# Collective Modes of an Ultracold6Li-40K Mixture in an Optical Lattice

Authors: Z. G. Koinov
Journal: Advances in Condensed Matter Physics
 (2015)

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## Abstract

A low-energy theory of the Nambu-Goldstone excitation spectrum and the corresponding speed of sound of an interacting Fermi mixture of Lithium-6 and
Potassium-40 atoms in a two-dimensional optical lattice at finite
temperatures with the Fulde-Ferrell order parameter has been formulated. It is assumed that the two-species interacting Fermi gas is described by the one-band Hubbard Hamiltonian with an attractive on-site interaction. The discussion is restricted to the BCS side of the Feshbach resonance where the Fermi atoms exhibit superfluidity. The quartic on-site interaction is decoupled via a Hubbard-Stratonovich transformation by introducing a four-component boson field which mediates the Hubbard interaction. A functional integral
technique and a Legendre transform are used to give a systematic derivation of the Schwinger-Dyson equations for the generalized single-particle Green’s function and the Bethe-Salpeter equation for the two-particle Green’s function and the associated collective modes. The numerical solution of the Bethe-Salpeter equation in the generalized random phase approximation shows that there exist two distinct sound
velocities in the long-wavelength limit. In addition to low-energy
(Goldstone) mode, the two-species Fermi gas has a superfluid phase revealed by two roton-like minima in the asymmetric collective-mode energy.


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## Body

## 1. Introduction

Optical lattices are formed by the interference of counter propagating laser beams. If the laser beams have equal frequencies, the gases of ultracold alkali atoms can be trapped in periodic potentials (microtraps) created by standing waves of laser light. Because of the Stark effect the ground-state alkali atoms couple to the electromagnetic field via an induced electric dipole moment. From theoretical point of view, the simplest approach to the trapped fermions is the tight-binding approximation, which requires sufficiently deep lattice potential. In the tight-binding limit, two alkali atoms of opposite pseudospins on the same site have an interaction energyU, while the probability to tunnel to a neighboring site is given by the hopping parameters. The hopping parameters as well as the interaction energy depend on the depth of the lattice potential and can be tuned by varying the intensity of the laser beams. We assume that the interacting fermions are in a sufficiently deep periodic lattice potential described by the Hubbard Hamiltonian. We restrict the discussion to the case of atoms confined to the lowest-energy band (single-band Hubbard model), with two possible states described by pseudospins σ. We consider different amounts of  Li6 and K40 atoms in each state (σ=↑=Li, σ=↓=K) achieved by considering different chemical potentials μ↑ and μ↓. There are M=M↑+M↓ atoms distributed along N sites, and the corresponding filling factors f↑,↓=M↑,↓/N are smaller than unity. The Hubbard Hamiltonian is defined as follows: (1)H=-∑i,j,σ‍Jσψi,σ†ψj,σ-U∑i‍n^i,↑n^i,↓-∑i,σ‍μσn^i,σ,where Jσ is the single electron hopping integral and n^i,σ=ψi,σ†ψi,σ is the density operator on site i. The Fermi operator ψi,σ† (ψi,σ) creates (destroys) a fermion on the lattice site i with pseudospin projection σ. The symbol ∑〈i,j〉‍ means sum over nearest-neighbor sites of the two-dimensional lattice. The first term in (1) is the usual kinetic energy term in a tight-binding approximation. All numerical calculations will be performed assuming that the hopping (tunneling) ratio JLi/JK≈0.15. In our notation the strength of the on-site interaction U>0 is positive, but the negative sign in front of the interaction corresponds to the Hubbard model with an attractive interaction. In the presence of an (effective) attractive interaction between the fermions, no matter how weak it is, the alkali atoms form bound pairs, also called the Cooper pairs. As a result, the system becomes unstable against the formation of a new many-body superfluid ground state. The superfluid ground state comes from the U(1) symmetry breaking and it is characterized by a nonzero order parameter, which in the population-balanced case is assumed to be a constant in space Δ0. Physically, it describes superfluid state of Cooper pairs with zero momentum. Superfluid state of Cooper pairs with nonzero momentum occurs in population-imbalanced case between a fermion with momentum k+q and spin ↑ and a fermion with momentum -k+q and spin ↓. As a result, the pair momentum is 2q. A finite pairing momentum implies a position-dependent phase of the order parameter, which in the Fulde-Ferrell [1] (FF) case varies as a single plane wave Δ(r)=Δqexp2ıq·r, where Δq is a real quantity. The order parameter also can be a combination of two plane waves as in the case of the Larkin-Ovchinnikov [2, 3] (LO) superfluid states. In both cases we are dealing with a spontaneous translational symmetry breaking and with an inhomogeneous superfluid state. When continuous and global symmetries are spontaneously broken the collective modes known as the Nambu-Goldstone (NG) modes appear [4, 5].Generally speaking, the single-particle excitations manifest themselves as poles of the single-particle Green’s function,G, while the two-particle (collective) excitations could be related to the poles of the two-particle Green’s function, K. The poles of these Green’s functions are defined by the solutions of the Schwinger-Dyson (SD) equation G-1=G(0)-1-Σ [6, 7] and the Bethe-Salpeter (BS) equation [K(0)-1-I]Ψ=0 [8], respectively. Here, G(0) is the free single-particle propagator, Σ is the fermion self-energy, I is the BS kernel, and the two-particle free propagator K(0)=GG is a product of two fully dressed single-particle Green’s functions. Since the fermion self-energy depends on the two-particle Green’s function, the positions of both poles must be obtained by solving the SD and BS equations self-consistently.Instead of solving the SD and BS equations self-consistently, it is widely accepted that the single-particle dispersion can be obtained in the mean-field approximation or by solving the Bogoliubov-de Gennes (BdG) equations in a self-consistent fashion, while the generalized random phase approximation (GRPA) is the one that can provide the collective excitations in a weak-coupling regime. In the GRPA, the single-particle excitations are replaced with those obtained by diagonalizing the Hartree-Fock (HF) Hamiltonian, while the collective modes are obtained by solving the BS equation in which the single-particle Green’s functions are calculated in HF approximation, and the BS kernel is obtained by summing ladder and bubble diagrams.From theoretical point of view, the corresponding expressions for Green’s functions cannot be evaluated exactly because the interaction part of the Hubbard Hamiltonian is quartic in the fermion fields. The simplest way to solve this problem is to apply the so-called mean-field decoupling of the quartic interaction. To go beyond the mean-field approximation, we apply the idea that we can transform the quartic term into quadratic form by making the Hubbard-Stratonovich transformation for the fermion operators. In contrast to the previous approaches, such that after performing the Hubbard-Stratonovich transformation the fermion degrees of freedom are integrated out; we decouple the quartic problem by introducing a model system which consists of a multicomponent boson fieldAα interacting with fermion fields ψ† and ψ.The functional-integral formulation of the Hubbard model requires the representation of the Hubbard interaction of (1) in terms of squares of one-body charge and spin operators. It is known that it may be possible to resolve the Hubbard interaction into quadratic forms of spin and electron number operators in an infinite number of ways. If no approximations were made in evaluating the functional integrals, it would not matter which of the ways is chosen. When approximations are taken, the final result depends on a particular form chosen. Thus, one should check that the results obtained with the Hubbard-Stratonovich transformation are consistent with the results obtained with the canonical mean-field approximation. It can be seen that our approach to the Hubbard-Stratonovich transformation provides results consistent with the results obtained with the mean-field approximation; that is, one can derive the mean-field gap equation using the collective-mode dispersion ω(Q) in the limit Q→0 and ω→0.There are three advantages of keeping both the fermion and the boson degrees of freedom. First, the approximation that is used to decouple the self-consistent relation between the fermion self-energy and the two-particle Green’s function automatically leads to conserving approximations because it relies on the fact that the BS kernel can be written as functional derivatives of the FockΣF and the Hartree ΣH self-energy I=Id+Iexc=δΣF/δG+δΣH/δG=δ2Φ/δGδG. As it is known, any self-energy approximation is conserving whenever: (i) the self-energy can be written as the derivative of a functional Φ[G], that is, Σ=δΦ[G]/δG, and (ii) the SD equation for G needs to be solved fully self-consistently for this form of the self-energy. Second, the collective excitations of the Hubbard model can be calculated in two different ways: as poles of the fermion Green’s function, K, and as poles of the boson Green’s function, D, or equivalently, as poles of the density and spin parts of the general response function, Π. Here, the boson Green’s function, D, is defined by the Dyson equation D=D(0)+D(0)ΠD(0) where D(0) is the free boson propagator. Third, the action which describes the interactions in the Hubbard model is similar to the action ψ†Aψ in quantum electrodynamics. This allows us to apply powerful field-theoretical methods, such as the method of Legendre transforms, to derive the SD and BS equations, as well as the vertex equation for the vertex function, Γ, and the Dyson equation for the boson Green’s function, D.The mean-field treatment of the FF and LO phases in a variety of systems shows that the FF and LO states compete with a number of other states, such as the Sarma (q=0) states and the superfluid-normal separation phase (also known as the phase separation phase). It turns out that in some regions of momentum space the FF (or LO) phase provides the minimum of the mean-field expression of the Helmholtz free energy. Phase diagrams for a Li6-K40 mixture at zero temperature were obtained in [9], but the calculations were limited to the emergence of insulating phases during the evolution of superfluidity from the BCS to the BEC regime, and the competition between the FF and Sarma phases was ignored. The polarization versus temperature diagrams in Figure 1 show that there are three phases: the Sarma phase, the FF phase, and the normal phase in which the Helmholtz free energy is minimized for gapless phase. The zero polarization line is the conventional Bardeen-Cooper-Schrieffer state. The phase diagram contains a Lifshitz point. When the interaction strength is increased, the Lifshitz point moves toward the higher temperatures and larger polarizations. Moreover, contrary to the phase diagram of population-imbalanced Li6 Fermi gas, where the phase separation appears for low polarizations, the existence of a polarization window for the FF phase was found. This means that as soon as the system is polarized it goes into the FF phase if the temperature is low enough. This polarization window is larger for a majority of K40 atoms compared to the majority of Li6 atoms.Figure 1
The phase diagrams of aLi6-K40 mixture in a square lattice [10]. The interaction strength is U=2J↑. The polarization is defined as P=(fK-fLi)/f, where the total filling is f=0.5 atoms/lattice site. Colors: Sarma states = blue (black), FF = red (dark grey), and normal gas = white.In what follows, the collective-mode dispersion of aLi6-K40 mixture loaded in a two-dimensional optical lattice is calculated numerically by solving the BS equation in the GRPA approximation.


## 2. Collective Modes of6Li-40K Mixture

The superfluid states can be described in terms of the Namby-Gor’kov single-particle Green’s function which is a thermodynamic average of theT^u-ordered tensor product of the four-component fermion fields:(2)Ψ^x=ψ↑xψ↓xψ↑†xψ↓†x,Ψ^†y=ψ↑†yψ↓†yψ↑yψ↓y.Here, we introduce composite variables y={ri,u}={i,u} and x={ri′,u′}={i′,u′}, where ri, ri′ are the lattice site vectors, and according to imaginary-time (Matsubara) formalism the variables u, u′ range from 0 to ħβ=ħ/(kBT). Throughout this paper we have assumed ħ=kB=1, the lattice constant a=1, and we use the summation-integration convention: that repeated variables are summed up or integrated over. The field operators allow us to define the generalized single-particle Green’s function which includes all possible thermodynamic averages:(3)G^x1;y2=-T^uΨ^x1⊗Ψ¯^y2-T^uψ↑x1ψ↑†y2T^uψ↑x1ψ↓†y2T^uψ↑x1ψ↑y2T^uψ↑x1ψ↓y2T^uψ↓x1ψ↑†y2T^uψ↓x1ψ↓†y2T^uψ↓x1ψ↑y2T^uψ↓x1ψ↓y2T^uψ↑†x1ψ↑†y2T^uψ↑†x1ψ↓†y2T^uψ↑†x1ψ↑y2T^uψ↑†x1ψ↓y2T^uψ↓†x1ψ↑†y2T^uψ↓†x1ψ↓†y2T^uψ↓†x1ψ↑y2T^uψ↓†x1ψ↓y2.In the GRPA, the generalized Green’s function (3) is replaced by its mean field approximation. The Fourier transform of the mean-field single-particle Green’s function is as follows:(4)G~^qk,ıωm=ıωm+ξ↓q-kAq,k,ıωm00ΔqAq,k,ıωm0ıωm+ξ↑q-kBq,k,ıωm-ΔqBq,k,ıωm00-ΔqBq,k,ıωmıωm-ξ↓q+kBq,k,ıωm0ΔqAq,k,ıωm00ıωm-ξ↑q+kAq,k,ıωm,Aq,k,ıωm=ıωm-ξ↑q+kıωm+ξ↓q-k-Δq2,Bq,k,ıωm=ıωm-ξ↓q+kıωm+ξ↑q-k-Δq2.Here, the FF vectorq, as well as the chemical potentials μ↑ and μ↓, and the gap Δq are defined by the solutions of following set of four mean-field equations (the number equations, the gap equation, and the q-equation) [10]: (5)f↑=1N∑k‍uq2kfω+k,q+vq2kf-ω-k,q,f↓=1N∑k‍uq2kfω-k,q+vq2kf-ω+k,q,1=UN∑k‍1-fω-k,q-fω+k,q2Eqk,0=1N∑k‍∂ηqk∂qxfω+k,q-fω-k,q=1N∑kwi‍+∂χqk∂qx1-χqkEqk=1N∑k‍+∂χqk∂qxW∂ηqk∂qxχqkEqk·1-fω+k,q-fω-k,q,χqk=12ξ↑q+k+ξ↓q-k,ηqk=12ξ↑q+k-ξ↓q-k,ω±k,q=Eqk±ηqk,uq2k=121+χqkEqk,vq2k=121-χqkEqk,Eqk=χq2k+Δq2.As was mentioned above, as soon as the system is polarized it goes into the FF phase, and therefore, it is natural to obtain the collective-mode dispersion using one point from the FF window shown in Figure 1. This point is defined by the following system parameters: fLi=0.225, fK=0.275, U/J↑=2, and T/J↑=0.01. The solution of the mean-field equations (5) provides the FF vector q=(qx,qy), the gap, and the chemical potentials. Since the formation of the FF superfluid state is driven by the mass and population imbalance, which leads to the distortion of the Fermi surfaces. There are many equivalent ways to deform the surfaces, and as a result, the direction of the single FF pairing momentum q is not specified. In what follows, we shall assume that the pairing momentum is along the x-axis. The corresponding mean-field solutions are as follows: q=(0.0489π/a,0), Δq/J↑=0.3668, μ↑/J↑=2.0909, and μ↓/J↑=0.5561. For the Sarma state the corresponding mean-field values are Δ/J↑=0.3635, μ↑/J↑=2.0466, and μ↓/J↑=0.5918. The FF phase is the most stable as it provides the minimum of the mean-field expression of the Helmholtz free energy (the ratio between the FF free energy and the Sarma free energy is about 0.9986).In the case when the order parameter is assumed to vary as a single plane wave, we have a broken translational invariance, and as a result, the normal and anomalous mean-field single-particle Green’s functions have phase factors associated with the FF quasimomentumq, which can be eliminated using the corresponding unitary transformation. In the appendix, we have shown that in the tight-binding approximation the BS equation (in the GRPA) can be reduced to a secular determinant, which determines the collective-mode dispersion. When the generalized Green’s function (3) is used, the BS approach provides a 16×16 secular determinant Z8, defined in the appendix.In Figure2, we have presented the dispersion relation ω(Q) calculated for the system parameters listed in Section 1. The FF vector Q is directed along the x-axis. The speed of sound, c, to the positive and negative directions of the Qx axis is defined by dω(Qx)/dQx at Qx→0. For the dispersions presented in Figure 2, we obtain c=0.614JLia/ħ in positive direction and c=0.534JLia/ħ in the negative direction. There are two roton minima at Qx=0.96π/a and at Qx=-0.76π/a.Figure 2
The collective-mode dispersion ofLi6-K40 mixture in 2D optical lattice along the positive (Qx,0) and negative (-Qx,0) directions. The filling factors are fLi=0.225 and fK=0.275. The interaction strength and the temperature are U/JLi=2 and T/JLi=0.01, respectively. The secular determinant Z8 provides the speed of sound to the positive direction c=0.614JLia/ħ, while the speeds of sound to the negative direction is c=0.534JLia/ħ.


## 3. Discussion

We have studied the Nambu-Goldstone excitation spectrum and the corresponding speed of sound of an interacting Fermi mixture of Lithium-6 and Potassium-40 atoms Fermi gases in deep optical lattices by using the Bethe-Salpeter equation in the GRPA. The generalized single-particle Green’s function, used in our numerical calculations, takes into account all possible thermodynamic averages. The chosen temperature and polarization correspond to a point from the polarization-temperature window for the FF phase in the phase diagram. The on-site attractive interaction corresponds to the weak-coupling regime, where the GRPA is valid.It is well known that at very low temperatures, the singularities of the integrals (A.41) in the appendix correspond physically to the possibility of depairing into two fermion excitations. At a zero temperature, the two fermion excitations have energies ω+(k+Q,q) and -ω-(k,q). The spectrum for this kind of excitation is known as the particle-hole continuum. At a zero temperature, the lower boundary of the particle-hole continuum is defined by the condition minkω+(k+Q,q)-ω-(k,q) where the minimum is to be taken over all the possible values of k. The collective-mode dispersion, presented in Figure 2, lies below the particle-hole continuum, and therefore, the possibility of depairing into two fermion excitations is not important.It is known that the superfluid ground state comes from theU(1) symmetry breaking and it is characterized by a nonzero order parameter. If the atoms have different masses and chemical potentials, the superfluid states of Cooper pairs are with nonzero momentum. A finite pairing momentum implies a position-dependent phase of the order parameter. Based on general symmetry principles, it is known that when continuous and global symmetries are spontaneously broken in Lorentz-invariant systems, the number of Nambu-Goldstone modes is always equal to the number of broken generators, and all of them have linear dispersion [11, 12]. In the FF and in the LO cases, we are dealing with a spontaneous translational symmetry breaking and with an inhomogeneous superfluid state; but in case of the FF order parameter, only one generator is spontaneously broken, because there is an unbroken combination of U(1) rotations and translations. Thus, the FF superfluid state is characterized by a single Nambu-Goldstone mode, and this statement is in accordance with the fact that the Z8 determinant in the appendix provides only one mode. It is worth mentioning that in the case of the LO order parameter, both the particle number U(1) symmetry and translations along a given direction are spontaneously broken, and therefore, there should exist two Nambu-Goldstone modes. The Bethe-Salpeter description of the collective modes in the case of LO superfluid states is expected to be more complicated because the phase factors of the nondiagonal elements of the single-particle Green’s function (see (A.32) in the appendix) one cannot eliminate as in the case of FF states.In view of the fact that most of the previous numerical calculations are based on the Nambu-Gor’kov single-particle Green’s function,(6)G^x1;y2=-T^uψ^x1⊗ψ¯^y2=-T^uψ↑x1ψ↑†y2T^uψ↑x1ψ↓y2T^uψ↓†x1ψ↑†y2T^uψ↓†x1ψ↓y2,which leads to the 4×4 secular determinant, one may well ask whether the collective-mode dispersion, defined by 4×4 determinant, is significantly different in comparison to the dispersion obtained with the secular determinant Z8.To answer this question, we have calculated the collective-mode dispersion relation of a population-imbalanced atomic Fermi gas. The dispersions obtained by means of the secular determinantZ8 and 4×4 are presented in Figure 3. In the range of small k, we find a difference of about 15% between the speed of sound obtained by means of Z8 secular determinant and the speed of sound, calculated by 4×4 secular determinant used in [13]. The figure also indicates that the difference between the two dispersion curves tends to increase with k, reaching 25% at k=0.5/a.Figure 3
The collective-mode dispersionω(k) (triangles) calculated by the Bethe-Salpeter formalism for a 2D system with q=qxex, qx=0.06π/a, μ↑=3.5J, μ↓=2.5J, Δ=0.27J, U=3J, and T=0.07J. The speed of sound to the direction of the x-axis is c=1.48Ja/ħ. The squire points are calculated by 4×4 secular determinant used in [13], and the corresponding speed of sound is c=1.29Ja/ħ.



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