Tensor Network Approach to Phase Transitions of a Non-Abelian Topological Phase

The non-Abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wave function with two tuning parameters dual with each other, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled ${\varphi }^2$-state Potts models, where $\varphi =(\sqrt{5}+1)/2$ is the golden ratio. By developing the tensor network representation of this wave function on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-Abelian Fibonacci topological phase is enclosed by three distinct nontopological phases and their dual phases of a single ${\varphi }^2$-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those nontopological phases.

Figure 1

The global phase diagram for the generic quantum-net phase with two dual parameters. The phase diagram is symmetric about the self-dual line $h=\widehat{h}$ and bounded by the elliptic curve of the two-decoupled Potts models. The $AC$ dot-dashed line represents a weak first-order transition and all other solid lines are continuous transitions. $C$ denotes a tricritical point and $D$ is a tetracritical point.

Figure 2

(a) Two local orthogonal quantum states $|1⟩$ and $|\tau ⟩$ and their dual local states $|\widehat{1}⟩$ and $|\widehat{\tau }⟩$. (b) The physical degrees of freedom located at the edges of the square lattice. The full lines denote the original lattice, while the dashed lines are dual lattice. $\mathcal{N}$ and $\mathcal{D}$ are two typical nets.

Figure 3

(a) The square lattice is transformed to a honeycomb lattice. (b) The triple-line local tensor $T$ in the string-net wave function. (c) Contracting two tensors $T$ yields the tensor $M$ on the square lattice.

Figure 4

(a) The correlation length $\xi$ obtained by the VUMPS method with bond dimensions $D$ along $h$ axis of the phase diagram. (b) The correlation length deduced from the CTM method with bond dimensions $D$ along the cut $\widehat{h}=-0.8-h$. $h=-0.4$ sits on the self-dual line. (c) The entanglement entropy $S$ at the point $(h,\widehat{h})=(-0.6,0)$. The blue line is $S=(c/6)\log \xi +S_0$ with the central charge $c=7/5$.