Broken translational symmetry at edges of high-temperature superconductors

Broken translational symmetry at edges of high-temperature superconductors ARTICLE DOI: 10.1038/s41467-018-04531-y OPEN Broken translational symmetry at edges of high-temperature superconductors 1 2 1 1 P. Holmvall , A.B. Vorontsov , M. Fogelström & T. Löfwander Flat bands of zero-energy 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 high-temperature super- conductors hosting flat bands of midgap Andreev surface states. In a second-order phase transition at roughly a fifth of the superconducting transition temperature, time-reversal 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 , that forms a planar vector field with defects, including edge sources and sinks. The critical points of the vector field satisfy a generalized Poincaré-Hopf theorem, relating the sum of Poincaré indices to the Euler characteristic of the system. 1 2 Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, SE-41296 Göteborg, Sweden. Department of Physics, Montana State University, Bozeman, MT 59717, USA. Correspondence and requests for materials should be addressed to T.Löf. (email: tomas.lofwander@chalmers.se) NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 1 | | | 1234567890():,; ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y uperconducting devices are often experimentally realized as interaction term in the Hamiltonian. Instead, as we will discuss thin-film circuits or hybrid structures operating in the below, it relies on the development of a texture in the gradient of 1–4 Smesoscopic regime . At this length scale, where the size of the d-wave order parameter phase χ, or more precisely in the the circuit elements becomes comparable with the super- gauge invariant superfluid momentum conducting coherence length, the nature of the superconducting h e ð1Þ state may be dictated by various finite-size or surface/interface p ðRÞ¼ ∇χðRÞ AðRÞ; 2 c effects . This holds true in particular for unconventional super- conductors, such as the high-temperature superconductors with where ħ is Planck’s constant, e the charge of the electron, and c an order parameter of d symmetry that changes the sign 2 2 x y the speed of light. This superfluid momentum spontaneously around the Fermi surface. Scattering at surfaces, or defects, leads takes the form of a planar vector field with a chain of sources and to substantial pair breaking and formation of Andreev states with sinks along the boundary and saddle points in the interior, see 6,7 energies within the superconducting gap . Today, the material Fig. 1. The vector field is illustrated by arrows showing the local control of high-temperature superconducting films is sufficiently unit vectors p ðRÞ, while the color scale illustrates the magnitude good that many advanced superconducting devices can work at p (R). An interior critical point at R is characterized by a s 0 8,9 elevated temperatures . This raises the question how the specific 28,29 Poincaré index defined as surface physics of d-wave superconductors influences devices. From a theory point of view, the physics at specular pair- I ¼ dθ; ð2Þ breaking surfaces of d-wave superconductors is rich and inter- 2π esting. The reason is the formation of zero-energy (midgap) where θ = arctan(p /p ) is the angle of p ðRÞ on the Jordan Andreev states due to the sign change of the d-wave order sy sx 6,7,10 curve Γ encircling R . Internal sources and sinks have I=+1, parameter for quasiparticles scattered at the surface .In 0 while saddle points have I = −1. Although the special points on modern terms, there is a flat band of spin-degenerate zero-energy the boundary have to be treated with care, there is a sum rule surface states as a function of the parallel component of the (Eq. (3)) for the Poincaré indices, as we will discuss below. We momentum, p , which is a good quantum number for a specular || 11,12 identify the p vector field as the order parameter of the surface. A topological invariant has been identified , that symmetry-broken phase, motivated by the fact that the free guarantees the flat band for a time-reversal symmetric super- energy is lowered by a large split of the flat band of Andreev states conducting order parameter and p conserved. However, the large || by a Doppler shift v · p , where v is the Fermi velocity. This free spectral weight of these states exactly at zero energy (i.e., at the F s F energy gain is maximized by maximizing the magnitude of p , Fermi energy) is energetically unfavorable. Different scenarios s which is achieved by the peculiar vector field in Fig. 1. The bal- have been proposed, within which there is a low-temperature ance of the Doppler shift gain and the energy cost in setting up instability and a phase transition into a time-reversal symmetry- the vector field with critical points where ∇ × p ≠ 0 and the broken phase where the flat band is split to finite energies, thus s splay patterns between them leads to a high T ≈ 0.18T . The lowering the free energy of the system. One scenario is the pre- c inhomogeneous vector field induces a chain of loop-currents at sence of a subdominant pairing interaction and appearance of the edge circulating clockwise and anti-clockwise. The induced another order parameter component π/2 out of phase with the 13–15 magnetic fluxes of each loop are a fraction of the flux quantum dominant one , for instance a subdominant s-wave resulting and form a chain of fluxes with alternating signs along the edge. in an order parameter combination Δ + iΔ . The phase transition d s Here we clarify the structure of the order parameter of the is driven by a split of the flat band of Andreev states to ±Δ . The symmetry-broken phase, i.e., p , and study the thermodynamics split Andreev states carry current along the surface, which results s of this phase under the influence of an external magnetic field, in a magnetic field that is screened from the bulk. In a second explicitly breaking time-reversal symmetry. scenario, exchange interactions drive a ferromagnetic transition at 16,17 the edge where the flat Andreev band is instead spin split .A third scenario involves spontaneous appearance of super- Results 18–20 currents that Doppler shifts the Andreev states and thereby Translational symmetry breaking in a magnetic field. In Fig. 2, lowers the free energy. Here the electrons couple to the electro- we show the influence of a rather weak external magnetic field, B magnetic gauge field A(R), and this mechanism was first con- = 0.5B , applied to the d-wave superconducting grain with pair- g1 sidered theoretically for a translationally invariant edge. In this breaking edges for varying temperature near the phase transition case, the transition is a result of the interplay of weakly Doppler temperature T . The scale B = Φ /A corresponds to one flux g1 0 shifted surface bound states, decaying away from the surface on quantum threading the grain area A, see the Methods section. the scale of the superconducting coherence length ξ , and weak The left and right columns show the currents and the magnetic diamagnetic screening currents, decaying on the scale of the field densities, respectively, induced in response to the applied penetration depth λ. The resulting transition temperature is very field. To be concrete, we discuss a few selected sets of model * * low, of order T ~(ξ /λ)T , where T is the d-wave super- parameters, as listed in Table 1. First, for T > T (parameter set I), 0 c c conducting transition temperature. Later, the transition tem- the expected diamagnetic response of the condensate in the inner 21–25 perature was shown to be enhanced in a film geometry part of the grain is present, see Fig. 2a, e. On the other hand, where two parallel pair-breaking edges are separated by a distance midgap quasiparticle Andreev surface states respond para- of the order of a few coherence lengths. The suppression of the magnetically. This situation is well established theoretically and order parameter between the pair-breaking edges can be viewed experimentally through measurements of the competition as an effective Zeeman field that splits the Andreev states and between the diamagnetic and paramagnetic responses seen as a 18,31,32 enhances the transition temperature. The mechanism does not low-temperature up-turn in the penetration depth . Upon involve subdominant channels or coupling to magnetic field, but lowering the temperature to T ≳ T (parameter set II), see Fig. 2b, depends on film thickness D, and the transition temperature f, the paramagnetic response at the edge becomes locally sup- decays rapidly with increasing thickness as T ~(ξ /D)T . pressed and enhanced, forming a sequence of local minima and 0 c 26,27 In this paper, we consider a peculiar scenario where maxima in the induced currents and fields. The bulk response is, spontaneous supercurrents also break translational symmetry on the other hand, relatively unaffected. Finally, as T < T along the edge. This scenario too does not rely on any additional (parameter set III), see Fig. 2c, g, the regions of minimum current 2 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE ⎥ p ⎥ /k T F s B c 0 0.22 T = 0.1T < T *, B = 0 c ext 0 20 40 60 x / Fig. 1 Superfluid momentum as a vector field. The superfluid momentum p forms a non-trivial planar vector field with a regular chain of sources and sinks along the edge, thereby breaking local continuous translational symmetry along the edge. Several critical points, including saddle points, sources, and sinks, are formed in the interior. The Poincaré indices of the critical points add up to fulfill the generalized Poincaré-Hopf theorem in Eq. (3). The magnetic field is zero, B = 0, while the temperature is T = 0.1T . Since T is well below T , the splay patterns are rather stiff, leading to triangular shapes near the edges. ext c The stiffness is clear from the magnitude variation shown in color scale. The inset shows one period of the edge structure turns into regions with reversed currents. The resulting loop the magnitude of p is large, much larger than in the interior part currents with clock-wise and anti-clockwise circulations induce of the grain still experiencing diamagnetism. In a magnetic field, magnetic fluxes along the surface with opposite signs between the vector field far from the surface has a preferred direction neighboring fluxes. The situation for T < T in an external mag- reflecting the diamagnetic response of the interior grain. This netic field can be compared with the one in zero magnetic field shifts the sources and sinks along the surface, as compared with displayed in Fig. 2d, h. In the presence of the magnetic field, there the regular chain for zero field in Fig. 1, and moves the saddle is an imbalance between positive and negative fluxes, while in points to the surface region. zero external magnetic field, the total induced flux integrated over The superflow pattern of sources, sinks, and saddle points the grain area is zero. satisfy a certain sum rule related to the topology of the sample. This relation also ties the special points of the p field on the edge of the sample with critical points in its bulk. The generalized 33,34 Topology of the superfluid momentum vector field. Let us Poincaré-Hopf theorem for manifolds with boundaries quantify the symmetry-broken phase in a magnetic field by connects the properties of a vector field v inside a manifold M, plotting the superfluid momentum defined in Eq. (1), see Fig. 3. and on its boundary ∂M, with the Euler characteristic of the For T ≳ T (parameter set II), the amplitude of p varies along the manifold χ(M). Using the formulation presented in ref. ,we edge (coordinate x), see Fig. 3a, reflecting the varying para- write magnetic response in Fig. 2b, f. For T < T (parameter set III), the hi sources and sinks have appeared pairwise together with a saddle ð3Þ Ind ðvÞþ Ind ðv Þ Ind ðv Þ ¼ χðMÞ; M ∂ M jj ∂ M jj 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 where Ind (v) is the total Poincaré index of critical points of the chain of sources, sinks, and saddle points are well established and field v internal to M, Ind ðv Þ is the total Poincaré index of ∂ M jj NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 3 | | | y/ 0 2 2 2 2 –6 –6 –6 –6 B / ( / ) (×10 ) B / ( / ) (×10 ) B / ( / ) (×10 ) B / ( / ) (×10 ) ind 0 0 ind 0 0 ind 0 0 ind 0 0 j /j j /j j /j j /j d d d d ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y 0.016 7 60 60 (I) : T = 0.182T > T *, B = 0.5B ae (I) : T = 0.182T > T *, B = 0.5B c ext g1 c ext g1 40 40 0 0 0 –7 0.027 7 60 > 60 > (II) : T = 0.176T T *, B = 0.5B (II) : T = 0.176T T *, B = 0.5B c ~ ext g1 f c ~ ext g1 40 40 20 20 0 0 0 –7 0.062 7 < < c (III) : T = 0.17T T *, B = 0.5B g (III) : T = 0.17T T *, B = 0.5B c ext g1 c ext g1 40 40 20 20 0 0 –7 0.056 60 60 < < (IV) : T = 0.17T T *, B = 0 (IV) : T = 0.17T T *, B = 0 d c ext c ext 40 40 20 20 0 0 0 –7 020 40 60 020 40 60 x / x / 0 0 Fig. 2 Spontaneously formed currents and induced magnetic field. a–d Total current magnitude and e–h induced magnetic flux density for different temperatures and external fields (see annotations). Lines and arrows have been added to illustrate the flow of the currents critical points of the tangent vector v || ∂M on the boundary. The (∂ M) / outside (∂ M)of M, come with positive/negative signs. || − + theorem applies when the boundary ∂M does not go through any In Supplementary Note 1, we demonstrate in detail how the sum critical points of v. Boundary indices where field points inside rule works for the vector field in Fig. 1, redrawn as a streamline 4 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | y/ y/ y/ y/ 0 0 0 0 y/ y/ y/ y/ 0 0 0 0  p /k T  p /k T  p /k T F s B c F s B c F s B c NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE plot in Supplementary Fig. 2. We also provide other examples of loop currents . We therefore conclude that the edge loop-current grain geometries in Supplementary Figs. 3–10. We utilize the sum phase established for T < T should survive into the mixed state, rule as a tool to verify that the calculations are correct. but a complete investigation of the geometry-dependent phase In a magnetic field, as in Fig. 3, a motif with one edge source, diagram for large fields is beyond the scope of this paper. one edge sink, and one saddle point annihilate at T . In the same fashion, increasing the magnetic field strength, the motif gets Induced currents and magnetic fields. Let us investigate further smaller as the defects are forced toward each other to match the how the currents and magnetic fields are induced at T .Aswe superflow in the bulk. However, the magnitude of p near the s have seen, the paramagnetic response and the spontaneously surface due to Meissner screening of the bulk is not large enough appearing edge loop currents compete, as they both lead to shifts to force an annihilation of the motifs. The broken symmetry of midgap Andreev states. As the temperature is lowered, the phase therefore survives the application of an external magnetic strength of the paramagnetic response increases slowly and lin- field within the whole Meissner state, b ∈ [0, 1]. early, while the strength of the loop currents increases highly For higher fields, when Abrikosov vortices start to enter the non-linearly. This is illustrated in Fig. 4, by plotting the area- grain, the problem quickly becomes complicated by the interplay averaged current magnitude of the Abrikosov vortex lattice formation and finite grain size effects. The free energy landscape is very flat and it is possible to j ¼ d Rjj jðRÞ ; ð4Þ find multiple metastable configurations. For a variety of grain sizes and magnetic field strengths, we have established coex- istence of Abrikosov vortices and the spontaneously formed edge as a function of temperature for the cases when B = 0 (solid ext line), B = 0.5B (dashed line), and for comparison also for a ext g1 system without pair-breaking edges having only a diamagnetic Table 1 Sets of parameters used for presenting results response at B = 0.5B (dash-dotted line). The paramagnetic ext g1 response is fully suppressed at low temperatures T < T . Such a Set Temperature External magnetic field sudden disappearance of the paramagnetic response at a tem- (I) T = 0.182T > T* B = 0.5B c ext g1 perature T should be experimentally measurable, for example in (II) T = 0.176T ≳ T* B = 0.5B c ext g1 36,37 the penetration depth or by using nano-squids . (III) T = 0.17T < T* B = 0.5B c ext g1 We show in Fig. 5a the total induced magnetic flux through the (IV) T = 0.17T < T* B = 0 c ext grain The field scale B = Φ /A corresponds to an external magnetic flux through the grain area g1 0 exactly equal to one flux quantum Φ ¼ d RB ðRÞ; ð5Þ ind ind 0.06 T = 0.176T T *, B = 0.5B c ~ ext g1 0 0.03 0.09 T = 0.17T < T *, B = 0.5B b c ext g1 0 0 0.28 T = 0.1T < T *, B = 0.5B c ext g1 4 ⎥ 0 0 20 25 30 35 40 x / Fig. 3 Superfluid momentum for varying temperature. a The superfluid momentum induced in an external magnetic field of B = 0.5B for a temperature ext g1 slightly above the transition temperature T reflects the paramagnetic response. b At the phase transition, source–sink–saddle-point motifs appear and separate along the edge breaking translational invariance along the edge coordinate x. c For lower temperature, the magnitude p grows large. Note that different color scales are used in the subfigures in order to enhance visibility NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 5 | | | y / y / y / 0 0 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y 2 Phase transition and thermodynamics. The sudden changes B = 0 ext with a discontinuity in the derivative as a function of temperature B = 0.5B ext g1 of the total induced current, the magnetic flux, as well as the B = 0.5B , (no p.b.) ext g1 Loop currents order parameter (Figs. 4 and 5) indicate that there is a phase SC transition occurring at the temperature T . In zero external (III) magnetic field, there is a second-order phase transition at T , (II) (IV) where both time-reversal symmetry and continuous translational (I) symmetry along the edge are spontaneously broken . Let us now investigate the thermodynamics in an external magnetic field 0.12 0.15 0.18 0.21 0.24 already explicitly breaking time-reversal symmetry. T /T In Fig. 6a, we plot the free energy difference between the superconducting and normal states Ω − Ω ,defined in Eq. (29), S N Fig. 4 Current as a function of temperature. The area-averaged current for external field B = 0.5B (red dashed line) and for zero field g1 magnitude, defined in Eq. (4), is plotted for zero external magnetic field (solid black line). For comparison, we show the free energy (solid line), with an external magnetic field of magnitude B = 0.5B ext g1 difference for a purely real order parameter in zero field (gray fine (dashed line), and for a system without pair-breaking edges at B = 0.5B ext g1 line), i.e., without the symmetry breaking edge loop currents. For (dash-dotted line). In the latter case, the system only displays a * T < T , this solution is not the global minimum of the free energy, diamagnetic response. Letters (I)–(IV) indicate the parameter values and we therefore refer to it as a metastable state. To enhance the corresponding to the fields in Fig. 2, see Table 1 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 − Ω . The small slope in S ms the red dashed line at T > T in Fig. 6b is caused by the shift of ab ×0.4 No p.b. midgap Andreev states due to the paramagnetic response, which B = 0.5B ext g1 increases as T decreases. The phase transition temperature T for 6 1.36 –4 6 × 10 the second-order phase transition can be identified with the ×0.9 5 “knee” in the entropy difference defined in Eq. (31), see Fig. 6c, d. Since time-reversal symmetry is already explicitly broken by the external magnetic field, the phase transition signals breaking of local continuous translational symmetry and establishment of the 3 1.35 vector field p 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 second-order phase transitions, p (T) ∝ 1 s * β (1 − T/T ) with β = 1/2, as shown in the inset of Fig. 6d. 0 1.34 However, the temperature range within which the scaling law 0.12 0.18 0.24 0.12 0.18 0.24 holds is very limited and non-linear terms play an important role T /T T /T c c for lower temperatures T < T . Fig. 5 Magnetic flux as a function of temperature. a Temperature The knee in the entropy leads to a jump in the specific heat, as dependence of the induced magnetic flux, defined in Eq. (5). The solid lines shown in Fig. 6e, f. The heat capacity is expressed in units of the indicate, from bottom to top (colors purple to red), the external field heat capacity jump at the normal-superconducting phase magnitude from B =0to B = 0.5B in steps of 0.05B . The line ext ext g1 g1 transition at T for a bulk d-wave system corresponding to zero field lies exactly at zero since there is an equal 2α amount of positive and negative fluxes induced in this case, see Fig. 2. Panel 2 ΔC ¼ Ak T N ; ð7Þ d B c F b shows the area-averaged order parameter magnitude defined in Eq. (6) versus temperature. Results are also shown for a system without pair- breaking edges (dash-dotted line) at B = 0.5B , but scaled with a factor ext g1 where α = 8π /[7ζ(3)], with ζ being the Riemann-zeta function. 0.4 and 0.9 in a and b, respectively The jump in heat capacity at the phase transition is an edge-to- area effect, and grows linearly as the sample becomes smaller. The jump is roughly 4.5% of ΔC for the mesoscopic A¼ 60 ´ 60ξ grain considered here, and grows as the size of the grain is and in Fig. 5b the area-averaged order parameter magnitude reduced. The phase transition temperature T is extracted as a function of B as the midpoint temperature of the jump in the 1 ext Δ ¼ d Rjj Δ ðRÞ ; ð6Þ d d 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 both as functions of temperature for different values of B . The extracted as the minimum (the “kink”, see Fig. 5a) in the induced ext figures also show results for a d-wave grain without pair-breaking flux. The small lowering of T with increased B is caused by the ext edges at B = 0.5B (dash-dotted line). For better visibility, the competing paramagnetic response. ext g1 latter results have been scaled by a factor 0.4 and 0.9 in (a) and From the above, it is clear that the phase with edge loop (b), respectively. Two different trends are distinguishable in the currents shows extreme robustness against an external magnetic * * observables for T < T and T > T , separated by a “kink”. The field in the whole Meissner region (B ≤ B ). The magnitude of ext g1 induced magnetic flux through the grain area decreases as T the spontaneously formed superfluid momentum p at the edge * * decreases down to T due to the increasing paramagnetic grows non-linearly to be very large for T < T , fueled by the response that competes with the diamagnetic one. At T , the lowering of the free energy by Doppler shifts of the flat band of inhomogeneous edge state appear and starts competing with the Andreev surface states. The corresponding correction to p , due paramagnetic response. Thus, the total magnetic flux increases to the process of screening of the external magnetic field, is in again. At the same time, the order parameter is partially healed. comparison small. Thereby, T is not dramatically shifted in a 6 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | –3 / (×10 ) ind 0 –2 j /j (×10 ) /  T d B c NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE ac e –0.07 –0.08 –0.09 –0.10 –0.11 –0.12 –0.13 –0.14 –0.15 0.15 0.18 0.21 0.15 0.18 0.21 0.15 0.18 0.21 T /T T /T T /T c c c b d f 0.05 0.04 ~√ 1 – T /T * 0.03 sx 0.02 B = 0, (ms) ext sy B = 0 ext 0.01 B = 0.5B ext g1 0.17 T /T 0.18 0.15 0.18 0.21 0.15 0.18 0.21 0.15 0.18 0.21 T /T T /T T /T c c c Fig. 6 Thermodynamics and phase transition. a, b free energy, c, d entropy, and e, f specific heat capacity, versus temperature. The lines correspond to a system with purely real order parameter without edge currents (gray fine line), a system with spontaneous edge currents in zero magnetic field (black solid line), and in a finite external field B = 0.5B (red dashed line). In the lower panels b, d, and f, the quantities have been subtracted by the corresponding g1 values of the system with a purely real order parameter, the metastable (ms) state. The heat capacity is normalized by the heat capacity jump at the normal-superconducting phase transition for a bulk d-wave system, ΔC in Eq. (7). The inset in (d) shows the temperature dependence of the superfluid momentum near T , averaged over a few source–sink unit cells at one edge. It follows the expected temperature dependence for the order parameter at a mean-field second-order phase transition Hamiltonian for the other scenarios would have to be sufficiently large in order to compete. It is even possible that one or another Loop currents scenario wins in different parts of the material’s phase diagram . 0.8 SC We note that the phase transition at T means that the initially T * from ΔC 0.6 topologically protected flat band of zero energy surface states is T * from B ind shifted away from the Fermi energy. Such fragility of topologically 0.4 protected states has been studied recently e.g., for topological insulators supporting the quantum spin-Hall state. In that case, 0.2 an edge reconstruction due to Coulomb interactions leads to breaking of time-reversal symmetry. In the d-wave super- 0.15 0.16 0.17 0.18 0.19 0.20 0.21 conductor case, although the bulk Hamiltonian still maintains T *(B )/T ext c required symmetries, a local instability at the surface violates * these symmetries spontaneously and moves the flat band of Fig. 7 Phase diagram. The transition temperature T to a state with bound states to finite energies. The spontaneously broken trans- spontaneously broken continuous translational symmetry is plotted as a lational symmetry allows for a larger shift from zero energy and a function of the external magnetic flux density. The crosses show T high T . extracted from the jump in the specific heat in Fig. 6e, while the open From an experimental point of view, the surface physics of d- circles show T extracted from the minimum of the total induced magnetic wave superconductors is complicated by, for instance, surface flux in Fig. 5a 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 experi- magnetic field and the symmetry-broken phase below T is ments, in agreement with the expectation for d-wave symmetry of robust. 6,7 the order parameter, as reviewed in refs. . 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 Discussion establishment of a time-reversal symmetry breaking phase Which of the scenarios outlined in the introduction wins will 40,41 remains under discussion, see for instance refs. . Several ultimately depend on the material properties of a specific high- 42–44 tunneling experiments on YBCO show a split of the zero- temperature superconducting sample, or the material properties 45,46 38 bias conductance peak, while others do not . Other probes of other candidate d-wave superconductors, e.g., FeSe . In the indicating time-reversal symmetry breaking include thermal scenario studied here, the resulting transition temperature is high, 47 5 * conductivity , Coulomb blockade in nanoscale islands , and T ~ 0.18T . It means that the interaction terms in the NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 7 | | | B /B ext g1 –  (a.u.)  –  (a.u.) S ms S N S – S (a.u.) S – S (a.u.) S ms S N p (a.u.) (C – C ) /ΔC S N d (C – C ) /ΔC S ms d 2 –6 B / ( / ) (×10 ) ind 0 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y STM tunneling at grain boundaries in FeSe . As we argued in 60 6 26,27 b aˆ refs. within the scenario with spontaneous loop currents, the split of the Andreev band might be difficult to resolve in a tun- neling 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 temperature-independent, 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 ext magnetic field, that induces screening supercurrents, and the 0 –6 phase transition at T into a state with the spontaneous loop 020 40 60 currents at the edges. We have shown that the phase should be x / quantified in terms of its order parameter, the vector field p (R), Fig. 8 Grain geometry. The system consists of a d-wave superconducting which contains edge sources and sinks, as well as saddle points. grain exposed to an external magnetic field B = B b z. The crystal ab- At all these critical points, ∇ × p ≠ 0. The p vector field drives ext s s ext axes are rotated 45° relative to the grain edges, inducing pair breaking at the loop currents with opposite circulations in neighboring loops. the edges of the system. The color scale shows the magnetic field B The loop-current strength increases highly non-linearly, sup- ind induced in response to an external field of size B = Φ /2A at a pressing the paramagnetic response present for T > T . As the ext 0 temperature T = 0.2T . There is a diamagnetic response carried by the strength of the external magnetic field increases, the size of the c condensate in the interior, and a paramagnetic response carried by midgap Doppler shift due to the paramagnetic response grows linearly. surface Andreev states at the edges Therefore, T decreases slightly as the magnitude of the external field increases. The influence of the external field, and in parti- 52–54 cular the sudden disappearance of the paramagnetic response, Quasiclassical theory. We utilize the quasiclassical theory of superconductivity , 55–58 which is a theory based on a separation of scales . For instance, the atomic scale is leads to observables which we argue should be visible in experi- assumed small compared with the superconducting coherence length, h=p  ξ . * F 0 ment. For example the “kink” in the total induced flux at T . The This separation of scales makes it possible to systematically expand all quantities in magnetic fluxes induced by the loop currents should be directly small parameters such as ħ/p ξ , Δ/ϵ ,and k T /ϵ ,where Δ is the superconducting F 0 F B c F 36,37 observable with recently developed scanning probes , and the order parameter, p is the Fermi momentum, and ϵ is the Fermi energy. In equili- brium, the central object of the theory is the quasiclassical Green’sfunction sudden disappearance of the paramagnetic response should be ^gðp ; R; zÞ, which is a function of quasiparticle momentum on the Fermi surface p , observable with nano-SQUIDS and possibly in penetration-depth the quasiparticle center-of-mass coordinate R, and the quasiparticle energy z.The + + experiments. Furthermore, the large jump in heat capacity at the latter is real z= ϵ + i0 with an infinitesimal imaginary part i0 for the retarded phase transition should be observable with nanocalorimetry . Green’s function, or an imaginary Matsubara energy z= iϵ = iπk T(2n+ 1) in the Matsubara technique (n is an integer). To keep the notation compact, the dependence The identification of the order parameter p (R), with its on the parameters p , R,and z will often not be written out. The hat on ^g denotes topological textures, leads to similarities with other systems, Nambu (electron-hole) space 33 49 50 including general relativity , fluid dynamics , liquid crystals , gf 3 51 and superfluid He . An interesting difference is that in those ^g ¼ ; ð10Þ f g systems, there is typically a transition in a preexisting vector field to a state with topological textures. Here, instead, we have a where g and f are the quasiparticle and pair propagators, respectively. The tilde singlet d-wave superconductor that spontaneously establishes operation denotes particle-hole conjugation p (R) with topological textures different than the traditional α ~ðp ; R; zÞ¼ α ðp ; R; z Þ: ð11Þ F F Abrikosov vortices. The quasiclassical Green’s function is parameterized in terms of two scalar coherence 59–65 functions, γ(p , R; z)and ~γ(p , R; z), as Methods F F Model and grain geometry. Our aim is to investigate the ground state of clean 1  γ~γ 2γ iπ mesoscopic d-wave superconducting grains in an external magnetic field applied ^g ¼ : ð12Þ 1 þ γ~γ 2~γ 1 þ γ~γ perpendicular to the crystal ab-plane, as shown in Fig. 8. As a typical geometry, we consider a square grain with side lengths D = 60ξ , where ξ = ħv /(2πk T ) is the 0 0 F B c Note that with this parameterization, the Green’s function is automatically normalized zero-temperature superconducting coherence length. Here, v is the normal state 2 2 to ^g ¼π 1. The coherence functions obey two Riccati equations: Fermi velocity, and k the Boltzmann constant. The sides of the system are assumed to be misaligned by a 45° rotation with respect to the crystal ab-axes, e ðihv  ∇ þ 2z þ 2 v  AÞγ ¼Δγ  Δ; ð13Þ F F inducing maximal pair-breaking at the edges. The external field is directed perpendicular to the xy-plane, ~ ~ ðihv  ∇  2z  2 v  AÞγ ¼Δγ  Δ; ð14Þ F F B ¼B b zjjbc: ð8Þ ext ext c where A is the vector potential. These first-order non-linear differential equations are We shall consider rather small external fields, and will use a field scale B = g1 solved by integration along straight (ballistic) quasiparticle trajectories. Quantum Φ /A, corresponding to one flux quantum threading the grain of area A = D = 0 coherence is retained along these trajectories, but not between neighboring trajec- 60ξ ×60ξ . The flux quantum Φ = hc/(2|e|) is given in Gaussian CGS units. The 0 0 0 tories. A clean superconducting grain in vacuum is assumed by imposing the field B is larger than the lower critical field B / Φ =λ , where vortices can enter g1 boundary condition of perfect specular reflection of quasiparticles along the edges of c1 0 0 a macroscopically large superconductor, since the grain side length is smaller than the system. the penetration depth. We assume that λ = 100ξ , relevant for YBCO. The upper 0 0 The superconducting order parameter is assumed to have pure d-wave critical field B / Φ =ξ is much larger than any field we include in this study. To c2 0 0 symmetry be precise, we parameterize the field strength as Δðp ; RÞ¼ Δ ðRÞη ðθÞ; ð15Þ F d d ð9Þ B ¼ bB ; B  ; ext g1 g1 where θ is the angle between the Fermi momentum p and the crystal b a-axis, and η (θ) is the d-wave basis function: pffiffiffi ð16Þ η ðθÞ¼ 2cosð2θÞ; and we will consider b ∈ [0, 1]. d 8 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | y / 0 NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE fulfilling the normalization condition and put that into the Riccati equation. We obtain dθ 2 ihv  ∇ þ 2ðz  v  p Þ γ ¼jj Δ η ðγ þ 1Þ; ð26Þ η ðθÞ ¼ 1: ð17Þ F F s 0 d d 0 2π where p is defined in Eq. (1). The order parameter amplitude satisfies the gap equation dθ Observables. The current density is computed within the Matsubara technique Δ ðRÞ¼ λ N k T η ðθÞf ðp ; R; ϵ Þ; d d F B d F n ð18Þ 2π through the formula ϵ Ω jj n c dθ jðRÞ¼ 2πeN k T v gðp ; R; ϵ Þ: F B F F n ð27Þ where λ is the pairing interaction, N is the density of states at the Fermi level in 2π d F the normal state, and Ω is a cutoff energy. The pairing interaction and the cutoff energy are eliminated in favor of the superconducting transition temperature T In the results section, we shall show this current density in units of the depairing (see for example ref. )as current 1 T 1 ¼ ln þ : j  4πjj e k T N v : ð28Þ ð19Þ d B c F F λ N T n þ d F c n0 2 The free-energy difference between the superconducting and the normal states The above equations are solved self-consistently with respect to γ, ~γ, and Δ .As d is calculated with the Eilenberger free-energy functional an initial guess, we assume a homogenous superconductor with a small modulation R 2 B ðRÞ 2 ind of the phase. The coherence functions on the boundaries have to be updated in Ω ðB; TÞ Ω ðB; TÞ¼ dR þjj Δ ðRÞ N ln S N 8π d F T each iteration, taking into account the specular boundary condition. The starting ) hi ð29Þ P 2 guess is the local homogeneous solution. After several iterations, the information of jj Δ ðRÞ þ2πN k T þ iIðR; ϵ Þ ; F B ϵ n the initial guess for the coherence functions is lost . n ϵ >0 We choose an electromagnetic gauge where the vector potential has the form dθ ð20Þ A ðRÞ¼ B ´ R: ext ext IðR; ϵ Þ¼ Δðp ; RÞγðp ; R; ϵ Þ Δðp ; RÞ~γðp ; R; ϵ Þ : ð30Þ 2 n F F n F F n 2π The total vector potential A(R), that enters Eqs. (13) and (14), is given by A (R) ext We have verified that this form of the free energy gives the same results as the and the field A (R) induced by the currents j(R) in the superconductor (Eq. (27) ind 26,55,64 Luttinger-Ward functional . The entropy and specific heat capacity are below): obtained from the thermodynamic definitions AðRÞ¼ A ðRÞþ A ðRÞ: ð21Þ ext ind ∂Ω S ¼ ; ð31Þ ∂T The vector potential A (R) should be solved from Ampère’s circuit law ind 4π 2 ∂S ∂ Ω ð22Þ ∇ ´ ∇ ´ A ðRÞ¼ jðRÞ ; ind ð32Þ C ¼ T ¼T : ∂T ∂T with appropriate boundary conditions for the induced field inside and outside the sample. To take the full electrodynamics into account, A (R) also needs to be ind Data availability. All relevant data are available from the authors. computed self-consistently in each iteration. However, the strength of the −2 electrodynamic back-coupling scales as κ , where κ ≡ λ /ξ is the dimensionless 0 0 Ginzburg-Landau parameter. The electrodynamic back-coupling is therefore a very Received: 13 December 2017 Accepted: 1 May 2018 −1 −2 small effect for type II superconductors (typically κ ≈ 10 for the cuprates). We have verified through fully self-consistent calculations that for grains with side lengths D < λ, as we limit ourselves to in this paper, it is always safe to neglect this back-coupling. For large system sizes, D λ, back-coupling 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 inter-vortex distances of order λ for moderate fields b > 1, corresponding to field strengths of order H . References c1 Since the spontaneous fields appearing below T are located within a small distance 1. Gol'tsman, G. N. et al. Picosecond superconducting single-photon optical of order ξ  λ from the boundary, the effect of back-coupling is small also in detector. Appl. Phys. Lett. 79, 705–707 (2001). these cases. Only in very high fields, approaching H , where inter-vortex distances c2 2. 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Search for broken time-reversal symmetry near the surface P.H. carried out the numerical calculations. P.H., A.B.V., M.F., and T.L. analyzed the of superconducting YBa Cu O films using β-detected nuclear magnetic 2 3 7−δ results. P.H. and T.L. wrote the paper with contributions from A.B.V. and M.F. resonance. Phys. Rev. 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Broken translational symmetry at edges of high-temperature superconductors

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ARTICLE DOI: 10.1038/s41467-018-04531-y OPEN Broken translational symmetry at edges of high-temperature superconductors 1 2 1 1 P. Holmvall , A.B. Vorontsov , M. Fogelström & T. Löfwander Flat bands of zero-energy 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 high-temperature super- conductors hosting flat bands of midgap Andreev surface states. In a second-order phase transition at roughly a fifth of the superconducting transition temperature, time-reversal 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 , that forms a planar vector field with defects, including edge sources and sinks. The critical points of the vector field satisfy a generalized Poincaré-Hopf theorem, relating the sum of Poincaré indices to the Euler characteristic of the system. 1 2 Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, SE-41296 Göteborg, Sweden. Department of Physics, Montana State University, Bozeman, MT 59717, USA. Correspondence and requests for materials should be addressed to T.Löf. (email: tomas.lofwander@chalmers.se) NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 1 | | | 1234567890():,; ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y uperconducting devices are often experimentally realized as interaction term in the Hamiltonian. Instead, as we will discuss thin-film circuits or hybrid structures operating in the below, it relies on the development of a texture in the gradient of 1–4 Smesoscopic regime . At this length scale, where the size of the d-wave order parameter phase χ, or more precisely in the the circuit elements becomes comparable with the super- gauge invariant superfluid momentum conducting coherence length, the nature of the superconducting h e ð1Þ state may be dictated by various finite-size or surface/interface p ðRÞ¼ ∇χðRÞ AðRÞ; 2 c effects . This holds true in particular for unconventional super- conductors, such as the high-temperature superconductors with where ħ is Planck’s constant, e the charge of the electron, and c an order parameter of d symmetry that changes the sign 2 2 x y the speed of light. This superfluid momentum spontaneously around the Fermi surface. Scattering at surfaces, or defects, leads takes the form of a planar vector field with a chain of sources and to substantial pair breaking and formation of Andreev states with sinks along the boundary and saddle points in the interior, see 6,7 energies within the superconducting gap . Today, the material Fig. 1. The vector field is illustrated by arrows showing the local control of high-temperature superconducting films is sufficiently unit vectors p ðRÞ, while the color scale illustrates the magnitude good that many advanced superconducting devices can work at p (R). An interior critical point at R is characterized by a s 0 8,9 elevated temperatures . This raises the question how the specific 28,29 Poincaré index defined as surface physics of d-wave superconductors influences devices. From a theory point of view, the physics at specular pair- I ¼ dθ; ð2Þ breaking surfaces of d-wave superconductors is rich and inter- 2π esting. The reason is the formation of zero-energy (midgap) where θ = arctan(p /p ) is the angle of p ðRÞ on the Jordan Andreev states due to the sign change of the d-wave order sy sx 6,7,10 curve Γ encircling R . Internal sources and sinks have I=+1, parameter for quasiparticles scattered at the surface .In 0 while saddle points have I = −1. Although the special points on modern terms, there is a flat band of spin-degenerate zero-energy the boundary have to be treated with care, there is a sum rule surface states as a function of the parallel component of the (Eq. (3)) for the Poincaré indices, as we will discuss below. We momentum, p , which is a good quantum number for a specular || 11,12 identify the p vector field as the order parameter of the surface. A topological invariant has been identified , that symmetry-broken phase, motivated by the fact that the free guarantees the flat band for a time-reversal symmetric super- energy is lowered by a large split of the flat band of Andreev states conducting order parameter and p conserved. However, the large || by a Doppler shift v · p , where v is the Fermi velocity. This free spectral weight of these states exactly at zero energy (i.e., at the F s F energy gain is maximized by maximizing the magnitude of p , Fermi energy) is energetically unfavorable. Different scenarios s which is achieved by the peculiar vector field in Fig. 1. The bal- have been proposed, within which there is a low-temperature ance of the Doppler shift gain and the energy cost in setting up instability and a phase transition into a time-reversal symmetry- the vector field with critical points where ∇ × p ≠ 0 and the broken phase where the flat band is split to finite energies, thus s splay patterns between them leads to a high T ≈ 0.18T . The lowering the free energy of the system. One scenario is the pre- c inhomogeneous vector field induces a chain of loop-currents at sence of a subdominant pairing interaction and appearance of the edge circulating clockwise and anti-clockwise. The induced another order parameter component π/2 out of phase with the 13–15 magnetic fluxes of each loop are a fraction of the flux quantum dominant one , for instance a subdominant s-wave resulting and form a chain of fluxes with alternating signs along the edge. in an order parameter combination Δ + iΔ . The phase transition d s Here we clarify the structure of the order parameter of the is driven by a split of the flat band of Andreev states to ±Δ . The symmetry-broken phase, i.e., p , and study the thermodynamics split Andreev states carry current along the surface, which results s of this phase under the influence of an external magnetic field, in a magnetic field that is screened from the bulk. In a second explicitly breaking time-reversal symmetry. scenario, exchange interactions drive a ferromagnetic transition at 16,17 the edge where the flat Andreev band is instead spin split .A third scenario involves spontaneous appearance of super- Results 18–20 currents that Doppler shifts the Andreev states and thereby Translational symmetry breaking in a magnetic field. In Fig. 2, lowers the free energy. Here the electrons couple to the electro- we show the influence of a rather weak external magnetic field, B magnetic gauge field A(R), and this mechanism was first con- = 0.5B , applied to the d-wave superconducting grain with pair- g1 sidered theoretically for a translationally invariant edge. In this breaking edges for varying temperature near the phase transition case, the transition is a result of the interplay of weakly Doppler temperature T . The scale B = Φ /A corresponds to one flux g1 0 shifted surface bound states, decaying away from the surface on quantum threading the grain area A, see the Methods section. the scale of the superconducting coherence length ξ , and weak The left and right columns show the currents and the magnetic diamagnetic screening currents, decaying on the scale of the field densities, respectively, induced in response to the applied penetration depth λ. The resulting transition temperature is very field. To be concrete, we discuss a few selected sets of model * * low, of order T ~(ξ /λ)T , where T is the d-wave super- parameters, as listed in Table 1. First, for T > T (parameter set I), 0 c c conducting transition temperature. Later, the transition tem- the expected diamagnetic response of the condensate in the inner 21–25 perature was shown to be enhanced in a film geometry part of the grain is present, see Fig. 2a, e. On the other hand, where two parallel pair-breaking edges are separated by a distance midgap quasiparticle Andreev surface states respond para- of the order of a few coherence lengths. The suppression of the magnetically. This situation is well established theoretically and order parameter between the pair-breaking edges can be viewed experimentally through measurements of the competition as an effective Zeeman field that splits the Andreev states and between the diamagnetic and paramagnetic responses seen as a 18,31,32 enhances the transition temperature. The mechanism does not low-temperature up-turn in the penetration depth . Upon involve subdominant channels or coupling to magnetic field, but lowering the temperature to T ≳ T (parameter set II), see Fig. 2b, depends on film thickness D, and the transition temperature f, the paramagnetic response at the edge becomes locally sup- decays rapidly with increasing thickness as T ~(ξ /D)T . pressed and enhanced, forming a sequence of local minima and 0 c 26,27 In this paper, we consider a peculiar scenario where maxima in the induced currents and fields. The bulk response is, spontaneous supercurrents also break translational symmetry on the other hand, relatively unaffected. Finally, as T < T along the edge. This scenario too does not rely on any additional (parameter set III), see Fig. 2c, g, the regions of minimum current 2 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE ⎥ p ⎥ /k T F s B c 0 0.22 T = 0.1T < T *, B = 0 c ext 0 20 40 60 x / Fig. 1 Superfluid momentum as a vector field. The superfluid momentum p forms a non-trivial planar vector field with a regular chain of sources and sinks along the edge, thereby breaking local continuous translational symmetry along the edge. Several critical points, including saddle points, sources, and sinks, are formed in the interior. The Poincaré indices of the critical points add up to fulfill the generalized Poincaré-Hopf theorem in Eq. (3). The magnetic field is zero, B = 0, while the temperature is T = 0.1T . Since T is well below T , the splay patterns are rather stiff, leading to triangular shapes near the edges. ext c The stiffness is clear from the magnitude variation shown in color scale. The inset shows one period of the edge structure turns into regions with reversed currents. The resulting loop the magnitude of p is large, much larger than in the interior part currents with clock-wise and anti-clockwise circulations induce of the grain still experiencing diamagnetism. In a magnetic field, magnetic fluxes along the surface with opposite signs between the vector field far from the surface has a preferred direction neighboring fluxes. The situation for T < T in an external mag- reflecting the diamagnetic response of the interior grain. This netic field can be compared with the one in zero magnetic field shifts the sources and sinks along the surface, as compared with displayed in Fig. 2d, h. In the presence of the magnetic field, there the regular chain for zero field in Fig. 1, and moves the saddle is an imbalance between positive and negative fluxes, while in points to the surface region. zero external magnetic field, the total induced flux integrated over The superflow pattern of sources, sinks, and saddle points the grain area is zero. satisfy a certain sum rule related to the topology of the sample. This relation also ties the special points of the p field on the edge of the sample with critical points in its bulk. The generalized 33,34 Topology of the superfluid momentum vector field. Let us Poincaré-Hopf theorem for manifolds with boundaries quantify the symmetry-broken phase in a magnetic field by connects the properties of a vector field v inside a manifold M, plotting the superfluid momentum defined in Eq. (1), see Fig. 3. and on its boundary ∂M, with the Euler characteristic of the For T ≳ T (parameter set II), the amplitude of p varies along the manifold χ(M). Using the formulation presented in ref. ,we edge (coordinate x), see Fig. 3a, reflecting the varying para- write magnetic response in Fig. 2b, f. For T < T (parameter set III), the hi sources and sinks have appeared pairwise together with a saddle ð3Þ Ind ðvÞþ Ind ðv Þ Ind ðv Þ ¼ χðMÞ; M ∂ M jj ∂ M jj 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 where Ind (v) is the total Poincaré index of critical points of the chain of sources, sinks, and saddle points are well established and field v internal to M, Ind ðv Þ is the total Poincaré index of ∂ M jj NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 3 | | | y/ 0 2 2 2 2 –6 –6 –6 –6 B / ( / ) (×10 ) B / ( / ) (×10 ) B / ( / ) (×10 ) B / ( / ) (×10 ) ind 0 0 ind 0 0 ind 0 0 ind 0 0 j /j j /j j /j j /j d d d d ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y 0.016 7 60 60 (I) : T = 0.182T > T *, B = 0.5B ae (I) : T = 0.182T > T *, B = 0.5B c ext g1 c ext g1 40 40 0 0 0 –7 0.027 7 60 > 60 > (II) : T = 0.176T T *, B = 0.5B (II) : T = 0.176T T *, B = 0.5B c ~ ext g1 f c ~ ext g1 40 40 20 20 0 0 0 –7 0.062 7 < < c (III) : T = 0.17T T *, B = 0.5B g (III) : T = 0.17T T *, B = 0.5B c ext g1 c ext g1 40 40 20 20 0 0 –7 0.056 60 60 < < (IV) : T = 0.17T T *, B = 0 (IV) : T = 0.17T T *, B = 0 d c ext c ext 40 40 20 20 0 0 0 –7 020 40 60 020 40 60 x / x / 0 0 Fig. 2 Spontaneously formed currents and induced magnetic field. a–d Total current magnitude and e–h induced magnetic flux density for different temperatures and external fields (see annotations). Lines and arrows have been added to illustrate the flow of the currents critical points of the tangent vector v || ∂M on the boundary. The (∂ M) / outside (∂ M)of M, come with positive/negative signs. || − + theorem applies when the boundary ∂M does not go through any In Supplementary Note 1, we demonstrate in detail how the sum critical points of v. Boundary indices where field points inside rule works for the vector field in Fig. 1, redrawn as a streamline 4 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | y/ y/ y/ y/ 0 0 0 0 y/ y/ y/ y/ 0 0 0 0  p /k T  p /k T  p /k T F s B c F s B c F s B c NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE plot in Supplementary Fig. 2. We also provide other examples of loop currents . We therefore conclude that the edge loop-current grain geometries in Supplementary Figs. 3–10. We utilize the sum phase established for T < T should survive into the mixed state, rule as a tool to verify that the calculations are correct. but a complete investigation of the geometry-dependent phase In a magnetic field, as in Fig. 3, a motif with one edge source, diagram for large fields is beyond the scope of this paper. one edge sink, and one saddle point annihilate at T . In the same fashion, increasing the magnetic field strength, the motif gets Induced currents and magnetic fields. Let us investigate further smaller as the defects are forced toward each other to match the how the currents and magnetic fields are induced at T .Aswe superflow in the bulk. However, the magnitude of p near the s have seen, the paramagnetic response and the spontaneously surface due to Meissner screening of the bulk is not large enough appearing edge loop currents compete, as they both lead to shifts to force an annihilation of the motifs. The broken symmetry of midgap Andreev states. As the temperature is lowered, the phase therefore survives the application of an external magnetic strength of the paramagnetic response increases slowly and lin- field within the whole Meissner state, b ∈ [0, 1]. early, while the strength of the loop currents increases highly For higher fields, when Abrikosov vortices start to enter the non-linearly. This is illustrated in Fig. 4, by plotting the area- grain, the problem quickly becomes complicated by the interplay averaged current magnitude of the Abrikosov vortex lattice formation and finite grain size effects. The free energy landscape is very flat and it is possible to j ¼ d Rjj jðRÞ ; ð4Þ find multiple metastable configurations. For a variety of grain sizes and magnetic field strengths, we have established coex- istence of Abrikosov vortices and the spontaneously formed edge as a function of temperature for the cases when B = 0 (solid ext line), B = 0.5B (dashed line), and for comparison also for a ext g1 system without pair-breaking edges having only a diamagnetic Table 1 Sets of parameters used for presenting results response at B = 0.5B (dash-dotted line). The paramagnetic ext g1 response is fully suppressed at low temperatures T < T . Such a Set Temperature External magnetic field sudden disappearance of the paramagnetic response at a tem- (I) T = 0.182T > T* B = 0.5B c ext g1 perature T should be experimentally measurable, for example in (II) T = 0.176T ≳ T* B = 0.5B c ext g1 36,37 the penetration depth or by using nano-squids . (III) T = 0.17T < T* B = 0.5B c ext g1 We show in Fig. 5a the total induced magnetic flux through the (IV) T = 0.17T < T* B = 0 c ext grain The field scale B = Φ /A corresponds to an external magnetic flux through the grain area g1 0 exactly equal to one flux quantum Φ ¼ d RB ðRÞ; ð5Þ ind ind 0.06 T = 0.176T T *, B = 0.5B c ~ ext g1 0 0.03 0.09 T = 0.17T < T *, B = 0.5B b c ext g1 0 0 0.28 T = 0.1T < T *, B = 0.5B c ext g1 4 ⎥ 0 0 20 25 30 35 40 x / Fig. 3 Superfluid momentum for varying temperature. a The superfluid momentum induced in an external magnetic field of B = 0.5B for a temperature ext g1 slightly above the transition temperature T reflects the paramagnetic response. b At the phase transition, source–sink–saddle-point motifs appear and separate along the edge breaking translational invariance along the edge coordinate x. c For lower temperature, the magnitude p grows large. Note that different color scales are used in the subfigures in order to enhance visibility NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 5 | | | y / y / y / 0 0 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y 2 Phase transition and thermodynamics. The sudden changes B = 0 ext with a discontinuity in the derivative as a function of temperature B = 0.5B ext g1 of the total induced current, the magnetic flux, as well as the B = 0.5B , (no p.b.) ext g1 Loop currents order parameter (Figs. 4 and 5) indicate that there is a phase SC transition occurring at the temperature T . In zero external (III) magnetic field, there is a second-order phase transition at T , (II) (IV) where both time-reversal symmetry and continuous translational (I) symmetry along the edge are spontaneously broken . Let us now investigate the thermodynamics in an external magnetic field 0.12 0.15 0.18 0.21 0.24 already explicitly breaking time-reversal symmetry. T /T In Fig. 6a, we plot the free energy difference between the superconducting and normal states Ω − Ω ,defined in Eq. (29), S N Fig. 4 Current as a function of temperature. The area-averaged current for external field B = 0.5B (red dashed line) and for zero field g1 magnitude, defined in Eq. (4), is plotted for zero external magnetic field (solid black line). For comparison, we show the free energy (solid line), with an external magnetic field of magnitude B = 0.5B ext g1 difference for a purely real order parameter in zero field (gray fine (dashed line), and for a system without pair-breaking edges at B = 0.5B ext g1 line), i.e., without the symmetry breaking edge loop currents. For (dash-dotted line). In the latter case, the system only displays a * T < T , this solution is not the global minimum of the free energy, diamagnetic response. Letters (I)–(IV) indicate the parameter values and we therefore refer to it as a metastable state. To enhance the corresponding to the fields in Fig. 2, see Table 1 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 − Ω . The small slope in S ms the red dashed line at T > T in Fig. 6b is caused by the shift of ab ×0.4 No p.b. midgap Andreev states due to the paramagnetic response, which B = 0.5B ext g1 increases as T decreases. The phase transition temperature T for 6 1.36 –4 6 × 10 the second-order phase transition can be identified with the ×0.9 5 “knee” in the entropy difference defined in Eq. (31), see Fig. 6c, d. Since time-reversal symmetry is already explicitly broken by the external magnetic field, the phase transition signals breaking of local continuous translational symmetry and establishment of the 3 1.35 vector field p 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 second-order phase transitions, p (T) ∝ 1 s * β (1 − T/T ) with β = 1/2, as shown in the inset of Fig. 6d. 0 1.34 However, the temperature range within which the scaling law 0.12 0.18 0.24 0.12 0.18 0.24 holds is very limited and non-linear terms play an important role T /T T /T c c for lower temperatures T < T . Fig. 5 Magnetic flux as a function of temperature. a Temperature The knee in the entropy leads to a jump in the specific heat, as dependence of the induced magnetic flux, defined in Eq. (5). The solid lines shown in Fig. 6e, f. The heat capacity is expressed in units of the indicate, from bottom to top (colors purple to red), the external field heat capacity jump at the normal-superconducting phase magnitude from B =0to B = 0.5B in steps of 0.05B . The line ext ext g1 g1 transition at T for a bulk d-wave system corresponding to zero field lies exactly at zero since there is an equal 2α amount of positive and negative fluxes induced in this case, see Fig. 2. Panel 2 ΔC ¼ Ak T N ; ð7Þ d B c F b shows the area-averaged order parameter magnitude defined in Eq. (6) versus temperature. Results are also shown for a system without pair- breaking edges (dash-dotted line) at B = 0.5B , but scaled with a factor ext g1 where α = 8π /[7ζ(3)], with ζ being the Riemann-zeta function. 0.4 and 0.9 in a and b, respectively The jump in heat capacity at the phase transition is an edge-to- area effect, and grows linearly as the sample becomes smaller. The jump is roughly 4.5% of ΔC for the mesoscopic A¼ 60 ´ 60ξ grain considered here, and grows as the size of the grain is and in Fig. 5b the area-averaged order parameter magnitude reduced. The phase transition temperature T is extracted as a function of B as the midpoint temperature of the jump in the 1 ext Δ ¼ d Rjj Δ ðRÞ ; ð6Þ d d 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 both as functions of temperature for different values of B . The extracted as the minimum (the “kink”, see Fig. 5a) in the induced ext figures also show results for a d-wave grain without pair-breaking flux. The small lowering of T with increased B is caused by the ext edges at B = 0.5B (dash-dotted line). For better visibility, the competing paramagnetic response. ext g1 latter results have been scaled by a factor 0.4 and 0.9 in (a) and From the above, it is clear that the phase with edge loop (b), respectively. Two different trends are distinguishable in the currents shows extreme robustness against an external magnetic * * observables for T < T and T > T , separated by a “kink”. The field in the whole Meissner region (B ≤ B ). The magnitude of ext g1 induced magnetic flux through the grain area decreases as T the spontaneously formed superfluid momentum p at the edge * * decreases down to T due to the increasing paramagnetic grows non-linearly to be very large for T < T , fueled by the response that competes with the diamagnetic one. At T , the lowering of the free energy by Doppler shifts of the flat band of inhomogeneous edge state appear and starts competing with the Andreev surface states. The corresponding correction to p , due paramagnetic response. Thus, the total magnetic flux increases to the process of screening of the external magnetic field, is in again. At the same time, the order parameter is partially healed. comparison small. Thereby, T is not dramatically shifted in a 6 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | –3 / (×10 ) ind 0 –2 j /j (×10 ) /  T d B c NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE ac e –0.07 –0.08 –0.09 –0.10 –0.11 –0.12 –0.13 –0.14 –0.15 0.15 0.18 0.21 0.15 0.18 0.21 0.15 0.18 0.21 T /T T /T T /T c c c b d f 0.05 0.04 ~√ 1 – T /T * 0.03 sx 0.02 B = 0, (ms) ext sy B = 0 ext 0.01 B = 0.5B ext g1 0.17 T /T 0.18 0.15 0.18 0.21 0.15 0.18 0.21 0.15 0.18 0.21 T /T T /T T /T c c c Fig. 6 Thermodynamics and phase transition. a, b free energy, c, d entropy, and e, f specific heat capacity, versus temperature. The lines correspond to a system with purely real order parameter without edge currents (gray fine line), a system with spontaneous edge currents in zero magnetic field (black solid line), and in a finite external field B = 0.5B (red dashed line). In the lower panels b, d, and f, the quantities have been subtracted by the corresponding g1 values of the system with a purely real order parameter, the metastable (ms) state. The heat capacity is normalized by the heat capacity jump at the normal-superconducting phase transition for a bulk d-wave system, ΔC in Eq. (7). The inset in (d) shows the temperature dependence of the superfluid momentum near T , averaged over a few source–sink unit cells at one edge. It follows the expected temperature dependence for the order parameter at a mean-field second-order phase transition Hamiltonian for the other scenarios would have to be sufficiently large in order to compete. It is even possible that one or another Loop currents scenario wins in different parts of the material’s phase diagram . 0.8 SC We note that the phase transition at T means that the initially T * from ΔC 0.6 topologically protected flat band of zero energy surface states is T * from B ind shifted away from the Fermi energy. Such fragility of topologically 0.4 protected states has been studied recently e.g., for topological insulators supporting the quantum spin-Hall state. In that case, 0.2 an edge reconstruction due to Coulomb interactions leads to breaking of time-reversal symmetry. In the d-wave super- 0.15 0.16 0.17 0.18 0.19 0.20 0.21 conductor case, although the bulk Hamiltonian still maintains T *(B )/T ext c required symmetries, a local instability at the surface violates * these symmetries spontaneously and moves the flat band of Fig. 7 Phase diagram. The transition temperature T to a state with bound states to finite energies. The spontaneously broken trans- spontaneously broken continuous translational symmetry is plotted as a lational symmetry allows for a larger shift from zero energy and a function of the external magnetic flux density. The crosses show T high T . extracted from the jump in the specific heat in Fig. 6e, while the open From an experimental point of view, the surface physics of d- circles show T extracted from the minimum of the total induced magnetic wave superconductors is complicated by, for instance, surface flux in Fig. 5a 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 experi- magnetic field and the symmetry-broken phase below T is ments, in agreement with the expectation for d-wave symmetry of robust. 6,7 the order parameter, as reviewed in refs. . 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 Discussion establishment of a time-reversal symmetry breaking phase Which of the scenarios outlined in the introduction wins will 40,41 remains under discussion, see for instance refs. . Several ultimately depend on the material properties of a specific high- 42–44 tunneling experiments on YBCO show a split of the zero- temperature superconducting sample, or the material properties 45,46 38 bias conductance peak, while others do not . Other probes of other candidate d-wave superconductors, e.g., FeSe . In the indicating time-reversal symmetry breaking include thermal scenario studied here, the resulting transition temperature is high, 47 5 * conductivity , Coulomb blockade in nanoscale islands , and T ~ 0.18T . It means that the interaction terms in the NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 7 | | | B /B ext g1 –  (a.u.)  –  (a.u.) S ms S N S – S (a.u.) S – S (a.u.) S ms S N p (a.u.) (C – C ) /ΔC S N d (C – C ) /ΔC S ms d 2 –6 B / ( / ) (×10 ) ind 0 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y STM tunneling at grain boundaries in FeSe . As we argued in 60 6 26,27 b aˆ refs. within the scenario with spontaneous loop currents, the split of the Andreev band might be difficult to resolve in a tun- neling 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 temperature-independent, 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 ext magnetic field, that induces screening supercurrents, and the 0 –6 phase transition at T into a state with the spontaneous loop 020 40 60 currents at the edges. We have shown that the phase should be x / quantified in terms of its order parameter, the vector field p (R), Fig. 8 Grain geometry. The system consists of a d-wave superconducting which contains edge sources and sinks, as well as saddle points. grain exposed to an external magnetic field B = B b z. The crystal ab- At all these critical points, ∇ × p ≠ 0. The p vector field drives ext s s ext axes are rotated 45° relative to the grain edges, inducing pair breaking at the loop currents with opposite circulations in neighboring loops. the edges of the system. The color scale shows the magnetic field B The loop-current strength increases highly non-linearly, sup- ind induced in response to an external field of size B = Φ /2A at a pressing the paramagnetic response present for T > T . As the ext 0 temperature T = 0.2T . There is a diamagnetic response carried by the strength of the external magnetic field increases, the size of the c condensate in the interior, and a paramagnetic response carried by midgap Doppler shift due to the paramagnetic response grows linearly. surface Andreev states at the edges Therefore, T decreases slightly as the magnitude of the external field increases. The influence of the external field, and in parti- 52–54 cular the sudden disappearance of the paramagnetic response, Quasiclassical theory. We utilize the quasiclassical theory of superconductivity , 55–58 which is a theory based on a separation of scales . For instance, the atomic scale is leads to observables which we argue should be visible in experi- assumed small compared with the superconducting coherence length, h=p  ξ . * F 0 ment. For example the “kink” in the total induced flux at T . The This separation of scales makes it possible to systematically expand all quantities in magnetic fluxes induced by the loop currents should be directly small parameters such as ħ/p ξ , Δ/ϵ ,and k T /ϵ ,where Δ is the superconducting F 0 F B c F 36,37 observable with recently developed scanning probes , and the order parameter, p is the Fermi momentum, and ϵ is the Fermi energy. In equili- brium, the central object of the theory is the quasiclassical Green’sfunction sudden disappearance of the paramagnetic response should be ^gðp ; R; zÞ, which is a function of quasiparticle momentum on the Fermi surface p , observable with nano-SQUIDS and possibly in penetration-depth the quasiparticle center-of-mass coordinate R, and the quasiparticle energy z.The + + experiments. Furthermore, the large jump in heat capacity at the latter is real z= ϵ + i0 with an infinitesimal imaginary part i0 for the retarded phase transition should be observable with nanocalorimetry . Green’s function, or an imaginary Matsubara energy z= iϵ = iπk T(2n+ 1) in the Matsubara technique (n is an integer). To keep the notation compact, the dependence The identification of the order parameter p (R), with its on the parameters p , R,and z will often not be written out. The hat on ^g denotes topological textures, leads to similarities with other systems, Nambu (electron-hole) space 33 49 50 including general relativity , fluid dynamics , liquid crystals , gf 3 51 and superfluid He . An interesting difference is that in those ^g ¼ ; ð10Þ f g systems, there is typically a transition in a preexisting vector field to a state with topological textures. Here, instead, we have a where g and f are the quasiparticle and pair propagators, respectively. The tilde singlet d-wave superconductor that spontaneously establishes operation denotes particle-hole conjugation p (R) with topological textures different than the traditional α ~ðp ; R; zÞ¼ α ðp ; R; z Þ: ð11Þ F F Abrikosov vortices. The quasiclassical Green’s function is parameterized in terms of two scalar coherence 59–65 functions, γ(p , R; z)and ~γ(p , R; z), as Methods F F Model and grain geometry. Our aim is to investigate the ground state of clean 1  γ~γ 2γ iπ mesoscopic d-wave superconducting grains in an external magnetic field applied ^g ¼ : ð12Þ 1 þ γ~γ 2~γ 1 þ γ~γ perpendicular to the crystal ab-plane, as shown in Fig. 8. As a typical geometry, we consider a square grain with side lengths D = 60ξ , where ξ = ħv /(2πk T ) is the 0 0 F B c Note that with this parameterization, the Green’s function is automatically normalized zero-temperature superconducting coherence length. Here, v is the normal state 2 2 to ^g ¼π 1. The coherence functions obey two Riccati equations: Fermi velocity, and k the Boltzmann constant. The sides of the system are assumed to be misaligned by a 45° rotation with respect to the crystal ab-axes, e ðihv  ∇ þ 2z þ 2 v  AÞγ ¼Δγ  Δ; ð13Þ F F inducing maximal pair-breaking at the edges. The external field is directed perpendicular to the xy-plane, ~ ~ ðihv  ∇  2z  2 v  AÞγ ¼Δγ  Δ; ð14Þ F F B ¼B b zjjbc: ð8Þ ext ext c where A is the vector potential. These first-order non-linear differential equations are We shall consider rather small external fields, and will use a field scale B = g1 solved by integration along straight (ballistic) quasiparticle trajectories. Quantum Φ /A, corresponding to one flux quantum threading the grain of area A = D = 0 coherence is retained along these trajectories, but not between neighboring trajec- 60ξ ×60ξ . The flux quantum Φ = hc/(2|e|) is given in Gaussian CGS units. The 0 0 0 tories. A clean superconducting grain in vacuum is assumed by imposing the field B is larger than the lower critical field B / Φ =λ , where vortices can enter g1 boundary condition of perfect specular reflection of quasiparticles along the edges of c1 0 0 a macroscopically large superconductor, since the grain side length is smaller than the system. the penetration depth. We assume that λ = 100ξ , relevant for YBCO. The upper 0 0 The superconducting order parameter is assumed to have pure d-wave critical field B / Φ =ξ is much larger than any field we include in this study. To c2 0 0 symmetry be precise, we parameterize the field strength as Δðp ; RÞ¼ Δ ðRÞη ðθÞ; ð15Þ F d d ð9Þ B ¼ bB ; B  ; ext g1 g1 where θ is the angle between the Fermi momentum p and the crystal b a-axis, and η (θ) is the d-wave basis function: pffiffiffi ð16Þ η ðθÞ¼ 2cosð2θÞ; and we will consider b ∈ [0, 1]. d 8 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | y / 0 NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE fulfilling the normalization condition and put that into the Riccati equation. We obtain dθ 2 ihv  ∇ þ 2ðz  v  p Þ γ ¼jj Δ η ðγ þ 1Þ; ð26Þ η ðθÞ ¼ 1: ð17Þ F F s 0 d d 0 2π where p is defined in Eq. (1). The order parameter amplitude satisfies the gap equation dθ Observables. The current density is computed within the Matsubara technique Δ ðRÞ¼ λ N k T η ðθÞf ðp ; R; ϵ Þ; d d F B d F n ð18Þ 2π through the formula ϵ Ω jj n c dθ jðRÞ¼ 2πeN k T v gðp ; R; ϵ Þ: F B F F n ð27Þ where λ is the pairing interaction, N is the density of states at the Fermi level in 2π d F the normal state, and Ω is a cutoff energy. The pairing interaction and the cutoff energy are eliminated in favor of the superconducting transition temperature T In the results section, we shall show this current density in units of the depairing (see for example ref. )as current 1 T 1 ¼ ln þ : j  4πjj e k T N v : ð28Þ ð19Þ d B c F F λ N T n þ d F c n0 2 The free-energy difference between the superconducting and the normal states The above equations are solved self-consistently with respect to γ, ~γ, and Δ .As d is calculated with the Eilenberger free-energy functional an initial guess, we assume a homogenous superconductor with a small modulation R 2 B ðRÞ 2 ind of the phase. The coherence functions on the boundaries have to be updated in Ω ðB; TÞ Ω ðB; TÞ¼ dR þjj Δ ðRÞ N ln S N 8π d F T each iteration, taking into account the specular boundary condition. The starting ) hi ð29Þ P 2 guess is the local homogeneous solution. After several iterations, the information of jj Δ ðRÞ þ2πN k T þ iIðR; ϵ Þ ; F B ϵ n the initial guess for the coherence functions is lost . n ϵ >0 We choose an electromagnetic gauge where the vector potential has the form dθ ð20Þ A ðRÞ¼ B ´ R: ext ext IðR; ϵ Þ¼ Δðp ; RÞγðp ; R; ϵ Þ Δðp ; RÞ~γðp ; R; ϵ Þ : ð30Þ 2 n F F n F F n 2π The total vector potential A(R), that enters Eqs. (13) and (14), is given by A (R) ext We have verified that this form of the free energy gives the same results as the and the field A (R) induced by the currents j(R) in the superconductor (Eq. (27) ind 26,55,64 Luttinger-Ward functional . The entropy and specific heat capacity are below): obtained from the thermodynamic definitions AðRÞ¼ A ðRÞþ A ðRÞ: ð21Þ ext ind ∂Ω S ¼ ; ð31Þ ∂T The vector potential A (R) should be solved from Ampère’s circuit law ind 4π 2 ∂S ∂ Ω ð22Þ ∇ ´ ∇ ´ A ðRÞ¼ jðRÞ ; ind ð32Þ C ¼ T ¼T : ∂T ∂T with appropriate boundary conditions for the induced field inside and outside the sample. To take the full electrodynamics into account, A (R) also needs to be ind Data availability. All relevant data are available from the authors. computed self-consistently in each iteration. However, the strength of the −2 electrodynamic back-coupling scales as κ , where κ ≡ λ /ξ is the dimensionless 0 0 Ginzburg-Landau parameter. The electrodynamic back-coupling is therefore a very Received: 13 December 2017 Accepted: 1 May 2018 −1 −2 small effect for type II superconductors (typically κ ≈ 10 for the cuprates). We have verified through fully self-consistent calculations that for grains with side lengths D < λ, as we limit ourselves to in this paper, it is always safe to neglect this back-coupling. For large system sizes, D λ, back-coupling 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 inter-vortex distances of order λ for moderate fields b > 1, corresponding to field strengths of order H . References c1 Since the spontaneous fields appearing below T are located within a small distance 1. Gol'tsman, G. N. et al. Picosecond superconducting single-photon optical of order ξ  λ from the boundary, the effect of back-coupling is small also in detector. Appl. Phys. Lett. 79, 705–707 (2001). these cases. Only in very high fields, approaching H , where inter-vortex distances c2 2. 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Search for broken time-reversal symmetry near the surface P.H. carried out the numerical calculations. P.H., A.B.V., M.F., and T.L. analyzed the of superconducting YBa Cu O films using β-detected nuclear magnetic 2 3 7−δ results. P.H. and T.L. wrote the paper with contributions from A.B.V. and M.F. resonance. Phys. Rev. B 83, 054504 (2011). 10 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04531-y ARTICLE Additional information Open Access This article is licensed under a Creative Commons Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- Attribution 4.0 International License, which permits use, sharing, 018-04531-y. adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Competing interests: The authors declare no competing interests. Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless Reprints and permission information is available online at http://npg.nature.com/ indicated otherwise in a credit line to the material. If material is not included in the reprintsandpermissions/ article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in the copyright holder. To view a copy of this license, visit http://creativecommons.org/ published maps and institutional affiliations. licenses/by/4.0/. © The Author(s) 2018 NATURE COMMUNICATIONS (2018) 9:2190 DOI: 10.1038/s41467-018-04531-y www.nature.com/naturecommunications 11 | | |

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