Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions

Chao Xu, Yixin Ma, Shenghan Jiang

arXiv:2308.06543Reports on Progress in PhysicsAug 16, 20245 citations

A unified diagrammatic approach to topological fixed point models

Andreas Bauer, Jens Eisert, Carolin Wille

arXiv:2011.12064SciPost Physics CoreJul 25, 20229 citations

arXiv:2309.13118

Topological dualities via tensor networks

Carolin Wille, Jens Eisert, Alexander Altland

Published Sep 22, 2023• 0 citations

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Abstract

DOI: 10.17169/refubium-43262

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Topological dualities via tensor networks

Carolin Wille, Jens Eisert, Alexander Altland

The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable in as much as it connects systems commonly associated to different areas of physics—that of long-range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically nonlocal, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this paper, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed-matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs. Published by the American Physical Society 2024The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable in as much as it connects systems commonly associated to different areas of physics—that of long-range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically nonlocal, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this paper, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed-matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs. Published by the American Physical Society 2024

Physical Review ResearchMar 19, 20245 citations

Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases

Meng Sun, Zongyuan Wang, Nathanan Tantivasadakarn +1 more

Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.

quant-pharXiv:2606.25048Jun 25, 20260 citations

A minimal tensor network beyond free fermions

Carolin Wille, Maksimilian Usoltcev, Jens Eisert +1 more

This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a multicritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a multicritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.

arXiv:2412.04216SciPost PhysicsJun 19, 20251 citation

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

Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions

Chao Xu, Yixin Ma, Shenghan Jiang

Abstract The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations and non-chiral topological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.Abstract The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations and non-chiral topological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

arXiv:2308.06543Reports on Progress in PhysicsAug 16, 20245 citations

A unified diagrammatic approach to topological fixed point models

Andreas Bauer, Jens Eisert, Carolin Wille

We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories, but is formalised and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism represent discrete path integrals in Euclidean space-time. This language is more directly related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We introduce our formalism by simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories, but is formalised and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism represent discrete path integrals in Euclidean space-time. This language is more directly related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We introduce our formalism by simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.

arXiv:2011.12064SciPost Physics CoreJul 25, 20229 citations

\mathbb Z_2

topological order in

(2{+}1)

dimensions, using

\mathbb Z_8

Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for

\mathbb Z_D^F

symmetries for

D

even. Using this, we realize a stabilizer model for a nontrivial

\mathbb Z_8^F

fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.

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.