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

Abstract

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 wavefunction with two tuning parameters dual with each other, and the norm can be exactly mapped into a partition function of the two-coupled φ 2state Potts models, where φ = ( √ 5 + 1)/2 is the golden ratio. By developing the tensor network representation of this wavefunction 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 non-topological phases and their dual phases of a single φ 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 non-topological phases.arXiv:1912.07836v2 [cond-mat.str-el]

Detecting transition between Abelian and non-Abelian topological orders through symmetric tensor networks

“…For example, the phases and phase transitions of the two dimensional (2D) toric code (TC) model with finite string tension, whose ground state is represented by a Z 2 -injective PEPS [18], can be fully understood within this framework. Similar idea of detecting topological phase transitions has also been generalized to non-Abelian cases recently [23][24][25].…”

Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions

“…which can be regarded as a generic DS wavefunction in an expanded parameter space. Actually, the similar deformation has been used to express the generic TC wavefunction [23][24][25] and Fibonacci quantum-net wavefunction 26 . For convenience, we define h x h cos θ and h z h sin θ, where h expresses the loop tension and θ is the spin angle.…”

Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters

“…Here, the topological order is accompanied by the presence of certain group or Matrix Product Operator (MPO) symmetries in the entanglement degrees of freedom of the tensor, which can be used to parametrize the ground space manifold and anyonic excitations alike [7][8][9]. The description of topologically ordered systems as PEPS based on entanglement symmetries suggests a natural way to construct and study topological phase transitions within PEPS, by applying deformations to the physical degrees of freedom which drive the system to a different phase (such as a trivial product state) [10][11][12][13][14][15][16][17]. In this language, the entanglement symmetry in the tensor is preserved throughout the path, but at some point, it no longer manifests itself in topological order.…”