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Anisotropic Ginzburg-Landau and Lawrence-Doniach Models for Layered Ultracold Fermi Gases

We study the anisotropic Ginzburg-Landau and Lawrence-Doniach models describing a layered superfluid ultracold Fermi gas in optical lattices. We derive the coefficients of the anisotropic Ginzburg-Landau and the mass tensor as a function of anisotropy, filling and interaction, showing that near the unitary limit the effective anisotropy of the masses is significantly reduced. The anisotropy parameter is shown to vary in realistic setups in a wide range of values. We also derive the Lawrence-Doniach model - often used to describe the 2D-3D dimensional crossover in layered superconductors - for a layered ultracold Fermi gas, obtaining a relation between the interlayer Josephson couplings and the Ginzburg-Landau masses. Comparing to the Ginzburg-Landau description, we find that the region of validity of the Lawrence-Doniach model is near the unitary limit.

The presence and nature of the superfluid and superconducting transition in systems of different dimensionality is a widely studied and interesting topic. It is well known that the transition in two and three dimensions is radically different, as the first is governed by a Kosterlitz-Thouless mechanism of topological origin (the unbinding of vortices) and the second is the paradigmatic example of the symmetry breaking well described, at least qualitatively, by mean-field theories. This is the main reason of the wide interest in the study of materials with properties in-between the two, i.e. layered superfluids and superconductors.
Their phenomenological properties are usually well described by anisotropic effective models: a successful example is given by the Lawrence-Doniach (LD) model [1] (in which the discrete layers are weakly coupled through Josephson terms) describing the electrodynamics of layered superconductors and copper oxides [2]. Another related way to introduce the effect of the layering is to consider a Ginzburg-Landau (GL) theory having anisotropic masses (eventually phenomenologically determined) [2]. A well known feature of these phenomelogical models is the fact that they work well even if the underlying microscopic mechanisms are not known (as it occurs for high-T c cuprates).
In this paper we study and compare the anisotropic GL and LD models describing a layered superfluid ultracold Fermi gas in optical lattices: the coefficient of these phenomenological models can be microscopically derived and the parameters, like the strength of the optical lattices, can be tuned with high accuracy in realistic experimental setups. We determine the mass tensor in the GL theory as a function of anisotropy, filling and interaction, showing that near the unitary limit the effective anisotropy of the masses is significantly reduced. By tuning the strength of the optical lattices, the dimensionless anisotropy parameter γ in realistic setups can vary in a wide range of values: γ ∼ 1 − 200 (for comparison with layered superconductors, we remind that γ ∼ 7 for YBCO and γ 150 for BSCCO).
We also derive the LD model, discussing a relation between the interlayer Josephson energy couplings in the LD model and the masses in the GL model. Using this result, the LD model with only couplings between adjacent planes is shown to be equivalent to the anisotropic GL theory in the BEC limit, while in the BCS side (where the size of the pairs is larger than the lattice spacing) next-nearest-neighbour couplings have to be taken into account.
Layered superconductors and superfluids: The first example of a layered superconductor was found in a naturally occurring compound of transition-metal dichalcogenides layers (such as TaS 2 ) intercalated with organic molecules [3], where, for temperatures of a few Kelvin, the metallic planes become superconductors. The discovery was followed later by the creation of artificial samples with alternating layers of different metals, having a different transition temperature [4]. The nature of the transition was found to be three dimensional [5], and the system was shown to be very well described by an anisotropic GL free energy at and above the transition, where the coherence length diverges as ξ ∝ (T − T c ) −1/2 . At lower temperatures where the length ξ becomes comparable with the plane separation, the presence of a layer structure becomes important. To describe this Lawrence and Doniach proposed a tight-binding approximation [1] able to describe the crossover between the 2D behaviours of essentially decoupled planes to the 3D.
Another very important class of superconductors having a layered structure is the one of cuprates, discovered in 1986 by Bednorz and Müller [6]. Their characteristic structure presents a layer of one or more conducting copper oxide planes alternated to an insulating lattice with a perovskite-like structure. The layered structure makes the behaviour of cuprates strongly anisotropic and the dimensional crossover temperature is in general very near the critical temperature for the resistive transition [2]: despite the lack of a definitive consensus on the mechanism responsible for the superconductivity in the cuprates, it is by now established that the LD model successfully describe the transition and the electromagnetic response of cuprates [2,7,8].
The study of the dimensional crossover in cuprates is not in general an easy task: on one hand, the smallest barrier attainable is that of a single plane of an insulating material, on the other hand inserting several planes is a delicate process as the separation can become nonuniform [8].
From this point of view, ultracold atoms (and in particular ultracold fermions) in optical lattices provide a favourable setting: the tunnelling rate t can be controlled by adjusting the optical lattice parameters [9,10] and the on-site energy coefficient U can be controlled through Feshbach resonances [11,12]. The ratio U/t can be then controlled with high precision, as well as the geometry and the anisotropy of the system (e.g., optical lattices with different strengths can be superimposed in different directions). For ultracold bosons, optical lattices has been used to induce a Mott-insulator/superfluid quantum phase transition [13] and to study Josephson dynamics in many-and two-wells settings [14,15]. The study of ultracold fermions in optical lattices also stimulated a huge amount of experimental work [16][17][18][19][20][21] and they provide a field of research in which even more far-reaching experiments are expected to take place. The low energy properties of ultracold fermions in deep optical lattices are described by Hubbard-like models [22]: metallic and insulating phases in 3D lattices have been studied [19,20]. in presence of attractive interaction (U < 0) one has then a physical realization of the attractive Hubbard model [23][24][25][26][27].
Superimposing a 1D optical lattice to an ultracold Fermi gas gives raise to a stack of two-dimensional layers: very recently, the evolution of fermion pairing from 3D to 2D was experimentally studied, showing the opening of a gap in the radiofrequency spectrum when the dimensionality is reduced [21]. The confinement of Fermi mixtures in 2D was also experimentally investigated [28][29][30][31][32][33], showing properties different from the 3D case [34]. The observation of a pairing pseudogap in a 2D Fermi gas, using momentum-resolved photoemission spectroscopy, has been reported [33]. An important advantage of the use of an optical lattice is that when it is applied (say along the z direction) then the interlayer coupling t between 2D pancakes along z can be easily modified by varying the strength of the optical lattice. Furthermore, if two additional optical lattices are added in the xy plane, then the tunnelling t ⊥ can be tuned separately from t and the layering can be tuned by varying the ratio t /t ⊥ .
Effective models: Near the critical temperature, the superfluid wavefunction Ψ is small, so that the free energy difference δF between the normal and superfluid state can be expanded in powers of |Ψ| 2 [35]: for a layered superconductor/superfluid δF reads then The phenomenological coefficients α and β appearing in (1) can be derived from experiments or from the microscopic Hamiltonian; the masses m ,⊥ are those of the Cooper pairs. The GL equations describing the equilibrium state can be found by minimizing (1) with respect to Ψ. For α > 0 there is only the minimum Ψ = 0 corresponding to the normal state, while for α < 0 there are uniform solutions Ψ = |Ψ 0 |e iθ , hence the critical temperature is given by α(T c ) = 0. In the LD tight-binding approach one has a 2D superfluid wavefunction Ψ n (x, y) for in the nth plane along z and the term with transverse derivative becomes the interlayer separation, i.e. the lattice spacing of the periodic potential along z).
In layered superconductors, the effects and the penetration of a magnetic field have been thoroughly studied [2,7,36]: such effects can be incorporated via the minimal substitution is the vector potential) and adding to the free energy the magnetization energy F H = ( B − H) 2 /8π. The penetration of the magnetic field in anisotropic type-II superconductors and the response of a layered superconductor to an electromagnetic field have been an established field of research for decades [2,7,36]. In the following we show that the layered GL theory (1) holds for a layered ultracold Fermi superfluid gas: then, to reproduce the phenomenology of the electrodynamics of layered superconductors in the context of ultracold atoms one can use a layered Fermi mixture in an fictitious magnetic field realized in a synthetic way using spatially dependent optical couplings between internal states of the atoms [37] or by rotating the trapping potential [38].
The Hamiltonian describing a two-component Fermi gas in a layered optical lattice readŝ H − µN , with the HamiltonianĤ given bŷ iσφiσ is the number operator, µ is the chemical potential, i, j are site indices, σ = ↑, ↓ is the spin index andφ † iσ is the fermionic operator (the sum is on distinct pairs of nearest-neighbours). The interaction is assumed attractive, U > 0, and the total number of fermions per site is denoted by n. The physical system modelled by (2) considers two optical lattices in the x, y directions (determining the tunnelling rate t between the sites in the xy planes) and a different lattice in the z direction, with interlayer coupling t ⊥ .
Notice that in our subsequent determination of the coefficients of the anisotropic GL and LD models the crucial ingredient is to have the optical lattice along z and the mechanism of layering in the form of tunnelling between adjacent layers: if the optical lattices in the that in each minimum of the lattice there is a total number of fermions n ≤ 2 (i.e., only the lowest band of the periodic potential is occupied): otherwise, we would have a multi-band Hubbard model and again (1) would be retrieved.
To derive the coefficients of the anisotropic GL we proceed in a path-integral setting [39][40][41]: performing the Hubbard-Stratonovich transformation and integrating out the fermionic field, a bosonic effective action is obtained (the critical temperature T c is the one at which the order parameter is vanishing). Details are in Appendix A: the coefficients of the GL free energy (1) are found to be in (3) a = x, y, z denotes the direction and the mass parameters M a are related to the masses It is also possible to derive the LD tunnelling coefficients: in the LD model, the xy layers are weakly coupled along the z direction through Josephson terms. Usually, only Josephson couplings between adjacent planes are considered. Here we consider the more general possibility of longer range tunnelling: this amounts to consider a Josephson contribution − n,m J n,m | ψ n − ψ m | 2 to the energy, where n, m are denoting the different planes (i.e., the minima along z). We find where the expression for χ(q) = χ(iω; q) is reported in Appendix A, Eq.(A10). In presence of lattices in the x, y directions, one can similarly define Josephson couplings J n,m . It is possible to determine a useful relation between the coefficients J n,m and the mass m ⊥ of the anisotropic GL model: one finds where we used the notation J p ≡ J n,n+p (we are assuming translational invariance along z): Using the critical temperature T c , we can compute the mass anisotropy parameter γ usually defined [2] as The anisotropy parameter γ plays a crucial role in the electrodynamics of layered superconductors since a formula giving the value of a quantity for a general anisotropy parameter γ given its value for γ = 1 holds [42]. Values of γ for several superconducting compounds has been determined from experimental data [2]. In layered Fermi superfluids, the value of γ affects the critical value of a synthetic magnetic field (this is the atomic counterpart of the effect of a magnetic field on a layered superconductor [2]): supposing to have a general rotation Ω, the critical values of the rotation for the vortex melting obey the relation Ω c2 /Ω c2⊥ = γ.
The mass ratio could be experimentally determined from momentum-resolved photoemission spectroscopy [33] or from the measurement of the frequency of the collective excitations [34].
The results for the anisotropy parameter γ of a layered ultracold Fermi gas are shown in value of t /t ⊥ smaller than one, we studied an effective anisotropy defined as γ 2 ef f = γ 2 ·t ⊥ /t . The results are drawn in Fig.6: it is seen that in the BCS side the enhancement is more pronounced with respect to the BEC side. Again, the effective enhancement is reduced in the unitary limit. We also checked that in the BEC limit the quantity γ 2 · (t ⊥ /t ) 2 is found to approach 1 for different fillings, see Fig.7.
can be seen in Fig.4, the validity region of the LD model is narrower for small layering (i.e., t ⊥ /t close to 1) and it is centered around the interval in which the critical temperature reaches its maximum value. This result is pictorially summarized in Fig.8.

Conclusions:
In this paper we studied the superfluid phase of an ultracold layered Fermi gas in the presence of an optical lattice near the critical temperature. This system is described by an anisotropic Ginzburg-Landau theory (that is also used for other systems such as layered superconductors). We derived the coefficients of the Ginzburg-Landau equation: the anisotropy was shown to be enhanced by the superfluid, and the anisotropy parameter whereφ † iσ is the fermionic operator, t ij denote the tunnelling rates, µ is the chemical potential and σ = ↑, ↓ is the spin index. Furthermore i, j are site indices andN = iσφ † iσφiσ is the total number operator. The interaction is assumed attractive, U > 0, and we denote by n the total number of fermions per site (n ≤ 2). The tunnelling rate between the sites in the xy planes is denoted by t , while the tunnelling rate along the z axis is denoted by t ⊥ .
As discussed in the main text, in our study of the anisotropic GL and LD coefficients the crucial ingredient is to have the optical lattice along z and the mechanism of layering in the form of tunnelling between adjacent layers: with small (or absent) optical lattices in the x, y directions, then the anisotropic GL (1) would be obtained. In writing (A1) we also that in each site of the lattice there is a total number of fermions n ≤ 2: otherwise, one has a multi-band Hubbard description and again the anisotropic GL would be retrieved.
In a path-integral setting the fermionic atoms are described by the independent Grassmann fields φ andφ [39-41, 43, 44]: the partition function is given by with S f the fermionic action in imaginary time [40]. First, the Hubbard-Stratonovich transformation is performed by completing the Gaussian integral: the theory has then the form of a (free) fermion field interacting with the bosonic field ∆. The corresponding action reads and k = −2t (cos k x +cos k y )− 2t ⊥ cos k z denote the single-particle energies (in this Appendix the lattice spacing is set to 1). Furthermore we used the notation k = (iω; k) where the wavevectors k belong to the Brillouin zone [−π; π] 3 and ω = π/β are the Matsubara frequencies (with even or odd in the case of bosons and fermions respectively). The sum over k corresponds to an integral over k over the Brillouin zone and sum over the integers ; furthermore, φ jσ (τ ) = k e i( k· r j −ω τ ) φ σ (iω ; k) ( r j is the position of the site j).

13
The fermionic degrees of freedom can be integrated out obtaining the bosonic effective The critical temperature T c is the temperature for which the solution has ∆ 0 = 0 where ∆ 0 = ∆ . When the interaction is weak, only the atoms around the Fermi level form Cooper pairs (BCS limit) [34]. In the BCS limit Cooper pairs have zero relative momentum, i.e. ∆ k ≈ 0 for k = 0, and the chemical potential is not changed from the Fermi energy E F . In this limit the phase-space integration can be done by saddle point approximation leading to the mean-field equations. The result is that T c T F , where T F is the Fermi temperature [34,41]. When this is not the case, one has to consider the effects of higher order fluctuations [40] and solve two coupled equations to obtain µ and ∆ 0 as a function of the interaction U , the hopping rates t and t ⊥ , the particle density n and the temperature T . The first equation is the saddle point equation for the order parameter ∆ 0 : The other is the number equation where (apart from a constant) is the free energy and χ(q) is the pair polarization function, which for ∆ 0 = 0 reads with G ± k = iω ± k /(ω 2 + 2 k ). To determine the critical temperature T c one has to solve the two Eqs. (A7)-(A8) with ∆ 0 = 0: in this way one obtains T c and µ as a function of the tunneling rates t and t ⊥ , the particle density n and the interaction U . We observe that the numerical solution of (A7)-(A8) involves for the lattice case the explicit evaluation of (derivatives of) integrals over six variables, while in the 3D continuum case [39] one can exploit the rotational invariance and actually evaluate integrals in three variables. A simplification in the numerics is obtained noticing that after transforming the sum over Matsubara frequencies into a contour integral a first order pole appears that has to be corrected by either a limiting procedure putting iω +0 + , or by subtracting the contour integral at infinity which can be obtained analytically.
The solution of Eqs. (A7)-(A8) for T c vs. U for different fillings and anisotropies is shown in Fig.1. One also observes that, as in the continuum case, the unitary point is different from the maximum T c point. If the system is near the critical temperature T c , the average value of ∆ is small and the effective action can be expanded in series [39,43]. Stopping at the fourth order gives where S 0 = ln det(G −1 ) − βµ describes a non-interacting Fermi gas. The last step in finding a GL form for the action is to expand each coefficient around q = 0 (time-dependence can be also included by expanding in iω). The resulting action is of the same form as the GL free energy (1), with the GL coefficients m ⊥ , m , α, β given in Eqs. (3)-(4).
We remark that the action just derived is also the free energy when the value ∆ is fixed.
A natural question arises whether this functional corresponds to the original GL free energy, that is a functional of the anomalous correlation function. It can be seen that the minimum condition ∂ ∆ F [∆] = 0 gives φ i↓ φ ↓ φ ↑ = ∆/U , so that the free energy just found gives the correct equilibrium values for the observables.
Once that the GL coefficients m ⊥ , m , α, β are obtained (and their explicit evaluation requires the previous determination of T c and µ), it is then possible to evaluate the anisotropy parameter γ = m ⊥ /m . Our results for γ as a function of the interaction U at a fixed filling n are plotted in Fig.2. We also studied the influence of the filling on the anisotropy parameter: our findings are reported in Fig.5. The filling modifies γ on the BCS side, where the dynamics is determined by the atoms at the Fermi surface. On the other side of the unitary limit, the dynamics interests all the energy scales and is less susceptible to filling.
In particular while the critical temperature for a given interaction value U increases with filling, the anisotropy on the BEC side remains virtually identical.
A natural question is whether the collective behaviour enhances or not the anisotropy that is artificially tuned by imposing a layering through a value of t /t ⊥ smaller than one: for this reason we define an effective anisotropy γ ef f defined through the relation The reason behind the choice of the definition (A12) is that we expect that the effective mass along the direction a is ∼ 1/t a (where a = x, y, z). Then a value of γ ef f larger than one signals an effective enhancement of the anisotropy (with respect to the "external" anisotropy t /t ⊥ ). The results are drawn in Fig.6: it is seen that in the BCS side the enhancement is more pronounced with respect to the BEC side. Again, the effective enhancement is reduced in the unitary limit.
The pair tunnelling in the BEC limit is expected to approach the strong coupling value t 2 ,⊥ /U for each direction. Indeed in the BEC limit the quantity γ 2 · (t ⊥ /t ) 2 is found to approach 1 (with the convergence to the BEC limit being slower for systems with stronger layering), as shown in Fig.7.
Finally, in Fig.8 we pictorially depict the physical regimes of a layered ultracold Fermi gas.