{'fraction_non_alphanumeric': 0.04395604395604396, 'fraction_numerical': 0.009802404168601352, 'mean_word_length': 4.520934206778696, 'pattern_counts': {'":': 1, '<': 4, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nonperturbative XY-model approach to strong coupling superconductivity in two and three dimensions

For an electron gas with delta-function attraction we investigate the crossover from weak- to strong-coupling supercoductivity in two and three dimensions. We derive analytic expressions for the stiffness of phase fluctuations and set up effective XY-models which serve to determine nonperturbatively the temperature of phase decoherence where superconductivity breaks down. We find the transition temperature T_c as a monotonous function of the coupling strength and carrier density both in two and three dimensions, and give analytic formulas for the merging of the temperature of phase decoherence with the temperature of pair formation in the weak-coupling limit.


Introduction
The crossover from BCS superconductors to a Bose-Einstein condensate of tightly bound fermion pairs was first studied many years ago in Refs. [1]- [3] in a model with δ-function attraction. This crossover has recently raised renewed interest [4]- [34], especially after the doping dependence of the critical temperature (T c ) and non-Fermi-liquid behavior above T c were combined in a phenomenological discussion of High-T c cuprates [20], [14]. In this paper we present a detailed study of the crossover based on a nonperturbative procedure in which derivative expansion is used to set up an effective XY-model whose well-known nonperturbative properties render information on the entire crossover regime of the above model.
Physically, the most important distinctions between conventional (BCS) and strongcoupling (Bose-Einstein) regime lies in the fact that in the former only a small fraction of the conduction electrons is paired with the superfluid density involving all pairs, whereas in the latter practically all carriers are paired below a certain temperature T * , although not condensed. The temperature has to be lowered further below some critical temperature T c < T * to make these pairs condensed and establish phase coherence, which leads to superconductive behavior.
In the model to be investigated in this paper, the crossover from BCS-type to Bosetype superconductivity will take place either by varying the coupling strength, or by decreasing the carrier density.
Since in the BCS theory pair coupling is weak, it can well be described by mean-field theory for the pair fields. In the opposite limit of strong pair binding, on the other hand, superconductivity set in as a macroscopic occupationis of q = 0 level and we are obliged to go beyond mean field level to describe such an effect. In three dimensions, crossover from BCS superconductivity to the Bose-Einstein condensation of tightly bound fermion pairs was first investigated in Ref.
[2] by summing particle-particle ladder diagrams which correspond to Gaussian fluctuations around the mean field, in the functional integral formalism it was studied in Ref. [5]. In both papers, fluctuation corrections were retained in the number equation which was solved together with mean-field gap equation. In these papers, starting from fermionic system, gas of electron pairs was maped in the strong-coupling limit to the ideal Bose gas and critical temperature was shown to becomes asymptoticaly temperature of Bose-Einstein condensation of ideal Bose gas of particles with mass 2m and density n/2, where m and n are fermion mass and density. In this aproximation T c has an arificial maximum at intermediate coupling strength thus approaching limiting value in the strong-coupling limit from above. This artifact was removed in the generalized self-consistent Green's function numerical approach in [9].
In this paper we shall study the properties of collective modes with help of the lowest gradient terms governing the Gaussian fluctuations around the mean-field solution. These are most violent in the phase of the order parameter. Phase transitions in a system with these fluctuations are well understood in two and three dimensions from extensive studies of the XY-model. By setting up an equivalent XY-model we are therefore able to describe well the onset and disappearance of superconductivity in the entire crossover regime. In this way, we shall obtain simple formulas for the critical temperature T c as a monotonously increasing function of the coupling strength and carrier density in both two and three dimensions. In the weak-coupling limit, we give simple explicit formulas which show how the temperature of the XY -model transition converges to the transition temperature in the BCS theory.
The weak-to strong-coupling superconductivity crossover in two dimensions was studied via the Kosterlitz-Thouless theory in Refs. [27], [25]. In Ref. [25] it was investigated within the same model as ours at a fixed carrier density, but only numerically 3 . The different properties of size and phase fluctuations was also exploited in Ref. [37].

Model
The Hamiltonian of our model is the typical BCS Hamiltonian in D dimensions (h = 1) where ψ σ (x) is the Fermi field operator, σ =↑, ↓ denotes the spin components, m is the effective fermionic mass, and g < 0 the strength of an attractive potential gδ(x − x ′ ). The mean-field equations for the gap parameter ∆ and the chemical potential µ are obtained in the standard way (see for example [36]): where the sum runs over all wave vectors k, N is the total number of fermions, V the volume of the system, and are the energies of single-particle excitations. The δ-function potential produces an artificial divergence and requires regularization. A BCS superconductor possesses a natural cutoff supplied by the Debye frequency ω D . For the crossover problem to be treated here this is no longer a useful quantity, since in the strong-coupling limit all fermions participate in the interaction, not only those in a thin shell of width ω D around the Fermi surface. To be applicable in this regime, we renormalize the gap equation in three dimensions with the help of the experimentally observable s-wave scattering length a s , for which the low-energy limit of the two-body scattering process gives an equally divergent expression [9]- [13]: Eliminating g from (5) and (2) we obtain a renormalized gap equation in which 1/k F a s plays the role of a dimensionless coupling constant which monotonically increases from −∞ to ∞ as the bare coupling constant |g| runs from small (BCS limit) to large values (BE limit). This equation is to be solved simultaneously with (3). These mean-field equations were first analyzed at a fixed carrier density in Refs. [5] and [7]. Here we shall first reproduce some of the obtained estimates for T * and µ.
In the BCS limit, the chemical potential µ does not differ much from the Fermi energy ǫ F , whereas with increasing interaction strength, the distribution function n k broadens and µ decreases. In the BE limit we have tightly bound pairs and nondegenerate fermions with a large negative chemical potential |µ| ≫ T . In the strong coupling limit Eq. (6) provide us an estimation for T * -characteristic temperature of the thermal pair breaking [5], whereas Eq. (3) determines µ. From Eq . (3) we obtain that in the BE limit s is the binding energy of the bound pairs. In the BE limit, we can estimate that the pseudogap sets in at which shows at strong couplings T * is indeed related to pair formation [5], [6] (which in the strong-coupling regime lies above the temperature of the onset of phase coherence [2]- [33]).
The gap in the spectrum of single-particle excitations has a special feature [3], [1], [7] when the chemical potential changes its sign. The sign change occurs at the minimum of the Bogoliubov quasiparticle energy E k where this energy defines the gap energy in the quasiparticle spectrum: Thus, for positive chemical potential, the gap energy is given directly by the gap function ∆, whereas for negative chemical potential, it is larger than that: In two dimensions, a nonzero bound state energy ǫ 0 exists for any coupling strength. The cutoff can therefore be eliminated by subtracting from the two-dimensional zerotemperature gap equation [10]- [12] After performing the elementary integrals, one finds: In the next sectioins we will work at finite tempetrature, in doing so, we do not fix the carrier density but assume the presence of a reservoir which provides us with a temperatureindependent chemical potential µ = µ(1/k F a s ; T = 0). Such a fixed µ will be most convenient for deriving simple analytic results for the finite-temperature behavior of the system. In this fixed µ model carrier density becomes temperature dependent. 4 For experimental measurements of µ see [38] 3

Phase fluctuations in Two Dimensions and Kosterlitz-Thouless Transition
In this section we make use of derivative expansion to determine the relevant stiffness parameter for the study of phase fluctuations, which in two dimensions determines the temperature of the Kosterlitz-Thouless transition. In two-dimensional system, the phase fluctuations are most violent causing the strongest modifications of the mean-field properties. The Coleman-Mermin-Wagner-Hohenberg theorem [39] forbids the existence of a strict long-range order, but there is quasi-long-range order manifesting itself in a power behavior of the correlation functions at all temperatures below T KT . We shall neglect the coupling to the magnetic vector potential throughout the upcoming discussion, so that the phase coherence below T c can be of long or quasi-long range, unspoiled by the Meissner effect which would reduce the range to a finite penetration depth. The effective Hamiltonian from which we deduce the stiffness of the phase fluctuations was derived in [42], [25]. In this Section we summarize a few important aspects of it, with a reminder of its derivation given below. Writing the spacetime-dependent order parameter as ∆(x)e iθ(x) , where x denotes the four-vector x = (τ, x) formed from imaginary time and position vector, the partition function may be written as a functional integral [42,40,41] Z(µ, T ) = ∆ D∆ Dθ exp [−βΩ(µ, T, ∆(x), ∂θ(x))], where βΩ(µ, T, ∆(x), ∂θ(x)) = 1 g is the one-loop effective action, containing the inverse Green function of the fermions in the collective pair field Here τ 1 , τ 3 are the usual Pauli matrices, and G 0 = G| µ,∆,θ=0 is added for regularization.
Let us now assume that phase gradients are small. Then Ω(µ, T, ∆(x), ∂θ(x)) can be approximated as follows: Ω(µ, ∆(x), ∂θ(x)) ≃ Ω kin (µ, T, ∆, ∂θ(x)) + Ω pot (µ, T, ∆), with the "kinetic" term (see [42]) (16) and the "potential" term The latter coincides with the mean-field energy, determining the modulus of ∆(µ, T ) and thus the stiffness of phase fluctuations. The kinetic part Ω kin contains gradient terms whose size is determined by the modulus of ∆(µ, T ). Given the stiffness, one may immediately set up an equivalent XY-model. Both Ω kin and Ω pot are expressed in terms of the Green function of the fermions, which solves the equation and The gradient expansion that we use to determine stiffness was first made in Ref. [42] at zero temperature. In Ref. [25], the kinetic term Ω kin was calculated in two dimensions at finite temperature for arbitrary chemical potential retaining terms with n = 1, 2 in the expansion (16). The result is where J(µ, T, ∆) is the stiffness coefficient whose explicit form is The other coefficents are: and n(µ, T, ∆) is the density of fermions (3) which varies with temperature in our model. At the temperature T * where the modulus of ∆ vanishes, also the stiffness disappears.
We are now ready to specify the effective XY-model governing the phase fluctuations. The model Hamiltonian corresponding to the gradient term in θ(x) is [43], [8]: In contrast to the standard XY-model, the stiffness parameter is not a constant but depends on temperature via the solution of gap and number equations (2) and (3). Clearly, Kosterlitz-Thouless transition always take place below T * . For vortices of a high fugacity, the temperature of the phase transition is determined by the well-known formula [44] which follows from the divergence of the average square size of a vortex-antivortex pair.
Since these attract each other by a Coulomb potential v(r) = 2πJ log(r/r 0 ), the average square distance is which diverges indeed at the temperature (24). In our case T KT should be determined self-consistently: From (21), (3) and (24) it is easily seen that T KT indeed tends to zero when the pair attraction vanishes in which case ∆(T = 0) = 0. In general, the behavior of T KT for strong and weak couplings is found by the following considerations. We observe that the particle number n does not vary appreciably in these limits with temperature in the range 0 < T < T * , so that weak-coupling estimates for T KT derived within the model with temperature-independent chemical potential (i.e., when the system is coupled to a large reservoir ) practically coincide with those derived from a fixed fermion density. Further it is immediately realized that in the weak-coupling limit ∆(T KT , µ)/T KT is a small parameter. At zero coupling, the stiffness J(µ, T KT , ∆(µ, T KT )) vanishes identically, such that an estimate of J at weak couplings requires calculating a lowest-order correction to the second term of eq.(21) proportional to ∆(T KT , µ)/T KT . Thus weak-coupling expression for stiffness reads: Equating this with the stiffness in (24) we obtain the weak-coupling equation for T KT : where ǫ F = (π/m)n is the Fermi energy of free fermions. It is useful to introduce reduced dimensionless temperaturesT KT ≡ T KT /ǫ F andT * = T * /ǫ F which are small in the weakcoupling limit. Then we rewrite Eq. (28) as For smallT * we may expandT This equation shows explicitly how for decreasing coupling strength T KT merges with T * . For weak coupling strengths, T KT behaves like The merging of the two temperatures in the weak-coupling regime is displayed in Fig. 1. Consider now the opposite limit of strong couplings. There we see from Eqs. (26), (2) (3), and (21) for T KT , n(T, µ), and ∆(T, µ) that T KT tends to a constant value. We can observe that in the strong-coupling limit ∆(T KT ) is always situated close to the zerotemperature value of ∆(T KT , µ) ≈ ∆(T = 0, µ). Taking this into the account we derive an estimate for the second term in (21), thus obtaining the strong-coupling equation for T KT : With the approximation ∆(T KT , µ) ≈ ∆(T = 0, µ) we find that the first term in the exponent tends in the strong-coupling limit to a constant, ∆ 2 (T KT , µ)/2µT KT → −4, whereas the first term in the brackets tends to −∞, so that Eq. (32) has the limiting form Thus for increasing coupling strength, the phase-decoherence temperature T KT tends very quickly towards a constant: In this limit we know from Eq. (3) that the difference in the carrier density at zero temperature, n(T = 0), becomes equal to n(T = T KT ), so that our limiting result coincides with that obtained in the "fixed carrier density model": where we have inserted ǫ F (n) = (π/m)n for the Fermi energy of free fermions at the carrier density n 0 = n(T = 0). From the above asymtotical formulas for weak-and strong-coupling limits we see that the temperature of the Kosterlitz-Thouless transition is a monotonous function of coupling strength and carrier density. The crossover takes place in a very narrow region µ/∆(0) ∈ (−1, 1). It is also observed in the behavior of three-dimensional condensation temperature T c of the gas of tightly bound, almost free composite bosons. In Refs. [2], [5] which include only quadratic fluctuations around the mean field (corresponding to ladder diagrams), T c was shown to tend to a constant free Bose gas value T c = [n/2ζ(3/2)] 2/3 π/m, with no dependence on the internal structure of the boson.
Here we find a similar result in two dimensions, where T KT tends to a constant depending only on the mass 2m and the density n/2 of the pairs. No dependence on the coupling strength is left. The only difference with respect to the three-dimensional case is that here the transition temperature T c = T KT is linear in the carrier density n, while growing like n 2/3 in three dimensions. Our limiting result (35) agrees with Ref. [27] and [25]. There exists a corresponding equation for the temperature T * in the strong-coupling limit ǫ 0 ≫ ǫ F :

Phase Fluctuations in Three Dimensions
In this section we discuss in a completely analogous way the fluctuations in three dimensions, where stiffness is, for small temperatures where ∆(T ) is close to ∆(0): governing the phase fluctuations via an effective XY-model The temperature of the phase transition in this model can reasonably be estimated using mean-field methods for the lattice 3D XY-Model [8]: is the lattice spacing of the theory [8] where n b is number of pairs. In the weak-coupling limit, the stiffness coefficent may be derived with the help of Gorkov's well-known method (setting T c ≈ T * ): This is precisely the coefficient of the gradient term in the Ginzburg-Landau expansion, In the weak-coupling limit the two temperatures merge according to the formula: which contains a larger power ofT * in the second term as well as a smaller prefactor α = (2π 2 ) 2/3 /2 ≈ 3.65, as compared with the two-dimensional separation formula (30). The merging behavior is displayed in Fig. (2). In the strong-coupling limit of the theory where we have tightly bound composite bosons, the phase stiffness tends asymptotically to: It obviously tends in this limit quickly to An estimate for the critical temperature, obtained via the mean-field treatment of the 3D XY-model on the lattice reads in this limit: This quickly tends from below to the value: This result is very close to the temperature of the condensation of bosons of mass 2m and density n/2, which, as it was discussed in the introduction was obtained including the effect of Gaussian fluctuations into the mean-field equation for the particle number [2,5] yielding 5 T Bosons c = [n/2ζ(3/2)] 2/3 π/m = 0.218ǫ F .

Conclusion
We have studied the crossover from BCS-type to Bose-type superconductivity. For this purpose we have used the gradient expansion of the effective energy functional to set up an equivalent XY-model which allows us to investigate the onset of long-range order in the phase fluctuations. In two dimensions, we have given a simple analytic expression which shows how the resulting Kosterlitz-Thouless temperature T KT at which quasi-long-range order sets in moves towards the pair-binding temperature T * , and merges with it in the weak-coupling limit. A similar expression was found in three dimensions. In the strong coupling limit we find that critical temperature tends in both two and three dimensions to a constant value as the chemical potential changes it sign. Let us finally remark that the separation of T * and T c has an analogy in the ferroelectrics and magnets which also contain two separate characteristic temperatures, for example in the latter case-the Stoner-and the Curie-temperature. It also can be studied more precisely in a simple field theoretic model in 2+ǫ dimensions with an O(n) symmetry for large n. In such a model, the existence of two small paramters ǫ and 1/n has permitted us recently to prove the existence of two transitions, and to exhibit clearly their different physical origins [45].