Quantum correlations in topological quantum phase transitions

We study the quantum correlations in a two-dimensional system that possesses a topological quantum phase transition. The quantumness of two-body correlations is measured by quantum discord. We calculate both the correlation of two local spins and that between an arbitrary spin and the rest of the lattice. It is notable that local spins are classically correlated, while the quantum correlation is hidden in the global lattice. This is different from other systems which are not topologically ordered. Moreover, the mutual information and global quantum discord show critical behavior in the topological quantum phase transition.

Figure 1

(Color online). Demonstration of the model. Spins lie on the edges (like $i,i^{\prime },i^{″})$. The nearest spins $i,i^{\prime }$ become next-nearest in the dual lattice (the dashed line). The red plaquette and the blue cross represent the plaquette and star operators. The red heavy line across the lattices represents a product of ${\sigma }_i^z$ along the nontrivial loop on a torus. The system is invariant under this transformation.

Figure 2

Mutual information of two nearest spins ${\sigma }_i^z{\sigma }_{i^{″}}^z$ (solid line) and ${\sigma }_i^z{\sigma }_{i^{\prime }}^z$ (dashed line) as shown in Fig. 1. Critical change happens at ${\beta }_c=(1/2)\ln (\sqrt{2}+1)$, which is in accord with previous studies. The quantum discord of the two spins is always zero.

Figure 3

Global correlations between a local spin and the rest of the whole lattice. Here, the quantum discord (solid line) is equal to the entropy of entanglement, and just one-half of the mutual information (dashed line) of the pairwise system. Both the quantum discord and the mutual information show critical change at the phase transition point.