Topological Transitions from Multipartite Entanglement with Tensor Networks: A Procedure for Sharper and Faster Characterization

Topological order in two-dimensional (2D) quantum matter can be determined by the topological contribution to the entanglement Rényi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here, we show how topological phase transitions in 2D systems can be much better assessed by multipartite entanglement, as measured by the topological geometric entanglement of blocks. Specifically, we present an efficient tensor network algorithm based on projected entangled pair states to compute this quantity for a torus partitioned into cylinders and then use this method to find sharp evidence of topological phase transitions in 2D systems with a string-tension perturbation. When compared to tensor network methods for Rényi entropies, our approach produces almost perfect accuracies close to criticality and, additionally, is orders of magnitude faster. The method can be adapted to deal with any topological state of the system, including minimally entangled ground states. It also allows us to extract the critical exponent of the correlation length and shows that there is no continuous entanglement loss along renormalization group flows in topological phases.

Figure 1

Left: phase transition for a topological 2D model with string tension. The black line corresponds to $|E_{\gamma }|$ as computed in Fig. 4 for $n_b\to \infty$ using GE (to be explained later). The rest of the lines correspond to the best achievable calculation by the authors of the topological term $|S_{\gamma }^{(n)}|$ of the $n$th Rényi entropies $S^{(n)}=(1-n{)}^{-1}\log [\mathrm{tr}({\rho }^n)]$ of half an infinite cylinder for $n=2,3$ and $n=\infty$ (i.e., the single-copy entanglement [8]), using the methods explained in the Supplemental Material [9]. Compare also to similar calculations with tensor networks in, e.g., Ref. [7]. Typical sizes of string nets populating the ground state for each phase are also represented. Right: average computation time ratio with respect to $|E_{\gamma }|$, for the different topological contributions.

Figure 2

(a) PEPS $|\mathrm{\Psi }(n,L)⟩$ wrapped around a torus. (b) Product state $|\mathrm{\Phi }⟩$ of MPS with PBC (cylinders with $l=1$). (c) Contraction to compute the optimal state for a given cylinder. (d) Exact result of the contraction in (c) in terms of the environment tensor $E$. (e) Approximation of $E$ by an effective environment $\tilde{E}$ described by an MPO. (f) Resulting optimal MPS for the cylinder in an iteration step (see main text).

Figure 3

$E_G(n,L)$ for the toric code with string tension on the square lattice (similar results are also obtained for the honeycomb lattice). (a) Case $g=0.4$. The extrapolation of the linear fit (red dashed line) for, e.g., $n=20$ hits $L=0$ around $\sim -1$ (red dot), as expected in the topological phase. (b) Case $g=0.6$. The same calculation yields $\sim 0$, as expected in the polarized phase.

Figure 4

Absolute value of $E_{\gamma }$ extrapolated from the scaling with $L$ as in Fig. 3, as a function of the string tension $g$ [for (a)–(c)] and the number of blocks $n_b$ [for (a)– (d)]. Plots are for the perturbed (a) $|0,0⟩$ state on the square lattice, (b) $|0,0⟩$ state on the honeycomb lattice, and (c) $|+,+⟩$ state on the honeycomb lattice. Notice that (b) and (c), though being ground states on the same lattice, have different transition points. This is because the string tension $g$ was applied in different bases, hence, corresponding to different physical perturbations. However, we have also checked that when the same perturbation is applied to different ground states on the same lattice the topological phase transition takes place at the same critical point, showing that the transitions in $E_{\gamma }$ do not depend on the specific choice of ground state. (d) Unperturbed toric code on a square lattice for two different ground states: an MES $|{\mathrm{\Xi }}_0⟩$ and a non-MES $|0,0⟩$.