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Microscopic theory of strain-controlled split superconducting and time-reversal symmetry breaking transitions in s+id superconductors

Anton Talkachov1,* and Egor Babaev1,2

  • *Contact author: anton.talkachov@gmail.com
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Phys. Rev. B 113, 144515 – Published 28 April, 2026

DOI: https://doi.org/10.1103/n81x-kx54

Abstract

We study conditions of the appearance of U(1)×Z2 superconducting states that spontaneously break time-reversal symmetry (BTRS) on a square lattice as a function of applied strain. Calculations show that if critical temperatures coincide at zero strain, they exhibit a linear kink and no kink otherwise for uniaxial and isotropic strain. Linear kink is absent for shear strain. We find that in general, the microscopic calculations show a complex phase diagram, for example, nonmonotonic behavior of BTRS transition. Another beyond-Ginzburg-Landau theory result is that U(1) critical temperature can decrease under compressional [100] uniaxial strain for small Poisson ratio materials. In the second part of the paper, we consider the effects of boundaries and finiteness of the sample on the strain-induced splitting of TcU(1) and TcZ2 transitions. A finite sample has BTRS boundary states with persistent superconducting currents over a wide range of band filling. Overall, the BTRS dome occupies a larger band-filling-temperature phase space region for a mesoscopic sample with [110] surface compared to an infinite system. Hence, the presence of boundaries helps to stabilize the BTRS phase.

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Phys. Rev. B 113, 144515– Published 28 April, 2026
  • Received 14 October 2025
  • Revised 30 January 2026
  • Accepted 11 February 2026
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