Superfluid phases of ultracold Fermi gases on a checkerboard superlattice

# Superfluid phases of ultracold Fermi gases on a checkerboard superlattice

M. Iskin Department of Physics, Koç University, Rumelifeneri Yolu, 34450 Sarıyer, Istanbul, Turkey.
July 14, 2019
###### Abstract

We analyze the ground-state phase diagram of two-component Fermi gases loaded into a two-dimensional checkerboard superlattice, i.e. a double-well optical lattice, potential within the BCS mean-field theory. We show that, by coupling the two -wave sublattice superfluid order parameters, a checkerboard potential gives rise to a Hamiltonian that has the form of a two-band superfluidity with three (two intraband and an interband) nonlocal order parameters. We study the evolution of these order parameters as a function of particle filling, interaction strength and checkerboard potential, and find that the system always prefers the -phase solutions, i.e. the phase difference between sublattice order parameters is 0, but never the -phase one. In addition, we find that the ground-state of the system undergo a superfluid-normal quantum phase transition at half fillings beyond a critical checkerboard potential , the threshold of which is precisely determined by the magnitude of the order parameter at , and that the normal state rapidly turns into a checkerboard insulator as increases.

###### pacs:
05.30.Fk, 03.75.Ss, 03.75.Hh

## I Introduction

Optical superlattices sebby06 (); lee06 (); blochdw (); blochdwgauge () have been of interest to the cold atom community for a long time now, and they have recently regained more interest in the past few years since they may allow for the studies of topological phases of matter in the cold-atom context moller (); dalibard (); smitha (); smithb (). One way to realize a checkerboard super optical lattice is to superimpose two independent optical standing waves differing in period by a factor of two, e.g. and , and tunable intensities and relative phases. In particular, a double-well optical lattice can be produced by arranging the shorter-wavelength () lattice potential in such a way to split the each potential well of the longer-wavelength () lattice into two. In addition, the energy difference between the wells of the resultant double-well potential can also be controlled by tuning the relative phase of the optical potentials sebby06 (); lee06 (); blochdw (); blochdwgauge ().

In spite of serious challenges in producing fermion superfluids in earlier optical lattice experiments inguscio (); modugno (); kohl (); stoferle (); bongs (), there is some recent experimental evidence for superfluid, metallic and insulating phases mit-lattice (); schneider (); jordens () (see also the recent review FHreview ()). Motivated by these experiments. in this paper, we investigate the ground-state phases of two-component Fermi gases loaded into a two-dimensional checkerboard superlattice. For this purpose, we study a Fermi-Hubbard type lattice model which includes, in addition to the usual nearest-neighbor hopping and onsite (attractive) interaction, an onsite energy difference between sublattice sites, i.e. a staggered checkerboard potential. We note that the the phase diagram of the Bose-Hubbard versions of such a model have recently been studied for the hardcore hencb () and softcore iskincb () bosons.

Our main findings, within the single-band tight-binding BCS mean-field theory, are as follows. First, we show that the -wave sublattice order parameters, which are momentum independent in the original Hamiltonian, are coupled by the presence of a checkerboard potential, and this gives rise to a Hamiltonian that has the form of a two-band superfluidity with three (two intraband and an interband) nonlocal (momentum-dependent higher partial waves) order parameters in the basis where the single-particle Hamiltonian is diagonal. We study the evolution of these order parameters as a function of particle filling, interaction strength and checkerboard potential, and found that the system always prefers the -phase solutions, i.e. the phase difference between sublattice order parameters is , but never the -phase one. In addition, we show that the ground-state of the system undergo a superfluid-normal quantum phase transition at half fillings beyond a critical checkerboard potential , the threshold of which is precisely determined by the magnitude of the order parameter at , and that the normal state rapidly turns into a checkerboard insulator as increases.

The rest of the paper is organized as follows. We introduce the single- and many-body Hamiltonians in Sec. II, and derive the single-quasiparticle/hole excitation spectra of the system as well as the complete set of self-consistency (superfluid order parameters and total and imbalance number) equations. We solve the resultant equations in Sec. III, and give a detailed analysis of the obtained results mentioned above. A brief summary of our main findings is given in Sec. IV.

## Ii Hamiltonian

It is well-established that Hubbard-type discrete lattice models can be used to capture the physics of cold atoms loaded into optical lattice potentials jaksch (); greiner02 (); bloch08 (); bloch (); lin09a (); lin09b (). For example, much of the theoretical predictions based on the simplest Bose-Hubbard model fisher () have been successfully verified with ultracold Bose gases loaded into optical lattices. The prime examples are the realizations of superfluid and Mott insulator phases as well as the transition between the two jaksch (); greiner02 (); bloch08 (); bloch (); lin09a (); lin09b (). Motivated by this success, here we study a Fermi-Hubbard type lattice model to analyze the physics of ultracold Fermi gases loaded into a checkerboard superlattice potential, as described next.

### ii.1 Single-particle problem

Let us first discuss the single-particle problem on a two-dimensional checkerboard superlattice which consists of two interpenetrating square sublattices as illustrated in Fig. 1. Within the tight-binding approximation, the hopping Hamiltonian for such a lattice can be written as where labels the sublattices and labels the two components of the Fermi gas, is the tunneling (hopping) amplitude of fermions between lattice sites and , and operator creates (annihilates) a fermion at lattice site . In this paper, we limit ourselves to the simplest model where is nonzero for all nearest-neighbor lattice sites but 0 otherwise. However, we also allow the possibility of alternating hopping elements with a checkerboard pattern where may take two values. In particular, let’s assume () corresponds to the hopping amplitude from the sublattice to the sublattice in the positive (negative) and directions. We note that the next-nearest-neighbor hoppings are suppressed by an order of magnitude compared to in a typical atomic setting FHreview ().

The eigenvalues of such real-space hopping Hamiltonians can be obtained by taking advantage of the discrete translational symmetry, and transforming them to the Fourier (momentum) space. Therefore, we assume a single-band description, and introduce where is the number of sublattice sites, is the momentum, 1BZ is the corresponding 1st Brillouin zone, and is the position of the lattice site . Using this transformation, we rewrite the Hamiltonian in momentum space as where

 ϵkσ= −2(t1σ+t2σ)cos(k1d/2)cos(k2d/2) −2i(t1σ−t2σ)cos(k1d/2)sin(k2d/2) (1)

is in general a complex number. Here, corresponds to the projections of the momentum vector along the directions (see Fig. 1) where , () operator creates (annihilates) a fermion with momentum on sublattice, and H.c. is the Hermitian conjugate. Thus, the single-particle dispersion relations are where labels the two bands. Note that, since the translational symmetry is doubled in real space, the original space is halved but the number of bands is doubled, in such a way that this dispersion recovers the usual result, i.e. , in the absence of a sublattice structure when .

In addition to the hopping part, we include an onsite checkerboard lattice potential, i.e. an alternating energy off-set between sublattices, which is given by where is its strength and () for (). This term lowers (raises) the onsite energy of the () sublattice sites by . Using the Fourier transformation described above, we rewrite the total Hamiltonian in space and diagonalize it, leading to where are the single-quasiparticle/hole dispersion relations in the presence of a checkerboard potential, and () operator is the new quasiparticle creation (annihilation) operator in the transformed basis. Note that reduces to when as expected. We note that the and operators are related via a Bogoliubov transformation and where and are the components of the eigenvector that corresponds to the eigenvalue . The eigenvectors are orthonormal in such a way that Having discussed the single-particle Hamiltonian, next we move on to the many-particle problem.

### ii.2 Many-particle problem

For the many-particle problem, the effects of local (onsite) and attractive interparticle density-density interactions can be taken into account within the BCS mean-field approximation, which is known to work well for weak interactions at all temperatures and even for moderate interactions at  leggett (); nsr (); bcsbecreview (). For this purpose, we introduce sublattice-dependent superfluid order parameters and with -wave symmetry, as defined by in real or in space, where is the strength of the interaction and is a thermal average. Therefore, the interaction contribution to the -space Hamiltonian can be written as We also introduce a spin-dependent chemical potential term to the Hamiltonian, i.e. in real or in space, which allows us to fix the number of fermions independently of each other.

Thus, the total many-body mean-field Hamiltonian for the checkerboard superlattice can be compactly written in space as, where denotes the fermionic operators collectively, and the Hamiltonian matrix is

 (2)

In this paper, we consider equal hoppings for and fermions, i.e. , leading to The single-quasiparticle/hole excitation spectra of the interacting system are given by the eigenvalues of this Hamiltonian matrix, and they can be compactly expressed as

 Eks =−h+βs ⎷ε2k+μ2+∑α|Δα|22−(−1)s2√X, (3) X =4|ϵk|2(|ΔA|2+|ΔB|2−2|ΔAΔB|cosΦ)+16ε2kμ2 +(|ΔA|2−|ΔB|2)(|ΔA|2−|ΔB|2+8μC), (4)

where with labels the eigenvalues, is the difference and is the average chemical potential, , and is the phase difference. Here, we assume . Note that the particle-hole symmetry of the Hamiltonian implies simultaneous transformation of and . Note also that, after setting in the limit, this expression recovers the usual result which is doubly degenerate since the original space is halved.

Using the quasiparticle/hole excitation spectra, we obtain the corresponding mean-field thermodynamic potential for the total Hamiltonian as

 Ω=M2g∑α|Δα|2 +T∑ksln⎡⎢ ⎢⎣1+tanh(βsEks2T)2⎤⎥ ⎥⎦ −12∑ksβsEks−2∑kμ, (5)

where we set the Boltzmann constant to unity. Following the usual procedure, we find the lowest-energy state of the system by minimizing with respect to the amplitudes and phase difference , leading to a set of (three) nonlinearly-coupled equations. In addition, we may set the total and imbalance number fillings using the thermodynamic identities and , where . This procedure leads to a set of (five) self-consistency equations that needs to be solved simultaneously, and four of those can be explicitly written as

 |ΔA|g =12M∑ks∂Eks∂|ΔA|tanh(Eks2T), (6) |ΔB|g =12M∑ks∂Eks∂|ΔB|tanh(Eks2T), (7) n =12M∑ks[1+∂Eks∂μtanh(Eks2T)], (8) p =12M∑ks∂Eks∂htanh(Eks2T). (9)

Some of these partial derivatives are long and not particularly illuminating, and therefore none of them are shown. Similar to the expressions above, the remaining (fifth) phase-difference equation can be written as and since , we immediately conclude that is either or . However, our numerical results suggest that the -phase solution is never realized for the particular model Hamiltonian that we consider in this paper. Note that these -space summations over the 1BZ can be converted into the -space integrations via

Before attacking this problem via numerical means, we would like to gain some physical insight. For this purpose, we rewrite the mean-field Hamiltonian in the basis of and operators, i.e. the one where the single-particle checkerboard Hamiltonian is diagonal. Up to a constant term, the resultant four terms can be compactly written as where the coefficients are given by It is easy to show that the intraband coefficients vanish when as one may expect. This immediately reveals that has the form of a two-band superfluidity Hamiltonian as long as . Note that the coefficients and correspond, respectively, to the -dependent intraband, intraband and interband superfluid order parameters, and all of them are spin singlet with even parity, i.e. , as one may expect. Therefore, the starting -wave sublattice order parameters and , which are independent in the original Hamiltonian, are coupled by , and this gives rise to three nonlocal (-dependent higher partial waves) order parameters in the transformed basis. Having derived the self-consistency equations, next we are ready to present main findings of this work mentioned above in the introduction.

## Iii Numerical Results

In this section, we present our numerical results that are obtained by solving Eqs. (6)-(9) for a self-consistent set of , , and as a function of given , and values. Here, we consider only the ground states of population-balanced Fermi gases, and set and . Motivated by the success of earlier theoretical works on the BCS-BEC crossover problem leggett (); nsr (); bcsbecreview (), we emphasize that the given set of four mean-field equations (that are suitably generalized here to the checkerboard superlattice model) is expected to describe the qualitative physics well for weak interactions at all , and even for moderate and strong interactions at , as long as the single-band tight-binding approximation remains valid. In order to isolate the effects of a nonzero from that of checkerboard hopping, let us first analyze the uniform hopping case.

### iii.1 Uniform hopping: t1=t2=t

In Fig. 2, we set and show and as a function of for a set of values. When , is symmetric around half filling ( or ), which is a consequence of the particle-hole symmetry of the parent Hamiltonian, and its maximum value at half filling is a consequence of the lattice density of states effect. As mentioned in Sec. II, when , the particle-hole symmetry implies simultaneous transformation of and (or . This is clearly illustrated in all of our numerical results, and therefore, it is sufficient to restrict our discussion only to low (particle) fillings . We note that while increases for low fillings as a function of , it decreases . This is because since lowers (raises) the onsite energy of the () sublattice, the local chemical potential of the () sublattice is also lowered (raised) by . Therefore, the () sublattice is effectively becoming more and more strongly (weakly) interacting as a function of . This effect is clearly seen in Fig. 3(a), where we plot and as a function of for a set of values. In the limit, we expect and for , which is in perfect agreement with our numerical results.

In addition to these findings, we find at precisely the half filling that the system undergo a superfluid-normal quantum phase transition beyond a critical , as illustrated in Fig. 3(a). To gain intuitive understanding of this transition, we analyze the single quasiparticle/hole excitation spectra of the interacting system at , and therefore, set and in Eqs. (3) and (II.2). This gives and a doubly degenerate whose form is the same as the spectra if we identify . Here, we recall that the single-quasiparticle/hole dispersion relation of the noninteracting () system is simply given by (see Sec. II.1). Therefore, these results suggest that the system does not favor superfluidity and the normal state with becomes the ground state when , where is the value. Our numerical results shown in Fig. 3(b) are consistent with this analysis, where the the critical values exactly coincide with along the phase transition boundary. Given that in the strong-coupling () limit, by setting , we obtain as the asymptotic limit of the boundary, and this is in perfect agreement with our numerical results. When the critical , we find that the () sublattice has () so that it corresponds to a sublattice band insulator (fully-empty sublattice band) forming a checkerboard insulator. This intuitive result can also be obtained by noting that which reduces to at half-filling at zero temperature. Therefore, the ground-state of the half-filled system first changes from a superfluid to normal when at which point vanishes, and then the normal state rapidly turns into a checkerboard insulator as . For instance, becomes and when is approximately set to and , respectively, in the normal state. Since these numbers are independent of , the phase transition is almost (up to one percent deviation) directly from the superfluid to a checkerboard insulator when .

We note in passing that while having a nonzero leads to a staggered pattern not only in the sublattice order parameters, i.e. , but also in the sublattice number fillings, i.e. , we avoid calling the ground-states of the system a supersolid (when and ) or a charge-density-wave insulator (when and is an integer number). This is because since breaks the translational invariance of the lattice, directly causing such an alternating order parameter and filling patterns, we believe it is important to distinguish our superfluid and checkerboard insulator phases from the true supersolid and charge-density-wave insulator ones, for both of which the translational invariance is broken spontaneously due for instance to the presence of nearest-neighbor interactions.

### iii.2 Checkerboard hopping: t1≠t2

Before we present our concluding remarks, here we discuss the possibility of having an alternating hopping amplitudes. Equation (II.1) indicates that the effects of small deviations from the uniform hopping, i.e. when , can be taken into account (to a very good approximation) via changing the normalization factor that is used in the previous section to . However, in the asymptotic limit, Eq. (II.1) gives and one may expect major quantitative differences. For instance, the superfluid-normal phase transition boundary of a half-filled system is shown in Fig. 3(b) as a function of (blue-dotted line), and it is clearly shown that the phase boundary deviates substantially from that of the uniform hopping case. We again note that the normal state rapidly turns into a checkerboard insulator as , for which becomes and when is approximately set to and , respectively, in the normal state. Having discussed the numerical results, next we briefly summarize the main findings of this paper.

## Iv Conclusions

To summarize, here we studied the ground-state phases of Fermi gases loaded into a two-dimensional checkerboard superlattice potential, i.e. a double-well optical lattice, consisting of two interpenetrating square sublattices and . We described this system with a Fermi-Hubbard type lattice model which includes, in addition to the usual nearest-neighbor hopping and onsite (attractive) density-density interaction , a sublattice-dependent local (onsite) energy . Within the single-band tight-binding BCS mean-field theory, we reached the following conclusions for such a Hamiltonian. First, we showed that the -wave sublattice order parameters and , which are independent in the original Hamiltonian, are coupled by the presence of a checkerboard potential , and this gives rise to a Hamiltonian that has the form of a two-band superfluidity with three (two intraband and an interband) nonlocal (-dependent higher partial waves) order parameters in the basis where the single-particle Hamiltonian is diagonal. We studied the evolution of these order parameters as a function of particle filling, interaction strength and checkerboard potential, and found that the system always prefers the -phase () solutions but never the -phase () one. In addition, we found at precisely half fillings that the ground-state of the system undergo a superfluid-normal quantum phase transition beyond a critical , the threshold of which is precisely determined by the magnitude of the order parameter at , and that the normal state rapidly turns into a checkerboard insulator as increases. One may extend this work in many ways, and motivated by the ongoing experiments nistsoc (); chinasocb (); chinasocf (); mitsoc (); gaugeshaken (), we are especially interested in studying the effects of artificial gauge fields on the ground-state phases of the system, e.g. the so-called optical flux lattices.

## V Acknowledgments

This work is supported by the Marie Curie IRG Grant No. FP7-PEOPLE-IRG-2010-268239, TÜBTAK Career Grant No. 3501-110T839, and TÜBA-GEBP.

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