Tensor-network approach to phase transitions in string-net models

We use a recently proposed class of tensor-network states to study phase transitions in string-net models. These states encode the genuine features of the string-net condensate such as, e.g., a nontrivial perimeter law for Wilson loops expectation values, and a natural order parameter detecting the breakdown of the topological phase. In the presence of a string tension, a quantum phase transition occurs between the topological phase and a trivial phase. We benchmark our approach for ${\mathbb{Z}}_2$ string nets and capture the second-order phase transition which is well known from the exact mapping onto the transverse-field Ising model. More interestingly, for Fibonacci string nets, we obtain first-order transitions in contrast with previous studies but in qualitative agreement with mean-field results.

Figure 1

Variational results for the ${\mathbb{Z}}_2$ theory obtained for $\beta =0$ (red) and $\beta \ne 0$ (green). Top: order parameter $\alpha$ indicating a continuous transition from a topological phase ($\alpha =1$) to a trivial phase ($\alpha <1$). Bottom: ground-state energy per plaquette $e_0$ compared with exact diagonalization data (blue crosses) (see inset for a broader range). Dashed lines give the position of the transition point obtained from series expansions [39] (blue) and from the order parameter behavior [red [35] and green (this work)].

Figure 2

Variational results for the Fibonacci theory (same conventions as in Fig. 1). Blue dashed lines indicate the position of the transition point computed from series expansions [7].

Figure 3

Variational results for the Fibonacci theory (same conventions as in Fig. 1). Blue dashed lines indicate the position of the transition point computed from series expansions [7] which coincides with the one obtained with the Ansatz (8), i.e., $\theta \simeq -0.630$.