Spontaneous breakdown of time-reversal symmetry induced by thermal fluctuations

In systems with broken $U\left(1\right)$ symmetry, such as superfluids, superconductors, or magnets, the symmetry restoration is driven by the proliferation of topological defects in the form of vortex loops (unless the phase transition is strongly first order). Here we discuss that the proliferation of topological defects can, by contrast, lead to the breakdown of an additional symmetry. We demonstrate that this effect should take place in $s+is$ superconductors, which are widely discussed in connection with iron-based materials (although the mechanism is much more general). In these systems a vortex excitation can create a “bubble” of fluctuating $Z_2$ order parameter. The thermal excitation of vortices then leads to the breakdown of $Z_2$ time-reversal symmetry when the temperature is increased.

Figure 1

Ground state and mass spectrum of frustrated three-band GL model. Two of the model parameters, ${\eta }_{13}={\eta }_{23}$, are scaled on the $x$ axis. The others are given by ${\alpha }_1={\alpha }_2=-3/64,{\alpha }_3=2.673/64,{\beta }_1={\beta }_2=3/64,{\beta }_3=0.18/64,{\eta }_{12}=2.25/64$, and $e=1.8/8.$ (In this example the parameters were chosen such that the relevant length scales of the theory are larger than numerical lattice spacing and substantially smaller than the system size.) (a) gives the ground-state amplitudes of the fields while (b) gives the phases (${\phi }_3$ is fixed to 0). A critical point at ${\eta }_{13}={\eta }_{23}\approx -0.0578$ separates a region with broken $U\left(1\right)$ symmetry (left) from a region that also breaks time-reversal symmetry (right). At a second critical point ${\eta }_{13}={\eta }_{23}\approx -0.0542$, time-reversal symmetry is restored. The broken symmetries are indicated in (b). (c) gives the mass spectrum and thus the inverse of the coherence lengths (in units of lattice spacings) associated with the three fields. For detailed discussion of definition of coherence lengths/mass spectrum in this kind of models see Ref. [20]. One mass (5) becomes zero at the critical point, implying a diverging coherence length. The mode that corresponds to this coherence length is shown in (d). On the left side it describes a perturbation to the phases ${\phi }_1,{\phi }_2$, i.e., a Leggett mode, which becomes massless at the critical point. In the region to the right of the critical point the character of the mode changes, as it now describes a perturbation to both phase and amplitude (mainly $|{\psi }_3|$). The red dot indicates the parameters simulated below.

Figure 2

Vortex group solutions resulting from energy minimization of the model (1) on a two-dimensional grid. An initial configuration of 12 vortices was prepared and the energy minimization was carried out with the constraint that the positions of the centers of the vortex cores remain fixed. This problem has two solutions that share the same magnetic field $\nabla \times \mathbf{A}$, shown in (a). However, the solutions do not exhibit the same phase difference between the components. (b) and (c) display $|{\psi }_1\left|\right|{\psi }_2|sin({\phi }_1-{\phi }_2)$ for these two cases. The GL parameter set is given by the red dot in Fig. 1 and features a ground state that breaks $U\left(1\right)$ symmetry only, implying that in the ground state, $sin({\phi }_1-{\phi }_2)=0$. The presence of vortices, however, induces a phase difference between the components ${\psi }_1$ and ${\psi }_2$ which is positive in the first solution (b) and negative in the second solution (c). Away from the vortices, the phase differences decay exponentially to zero. The group thus produces a bubble of $Z_2$ order parameter.

Figure 3

Summary of the simulation results. (a) The density of thermally induced vortices [${\rho }_V$, Eq. (11)]. It approaches zero at low temperature, but starts to increase substantially at $\beta \approx 1.5$. (b) The Fourier components of the magnetic field scaled by system size [$F_A(L,K)$, Eq. (10)] collapse onto the same curve, indicating that the system is nonsuperconducting for $\beta <{\beta }_c\approx 0.9$. (c) The order parameter of the broken time-reversal symmetry $O_{Z_2}$ [Eq. (7)] is zero at low temperature (except for finite size effects). As the temperature increases, an order emerges and reaches a maximum at $\beta \approx 1.35\text{–}1.4$. At higher temperatures this order starts to decay rapidly. (d) The correlation between the density of thermally induced vortices ${\rho }_V$ and the Ising order parameter $O_{Z_2}$ [Eq. (12)] is positive at low temperature and reaches a maximum of $\sim 0.72$, implying that breaking of the $Z_2$ symmetry is driven by vortex proliferation. At higher temperatures the correlation becomes negative, indicating that vortices contribute to restoring the symmetry.