Entanglement and topological entropy of the toric code at finite temperature

We calculate exactly the von Neumann and topological entropies of the toric code as a function of system size and temperature. We do so for systems with infinite energy scale separation between magnetic and electric excitations, so that the magnetic closed loop structure is fully preserved while the electric loop structure is tampered with by thermally excited electric charges. We find that the entanglement entropy is a singular function of temperature and system size, and that the limit of zero temperature and the limit of infinite system size do not commute. The two orders of limit differ by a term that does not depend on the size of the boundary between the partitions of the system, but instead depends on the topology of the bipartition. From the entanglement entropy we obtain the topological entropy, which is shown to drop to half its zero-temperature value for any infinitesimal temperature in the thermodynamic limit, and remains constant as the temperature is further increased. Such discontinuous behavior is replaced by a smooth decreasing function in finite-size systems. If the separation of energy scales in the system is large but finite, we argue that our results hold at small enough temperature and finite system size, and a second drop in the topological entropy should occur as the temperature is raised so as to disrupt the magnetic loop structure by allowing the appearance of free magnetic charges. We discuss the scaling of these entropies as a function of system size, and how the quantum topological entropy is shaved off in this two-step process as a function of temperature and system size. We interpret our results as an indication that the underlying magnetic and electric closed loop structures contribute equally to the topological entropy (and therefore to the topological order) in the system. Since each loop structure per se is a classical object, we interpret the quantum topological order in our system as arising from the ability of the two structures to be superimposed and appear simultaneously.

Figure 1

Qualitative behavior of the topological entropy as a function of temperature $T$ and number of degrees of freedom $2N$, for the generic case where the two coupling constants in the model are well separated, i.e., ${\lambda }_A⪡{\lambda }_B$. The exact shape of the first crossover is shown in Fig. 5.

Figure 2

Illustration of the spin-loop correspondence discussed in the text. All vertices with two positive and two negative ${\sigma }^x$ components on the adjacent links can be obtained via appropriate rotations of the ones shown in the figure.

Figure 3

Illustration of the four bipartitions used to compute the topological entropy in Ref. 5.

Figure 4

(Top) Limiting behavior of the entropy difference Eq. (54) in units of $\ln 2$ in the thermodynamic limit, as a function of $K_A{\Sigma }_{1{\mathcal{P}}_1}∕2$, where $K_A=-\ln \left[\tanh \right({\lambda }_A∕T\left)\right]$ and ${\Sigma }_{1{\mathcal{P}}_1}\sim (R-2r{)}^2$, the area of the inner square in Fig. 3. Notice the logarithmic scale on the horizontal axis. (Bottom) The same curve represented as a function of $T∕{\lambda }_A$, for three different values of ${\Sigma }_{1{\mathcal{P}}_1}=\mathrm{20,200},2000$ (from right to left).

Figure 5

Topological entropy as a function of ${\mathcal{K}}_{\mathcal{A}}$ for increasing system sizes $N={10}^3,{10}^6,{10}^9$. Notice the complete overlap between the different curves, due to the fact that the topological entropy Eq. (51) becomes a pure function of ${\mathcal{K}}_{\mathcal{A}}=K_{A}N$ when all $\Sigma$’s scale linearly with $N$. Notice the logarithmic scale on the horizontal axis.