Broken translational symmetry at edges of high-temperature superconductors

P Holmvall,A B Vorontsov,T Löfwander,M Fogelström

doi:10.1038/s41467-018-04531-y

Abstract

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.

Highlights

Flat bands of zero-energy states at the edges of quantum materials have a topological origin
The split Andreev states carry current along the surface, which results in a magnetic field that is screened from the bulk
We identify the ps vector field as the order parameter of the symmetry-broken phase, motivated by the fact that the free energy is lowered by a large split of the flat band of Andreev states by a Doppler shift vF · ps, where vF is the Fermi velocity

Summary

Introduction

Flat bands of zero-energy states at the edges of quantum materials have a topological origin. The electrons couple to the electromagnetic gauge field A(R), and this mechanism was first considered theoretically for a translationally invariant edge In this case, the transition is a result of the interplay of weakly Doppler shifted surface bound states, decaying away from the surface on the scale of the superconducting coherence length ξ0, and weak diamagnetic screening currents, decaying on the scale of the penetration depth λ. The mechanism does not involve subdominant channels or coupling to magnetic field, but depends on film thickness D, and the transition temperature decays rapidly with increasing thickness as T* ~ (ξ0/D)Tc. In this paper, we consider a peculiar scenario[26,27] where spontaneous supercurrents break translational symmetry along the edge. As we will discuss below, it relies on the development of a texture in the gradient of the d-wave order parameter phase χ, or more precisely in the gauge invariant superfluid momentum psðRÞ

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