Symmetric Topological Phases and Tensor Network States

We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on kagome lattice. We develop a method to keep track of symmetries in TNS update procedure and coarse-graining tensor renormalization group (TRG). The symmetric update generates symmetric TNS that has $SU(2)$ spin rotation, time reversal and translation symmetries. The variational energy (per site) is -0.4355 for TNS with bond dimension $D=20$ . The global SU(2) spin-rotation symmetry implies $\mathbb{Z}_2$ gauge symmetry in TNS. We use symmetric TRG to compute the modular matrices and confirm that $\mathbb{Z}_2$ gauge symmetry deconfines and the SU(2) spin-rotation symmetric TNS we find has the toric code topological order.We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on kagome lattice. We develop a method to keep track of symmetries in TNS update procedure and coarse-graining tensor renormalization group (TRG). The symmetric update generates symmetric TNS that has $SU(2)$ spin rotation, time reversal and translation symmetries. The variational energy (per site) is -0.4355 for TNS with bond dimension $D=20$. The global SU(2) spin-rotation symmetry implies $\mathbb{Z}_2$ gauge symmetry in TNS. We use symmetric TRG to compute the modular matrices and confirm that $\mathbb{Z}_2$ gauge symmetry deconfines and the SU(2) spin-rotation symmetric TNS we find has the toric code topological order.

Jun 30, 20162 citations

Quantum phase transition between symmetry enriched topological phases in tensor-network states

Lukas Haller, Wen-Tao Xu, Yu-Jie Liu +1 more

Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this paper, 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 nontrivially 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 ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$ symmetry protected topological phases in two dimensions.Quantum phase transitions between different topologically ordered phases exhibit rich structures and are generically challenging to study in microscopic lattice models. In this paper, 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 nontrivially 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 ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$ symmetry protected topological phases in two dimensions.

arXiv:2305.02432Physical Review ResearchOct 23, 202318 citations

Symmetry-Enriched Topological Order in Tensor Networks

Dominic J. Williamson, Nick Bultinck, Frank Verstraete

Bulletin of the American Physical SocietyMar 5, 20180 citations

1-Form symmetric projected entangled-pair states

Yi Tan, Wayne Zheng, Ji-Yao Chen +3 more

Higher-form symmetries acting on extended objects have reshaped our understanding of quantum phases and topological order, yet explicit incorporation of 1-form symmetry in tensor-network states remains unexplored. Here, we introduce a pull-through framework to embed 1-form symmetries into Projected Entangled-Pair States (PEPS), providing an operator-level definition of gauge flux and algebraic constraints on virtual symmetry matrices that encode anomalous braiding phases and enable efficient optimization of topologically ordered states. Our framework offers diagnostic tools for anomalies in topological phases and paves the way for more efficient tensor-network simulations in a variety of models with higher-form symmetries. Quantum many-body systems with symmetries acting on extended objects host exotic phases that are difficult to simulate. We develop a pull-through tensor-network framework that embeds 1-form symmetries in projected entangled-pair states, enabling symmetry-resolved optimization and diagnostics of topologically ordered states.Higher-form symmetries acting on extended objects have reshaped our understanding of quantum phases and topological order, yet explicit incorporation of 1-form symmetry in tensor-network states remains unexplored. Here, we introduce a pull-through framework to embed 1-form symmetries into Projected Entangled-Pair States (PEPS), providing an operator-level definition of gauge flux and algebraic constraints on virtual symmetry matrices that encode anomalous braiding phases and enable efficient optimization of topologically ordered states. Our framework offers diagnostic tools for anomalies in topological phases and paves the way for more efficient tensor-network simulations in a variety of models with higher-form symmetries. Quantum many-body systems with symmetries acting on extended objects host exotic phases that are difficult to simulate. We develop a pull-through tensor-network framework that embeds 1-form symmetries in projected entangled-pair states, enabling symmetry-resolved optimization and diagnostics of topologically ordered states.

arXiv:2407.16531Communications PhysicsJul 14, 20260 citations

SU(2)

spin rotation, time reversal and translation symmetries. The variational energy (per site) is -0.4355 for TNS with bond dimension

D=20

. The global SU(2) spin-rotation symmetry implies

\mathbb{Z}_2

gauge symmetry in TNS. We use symmetric TRG to compute the modular matrices and confirm that

\mathbb{Z}_2

gauge symmetry deconfines and the SU(2) spin-rotation symmetric TNS we find has the toric code topological order.

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 nontrivially 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 ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$ symmetry protected topological phases in two dimensions.