Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1912.00908 · Physical review. B./Physical review. B

Detecting a Z2 topologically ordered phase from unbiased infinite projected entangled-pair state simulations

S. P. G. Crone, Philippe Corboz

Published Mar 25, 2020• 29 citations
arXivHTMLPDF

Abstract

We present an approach to identify topological order based on unbiased infinite projected entangled-pair states simulations, i.e., where we do not impose a virtual symmetry on the tensors during the optimization of the tensor network ansatz. As an example we consider the ground state of the toric code model in a magnetic field exhibiting ${Z}_{2}$ topological order. The optimization is done by an efficient energy minimization approach based on a summation of tensor environments to compute the gradient. We show that the optimized tensors, when brought into the right gauge, are approximately ${Z}_{2}$ symmetric, and they can be fully symmetrized a posteriori to generate a stable topologically ordered state, yielding the correct topological entanglement entropy and modular S and U matrices. To compute the latter we develop a variant of the corner-transfer matrix method, which is computationally more efficient than previous approaches based on the tensor renormalization group.

DOI
10.1103/physrevb.101.115143

Related papers

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

Yu-Hsueh Chen, Ching-Yu Huang, Ying-Jer Kao

We propose a unified scheme to identify phase transitions out of the ${\mathbb{Z}}_{2}$ Abelian topological order, including the transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko, T. Okubo, and N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example, we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similar to the anyon condensation, continuous deformation of a ${\mathbb{Z}}_{2}$-injective projected entangled-pair state (PEPS) also allows us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $\ensuremath{\sigma}$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.

arXiv:2102.10980•Physical review. B./Physical review. B•Jul 20, 2021•4 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.

arXiv:2407.16531•Communications Physics•Jul 14, 2026•0 citations

Determining topological order from infinite projected entangled pair states

Anna Francuz, Jacek Dziarmaga, Guifré Vidal +1 more

We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators onto states with well-defined anyon flux are used to compute topological $S$ and $T$ matrices (encoding mutual and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.

arXiv:1910.09661•Physical review. B./Physical review. B•Jan 21, 2020•10 citations

Variational methods for characterizing matrix product operator symmetries

Anna Francuz, Laurens Lootens, Frank Verstraete +1 more

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga, Phys. Rev. B 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological $F$ symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the $F$ symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological $S$ and $T$ matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of ${Z}_{3}$, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of ${S}_{3}$ and $\mathrm{Rep}({S}_{3})$ string-net models.

arXiv:2107.05265•Physical review. B./Physical review. B•Nov 29, 2021•1 citation

Fermionic topological quantum states as tensor networks

Carolin Wille, Oliver Buerschaper, Jens Eisert

Tensor network states, and in particular projected entangled pair states, play an important role in the description of strongly correlated quantum lattice systems. They do not only serve as variational states in numerical simulation methods, but also provide a framework for classifying phases of quantum matter and capture notions of topological order in a stringent and rigorous language. The rapid development in this field for spin models and bosonic systems has not yet been mirrored by an analogous development for fermionic models. In this work, we introduce a tensor network formalism capable of capturing notions of topological order for quantum systems with fermionic components. At the heart of the formalism are axioms of fermionic matrix-product operator injectivity, stable under concatenation. Building upon that, we formulate a Grassmann number tensor network ansatz for the ground state of fermionic twisted quantum double models. A specific focus is put on the paradigmatic example of the fermionic toric code. This work shows that the program of describing topologically ordered systems using tensor networks carries over to fermionic models.

arXiv:1609.02574•Physical review. B./Physical review. B•Jun 21, 2017•30 citations

Modular matrices as topological order parameter by a gauge-symmetry-preserved tensor renormalization approach

Huan He, Heidar Moradi, Xiao-Gang Wen

Topological order has been proposed to go beyond Landau symmetry breaking theory for more than 20 years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in two-dimensional systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including $S$ and $T$ matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data ($S$ and $T$ matrices), without any nonuniversal contributions. Furthermore, it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, in which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.

arXiv:1401.5557•Physical Review B•Nov 10, 2014•55 citations
We present an approach to identify topological order based on unbiased infinite projected entangled-pair states simulations, i.e., where we do not impose a virtual symmetry on the tensors during the optimization of the tensor network ansatz. As an example we consider the ground state of the toric code model in a magnetic field exhibiting Z2{Z}_{2}Z2​ topological order. The optimization is done by an efficient energy minimization approach based on a summation of tensor environments to compute the gradient. We show that the optimized tensors, when brought into the right gauge, are approximately Z2{Z}_{2}Z2​ symmetric, and they can be fully symmetrized a posteriori to generate a stable topologically ordered state, yielding the correct topological entanglement entropy and modular S and U matrices. To compute the latter we develop a variant of the corner-transfer matrix method, which is computationally more efficient than previous approaches based on the tensor renormalization group.
We propose a unified scheme to identify phase transitions out of the Z2{\mathbb{Z}}_{2}Z2​ Abelian topological order, including the transition to a non-Abelian chiral spin liquid. Using loop gas and and string gas states [H.-Y. Lee, R. Kaneko, T. Okubo, and N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)] on the star lattice Kitaev model as an example, we compute the overlap of minimally entangled states through transfer matrices. We demonstrate that, similar to the anyon condensation, continuous deformation of a Z2{\mathbb{Z}}_{2}Z2​-injective projected entangled-pair state (PEPS) also allows us to study the transition between Abelian and non-Abelian topological orders. We show that the charge and flux anyons defined in the Abelian phase transmute into the $\ensuremath{\sigma}$ anyon in the non-Abelian topological order. Furthermore, we show that contrary to the claim in [Phys. Rev. B 101, 035140 (2020)], both the LG and SG states have infinite correlation length in the non-Abelian regime, consistent with the no-go theorem that a chiral PEPS has a gapless parent Hamiltonian.
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.
We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators onto states with well-defined anyon flux are used to compute topological SSS and TTT matrices (encoding mutual and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.
We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga, Phys. Rev. B 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological FFF symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the FFF symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological SSS and TTT matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of Z3{Z}_{3}Z3​, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S3{S}_{3}S3​ and Rep(S3)\mathrm{Rep}({S}_{3})Rep(S3​) string-net models.
Tensor network states, and in particular projected entangled pair states, play an important role in the description of strongly correlated quantum lattice systems. They do not only serve as variational states in numerical simulation methods, but also provide a framework for classifying phases of quantum matter and capture notions of topological order in a stringent and rigorous language. The rapid development in this field for spin models and bosonic systems has not yet been mirrored by an analogous development for fermionic models. In this work, we introduce a tensor network formalism capable of capturing notions of topological order for quantum systems with fermionic components. At the heart of the formalism are axioms of fermionic matrix-product operator injectivity, stable under concatenation. Building upon that, we formulate a Grassmann number tensor network ansatz for the ground state of fermionic twisted quantum double models. A specific focus is put on the paradigmatic example of the fermionic toric code. This work shows that the program of describing topologically ordered systems using tensor networks carries over to fermionic models.
Topological order has been proposed to go beyond Landau symmetry breaking theory for more than 20 years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in two-dimensional systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including SSS and TTT matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data (SSS and TTT matrices), without any nonuniversal contributions. Furthermore, it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, in which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.