A. Rosch, Metallic and insulating phases of repulsivelyinteracting fermions in a 3D optical lattice, Science. 322(5907), 1520–1525,(2008).

[37] P. Barmettler and C. Kollath. Quantum Gases in Optical Lattices. In eds.P. Torma and K. Sengstock, Quantum Gas Experiments: Exploring Many-Body States, pp. 33–61. Imperial College Press, London, (2015).

[38] W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin, High-temperature superfluidity of fermionic atoms in optical lattices, Phys. Rev.Lett. 89, 220407 (Nov, 2002). doi: 10.1103/PhysRevLett.89.220407. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.89.220407.

[39] M. Randeria. Crossover from BCS theory to BEC. In eds. A. Gri�n, D. W.Snoke, and S. Stringari, Bose–Einstein Condensation, pp. 355–392. CambridgeUniversity Press, Cambridge, (1995).

[40] H. P. Buchler, Microscopic derivation of Hubbard parame-ters for cold atomic gases, Phys. Rev. Lett. 104, 090402(Mar, 2010). doi: 10.1103/PhysRevLett.104.090402. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.104.090402.

[41] R. Jordens, N. Strohmaier, K. Gunter, H. Moritz, and T. Esslinger, A Mottinsulator of fermionic atoms in an optical lattice, Nature. 455, 204, (2008).

[42] D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Short-rangequantum magnetism of ultracold fermions in an optical lattice., Science. 340(6138), 1307–10, (2013).

[43] R. Hart, P. Duarte, T.-L. Yang, X. Liu, T. Paiva, E. Khatami, R. Scalettar,N. Trivedi, D. Huse, and R. Hulet, Observation of antiferromagnetic correla-tions in the Hubbard model with ultracold atoms, Nature. 40(519), 211, (2015).

[44] A. Georges and T. Giamarchi. Strongly correlated bosons and fermions inoptical lattices. In Many-Body Physics with Ultracold Gases Lecture Notes ofthe Les Houches Summer School, chapter 1. Oxford University Press, (2010).

[45] F. Essler, H. Frahm, F. Gohmann, A. Klumper, and V. Korepin, The One-Dimensional Hubbard Model. (Cambridge University Press, 2005).

[46] M. Kohl, H. Moritz, T. Stoferle, K. Gunter, and T. Esslinger, Fermionic atomsin a three dimensional optical lattice: Observing Fermi surfaces, dynamics, andinteractions, Phys. Rev. Lett. 94(8), 80403, (2005).

[47] T. Esslinger, Fermi-Hubbard physics with atoms in an pptical lattice, AnnualReview of Condensed Matter Physics. 1(1), 129–152, (2010).

[48] C. Kollath, A. Iucci, I. P. McCulloch, and T. Giamarchi, Modulationspectroscopy with ultracold fermions in an optical lattice, Phys. Rev.A. 74, 041604 (Oct, 2006). doi: 10.1103/PhysRevA.74.041604. URLhttp://link.aps.org/doi/10.1103/PhysRevA.74.041604.

[49] D. Greif, L. Tarruell, T. Uehlinger, R. Jordens, and T. Esslinger, Probing

50


nearest-neighbor correlations of ultracold Fermions in an optical lattice, Phys.Rev. Lett. 106(14), 145302, (2011).

[50] R. Jordens, L. Tarruell, D. Greif, T. Uehlinger, N. Strohmaier, H. Moritz,T. Esslinger, L. De Leo, C. Kollath, A. Georges, V. Scarola, L. Pollet,E. Burovski, E. Kozik, and M. Troyer, Quantitative determination of tem-perature in the approach to magnetic order of ultracold Fermions in an opticallattice, Phys. Rev. Lett. 104(18), (2010).

[51] D. C. McKay and B. DeMarco, Cooling in strongly correlated optical lat-tices: prospects and challenges, Reports on Progress in Physics. 74(5), 054401,(2011).

[52] U. Schneider, L. Hackermuller, J. P. Ronzheimer, S. Will, S. Braun, T. Best,I. Bloch, E. Demler, S. Mandt, D. Rasch, and A. Rosch, Fermionic trans-port and out-of-equilibrium dynamics in a homogeneous Hubbard model withultracold atom, Nat. Phys. 8, 213, (2012). doi: 10.1038/nphys2205. URLhttp://dx.doi.org/10.1038/nphys2205.

[53] J. Kajala, F. Massel, and P. Torma, Expansion dynamics in theone-dimensional Fermi-Hubbard model, Phys. Rev. Lett. 106,206401 (May, 2011). doi: 10.1103/PhysRevLett.106.206401. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.106.206401.

[54] B. S. Chandrasekhar, A note on the maximum critical field of highfield super-conductors, Applied Physics Letters. 1(1), (1962).

[55] A. M. Clogston, Upper limit for the critical field in hard superconductors,Phys. Rev. Lett. 9, 266–267 (Sep, 1962). doi: 10.1103/PhysRevLett.9.266.URL http://link.aps.org/doi/10.1103/PhysRevLett.9.266.

[56] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Fermionicsuperfluidity with imbalanced spin populations, Science. 311(5760), 492–496,(2006). doi: 10.1126/science.1122318.

[57] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field,Phys. Rev. 135, A550, (1964).

[58] A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Zh.Eksp. Teor. Fiz. 47, 1136, (1964).

[59] D. E. Sheehy and L. Radzihovsky, BEC-BCS crossover in magnetized Feshbach-resonantly paired superfluids, Phys. Rev. Lett. 96, 060401, (2006).

[60] J. Kinnunen, L. M. Jensen, and P. Torma, Strongly interact-ing Fermi gases with density imbalance, Phys. Rev. Lett. 96,110403 (Mar, 2006). doi: 10.1103/PhysRevLett.96.110403. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.96.110403.

[61] T. K. Koponen, T. Paananen, J.-P. Martikainen, and P. Torma, Finite-temperature phase diagram of a polarized Fermi gas in an optical lattice, Phys.Rev. Lett. 99, 120403, (2007).

[62] S. Takada and T. Izuyama, Superconductivity in a molecular field. I, Prog.Theor. Phys. 41, 635, (1969).

[63] A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Sov.Phys. JETP. 20, 762, (1965).

51

the fermi-hubbard model - eth z · pdf filethe fermi-hubbard model ... they can be two...

Documents