Abstract
Flat bands of zeroenergy states at the edges of quantum materials have a topological origin. However, their presence is energetically unfavorable. If there is a mechanism to shift the band to finite energies, a phase transition can occur. Here we study hightemperature superconductors hosting flat bands of midgap Andreev surface states. In a secondorder phase transition at roughly a fifth of the superconducting transition temperature, timereversal symmetry and continuous translational symmetry along the edge are spontaneously broken. In an external magnetic field, only translational symmetry is broken. We identify the order parameter as the superfluid momentum p_{s}, that forms a planar vector field with defects, including edge sources and sinks. The critical points of the vector field satisfy a generalized PoincaréHopf theorem, relating the sum of Poincaré indices to the Euler characteristic of the system.
Introduction
Superconducting devices are often experimentally realized as thinfilm circuits or hybrid structures operating in the mesoscopic regime^{1,2,3,4}. At this length scale, where the size of the circuit elements becomes comparable with the superconducting coherence length, the nature of the superconducting state may be dictated by various finitesize or surface/interface effects^{5}. This holds true in particular for unconventional superconductors, such as the hightemperature superconductors with an order parameter of \(d_{x^2  y^2}\) symmetry that changes the sign around the Fermi surface. Scattering at surfaces, or defects, leads to substantial pair breaking and formation of Andreev states with energies within the superconducting gap^{6,7}. Today, the material control of hightemperature superconducting films is sufficiently good that many advanced superconducting devices can work at elevated temperatures^{8,9}. This raises the question how the specific surface physics of dwave superconductors influences devices.
From a theory point of view, the physics at specular pairbreaking surfaces of dwave superconductors is rich and interesting. The reason is the formation of zeroenergy (midgap) Andreev states due to the sign change of the dwave order parameter for quasiparticles scattered at the surface^{6,7,10}. In modern terms, there is a flat band of spindegenerate zeroenergy surface states as a function of the parallel component of the momentum, p_{}, which is a good quantum number for a specular surface. A topological invariant has been identified^{11,12}, that guarantees the flat band for a timereversal symmetric superconducting order parameter and p_{} conserved. However, the large spectral weight of these states exactly at zero energy (i.e., at the Fermi energy) is energetically unfavorable. Different scenarios have been proposed, within which there is a lowtemperature instability and a phase transition into a timereversal symmetrybroken phase where the flat band is split to finite energies, thus lowering the free energy of the system. One scenario is the presence of a subdominant pairing interaction and appearance of another order parameter component π/2 out of phase with the dominant one^{13,14,15}, for instance a subdominant swave resulting in an order parameter combination Δ_{d} + iΔ_{s}. The phase transition is driven by a split of the flat band of Andreev states to ±Δ_{s}. The split Andreev states carry current along the surface, which results in a magnetic field that is screened from the bulk. In a second scenario, exchange interactions drive a ferromagnetic transition at the edge where the flat Andreev band is instead spin split^{16,17}. A third scenario involves spontaneous appearance of supercurrents^{18,19,20} that Doppler shifts the Andreev states and thereby lowers the free energy. Here the electrons couple to the electromagnetic gauge field A(R), and this mechanism was first considered theoretically for a translationally invariant edge. In this case, the transition is a result of the interplay of weakly Doppler shifted surface bound states, decaying away from the surface on the scale of the superconducting coherence length ξ_{0}, and weak diamagnetic screening currents, decaying on the scale of the penetration depth λ. The resulting transition temperature is very low, of order T^{*} ~ (ξ_{0}/λ)T_{c}, where T_{c} is the dwave superconducting transition temperature. Later, the transition temperature was shown to be enhanced in a film geometry^{21,22,23,24,25} where two parallel pairbreaking edges are separated by a distance of the order of a few coherence lengths. The suppression of the order parameter between the pairbreaking edges can be viewed as an effective Zeeman field that splits the Andreev states and enhances the transition temperature. The mechanism does not involve subdominant channels or coupling to magnetic field, but depends on film thickness D, and the transition temperature decays rapidly with increasing thickness as T^{*} ~ (ξ_{0}/D)T_{c}.
In this paper, we consider a peculiar scenario^{26,27} where spontaneous supercurrents also break translational symmetry along the edge. This scenario too does not rely on any additional interaction term in the Hamiltonian. Instead, as we will discuss below, it relies on the development of a texture in the gradient of the dwave order parameter phase χ, or more precisely in the gauge invariant superfluid momentum
where ħ is Planck’s constant, e the charge of the electron, and c the speed of light. This superfluid momentum spontaneously takes the form of a planar vector field with a chain of sources and sinks along the boundary and saddle points in the interior, see Fig. 1. The vector field is illustrated by arrows showing the local unit vectors \(\hat {\bf{p}}_{\mathrm{s}}({\bf{R}})\), while the color scale illustrates the magnitude p_{s}(R). An interior critical point at R_{0} is characterized by a Poincaré index defined as^{28,29}
where θ = arctan(p_{sy}/p_{sx}) is the angle of \(\hat {\bf{p}}_{\mathrm{s}}({\bf{R}})\) on the Jordan curve Γ encircling R_{0}. Internal sources and sinks have I = +1, while saddle points have I = −1. Although the special points on the boundary have to be treated with care, there is a sum rule (Eq. (3)) for the Poincaré indices, as we will discuss below. We identify the p_{s} vector field as the order parameter of the symmetrybroken phase, motivated by the fact that the free energy is lowered by a large split of the flat band of Andreev states by a Doppler shift v_{F} · p_{s}, where v_{F} is the Fermi velocity. This free energy gain is maximized by maximizing the magnitude of p_{s}, which is achieved by the peculiar vector field in Fig. 1. The balance of the Doppler shift gain and the energy cost in setting up the vector field with critical points where^{30} ∇ × p_{s} ≠ 0 and the splay patterns between them leads to a high T^{*} ≈ 0.18T_{c}. The inhomogeneous vector field induces a chain of loopcurrents at the edge circulating clockwise and anticlockwise. The induced magnetic fluxes of each loop are a fraction of the flux quantum and form a chain of fluxes with alternating signs along the edge. Here we clarify the structure of the order parameter of the symmetrybroken phase, i.e., p_{s}, and study the thermodynamics of this phase under the influence of an external magnetic field, explicitly breaking timereversal symmetry.
Results
Translational symmetry breaking in a magnetic field
In Fig. 2, we show the influence of a rather weak external magnetic field, B = 0.5B_{g1}, applied to the dwave superconducting grain with pairbreaking edges for varying temperature near the phase transition temperature T^{*}. The scale B_{g1} = Φ_{0}/\({\cal A}\) corresponds to one flux quantum threading the grain area \({\cal A}\), see the Methods section. The left and right columns show the currents and the magnetic field densities, respectively, induced in response to the applied field. To be concrete, we discuss a few selected sets of model parameters, as listed in Table 1. First, for T > T^{*} (parameter set I), the expected diamagnetic response of the condensate in the inner part of the grain is present, see Fig. 2a, e. On the other hand, midgap quasiparticle Andreev surface states respond paramagnetically. This situation is well established theoretically and experimentally through measurements of the competition between the diamagnetic and paramagnetic responses seen as a lowtemperature upturn in the penetration depth^{18,31,32}. Upon lowering the temperature to T ≳ T^{*} (parameter set II), see Fig. 2b, f, the paramagnetic response at the edge becomes locally suppressed and enhanced, forming a sequence of local minima and maxima in the induced currents and fields. The bulk response is, on the other hand, relatively unaffected. Finally, as T < T^{*} (parameter set III), see Fig. 2c, g, the regions of minimum current turns into regions with reversed currents. The resulting loop currents with clockwise and anticlockwise circulations induce magnetic fluxes along the surface with opposite signs between neighboring fluxes. The situation for T < T^{*} in an external magnetic field can be compared with the one in zero magnetic field^{26} displayed in Fig. 2d, h. In the presence of the magnetic field, there is an imbalance between positive and negative fluxes, while in zero external magnetic field, the total induced flux integrated over the grain area is zero.
Topology of the superfluid momentum vector field
Let us quantify the symmetrybroken phase in a magnetic field by plotting the superfluid momentum defined in Eq. (1), see Fig. 3. For T ≳ T^{*} (parameter set II), the amplitude of p_{s} varies along the edge (coordinate x), see Fig. 3a, reflecting the varying paramagnetic response in Fig. 2b, f. For T < T^{*} (parameter set III), the sources and sinks have appeared pairwise together with a saddle point, see Fig. 3b. The left defects in the figure are not well developed because of the proximity to the corner. Finally, in Fig. 3c, we show the vector field at a lower temperature when the chain of sources, sinks, and saddle points are well established and the magnitude of p_{s} is large, much larger than in the interior part of the grain still experiencing diamagnetism. In a magnetic field, the vector field far from the surface has a preferred direction reflecting the diamagnetic response of the interior grain. This shifts the sources and sinks along the surface, as compared with the regular chain for zero field in Fig. 1, and moves the saddle points to the surface region.
The superflow pattern of sources, sinks, and saddle points satisfy a certain sum rule related to the topology of the sample. This relation also ties the special points of the p_{s} field on the edge of the sample with critical points in its bulk. The generalized PoincaréHopf theorem for manifolds with boundaries^{33,34} connects the properties of a vector field v inside a manifold M, and on its boundary ∂M, with the Euler characteristic of the manifold χ(M). Using the formulation presented in ref. ^{34}, we write
where Ind_{ M }(v) is the total Poincaré index of critical points of the field v internal to M, \({\mathrm{Ind}}_{\partial _ \pm M}(v_{})\) is the total Poincaré index of critical points of the tangent vector v_{}  ∂M on the boundary. The theorem applies when the boundary ∂M does not go through any critical points of v. Boundary indices where field points inside (∂_{−}M) / outside (∂_{+}M) of M, come with positive/negative signs. In Supplementary Note 1, we demonstrate in detail how the sum rule works for the vector field in Fig. 1, redrawn as a streamline plot in Supplementary Fig. 2. We also provide other examples of grain geometries in Supplementary Figs. 3–10. We utilize the sum rule as a tool to verify that the calculations are correct.
In a magnetic field, as in Fig. 3, a motif with one edge source, one edge sink, and one saddle point annihilate at T^{*}. In the same fashion, increasing the magnetic field strength, the motif gets smaller as the defects are forced toward each other to match the superflow in the bulk. However, the magnitude of p_{s} near the surface due to Meissner screening of the bulk is not large enough to force an annihilation of the motifs. The broken symmetry phase therefore survives the application of an external magnetic field within the whole Meissner state, b ∈ [0, 1].
For higher fields, when Abrikosov vortices start to enter the grain, the problem quickly becomes complicated by the interplay of the Abrikosov vortex lattice formation and finite grain size effects. The free energy landscape is very flat and it is possible to find multiple metastable configurations. For a variety of grain sizes and magnetic field strengths, we have established coexistence of Abrikosov vortices and the spontaneously formed edge loop currents^{35}. We therefore conclude that the edge loopcurrent phase established for T < T^{*} should survive into the mixed state, but a complete investigation of the geometrydependent phase diagram for large fields is beyond the scope of this paper.
Induced currents and magnetic fields
Let us investigate further how the currents and magnetic fields are induced at T^{*}. As we have seen, the paramagnetic response and the spontaneously appearing edge loop currents compete, as they both lead to shifts of midgap Andreev states. As the temperature is lowered, the strength of the paramagnetic response increases slowly and linearly, while the strength of the loop currents increases highly nonlinearly. This is illustrated in Fig. 4, by plotting the areaaveraged current magnitude
as a function of temperature for the cases when B_{ext} = 0 (solid line), B_{ext} = 0.5B_{g1} (dashed line), and for comparison also for a system without pairbreaking edges having only a diamagnetic response at B_{ext} = 0.5B_{g1} (dashdotted line). The paramagnetic response is fully suppressed at low temperatures T < T^{*}. Such a sudden disappearance of the paramagnetic response at a temperature T^{*} should be experimentally measurable, for example in the penetration depth or by using nanosquids^{36,37}.
We show in Fig. 5a the total induced magnetic flux through the grain
and in Fig. 5b the areaaveraged order parameter magnitude
both as functions of temperature for different values of B_{ext}. The figures also show results for a dwave grain without pairbreaking edges at B_{ext} = 0.5B_{g1} (dashdotted line). For better visibility, the latter results have been scaled by a factor 0.4 and 0.9 in (a) and (b), respectively. Two different trends are distinguishable in the observables for T < T^{*} and T > T^{*}, separated by a “kink”. The induced magnetic flux through the grain area decreases as T decreases down to T^{*} due to the increasing paramagnetic response that competes with the diamagnetic one. At T^{*}, the inhomogeneous edge state appear and starts competing with the paramagnetic response. Thus, the total magnetic flux increases again. At the same time, the order parameter is partially healed.
Phase transition and thermodynamics
The sudden changes with a discontinuity in the derivative as a function of temperature of the total induced current, the magnetic flux, as well as the order parameter (Figs. 4 and 5) indicate that there is a phase transition occurring at the temperature T^{*}. In zero external magnetic field, there is a secondorder phase transition at T^{*}, where both timereversal symmetry and continuous translational symmetry along the edge are spontaneously broken^{26}. Let us now investigate the thermodynamics in an external magnetic field already explicitly breaking timereversal symmetry.
In Fig. 6a, we plot the free energy difference between the superconducting and normal states Ω_{S} − Ω_{N}, defined in Eq. (29), for external field B = 0.5B_{g1} (red dashed line) and for zero field (solid black line). For comparison, we show the free energy difference for a purely real order parameter in zero field (gray fine line), i.e., without the symmetry breaking edge loop currents. For T < T^{*}, this solution is not the global minimum of the free energy, and we therefore refer to it as a metastable state. To enhance the visibility of the differences in free energy between the possible solutions, we show in Fig. 6b the free energy difference with respect to the metastable state, i.e., Ω_{S} − Ω_{ms}. The small slope in the red dashed line at T > T^{*} in Fig. 6b is caused by the shift of midgap Andreev states due to the paramagnetic response, which increases as T decreases. The phase transition temperature T^{*} for the secondorder phase transition can be identified with the “knee” in the entropy difference defined in Eq. (31), see Fig. 6c, d. Since timereversal symmetry is already explicitly broken by the external magnetic field, the phase transition signals breaking of local continuous translational symmetry and establishment of the vector field p_{s} with the chain of defects along the edge, as shown in Fig. 3. The magnitude of the order parameter follows the expected scaling law for secondorder phase transitions, p_{s}(T) ∝ (1 − T/T^{*})^{β} with β = 1/2, as shown in the inset of Fig. 6d. However, the temperature range within which the scaling law holds is very limited and nonlinear terms play an important role for lower temperatures T < T^{*}.
The knee in the entropy leads to a jump in the specific heat, as shown in Fig. 6e, f. The heat capacity is expressed in units of the heat capacity jump at the normalsuperconducting phase transition at T_{c} for a bulk dwave system
where α = 8π^{2}/[7ζ(3)], with ζ being the Riemannzeta function. The jump in heat capacity at the phase transition is an edgetoarea effect, and grows linearly as the sample becomes smaller. The jump is roughly 4.5% of ΔC_{d} for the mesoscopic \({\cal A} = 60 \times 60\xi _0^2\) grain considered here, and grows as the size of the grain is reduced. The phase transition temperature T^{*} is extracted as a function of B_{ext} as the midpoint temperature of the jump in the specific heat. Figure 7 shows a phase diagram where the T^{*}, extracted in this way from the specific heat, is plotted versus external field strength (crosses). We compare this with T^{*} extracted as the minimum (the “kink”, see Fig. 5a) in the induced flux. The small lowering of T^{*} with increased B_{ext} is caused by the competing paramagnetic response.
From the above, it is clear that the phase with edge loop currents shows extreme robustness against an external magnetic field in the whole Meissner region (B_{ext} ≤ B_{g1}). The magnitude of the spontaneously formed superfluid momentum p_{s} at the edge grows nonlinearly to be very large for T < T^{*}, fueled by the lowering of the free energy by Doppler shifts of the flat band of Andreev surface states. The corresponding correction to p_{s}, due to the process of screening of the external magnetic field, is in comparison small. Thereby, T^{*} is not dramatically shifted in a magnetic field and the symmetrybroken phase below T^{*} is robust.
Discussion
Which of the scenarios outlined in the introduction wins will ultimately depend on the material properties of a specific hightemperature superconducting sample, or the material properties of other candidate dwave superconductors, e.g., FeSe^{38}. In the scenario studied here, the resulting transition temperature is high, T^{*} ~ 0.18T_{c}. It means that the interaction terms in the Hamiltonian for the other scenarios would have to be sufficiently large in order to compete. It is even possible that one or another scenario wins in different parts of the material’s phase diagram^{16}.
We note that the phase transition at T^{*} means that the initially topologically protected flat band of zero energy surface states is shifted away from the Fermi energy. Such fragility of topologically protected states has been studied recently e.g., for topological insulators^{39} supporting the quantum spinHall state. In that case, an edge reconstruction due to Coulomb interactions leads to breaking of timereversal symmetry. In the dwave superconductor case, although the bulk Hamiltonian still maintains required symmetries, a local instability at the surface violates these symmetries spontaneously and moves the flat band of bound states to finite energies. The spontaneously broken translational symmetry allows for a larger shift from zero energy and a high T^{*}.
From an experimental point of view, the surface physics of dwave superconductors is complicated by, for instance, surface roughness, inhomogeneous stoichiometry, and presence of impurities. The formation of a band of Andreev states centered at zero energy is well established by numerous tunneling experiments, in agreement with the expectation for dwave symmetry of the order parameter, as reviewed in refs. ^{6,7}. One consistent experimental result is that the band is typically quite broad, with a width that saturates at low temperature. On the other hand, the establishment of a timereversal symmetry breaking phase remains under discussion, see for instance refs. ^{40,41}. Several tunneling experiments on YBCO^{42,43,44} show a split of the zerobias conductance peak, while others do not^{45,46}. Other probes indicating timereversal symmetry breaking include thermal conductivity^{47}, Coulomb blockade in nanoscale islands^{5}, and STM tunneling at grain boundaries in FeSe^{38}. As we argued in refs. ^{26,27} within the scenario with spontaneous loop currents, the split of the Andreev band might be difficult to resolve in a tunneling experiment because of the broken translational symmetry along the edge and associated variations in the superflow field. This leads to a smearing effect for tunnel contacts with an area larger than the coherence length and an expected wide, largely temperatureindependent, peak centered at zero energy. In fact, this would be consistent with most tunneling experiments.
With an eye to inspire a new generation of experiments, we have presented results for the interplay between an external magnetic field, that induces screening supercurrents, and the phase transition at T^{*} into a state with the spontaneous loop currents at the edges. We have shown that the phase should be quantified in terms of its order parameter, the vector field p_{s}(R), which contains edge sources and sinks, as well as saddle points. At all these critical points, ∇ × p_{s} ≠ 0. The p_{s} vector field drives the loop currents with opposite circulations in neighboring loops. The loopcurrent strength increases highly nonlinearly, suppressing the paramagnetic response present for T > T^{*}. As the strength of the external magnetic field increases, the size of the Doppler shift due to the paramagnetic response grows linearly. Therefore, T^{*} decreases slightly as the magnitude of the external field increases. The influence of the external field, and in particular the sudden disappearance of the paramagnetic response, leads to observables which we argue should be visible in experiment. For example the “kink” in the total induced flux at T^{*}. The magnetic fluxes induced by the loop currents should be directly observable with recently developed scanning probes^{36,37}, and the sudden disappearance of the paramagnetic response should be observable with nanoSQUIDS and possibly in penetrationdepth experiments. Furthermore, the large jump in heat capacity at the phase transition should be observable with nanocalorimetry^{48}.
The identification of the order parameter p_{s}(R), with its topological textures, leads to similarities with other systems, including general relativity^{33}, fluid dynamics^{49}, liquid crystals^{50}, and superfluid ^{3}He^{51}. An interesting difference is that in those systems, there is typically a transition in a preexisting vector field to a state with topological textures. Here, instead, we have a singlet dwave superconductor that spontaneously establishes p_{s}(R) with topological textures different than the traditional Abrikosov vortices.
Methods
Model and grain geometry
Our aim is to investigate the ground state of clean mesoscopic dwave superconducting grains in an external magnetic field applied perpendicular to the crystal abplane, as shown in Fig. 8. As a typical geometry, we consider a square grain with side lengths D = 60ξ_{0}, where ξ_{0} = ħv_{F}/(2πk_{B}T_{c}) is the zerotemperature superconducting coherence length. Here, v_{F} is the normal state Fermi velocity, and k_{B} the Boltzmann constant. The sides of the system are assumed to be misaligned by a 45° rotation with respect to the crystal abaxes, inducing maximal pairbreaking at the edges.
The external field is directed perpendicular to the xyplane,
We shall consider rather small external fields, and will use a field scale B_{g1} = Φ_{0}/\({\cal A}\), corresponding to one flux quantum threading the grain of area \({\cal A}\) = D^{2} = 60ξ_{0} × 60ξ_{0}. The flux quantum Φ_{0} = hc/(2e) is given in Gaussian CGS units. The field B_{g1} is larger than the lower critical field \(B_{{\mathrm{c}}1} \propto { {\Phi }}_0{\mathrm{/}}\lambda _0^2\), where vortices can enter a macroscopically large superconductor, since the grain side length is smaller than the penetration depth. We assume that λ_{0} = 100ξ_{0}, relevant for YBCO. The upper critical field \(B_{{\mathrm{c}}2} \propto { {\Phi }}_0{\mathrm{/}}\xi _0^2\) is much larger than any field we include in this study. To be precise, we parameterize the field strength as
and we will consider b ∈ [0, 1].
Quasiclassical theory
We utilize the quasiclassical theory of superconductivity^{52,53,54}, which is a theory based on a separation of scales^{55,56,57,58}. For instance, the atomic scale is assumed small compared with the superconducting coherence length, \(\hbar {\mathrm{/}}p_{\mathrm{F}} \ll \xi _0\). This separation of scales makes it possible to systematically expand all quantities in small parameters such as ħ/p_{F}ξ_{0}, Δ/\(\epsilon _{\mathrm{F}}\), and k_{B}T_{c}/\(\epsilon _{\mathrm{F}}\), where Δ is the superconducting order parameter, p_{F} is the Fermi momentum, and \(\epsilon _{\mathrm{F}}\) is the Fermi energy. In equilibrium, the central object of the theory is the quasiclassical Green’s function \(\hat g({\bf{p}}_{\mathrm{F}},{\bf{R}};z)\), which is a function of quasiparticle momentum on the Fermi surface p_{F}, the quasiparticle centerofmass coordinate R, and the quasiparticle energy z. The latter is real z = \(\epsilon\) + i0^{+} with an infinitesimal imaginary part i0^{+} for the retarded Green’s function, or an imaginary Matsubara energy z = i\(\epsilon _n\) = iπk_{B}T(2n + 1) in the Matsubara technique (n is an integer). To keep the notation compact, the dependence on the parameters p_{F}, R, and z will often not be written out. The hat on \(\hat g\) denotes Nambu (electronhole) space
where g and f are the quasiparticle and pair propagators, respectively. The tilde operation denotes particlehole conjugation
The quasiclassical Green’s function is parameterized in terms of two scalar coherence functions, γ(p_{F}, R; z) and \(\tilde \gamma\)(p_{F}, R; z), as^{59,60,61,62,63,64,65}
Note that with this parameterization, the Green’s function is automatically normalized to \(\hat g^2 =  \pi ^2\hat 1\). The coherence functions obey two Riccati equations:
where A is the vector potential. These firstorder nonlinear differential equations are solved by integration along straight (ballistic) quasiparticle trajectories. Quantum coherence is retained along these trajectories, but not between neighboring trajectories. A clean superconducting grain in vacuum is assumed by imposing the boundary condition of perfect specular reflection of quasiparticles along the edges of the system.
The superconducting order parameter is assumed to have pure dwave symmetry
where θ is the angle between the Fermi momentum p_{F} and the crystal \(\widehat {\bf{a}}\)axis, and η_{d}(θ) is the dwave basis function:
fulfilling the normalization condition
The order parameter amplitude satisfies the gap equation
where λ_{d} is the pairing interaction, N_{F} is the density of states at the Fermi level in the normal state, and Ω_{c} is a cutoff energy. The pairing interaction and the cutoff energy are eliminated in favor of the superconducting transition temperature T_{c} (see for example ref. ^{66}) as
The above equations are solved selfconsistently with respect to γ, \(\tilde \gamma\), and Δ_{d}. As an initial guess, we assume a homogenous superconductor with a small modulation of the phase. The coherence functions on the boundaries have to be updated in each iteration, taking into account the specular boundary condition. The starting guess is the local homogeneous solution. After several iterations, the information of the initial guess for the coherence functions is lost^{67}.
We choose an electromagnetic gauge where the vector potential has the form
The total vector potential A(R), that enters Eqs. (13) and (14), is given by A_{ext}(R) and the field A_{ind}(R) induced by the currents j(R) in the superconductor (Eq. (27) below):
The vector potential A_{ind}(R) should be solved from Ampère’s circuit law
with appropriate boundary conditions for the induced field inside and outside the sample. To take the full electrodynamics into account, A_{ind}(R) also needs to be computed selfconsistently in each iteration. However, the strength of the electrodynamic backcoupling scales as κ^{−2}, where κ ≡ λ_{0}/ξ_{0} is the dimensionless GinzburgLandau parameter. The electrodynamic backcoupling is therefore a very small effect for type II superconductors (typically κ^{−1} ≈ 10^{−2} for the cuprates). We have verified through fully selfconsistent calculations that for grains with side lengths D < λ, as we limit ourselves to in this paper, it is always safe to neglect this backcoupling. For large system sizes, \(D \gg \lambda\), backcoupling would ensure proper Meissner screening on the length scale λ in the interior for b < 1 and the establishment of a proper Abrikosov vortex lattice with intervortex distances of order λ for moderate fields b > 1, corresponding to field strengths of order H_{c1}. Since the spontaneous fields appearing below T^{*} are located within a small distance of order \(\xi _0 \ll \lambda\) from the boundary, the effect of backcoupling is small also in these cases. Only in very high fields, approaching H_{c2}, where intervortex distances become of order ξ_{0} may we expect a serious effect on T^{*}, but this is beyond the scope of this paper.
The induced magnetic flux density is computed as
We consider a layered superconductor with many weakly, for our purposes negligibly, coupled layers stacked in the caxis direction. This ensures translational invariance in that direction. Therefore, we neglect the problem of the field distribution around the superconductor and focus on the field induced at the abplane where we have simply B_{ind} = \(B_{{\mathrm{ind}}}\widehat {\bf{z}}\).
Gauge transformation
Once the Green’s function and the order parameter have been determined selfconsistently, we can perform a gauge transformation in order to make the order parameter a real quantity and in the process extract the superfluid momentum p_{s}. This can be illustrated by transforming the Riccati equation in Eq. (13). To begin with, the selfconsistently obtained order parameter is complex, i.e.,
We make the ansatz
and put that into the Riccati equation. We obtain
where p_{s} is defined in Eq. (1).
Observables
The current density is computed within the Matsubara technique through the formula
In the results section, we shall show this current density in units of the depairing current
The freeenergy difference between the superconducting and the normal states is calculated with the Eilenberger freeenergy functional^{52}
We have verified that this form of the free energy gives the same results as the LuttingerWard functional^{26,55,64}. The entropy and specific heat capacity are obtained from the thermodynamic definitions
Data availability
All relevant data are available from the authors.
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Acknowledgements
We thank the Swedish Research Council for financial support. It is a pleasure to thank Mikael Håkansson, Niclas Wennerdal, and Per Rudquist for valuable discussions.
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P.H. carried out the numerical calculations. P.H., A.B.V., M.F., and T.L. analyzed the results. P.H. and T.L. wrote the paper with contributions from A.B.V. and M.F.
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Holmvall, P., Vorontsov, A.B., Fogelström, M. et al. Broken translational symmetry at edges of hightemperature superconductors. Nat Commun 9, 2190 (2018). https://doi.org/10.1038/s4146701804531y
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