Extended Data Fig. 10: Fluctuation-induced phases in multiband models.
From: State with spontaneously broken time-reversal symmetry above the superconducting phase transition
Panels (a)-(d) show results for the two-component approximation of a three-band model in zero external magnetic field in the extreme type-II limit for various values of the mixed gradient coupling ν. (a) Phase diagram. The U(1) and Z2 transitions are clearly split apart for sufficiently large ν, giving rise to the quartic metal phase. The gray dashed line indicates the value ν = 0.6. (b) Binder cumulant U at ν = 0.6 for different system sizes as function of the inverse temperature β. (c) Helicity modulus for the phase sum, ϒ+, at ν = 0.6 for different system sizes versus β. (d) Illustrative example of the meaning of the two-component Ising order parameter m. Panels (e) -(i) Show results for a three-component Ginzburg-Landau model. (e) Heat capacity L−3 d〈E〉/dT versus β for the system with applied field that we consider. The heat capacity shows a signature of the Z2 transition to a non-superconducting state associated with the breaking of time-reversal symmetry. (f) Histogram of the Ising order parameter m for β = 4.1. For this inverse temperature the Z2 symmetry is clearly broken. (g) Illustration of the order parameter m for the three-component case. Structure factors for (h) the vorticity of ψ1 and (i) the magnetic field, at the same inverse temperature β = 4.1 as for the above histogram. Snapshots are shown to the left and thermal averages to the right. In the presence of a vortex lattice, the structure factors will have pronounced peaks. The absence of such peaks indicates that the system is in a resistive vortex-liquid state which spontaneously breaks Z2 symmetry due to nontrivial phase locking. (We remove the trivial zero-wave-vector components of the structure factors for clarity, and normalize the remaining components to the zero-wave-vector component.)
