Temperature-Induced Spontaneous Time-Reversal Symmetry Breaking on the Honeycomb Lattice

Phase transitions involving spontaneous time-reversal symmetry breaking are studied on the honeycomb lattice at finite hole doping with next-nearest-neighbor repulsion. We derive an exact expression for the mean-field equation of state in closed form, valid at temperatures much less than the Fermi energy. Contrary to standard expectations, we find that thermally induced intraband particle-hole excitations can create and stabilize a uniform metallic phase with broken time-reversal symmetry as the temperature is raised in a region where the ground state is a trivial metal.

Figure 1

The difference of the canonical free energy $\delta \mathcal{F}(\overline{\mathrm{\Delta }})$ defined in Eq. (11) is evaluated numerically, keeping the density ${\tilde{n}}_e$ fixed and plotted for $V<V_c$ as a function of $\overline{\mathrm{\Delta }}$ for $T=T_c$ and $T>T_c$. A clear minimum is observed as the temperature is raised above $T_c$ [see Eq. (15)], signifying the formation of a topological liquid phase at finite $T$. The evolution of the order parameter $\overline{\mathrm{\Delta }}(T)$ (the location of the minimum) is shown in the inset by the solid line; the dashed line is the approximate expression for $\overline{\mathrm{\Delta }}$ derived in Eq. (19). The filling is fixed at ${\epsilon }_F/{\epsilon }_{\mathrm{\Lambda }}=\sqrt{n_h/n}=0.1$ and $V_c=1.3$.