A growing number of superconducting materials display evidence for spontaneous time-reversal symmetry breaking (TRSB) below their critical transition temperatures. Precisely what this implies for the nature of the superconducting ground state of such materials, however, is often not straightforward to infer. We review the experimental status and survey different theoretical mechanisms for the generation of TRSB in superconductors. In cases where a TRSB complex combination of two superconducting order parameter components is realized, defects, dislocations and sample edges may generate superflow patterns that can be picked up by magnetic probes. However, even single-component condensates that do not break time-reversal symmetry in their pure bulk phases can also support signatures of magnetism inside the superconducting state. This includes, for example, the generation of localized orbital current patterns or spin-polarization near atomic-scale impurities, twin boundaries and other defects. Signals of TRSB may also arise from a superconductivity-enhanced Ruderman-Kittel-Kasuya-Yosida exchange coupling between magnetic impurity moments present in the normal state. We discuss the relevance of these different mechanisms for TRSB in light of recent experiments on superconducting materials of current interest.
We argue that that in their recent experiment in which they claim to have found evidence for a time-reversal symmetry broken state, Kaminski et al. overlooked small temperature dependent changes in the superstructure of Bi2212. These subtle changes may manifest themselves by changing the final state configurations of the photoemission process and thus invalidate their ultimate conclusions.
We study the behavior of spinless fermions in superconducting state, in which the phases of the superconducting order parameter depend on the direction of the link. We find that the energy of the superconductor depends on the phase differences of the superconducting order parameter. The solutions for the phases corresponding to the energy minimuma, lead to a topological superconducting state with the nontrivial Chern numbers. We focus our quantitative analysis on the properties of topological states of superconductors with different crystalline symmetry and show that the phase transition in th…
The robustness against local perturbations, as long as the symmetry of the system is preserved, is a distinctive feature of topological quantum states. Magnetic impurities and defects break time-reversal invariance and, consequently, time-reversal invariant (TRI) topological superconductors are fragile against this type of disorder. Non-magnetic impurities, however, preserve time-reversal symmetry and one naively expects a TRI topological superconductor to persist in the presence of non-magnetic impurities. In this work, we study the effect of non-magnetic disorder on a TRI topological superco…
Recent experiments have pointed to a chiral p-wave-like superconductivity in epitaxial Bi/Ni bilayers that are spontaneously time-reversal symmetry breaking (TRSB), making it a promising platform for exploring physics useful for topologically protected quantum computing. Quite intriguingly, evidence has emerged that in non-epitaxial Bi/Ni bilayers, superconductivity arises due to the formation of NiBi3, which has been reported to host coexisting ferromagnetic and superconducting orders at the surface. We perform high resolution surface magneto-optic Kerr effect (SMOKE) measurements using a Sag…
Spatial distributions of spontaneous magnetic fields near a surface of cuprate high-$T_C$ superconductor YBCO with broken time-reversal symmetry are calculated using the Ginzburg-Landau theory derived from the $t-J$ model. It is found that the magnetic field exists in a narrow region inside the superconductor, and it decays quickly outside the surface. Experimental approaches possible to detect such spontaneous magnetic fields are discussed.
We study the boundary Majorana modes for the single component p-wave weak topological superconductors or superfluids, which form zero energy flat bands protected by time-reversal symmetry in the orbital channel. However, due to the divergence of density of states, the band flatness of the edge Majorana modes is unstable under spontaneously generated spatial variations of Cooper pairing phases. Staggered current loops appear near the boundary and thus time-reversal symmetry is spontaneously broken in the orbital channel. This effect can appear in both condensed matter and ultra-cold atom system…
States of matter that break time-reversal symmetry are invariably associated with magnetism or circulating currents. Recently, one of us proposed a phase, the directional scalar spin chiral order (DSSCO), as an exception: it breaks time-reversal symmetry via chiral ordering of spins along a particular direction, but is spin-rotation symmetric. In this work, we prove the existence of this state via state-of-the-art density matrix renormalization group (DMRG) analysis on a spin-1 chain with nearest-neighbor bilinear-biquadratic interactions and additional third-neighbor ferromagnetic Heisenberg …
Broken time reversal symmetry (BTRS) in d wave superconductors is studied and is shown to yield current carrying surface states. The corresponding spontaneous magnetization is temperature independent near the critical temperature Tc for weak BTRS, in accord with recent data. For strong BTRS and thin films we expect a temperature dependent spontaneous magnetization with a paramagnetic anomaly near Tc. The Hall conductance is found to vanish at zero wavevector q and finite frequency w, however at finite q,w it has an unusual structure.
In a recent comment [1], Armitage and Hu have suggested that our experiment observing dichroism in angle resolved photoemission (ARPES) [2] could not be conclusively interpreted as arising from time reversal symmetry breaking, arguing that our observations are likely due to structural effects. The concerns expressed by Armitage and Hu that our results could be due to a change in the mirror plane are as important as they are obvious. In fact the first part of their comment merely restates the results of Simon and Varma [3] about the relationship and contrast of effects due to time reversal symm…
We present a study of the basic microscopic model of an $s$-wave superconductor with frustrated interband interaction. When frustration is strong, such an interaction gives raise to a $s+is$ state. This is a $s$-wave superconductor that spontaneously breaks time reversal symmetry. We show that, in addition to the known $s+is$ state, there is additional phase, where the system's bulk is a conventional $s$-wave state, but superconducting surface states break time reversal symmetry. Furthermore, we show that $s+is$ superconductors can have spontaneous boundary currents and spontaneous magnetic fi…
We study the physical properties of an $s$-wave -- normal metal -- $d$-wave junction in terms of the Andreev bound states in the normal metal layer. The phase dependence of bound states with different orientations leads to superconducting states with broken time reversal symmetry for generic orientations of the $d$-wave superconductor crystal. The occurrence of such a state and the associated spontaneous supercurrent along the junction is analyzed in the framework of Ginzburg-Landau theory and by the solution of the Bogolyubov-de Gennes equations.
Broken time reversal symmetry (BTRS) in d+id' as well as in d+is superconductors is studied and is shown to yield current carrying surface states. We evaluate the temperature and thickness dependence of the resulting spontaneous magnetization and show a marked difference between weak and strong BTRS. We also derive the Hall conductance which vanishes at zero wavevector q and finite frequency w, however at finite q,w it has an unusual structure. The chirality of the surface states leads to quantum Hall effects for spin and heat transport in d+id' superconductors.
Conventional superconductors are strong diamagnets that through the Meissner effect expel magnetic fields. It would therefore be surprising if a superconducting ground state would support spontaneous magnetics fields. Such time-reversal symmetry broken states have been proposed for the high-temperature superconductors, but their identification remains experimentally controversial. Here we show a route to a low-temperature superconducting state with broken time-reversal symmetry that may accommodate currently conflicting experiments. This state is characterised by an unusual vortex pattern in t…
It is argued that the $d_{x^2-y^2}$ superconductor exhibits the transition into $d_{x^2-y^2} + id_{xy}$ state in the presence of magnetic impurities in certain conditions. Linear coupling between impurity magnetization and the $d_{xy}$ component drives this transition. Violation of time-reversal symmetry and parity occurs spontaneously via the second order transition. In the ordered phase both impurity magnetization and $d_{xy}$ component of the order parameter develop and are proportional to each other.
We study theoretically the spatial distribution of spontaneous magnetic fields near a (110) surface of cuprate high-$T_C$ superconductor YBCO by treating a bilayer $t-J$ model with the Bogoliubov-de Gennes method. Near a (110) surface where $d_{x^2-y^2}$-wave superconductivity is strongly suppressed, flux phase order occurs locally leading to a time-reversal symmetry broken surface state. Since the spontaneous currents flowing in the flux phase in two layers are antiparallele, resulting spontaneous magnetic fields are quite small. They exist in a narrow region inside the superconductor, and es…
Recently, a theoretical study [Li {\it et al.},~npj Quantum Materials 6, 36 (2021)] investigated a model of a disordered $d$-wave superconductor, and reported local time-reversal symmetry breaking current loops for sufficiently high disorder levels. Since the pure $d$-wave superconducting state does not break time-reversal symmetry, it is surprising that such persistent currents arise purely from nonmagnetic disorder. Here we perform a detailed theoretical investigation of such disorder-induced orbital currents, and show that the occurrence of the currents can be traced to local extended $s$-w…
We propose a mechanism that helps stabilize a superconducting state with broken time-reversal symmetry, which was predicted to realize in a d-wave superconducting film [A. B. Vorontsov, Phys. Rev. Lett. 102, 177001 (2009)]. In this superconducting phase, the time-reversal symmetry breaking is accompanied by spontaneous breaking of the translation symmetry along the film surface. We examine how the normal-superconducting phase boundary in the thickness-temperature phase diagram of the film is modified depending on the Fermi surface shape. In particular, the nonuniform superconducting phase is f…
Superconductivity and magnetism are antagonistic states of matter. The presence of spontaneous magnetic fields inside the superconducting state is, therefore, an intriguing phenomenon prompting extensive experimental and theoretical research. In this review, we discuss recent experimental discoveries of unconventional superconductors which spontaneously break time-reversal symmetry and theoretical efforts in understanding their properties. We discuss the main experimental probes and give an extensive account of theoretical approaches to understand the order parameter symmetries and the corresp…
We present a systematic investigation of muon-stopping states in superconductors that reportedly exhibit spontaneous magnetic fields below their transition temperatures due to time-reversal symmetry breaking. These materials include elemental rhenium, several intermetallic systems and Sr$_2$RuO$_4$. We demonstrate that the presence of the muon leads to only a limited and relatively localized perturbation to the local crystal structure, while any small changes to the electronic structure occur several electron volts below the Fermi energy leading to only minimal changes in the charge density on…
We study surface states of high-$T_C$ cuprate superconductor YBCO using the bilayer $t-J$ model. Calculations based on the Bogoliubov de Gennes method show that a flux phase that breaks time-reversal symmetry (${\cal T}$) may arise near a (110) surface where the $d_{x^2-y^2}$-wave superconductivity is strongly suppressed. It is found that the flux phase in which spontaneous magnetic fields in two layers have opposite directions may be stabilized in a wide region of doping rate, and split peaks in the local density of states appear. Near the surface, spontaneous magnetic field may not be observ…
The induction of an s-wave component in a d-wave superconductor is considered. Near the {110}-oriented edges of such a sample, the induced s-wave order parameter together with d-wave component forms a complex combination d+e^{iϕ} s, which breaks the time reversal symmetry (BTRS) of the pairing state. As a result, the spontaneous current is created. We numerically study the current distribution and the formation of the spontaneous flux induced by the current. We show that the spontaneous flux formed from a number of defect lines with {110} orientation has a measurable strength. This result may …
In systems with broken $U(1)$ symmetry, such as superfluids, superconductors or magnets, the symmetry restoration is driven by proliferation of topological defects in the form of vortex loops. Here we discuss that in certain systems the proliferation of topological defects can, by contrast, lead to the breakdown of an additional symmetry. As a particular example we demonstrate that this effect should take place in $s+is$ superconductors, which are widely discussed in connection with the Iron-based materials. In these systems a vortex excitation can create a "bubble" of fluctuating $Z_2$ order …
It is shown that the patches of complex d_{xy} component are generated around magnetic impurity in the presence of the coupling between orbital moment of the condensate and impurity spin S. Locally induced d_{xy} gap leads to the fully gapped quasiparticle spectrum near impurity. The phase of the induced i d_{xy} component is determined by impurity spin and averages to zero at high temperatures. It is suggested that at low temperature the well defined patches of d_{xy} are formed and they can undergo a phase transition into phase locked state via Josephson effect. Violation of time-reversal sy…
We report the study of spontaneous magnetization (i.e., spin-polarization) for time-reversal symmetry (TRS)-breaking superconductors with unitary pairing potentials, in the absence of external magnetic fields or Zeeman fields. Spin-singlet ($Δ_s$) and spin-triplet ($Δ_t$) pairings can coexist in superconductors whose crystal structure lacks inversion symmetry. The TRS can be spontaneously broken once a relative phase of $\pmπ/2$ is developed, forming a TRS-breaking unitary pairing state ($Δ_s\pm iΔ_t$). We demonstrate that such unitary pairing could give rise to spontaneous spin-polarization w…
I study the effects of disorder on the superconductivity of a three-band model with repulsive interband pairing. Such a model can support several possible superconducting order parameters, including a complex time-reversal symmetry breaking (TRSB) state. Impurity scattering suppresses the critical temperature of all these states, but the complex state survives, and remains a part of the phase diagram of the model even in the presence of moderate amount of disorder. This means that the TRSB states could be experimentally accessible in multiband materials like iron pnictides and chalcogenides.
We report the results of a search for spontaneous magnetism due to a time reversal symmetry breaking phase in the superconducting state of (110)-oriented YBCO films, expected near the surface in this geometry. Zero field and weak transverse field measurements performed using the low-energy muon spin rotation technique with muons implanted few nm inside optimally-doped YBCO-(110) films show no appearance of spontaneous magnetic fields below the superconducting temperature down to 2.9 K. Our results give an upper limit of ~0.02 mT for putative spontaneous internal fields.
Discoveries of ordered quantum states of matter are of great fundamental interest, and often lead to unique applications. The most well known example -- superconductivity -- is caused by the formation and condensation of pairs of electrons. A key property of superconductors is diamagnetism: magnetic fields are screened by dissipationless currents. Fundamentally, what distinguishes superconducting states from normal states is a spontaneously broken symmetry corresponding to long-range coherence of fermion pairs. Here we report a set of experimental observations in hole doped Ba$_{1-x}$K$_x$Fe$_…
We demonstrate the ability to control the spontaneous emission from a superconducting qubit coupled to a cavity. The time domain profile of the emitted photon is shaped into a symmetric truncated exponential. The experiment is enabled by a qubit coupled to a cavity, with a coupling strength that can be tuned in tens of nanoseconds while maintaining a constant dressed state emission frequency. Symmetrization of the photonic wave packet will enable use of photons as flying qubits for transfering the quantum state between atoms in distant cavities.
Noncentrosymmetric superconductors with strong spin-orbit coupling and the B phase of ${}^3$He are possible realizations of topological superconductors with time-reversal symmetry. The nontrivial topology of these time- reversal invariant superconductors manifests itself at the material surface in the form of helical Majorana modes. In this paper, using extensive numerical simulations, we investigate the stability and properties of these Majorana states under strong surface disorder, which influences both bulk and surface states. To characterize the effects of strong disorder, we compute the l…
In this work we present a comprehensive study of the spontaneous currents in time-reversal symmetry breaking (TRSB) multi-component superconductors with cubic crystalline symmetry. We argue, not limiting to cubic lattices, that spontaneous current on certain high-symmetry surfaces can exist only if the TRSB pairing simultaneously breaks a certain pair of mirror symmetries. This is shown to have exact correspondence with the Gingzburg-Landau (GL) theory and is verified by numerical Bogoliubov de-Gennes (BdG) calculations. In the course we extend the BdG to include effects of gap anisotropy and …
We investigate the superconducting gap function of topological superconductor PbTaSe$_2$. Temperature, magnetic field, and three-dimensional (3D) field-angle dependences of the specific heat prove that the superconductivity of PbTaSe$_2$ is fully-gapped, with two isotropic $s$-wave gaps. The pair-breaking effect is probed by systematically increasing non-magnetic disorders through H$^+$-irradiations. The superconducting transition temperature, $T_{\rm{c}}$, is found to be robust against disorders, which suggests that the pairing should be sign-preserved rather than sign-reversed.
We solve the Bogoliubov-de Gennes equations self-consistently for the $d$-wave order parameter in nanoscale $d$-wave systems with [110] surfaces and show that spontaneous time-reversal symmetry (TRS) breaking occurs at low temperatures due to a spontaneously induced complex order parameter of extended $s$-wave symmetry. The Andreev surface bound states, which are protected by a one-dimensional (1D) topological invariant in the presence of TRS, are gapped by the emergence of this new order parameter. The extended $s$-wave order parameter is localized within a narrow region near the surfaces, wh…
One of the mysteries of modern condenced-matter physics is the nature of the pseudogap state of the superconducting cuprates. Kaminski et al.1 claimed to have observed signatures of time-reversal symmetry breaking in the pseudogap regime in underdoped Bi2Sr2CaCu2O8+d (Bi2212). Here we argue that the observed dichroism is due to the 5x1 superstructure replica of the electronic bands and therefore cannot be considered as evidence for the spontaneous time-reversal symmetry breaking in cuprates.
Twin boundaries are generic crystalline defects in noncentrosymmetric crystal structures. We study theoretically twin boundaries in time-reversal symmetric noncentrosymmetric superconductors that admit parity-mixed Cooper pairing. Twin boundaries support spin currents as a consequence of this parity mixing. If the singlet and triplet components of the superconducting order parameter are of comparable magnitude, the superconducting state breaks spontaneously the bulk time-reversal symmetry locally near the twin boundary. By self-consistently evaluating the Bogoliubov-de-Gennes equations and the…
Recent mean-field calculations suggest that the superconducting state of twisted bilayer graphene exhibits either a nematic order or a spontaneous breakdown of the time-reversal symmetry. The two-dimensional character of the material and the large critical temperature relative to the Fermi energy dictate that the material should have significant fluctuations. We study the effects of these fluctuations using Monte Carlo simulations. We show that in a model proposed earlier for twisted bilayer graphene there is a fluctuation-induced phase with quadrupling fermionic order for all considered param…
We solve the Bogoliubov-de Gennes equations self-consistently for d-wave superconductors with [110] surfaces. We find spontaneous breaking of time-reversal symmetry (TRS) caused by the spontaneous occurrence of a complex order parameter with an extended s-wave symmetry along the [110] surfaces. In the TRS-breaking phase, the d-wave order parameter itself becomes complex. We also show that vortex-antivortex pairs can appear along the [110] surfaces of nanoribbons. These pairs are detectable with surface sensitive probes.
Time-reversal symmetry breaking superconductivity in the presence of loop-current fluctuations
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）；Key Laboratory of Multiscale Spin Physics (Beijing Normal University), Ministry of Education（教育部多尺度自旋物理重点实验室（北京师范大学））；Beijing Computational Science Research Center（北京计算科学研究中心）；New Cornerstone Science Laboratory, Department of Physics, School of Science, Westlake University（西湖大学理学院物理系新基石科学实验室）；Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究院）；Key Laboratory for Quantum Materials of Zhejiang Province, School of Science, Westlake University（西湖大学理学院浙江省量子材料重点实验室）；Institute of Natural Sciences, Westlake Institute for Advanced Study（西湖高等研究院自然科学研究院）
We report muon spin relaxation measurements on the superconductor Sr2RuO4 that reveal the spontaneous appearance of an internal magnetic field below the transition temperature: the appearance of such a field indicates that the superconducting state in this material is characterized by the breaking of time-reversal symmetry. These results, combined with other symmetry considerations, suggest that superconductivity in Sr2RuO4 is of p-wave (odd-parity) type, analogous to superfluid 3He.
We propose a superconducting instability where microscopic supercurrent loops form spontaneously within a unit cell at the superconducting transition temperature with only uniform, onsite and intra-orbital singlet pairing. As a result of the circulating currents time-reversal symmetry is spontaneously broken in the superconducting state. Using Ginzburg-Landau theory, we describe in detail how these currents emerge in a toy model. We discuss the crystallographic symmetry requirements to realize such a state and show that they are met by the Re6X (X=Zr, Hf, Ti) family of time-reversal symmetry b…
We report the results of a zero-field muon spin relaxation (ZF-muSR) study of superconducting Ba1-xKxFe2As2 (0.5 < x < 0.9) in search of weak spontaneous internal magnetic fields associated with proposed time-reversal-symmetry breaking mixed pairing states. The measurements were performed on polycrystalline samples, which do not exhibit the mesoscopic phase separation previously observed in single crystals of Ba1-xKxFe2As2. No evidence of spontaneous internal magnetic fields is found in any of the samples at temperatures down to T ~ 0.02 K.
A major challenge to the chiral $p$-wave hypothesis for the pairing symmetry of the unconventional superconductor Sr$_2$RuO$_4$ is the null result of sensitive scanning magnetometry experiments designed to detect the expected spontaneous charge currents. Motivated by junction tunneling conductance measurements which indicate the quenching of superconductivity at the surfaces of even high-purity samples, we examine the spontaneous currents in a chiral $p$-wave superconductor near a normal metal / superconductor interface using the lattice Bogoliubov-de Gennes equations and Ginzburg-Landau theor…
It is widely believed that the perovskite Sr$_2$RuO$_4$ is an unconventional superconductor with broken time reversal symmetry. It has been predicted that superconductors with broken time reversal symmetry should have spontaneously generated supercurrents at edges and domain walls. We have done careful imaging of the magnetic fields above Sr$_2$RuO$_4$ single crystals using scanning Hall bar and SQUID microscopies, and see no evidence for such spontaneously generated supercurrents. We use the results from our magnetic imaging to place upper limits on the spontaneously generated supercurrents a…
We have studied the Zero Bias Conductance Peak (ZBCP) of the tunneling conductance measured on (1,1,0) oriented Y(1-y)Ca(y)Ba(2)Cu(3)O(7-x) thin films as a function of doping and of magnetic field. A spontaneous (zero field) split of the ZBCP was observed only in overdoped samples (either by O or by Ca). The magnitude of this split was found to be linear in doping. All samples exhibited a magnetic field splitting, also strongly doping dependent. The field susceptibility chi=d(delta)/dH diverges at the point at which spontaneous ZBCP splitting occurs, its inverse value, chi^(-1), following a li…
We present an exactly solvable model of a spin-triplet $f$-wave topological superconductor on the honeycomb lattice in the presence of the Hubbard interaction for arbitrary interaction strength. First we show that the Kane-Mele model with the corresponding spin-triplet $f$-wave superconducting pairings becomes a full-gap topological superconductor possessing the time-reversal symmetry. We then introduce the Hubbard interaction. The exactly solvable condition is found to be the emergence of perfect flat bands at zero energy. They generate infinitely many conserved quantities. It is intriguing t…
Superconductivity that spontaneously breaks time-reversal symmetry (TRS) has been found, so far, only in a handful of 3D crystals with bulk inversion symmetry. Here we report an observation of spontaneous TRS breaking in a 2D superconducting system without inversion symmetry: the epitaxial bilayer films of bismuth and nickel. The evidence comes from the onset of the polar Kerr effect at the superconducting transition in the absence of an external magnetic field, detected by the ultrasensitive loop-less fiber-optic Sagnac interferometer. Because of strong spin-orbit interaction and lack of inve…
We measure the local harmonic generation from superconducting thin films at microwave frequencies to investigate the intrinsic nonlinear Meissner effect near Tc in zero magnetic field. Both second and third harmonic generation are measured to identify time-reversal symmetry breaking (TRSB) and time-reversal symmetric (TRS) nonlinearities. We perform a systematic doping-dependent study of the nonlinear response and find that the TRS characteristic nonlinearity current density scale follows the doping dependence of the de-pairing critical current density. We also extract a spontaneous TRSB chara…
We present results of numerical studies of spin quantum Hall transitions in disordered superconductors, in which the pairing order parameter breaks time-reversal symmetry. We focus mainly on p-wave superconductors in which one of the spin components is conserved. The transport properties of the system are studied by numerically diagonalizing pairing Hamiltonians on a lattice, and by calculating the Chern and Thouless numbers of the quasiparticle states. We find that in the presence of disorder, (spin-)current carrying states exist only at discrete critical energies in the thermodynamic limit, …
The off-diagonal components of thermal conductance tensor, thermal Hall conductivities (THCs), have extensively been studied in recent condensed matter experiments to investigate fractionalized quantum spin liquids, and quantum Hall systems. Under zero magnetic field, THCs spontaneously become non-zero for time-reversal symmetry (TRS) broken systems, and can have contributions from topologically protected edge states. Here we focus on an additional bulk effect, the impurity mechanism in TRS broken superconductors. Inspired by $Sr_2 Ru O_4$, the low temperature THC was calculated [Sup. Sci. and…
There is considerable evidence that the superconducting state of Sr$_2$RuO$_4$ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking its onset temperature, $T_\text{TRSB}$, is generally found to match the critical temperature, $T_\text{c}$, within resolution. In combination with evidence for even parity, this result has led to consideration of a $d_{xz} \pm id_{yz}$ order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding $T_\text{TRSB} = T_\text{c}$, but it has a hard-to-explain horizontal line node at…
We show that multiband superconductors with broken time-reversal symmetry can produce spontaneous currents and magnetic fields in response to the local variations of pairing constants. Considering the iron pnictide superconductor Ba$_{1-x}$K$_x$Fe$_2$As$_2$ as an example we demonstrate that both the point-group symmetric $s+is$ state and the C$_4$-symmetry breaking $s+id$ states produce in general the same magnitudes of spontaneous magnetic fields. In the $s+is$ state these fields are polarized mainly in ab crystal plane, while in the $s+id$ state their ab-plane and c-axis components are of th…
We investigate the properties of spontaneous currents generated at surfaces and interfaces of d-wave superconductors using the self-consistent quasiclassical Eilenberger equations. The influence of the roughness and reflectivity of the boundaries on the spontaneous current are studied. We show that these have very different effects at the surfaces compared to the interfaces, which reflects the different nature of the time reversal symmetry breaking states in these two systems. We find a signature of the ``anomalous proximity effect'' at rough d-wave interfaces. We also show that the existence …
