Figure 1

Non-Hermitian toric-code model. (a) Spin-$1/2$ magnetic moments placed at the edges of a square lattice with $N\times N$ vertices. (b) Difference (scaled by $2N$) in the magnetization ${⟨M⟩}_k(\beta )$ between different ground states for $N=8$, 16, 24. Here $k\in \{ee,eo,oe,oo\}$ labels different topological sectors, where the first (second) letter represents the parity of the number of noncontractible loops of down spins on the dual lattice winding around the torus in the $x$ ($y$) direction. For $\beta <{\beta }_c$ ($\beta >{\beta }_c$), the difference in ${⟨M⟩}_k(\beta )$ tends to vanish (becomes of the order of $N$) with an increase in $N$, which suggests a topological (trivial) phase. Moreover, the different curves cross at the transition point $\beta ={\beta }_c$. (c) Magnetic susceptibility ${\chi }_M^k(\beta )=(d/d\beta )⟨M{⟩}_k(\beta )$ for $N=24$, which exhibits a singularity at $\beta ={\beta }_c$.