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: email@example.com) 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. 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