Topological order in a three-dimensional toric code at finite temperature

We study topological order in a toric code in three spatial dimensions or a $3+1\text{D}$ ${\mathbb{Z}}_2$ gauge theory at finite temperature. We compute exactly the topological entropy of the system and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature $T_c$, dropping to zero above $T_c$. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of fluctuating strings and membranes and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature and its quantum and classical aspects.

Figure 1

(Color online) Illustration of the Kitaev model in three dimensions, with explicit examples of a star operator $A_s=\prod {\sigma }_i^x$ at the lattice site $s$ and of three plaquette operators $B_p=\prod {\sigma }_i^z$ at the plaquette-dual sites $p_1$, $p_2$, and $p_3$. The $\sigma$ spin index $i$ labels, respectively, the six (red) spins around $s$ and the four (blue) spins around $p$ (connected by dashed lines).

Figure 2

(Color online) Two examples of the nonlocal operators needed to distinguish between the degenerate GS of the 3D Kitaev model.

Figure 3

(Color online) Two examples of the underlying structures of the 3D Kitaev model: the closed ${\sigma }^z$ loops along the edges of the cubic lattice, which satisfy ${\prod }_{\text{loop}}{\sigma }_i^z=1$ and the closed ${\sigma }^x$ membranes in the body-centered dual lattice, satisfying ${\prod }_{\text{membrane}}{\sigma }_i^x=1$.

Figure 4

Illustration of the two bipartition schemes used for the 3D Kitaev model: spherical (top) and donut shaped (bottom).

Figure 5

(Color online) Illustration of the collective operations in bipartitions 4 and 5 in the ${\sigma }^x$ basis, acting on subsystem $\mathcal{B}$ (top) and subsystem $\mathcal{A}$ (bottom), respectively.

Figure 6

(Color online) Qualitative illustration of the disruptive effect of two defects (solid red dots) in the 2D Kitaev model on a torus: two winding loops (black wavy lines) on either side of a defect (solid circle), read off opposite eigenvalues of the corresponding winding loop operator.

Figure 7

(Color online) Qualitative illustration of the reason why the topological information stored in the underlying ${\sigma }^z$ loop structure of the 3D Kitaev model is robust to thermal fluctuations: even in presence of sparse defects (solid red circles), any two winding loops (black wavy lines), with equal winding numbers, can be smoothly deformed one into the other without crossing any defects. (The wiggly lines represent qualitatively the confining strings between defect “pairs” discussed in the text.)

Figure 8

(Color online) Qualitative examples of terms in the loop expansion that appear with different signs in $Z_{\left\{k_i\right\}}^{\overline{4\mathcal{A}},-}$ and $Z_{\left\{k_i\right\}}^{\overline{4\mathcal{A}},+}$: closed loops that wind around the donut shape and open strings that connect boundary spins $S_i$ (which appear in the high-temperature expansion whenever the corresponding $k_i$ is odd).

Figure 9

(Color online) Schematic projected illustration of open strings between boundary spins. The locations of the spins 1–8 are given by the sites where $k_i$ is odd (recall that their total number must be even). One can verify that the parity of the number of intersections with the twist surface is fixed by the choice of the locations 1–8, up to exponentially small corrections such as the red dotted string in the figure, which vanish in the thermodynamic limit of $N_{\partial }\to \infty$. For example, consider the change upon reconnecting spin 5–8 via the dashed lines instead of the solid lines. (Notice that the case where, say, the points 1–4 are uniformly distributed on the boundary is exponentially suppressed by the probability ${\mathcal{P}}_{\left\{k_i\right\}}$.)