Topological Phase Transitions in the Golden String-Net Model

We examine the zero-temperature phase diagram of the two-dimensional Levin-Wen string-net model with Fibonacci anyons in the presence of competing interactions. Combining high-order series expansions around three exactly solvable points and exact diagonalizations, we find that the non-Abelian doubled Fibonacci topological phase is separated from two nontopological phases by different second-order quantum critical points, the positions of which are computed accurately. These trivial phases are separated by a first-order transition occurring at a fourth exactly solvable point where the ground-state manifold is infinitely many degenerate. The evaluation of critical exponents suggests unusual universality classes.

Figure 1

Ground-state energy per plaquette $e_0=E_0/N_p$ as a function of $\theta$. ED results (black line) for $N_p=\sqrt{13}\times \sqrt{13}$ plaquettes (see inset) are in excellent agreement with typical Padé approximants (white lines) computed from high-order series expansions [40].

Figure 2

First four excitation energies obtained from the ED results (black lines) for $N_p=3\times 3$ plaquettes compared with the low-energy gap computed from high-order bare series expansions (white lines) [40]. The topological degeneracy splitting is clearly observed in the vicinity of the critical points. For symmetry reasons, this splitting is only partial for the system considered here (see inset).

Figure 3

ED results for $N_p=2\times 2$ (dotted line), $3\times 3$ (dashed line), and $\sqrt{13}\times \sqrt{13}$ (solid line) plaquettes. Left and right panels: ${\partial }_{\theta }^2{e}_0$ decreases with the system size indicating second-order transitions at ${\theta }_1^c$ and ${\theta }_2^c$. Central panel: ${\partial }_{\theta }{e}_0$ displays a clear jump at $\theta =\pi$ indicating a first-order transition. Dips indicated by arrows in the DFib topological phase are due to (irrelevant and avoided) level crossings between the four lowest-energy levels that become degenerate in the thermodynamical limit on a torus.

Figure 4

Position of different quantities as a function of either the inverse order $n^{-1}$ of the corresponding series expansion [crossing points of $e_0$ (dots), crossing points of the gap (triangles)] or $N_p^{-1}$ [minimum of the gap (squares), minimum of ${\partial }_{\theta }^2{e}_0$ (diamonds)]. Lines are power-law fits obtained by choosing ${\theta }_1^c$ such that it maximizes the correlations for the crossing points of $e_0$ (solid lines) that are the most accurate results in this work. These crossing points have been obtained from order $n$ series around $\theta =0$ and order $2n-1$ series around $\theta =\pi /2$. For the crossing points of the gap, we used order $n$ series around both limits.