Shift of the critical temperature in superconductors: a self-consistent approach

Within the Ginzburg-Landau functional framework for the superconducting transition, we analyze the fluctuation-driven shift of the critical temperature. In addition to the order parameter fluctuations, we also take into account the fluctuations of the vector potential above its vacuum. We detail the approximation scheme to include the fluctuating fields contribution, based on the Hartree-Fock-Bogoliubov-Popov framework. We give explicit results for d = 2 and d = 3 spatial dimensions, in terms of easily accessible experimental parameters such as the Ginzburg-Levanyuk number Gi(d), which is related to the width of the critical region where fluctuations cannot be neglected, and the Ginzburg-Landau parameter κ, defined as the ratio between the magnetic penetration length and the coherence one.

= d 2 and = d 3 dimensions, where the shift turns out to depend only on the Ginzburg-Levanyuk number Gi d ( ) and the parameter κ, the latter being the ratio between the magnetic penetration length and the coherence one. Our analytical formulas for the shift of the critical temperature, directly obtained from the Ginzburg-Landau functional, are a generalization of familiar ones 20 , which take into account only the fluctuations of the order parameter.

Results
The Ginzburg-Landau formulation: general features. Concerning the normal-to-superconductor transition, close to the critical temperature, the free energy of a single-band superconductor can be split into s n    with n  the contribution of the disordered (i.e. normal) component, while  s models the contribution due to the emergence of an ordered phase (i.e. superconductor) characterized by an order parameter ψ r ( ) acquiring a non-zero value below a certain critical temperature to determine. The great intuition of Landau and Ginzburg consisted in writing down the latter in terms of few simple terms based on the symmetries to which the theory obeys. In our case, this implies 20 (2) is the gauge-invariant derivative coupling the order parameter ψ r ( ) with the vector potential A r ( ), with e* being the effective charge. In the equations above, the index µ labels the component of the vector A. One has also to recall that the order parameter ψ has to be assumed as a complex field if we aim to describe the normal-to-superconducting transition.
Concerning the couplings of the theory described by  s in Eq. (2), b and γ can be taken as positive constants, while the coefficient of the ψ 2 -term contains the temperature dependence. Simple considerations lead to the well-known conclusion that this dependence is linear, namely with α > 0 and k B the Boltzmann constant. In the equation above, T c0 is the mean-field critical temperature. It has to be remarked that Eq. (2) holds in absence of an external magnetic field. In order to model this situation,  ψ A [ ; ] s must be modified adding a term proportional to ∇ ∧ ⋅ A H ( ) . However, in this paper we limit our analysis to case of no applied external field.
It is immediate to show that, with the ansatz ψ ψ = r ( ) 0 and = A r 0 ( ) , one can find the usual saddle-point solution Thus, the order parameter acquires a non-zero value when a T ( ) changes its sign. Within the mean-field scheme, the phase transition then occurs at a temperature given by the solution of This simple approach lacks every information about the fluctuations of the order parameter and the vector potential. For conventional superconductors, the critical region where fluctuations are crucial is very small and cannot be observed.
However, with the discovery of novel materials displaying exotic superconductivity, it appears clear that deviations from the mean-field picture have to be explored very carefully.
Including the fluctuations: the HFBP scheme. The saddle-point equation in the HFBP approximation. Differently from the case of neutral superfluids, the Ginzburg-Landau functional for a superconducting system, as given by Eq. (2), depends on two different fields because of the minimal coupling with the electromagnetic field.
Thus, the saddle-point configuration of the system has to be determined by searching for the stationary trajectories of  ψ A [ , ] s . In other words, one has to solve the system s s www.nature.com/scientificreports www.nature.com/scientificreports/ where δ/δ(•) has to be intended in the sense of first variation. The equation resulting from the variation with respect to ψ ⁎ reads In both Eqs. (8) and (9) the spatial dependence of both fields is left intended. Obviously, assuming the ansatz ψ ψ = r ( ) 0 and = A 0 as in the previous section, the Ginzburg-Landau equation are solved by Eq. (5). It is then clear that a saddle-point configuration exists for a uniform order parameter and in absence of a vector potential.
In order to encode the thermal fluctuations in a saddle-point scheme, let us then split the order parameter where ψ 0 is a constant which can be assumed as real but it is not necessarily given by ψ 0 (mf) in Eq. (5). On the other hand, η r ( ) is the space dependent fluctuation field. The crucial feature concerning ψ r ( ) as defined above is that 0 and, as an immediate consequence, The thermal average … has to be intended as performed over a proper statistical ensemble where a global gauge symmetry can be spontaneously broken.
For the vector potential A r ( ), we only consider fluctuations A r ( ) above its vacuum 0, namely Now, we proceed by replacing Eq. (10) in Eq. (8), such that We are interested in how fluctuations modify the uniform background ψ 0 within the broken symmetry phase, i.e. for < T T c . We remark that, according to this framework, the critical temperature ≠ T T c c 0 , since Eq. (6) does not take into account the role of fluctuations.
In order to derive an equation for ψ 0 including a contribution due to fluctuations, both of the order parameter and the vector potential, we take the thermal average of Eq. (14). Because of Eq. (12), linear terms in η and η ⁎ are erased by default. One is then left with which still appears rather complicated. Up to this point, we still have not fixed the gauge for the vector potential. It is a well-known fact that the most natural choice is the Coulomb (or transverse) gauge, where Actually, besides the convenience matter, this fixing has profound physical consequences. Indeed, it has been shown that only with Eq. (16) the order parameter correlations ψ ψ r ( ) (0) acquires a long-ranged character (see, for instance 5 ). Concerning Eq. (15), let us notice that, through a Fourier transformation, the Coulomb gauge is equivalently given by , with q a wavevector in the reciprocal space. This implies that η ⋅ ∇ = A 0. Moving further, we simplify Eq. (15) by means of the following approximation scheme. First, we neglect the three-field correlations, namely η η  0 2 and η  A 0 2 . By drawing an analogy with the analysis of bosonic gases at finite temperatures, this corresponds to the Hartree-Fock-Bogoliubov (HFB) scheme 17 . Moreover, we also discard the anomalous average, namely η η =  ⁎ ( ) 0 2 2 , according to the so-called Popov approximation of the HFB framework (HFBP in the following).
Thus, Eq. (15) finally reads A whose solution, for < T T c , is given by www.nature.com/scientificreports www.nature.com/scientificreports/ From the equation is then immediate to derive the generalization of Eq. (6) for the critical temperature, i.e.
where … c implies that the average has to be computed at the critical point.
In contrast with Eq. (6), Eq. (19) takes into account the presence of fluctuations both in the order parameter and the vector potential, through the averages η 2 and A 2 .
Averages in the HFBP scheme. The shift of the critical temperature as in Eq. (19) requires the calculation of η 2 and A 2 at the criticality. As detailed in the Methods section, they can be computed by taking the GL equations for the fluctuating fields η r ( ) and A r ( ) as starting point (cfr. Eqs. (50) and (54)). From there, as outlined below, one can infer the corresponding Gaussian functional (i.e. a free energy, similarly to Eq. (2)) driving the thermodynamic properties of fluctuations. Indeed both Eqs. (50) and (54) in Methods are the first variation of a Gaussian functional. By working slightly above the critical temperature (i.e.
which is the trajectory stationarizing the functional with V being the large d-dimensional volume enclosing the system. Similarly, from Eq. (54), one can infer the corresponding functional for A, i.e.
are related to their corresponding partition function, from which it is usually easy to compute average values and correlations, since both of them are (functional) Gaussian integrals.
The presence of differential operators suggests that the calculation is easier in the Fourier space. Thus, with Equations (21) and (22) transform as, respectively, are Gaussian, it is immediate to infer that At the criticality Eq. (19) holds, therefore the equation above is simplified into Let us note that this result is the same one can derive in absence of the minimal coupling with the electromagnetic field (i.e. = A 0). The crucial point is the fact that, on the contrary, A c 2 is affected by the fluctuation of the order parameter. Indeed, from Eq. (26), one gets www.nature.com/scientificreports www.nature.com/scientificreports/ It worth to remember, at this point, the main results we have obtained by means of the HFBP approximation scheme. First, we have derived the equation describing the shift of the critical temperature (compared to the usual saddle-point result in Eq. (6)). In order to actually solve Eq. (19), we also need the average values (at the criticality) η c 2 and A c 2 , respectively in Eqs. (28) and (29). In the following, we are going to consider the continuum limit, namely (in spherical coordinates) is the whole solid d-dimensional solid angle. In the equation above, we have introduced both an ultraviolet and an infrared cutoff to keep eventual divergences under control.
In the following section, we consider the case of = d 2 and = d 3 spatial dimensions. An extremely interesting problem is represented by the dimensional crossover, i.e. the analysis of a thin film but with a finite thickness δ. This physical realization has been investigated in presence of a disordered environment [21][22][23] , reading an additional (logarithmic) shift of the critical temperature depending on δ and a properly defined diffusion coefficient.
Fluctuation-driven critical temperature shift. The case d = 2. In = d 2, by taking the continuum limit as in Eq. (30), Eqs. (28) and (29) easily lead us to According to 20 , a reasonable choice for the UV cutoff is Concerning the infrared cutoff q 0 , we define it in terms of the Ginzburg-Levanyuk number through The definition of the Ginzburg-Levanyuk number Gi d ( ) strongly depends on the system dimensionality. Again, according to 20 The UV cutoff is naturally defined by Eq. (33), which specifies the minimal size of spatial fluctuations. The choice for the infrared cutoff in Eq. (34) is less obvious and has to rely upon a renormalization-group (RG) argument 15 . Within the Wilson's standard framework of momentum-shell integration 4,5 , the flow equation for cutoff-dependent λ a T ( ) can be simplified by assuming that the parameter b does not flow (i.e. it is equal to its bare value). The cutoff λ, separating the slow from fast modes in Wilson's approach, has an upper (i.e. ultraviolet) limit equal to λ ξ → Λ = ∞ 1/ c with ξ c as in Eq. (33). In = d 2, this approximation the equation for λ a T ( ) reads a logarithmic solution diverging when λ → 0, naturally introducing the infrared cutoff q 0 as in Eq. (34). By solving simultaneously the RG equations for λ a and λ b the divergence disappears but, remarkably, it has been shown that the logarithmic approximation outlined above works really well for  Gi 1/2 (2) , which is exactly the range of values explored in this paper (cfr. Fig. 1).
Therefore, by following the cutoff prescriptions outlined above, Eqs. (31) and (32)  www.nature.com/scientificreports www.nature.com/scientificreports/ The parameter κ is the usual ratio between the magnetic penetration length λ T ( ) and the coherence length ξ T ( ), i.e. κ λ ξ = T T ( )/ ( ). We recall that κ is a crucial quantity in the GL theory for superconductors, since type-I superconductors all have κ < 1/ 2 . On the contrary, the condition κ > 1/ 2 characterizes type-II superconducting materials. By assuming It has to be remarked that the equation above still holds for = d 3. By replacing Eqs. (36) and (37) in Eq. (19), the shift of the critical temperature in = d 2, compared to the usual saddle-point picture, is given by (2) 2 Gi 1 Gi (2) 2 (2) We stress that only two parameters, the Ginzburg-Levanyuk number Gi (2) and the Ginzburg-Landau ratio κ, drive the shift of the critical temperature in two spatial dimensions. Equation (40) is one of the main results of the paper. The first addend of this equation, already known in literature 15,20 , takes into account the thermal fluctuations of the order parameter. The second addend takes instead into account the thermal fluctuations of the electromagnetic vector potential around its vacuum. In Fig. 1 we report the behaviour of the critical temperature shift given by Eq. (40) ) as a function of Gi (2) for three different values of κ. Let us remark that the displacement from the mean-field prediction T c0 is progressively reduced by moving towards to the type-II regime.
It is important to recall that two-dimensional physical systems require from us to also consider the eventual occurring of the Berezinskii-Kosterlitz-Thouless (BKT) transition [24][25][26] . Its peculiar behaviour can be understood in terms of phase fluctuations of the order parameter. Indeed, a complex field can be naturally characterized in terms of two real fields by means of the phase-amplitude representation In the equation above, Ψ 0 can be taken as the uniform solution of the GL equations. By neglecting the crucial role played by the phase field φ r ( ), the amplitude acquires a non-zero value for < T T c , with T c being the shifted (compared to the mean-field result) critical temperature in Eq. (40). On the other hand, the phase field φ r ( ) is obviously defined on a compact support. As a consequence, the system can display nontrivial topological excitations in form of quantized vortices.
In Methods we outline a procedure providing, in first approximation, the additional BKT shift 15,20 by means of the Nelson-Kosterlitz criterion 27 . According to this approach, T BKT is shifted with respect to the unrenormalized contribution in Eq. (40) by It is important to remark that, while the Eq. (42) has a very simple appearance, the fundamental input is the critical temperature computed in Eq. (40).

The case d = 3.
In the case = d 3, the integration over q as prescribed by Eq. (30) of Eqs. (28) and (29) gives us back No infrared divergence arises by performing the integration leading to the equations above, so we have safely taken the limit → q 0 0 . We also approximate the π … → arctan( ) /2 since it provides only a subleading contribution, compared to the coefficient in front of it. However, an ultraviolet divergence are still present: as for = d 2, we assume ξ Λ  1/ c , with ξ c given by Eq. (33), holding also in three spatial dimensions. The Ginzburg-Levanyuk number, on the contrary, reads 20 Consequently, is remarkably reduced. Again, by assuming the minimal coupling with the electromagnetic field, the Ginzburg-Levanyuk number Gi (3) and the Ginzburg-Landau ratio κ are all one needs to characterize the critical temperature shift from the usual mean-field scheme. Equation (48) is the another important result of our paper. The first addend of this equation is already known in literature 4,20 while the second addend, related to the fluctuations of the vector potential, is instead a novel result.

Discussion
In this paper, we have presented a self-consistent approach to include the fluctuations of order parameter and vector potential in an improved saddle-point equation. As a consequence, it has been possible to derive the fluctuation-driven shift of T c both in = d 3 and = d 2. Remarkably, this displacement from the usual mean-field results only depends from two combinations of the GL coupling constants: the Ginzburg-Levanyuk number Gi d ( ) , related to the width of the critical region, and the ratio κ λ ξ = T T ( )/ ( ), dividing type-I from type-II superconductors.
In presence of a wide critical region, namely a high value of Gi d ( ) , it is crucial to properly consider not only the role of fluctuations on the various thermodynamic functions, but also the eventual strategies apt to control them. This is not a purely academical question; on the contrary it can have relevant technological applications, since this kind of materials are the ones displaying pairing mechanisms leading to high-temperature superconductivity, such as electron-hole superfluidity [28][29][30] . Indeed, as already mentioned, an inter-band Josephson-like coupling has been proved to contain the detrimental effect of fluctuations and preserve an optimal superconductivity regime 15 , also in reduced dimensionality, where coherent behaviour is otherwise greatly suppressed 31 .
By adopting this point of view, in this paper we have considered a single-band structure but in presence of a minimal coupling to the electromagnetic field. As a consequence, it turns out the fluctuations of the vector potential A that the critical temperature acquires an additional shift depending from κ (cfr. Eqs. (40) and (48)). In other words, it depends on how much the material sample can be penetrated by the magnetic fields. A natural extension of this analysis is involving the presence of at least another band with a proper coupling, together with the possibility to introduce the particle statistics moving to a quantum framework which is closer, for instance, to the phenomenology of the BCS-BEC crossover 13,32 .

Methods
The GL equations for the fluctuating fields. In this section we detail the derivation of the GL equations for the fluctuations of the order parameter and the vector potential, namely η r ( ) and A r ( ). These equations then serve as the starting point to engineer a (Gaussian) functional accounting for the statistical properties of the fluctuating fields. This task can be fulfilled by looking back at Eq. (14). Here, the HFB scheme prescribes that all the terms more than linear in the fluctuating fields are decoupled with a couple-by-couple average. For instance, η η η η η η In addition, the Popov prescription imposes η  0 2 . We also assume that the order parameter and the vector potential fluctuates independently, so where the last equality is due to Eq. (12). This series of considerations leads us to the GL equation for the order parameter fluctuations, The corresponding equation for A can be easily derived by following the same steps detailed above, together with the Coulomb gauge. More in detail, by replacing Eqs. (10) and (13) in (9), together with the assumption in Eq. (49), this procedure transforms the left-hand-side (LHS) as Concerning the right-hand-side of Eq. (9), additional comments are in order. First, let us note that, because of Eq. (10), terms linear in η ∇ and η ∇ arise. They are linear in A in the Ginzburg-Landau functional whose Eq. (9) is a first variation. Because of the Coulomb gauge, terms of the kind η ⋅ ∇ A give a null contribution to the functional and consequently should not appear in the corresponding GL equation. The remaining term is simply Since η is complex, it can be equivalently rephrased as The last approximation we perform, together with the HFBP scheme and Eq. (49), is the statistical independence of the real and imaginary part of η. It can be shown that, up to the Gaussian order in the fluctuating fields, www.nature.com/scientificreports www.nature.com/scientificreports/ within the usual perturbative scheme 4 , this assumption holds. In the end, no contribution comes from the right-hand-side of Eq. (9), therefore one is left with the following equation for A In the end, within the HFBP we have derived three GL equations, one for the homogeneous background ψ 0 , Eq. (17), the other two for η and A, respectively Eqs. (50) and (54).
Including the BKT contribution. As a starting point, we take the field ψ r ( ) in its phase-amplitude representation given by Eq. (41). In order to compute the additional BKT shift, as a first approximation 15,20 one can consider the following phase-only functional The phase stiffness J T ( ) is defined in terms of the original GL parameters, i.e.
The key point in the Eqs. (55) and (56) consists in the benchmark temperature in J T ( ): it is no more T c0 as for a T ( ) in Eq. (4), but the fluctuation-shifted T c in Eq. (40). As already clear from the seminal papers [24][25][26] , the major step forward in the BKT understanding was the fact that it is actually a topological transition. Indeed, the compactness of the phase field φ r ( ) implies the possibility for the system to display excited configurations (compared to the uniform one) which cannot be reached by continuously deforming the order parameter 4 . In = d 2 these excitations are simply vortices and antivortices, depending on the sign of their (topological) charge ν ∈ Z, defined in terms of phase winding 2 Now, a standard approach consists in drawing an electrostatic analogy and treating vortices and antivortices as a Coulomb gas moving in a uniform and neutral background 4,5,33 . The BKT critical point then divides two different phase of this vortex gas: for < T T BKT , bound states of vortex-antivortex are predominant, while they are unbound above the critical temperature, destroying every global coherence property, such as superfluidity.
Remarkably, this appears in a discontinuous way. Indeed, the transition displays a universal jump in the phase stiffness (and, consequently, in the superfluid density). This peculiar feature of the transition can be used to compute the critical temperature, according to the so-called Nelson-Kosterlitz criterion 26 reading Eq. (42) in the main text.