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 superconductors 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 ps, 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.

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Background: A new mode of nuclear fission has been proposed by the FOBOS collaboration, called Collinear Cluster Tri-partition (CCT), suggesting that three heavy fission fragments can be emitted perfectly collinearly in low-energy fission. This claim is based on indirect observations via missing-energy events using the 2v2E method. This proposed CCT seems to be an extraordinary new aspect of nuclear fission. It is surprising that CCT escaped observation for so long given the relatively high reported yield, of roughly 0.5% relative to binary fission. These claims call for an independent verification with a different experimental technique.Purpose: Verification experiments based on direct observation of CCT fragments with fission fragment spectrometers require guidance with respect to the allowed kinetic energy range, which we present in this paper. Furthermore, we discuss corresponding model calculations which, if CCT is found in such verification experiments, could indicate how the breakups proceed. Since CCT refers to collinear emission, we also study the intrinsic stability of collinearity.Methods: Three different decay models are used that together span the timescales of three-body fission. These models are used to calculate the possible kinetic energy ranges of CCT fragments by varying fragment mass splits, excitation energies, neutron multiplicities and scission-point configurations. Calculations are presented for the systems 235 U(n th , f) and 252 Cf(sf), and the fission fragments previously reported for CCT, namely isotopes of the elements Ni, Si, Ca and Sn. In addition, we use semi-classical trajectory calculations with a Monte-Carlo method to study the intrinsic stability of collinearity.Results: CCT has a high net Q-value, but in a sequential decay, the intermediate steps are energetically and geometrically unfavorable or even forbidden. Moreover, perfect collinearity is extremely unstable, and broken by the slightest perturbation.Conclusions: According to our results, the central fragment would be very difficult to detect due to its low kinetic energy, raising the question of why other 2v2E experiments could not detect a missingmass signature corresponding to CCT. Considering the high kinetic energies of the outer fragments reported in our study, direct-observation experiments should be able to observe CCT. Furthermore, we find that a realization of CCT would require an unphysical fine-tuning of the initial conditions. Finally, our stability calculations indicate that, due to the pronounced instability of the collinear configuration, a prolate scission configuration does not necessarily lead to collinear emission, nor does equatorial emission necessarily imply an oblate scission configuration. In conclusion, our results enable independent experimental verification and encourage further critical theoretical studies of CCT.

Unconventional d-wave superconductors with pair-breaking edges are predicted to have ground states with spontaneously broken time-reversal and translational symmetries. We use the quasiclassical theory of superconductivity to demonstrate that such phases can exist at any single pair-breaking facet. This implies that a greater variety of systems, not necessarily mesoscopic in size, should be unstable to such symmetry breaking. The density of states averaged over the facet displays a broad peak centered at zero energy, which is consistent with experimental findings of a broad zero-bias conductance peak with a temperature-independent width at low temperatures.