Ising anyons with a string tension

We consider the string-net model on the honeycomb lattice for Ising anyons in the presence of a string tension. This competing term induces a nontrivial dynamics of the non-Abelian anyonic quasiparticles and may lead to a breakdown of the topological phase. Using high-order series expansions and exact diagonalizations, we determine the robustness of this doubled Ising phase, which is found to be separated from two gapped phases. An effective quantum dimer model emerges in the large tension limit, giving rise to two different translation symmetry-broken phases. Consequently, we obtain four transition points, two of which are associated with first-order transitions whereas the two others are found to be continuous and provide examples of recently proposed Bose condensation for anyons.

Figure 1

From top to bottom, ground-state energy per plaquette $e_0$ computed from series expansions at order 1–11 around $\theta =0$ (yellow) and at order 1–18 around $\theta =\pi /2$ (blue). For comparison, ED results (red) are also displayed for $N_{\mathrm{p}}=7,9,13$ (from bottom to top).

Figure 2

From top to bottom, low-energy gap $\Delta$ computed from series expansions at order 1–10 around $\theta =0$ (yellow) and at order 1–8 around $\theta =\pi /2$ (blue). For comparison, the first nine excitation energies obtained from ED (red) are shown for $N_{\mathrm{p}}=13$. The first excited level is eightfold degenerate. Inset: Minimum of ninth excitation energy as a function of $N_{\mathrm{p}}^{-1}$ computed from ED (squares); minimum of $\Delta$ as a function of $n^{-1}$ (circles) computed at the crossing point between order $n$ series performed around $\theta =0$ and $\theta =\pi /2$. Lines are power law fits consistent with ${lim}_{n,N_{\mathrm{p}}\to 0}\Delta =0$.

Figure 3

Low-energy gap $\Delta$ computed from resummed series expansions around $\theta =0$ [$d$log Padé approximants $[m,n]$ for $m,n\ge 3$ are displayed (yellow)] and from the expression (8), which is a good approximation of the gap around $\theta =3\pi /2^{+}$ (green). The first nine excitation energies obtained from ED (red) are shown for $N_{\mathrm{p}}=12$.

Figure 4

Sketch of the phase diagram. Four different gapped phases are identified among which one is topologically ordered (DIsing) and two break the translation symmetry (SC and PC). Diamonds (circles) indicate first-order (second-order) transition points (see text for more details).