[Paper Review] Quantum phase transition between symmetry enriched topological phases in tensor-network states

Lukas Haller, Wen-Tao Xu+2 more|arXiv (Cornell University)|May 3, 2023

Quantum many-body systems74 references4 citations

TL;DR

This paper proposes a tensor-network solvable model that realizes a continuous quantum phase transition between symmetry-enriched topological (SET) phases in a 2D decorated toric code with time-reversal symmetry. Using a bond dimension D=3 tensor network, it identifies three phases—SET toric code, trivial toric code, and topologically trivial—distinguished by topological entanglement entropy and a membrane order parameter, with a critical $O(2)$ loop gas point at the phase boundary governed by a $c=1$ conformal field theory.

ABSTRACT

Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this work, we propose a tensor-network solvable model that allows us to tune between different symmetry enriched topological (SET) phases. Concretely, we consider a decorated two-dimensional toric code model for which the ground state can be expressed as a two-dimensional tensor-network state with bond dimension $D=3$ and two tunable parameters. We find that the time-reversal (TR) symmetric system exhibits three distinct phases (i) an SET toric code phase in which anyons transform non-trivially under TR, (ii) a toric code phase in which TR does not fractionalize, and (iii) a topologically trivial phase that is adiabatically connected to a product state. We characterize the different phases using the topological entanglement entropy and a membrane order parameter that distinguishes the two SET phases. Along the phase boundary between the SET toric code phase and the toric code phase, the model has an enhanced $U(1)$ symmetry and the ground state is a quantum critical loop gas wavefunction whose squared norm is equivalent to the partition function of the classical $O(2)$ model. By duality transformations, this tensor-network solvable model can also be used to describe transitions between SET double-semion phases and between $\mathbb{Z}_2 imes\mathbb{Z}_2^T$ symmetry protected topological phases in two dimensions.

Motivation & Objective

• To construct a solvable microscopic model that realizes a continuous quantum phase transition between distinct symmetry-enriched topological (SET) phases.
• To characterize how time-reversal symmetry fractionalizes on anyons in different SET phases.
• To identify and analyze a critical point with enhanced $U(1)$ symmetry and $c=1$ conformal field theory behavior.
• To demonstrate the utility of tensor networks in capturing topological order, symmetry fractionalization, and phase transitions in 2D quantum systems.

Proposed method

• The ground state is constructed as a 2D tensor-network state (TNS) with bond dimension $D=3$, parameterized by two tunable parameters.
• The model is built by decorating loops in the toric code with 1D symmetry-protected topological (SPT) chains, enabling symmetry fractionalization on anyons.
• Topological entanglement entropy (TEE) and a membrane order parameter are computed numerically to distinguish phases.
• The critical point is analyzed via duality, mapping the wavefunction amplitudes to the partition function of the classical $O(2)$ loop model.
• The critical behavior is confirmed to be described by the compactified free boson CFT with central charge $c=1$.
• The TNS formalism enables exact calculation of the second Renyi entropy and TEE in the $N \to \infty$ limit using fixed-point transfer operators.

Experimental results

Research questions

• RQ1 Can a solvable tensor-network model realize a continuous quantum phase transition between distinct SET phases with time-reversal symmetry?
• RQ2 How does time-reversal symmetry fractionalize on anyons in the SET toric code phase compared to the trivial toric code phase?
• RQ3 What is the nature of the critical point separating the two SET phases, and does it exhibit enhanced symmetry?
• RQ4 Can the critical behavior be mapped to a known classical statistical model, and what is its conformal field theory description?
• RQ5 How do topological entanglement entropy and membrane order parameters distinguish the different phases in the model?

Key findings

• The model realizes three distinct phases: an SET toric code phase with non-trivial time-reversal fractionalization, a trivial toric code phase with no fractionalization, and a topologically trivial phase adiabatically connected to a product state.
• The phase transition between the SET toric code and trivial toric code phases is continuous and features an enhanced $U(1)$ symmetry at the critical point.
• The critical wavefunction is a quantum loop gas whose squared norm maps to the partition function of the classical $O(2)$ loop model with $c=1$ conformal field theory.
• The membrane order parameter successfully distinguishes the two SET phases, confirming different symmetry fractionalization patterns.
• Topological entanglement entropy (TEE) is quantized at $\gamma = \log 2$ in the SET toric code phase and vanishes in the trivial phase, consistent with topological order.
• The entanglement spectrum in the SET-TC phase exhibits even-fold degeneracy in the $\mathbf{e}$ and $\mathbf{f}$ anyon sectors due to Kramers' theorem, confirming time-reversal symmetry fractionalization.