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:2005.09668 · Physical Review Research

Protection of parity-time symmetry in topological many-body systems: Non-Hermitian toric code and fracton models

Henry Shackleton, Mathias S. Scheurer

Published Jul 6, 2020• 28 citations
arXivHTMLPDF

Abstract

This paper demonstrates that the ground state subspace of systems with topological order - such as the toric code and systems with fracton order - can stay real under a robust set of non-Hermitian perturbations. This preservation of the reality of eigenvalues is sensitive to the size of the system, the conditions for which can be formulated both algebraically and geometrically.

DOI
10.1103/physrevresearch.2.033022

Related papers

Non-Hermitian avalanche effect: Non-perturbative effect induced by local non-Hermitian perturbation on a Z 2 topological order

Cui-Xian Guo, Xiao-Ran Wang, Su-Peng Kou

In this paper, based on a non-Hermitian toric-code model we found the fragileness of intrinsic topological orders under non-Hermitian perturbations. The effect of non-Hermitian avalanche for a designed toric-code model is uncovered: for the designed toric-code model with special external fields, a tiny non-Hermitian perturbation (local imaginary state-selective dissipation) leads to anomalous topological degeneracy and breaks down the bulk-degeneracy correspondence (a correspondence between bulk quasi-particles and topologically protected degenerate ground states). To support our theoretical predictions, we calculated the ground-state degeneracy and fidelity susceptibility of the ground states based on the non-Hermitian toric-code model on lattice and that on lattice. The numerical results are consistent with theoretical prediction.

arXiv:2001.04209•Europhysics Letters (EPL)•Aug 24, 2020•14 citations

Many-body-localization protection of eigenstate topological order in two dimensions

Florian Venn, Thorsten B. Wahl, Benjámin Béri

Many-body localization (MBL) has been proposed to enable and protect topological order in all eigenstates, vastly expanding the traditional ground-state setting. However, for the most intriguing case of two-dimensional (2D) systems with anyons and topology-dependent degeneracies, the dense many-body spectrum challenges studying this MBL protection numerically. Here we use large-scale full-spectrum variational to demonstrate MBL-protected topological order in the disordered 2D toric code perturbed by magnetic fields. We show that the system has topological local integrals of motion (tLIOMs) for magnetic field strengths below h c ≈ 0.1 times the toric-code coupling scale. Combining tLIOMs with exact diagonalization, we also identify high-energy topological multiplets in the dense many-body spectrum. The phase diagram we find is consistent with toric-code and trivial MBL phases being separated by an intervening thermal phase. Published by the American Physical Society 2024

arXiv:2212.09775•Physical review. B./Physical review. B•Oct 21, 2024•9 citations

Robustness of Topologically Ordered Matter

Steven H. Simon

Abstract Topologically ordered matter is famously robust to small perturbations. If a Hamiltonian has a topologically ordered (TQFT) ground state with a gap, then adding a small pertrubation (on the scale of the gap) to the Hamiltonian cannot change its TQFT properties. We explore this robustness with the example of the toric code. We show that the robustness is guaranteed by the fact that the toric code has a protected code space. We further discuss how the properties of the quasiparticles are also unchanged under perturbations of the Hamiltonian. These properties might be unsurprising, given the rigidity of the properties of TQFTs. We define the notion of topological order and give a definition of a topological phase of matter.

•Sep 29, 2023•1 citation

Topological entanglement entropy of fracton stabilizer codes

Han Ma, Albert T. Schmitz, S. A. Parameswaran +2 more

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.

arXiv:1710.01744•Physical review. B./Physical review. B•Mar 1, 2018•111 citations

Ground state degeneracy on torus in a family of ZN toric code

Haruki Watanabe, Meng Cheng, Yohei Fuji

Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.

arXiv:2211.00299•Journal of Mathematical Physics•May 1, 2023•34 citations

Symmetry-deformed toric codes and the quantum dimer model

Jiaxin Qiao, Yoshito Watanabe, Simon Trebst

Motivated by the recent introduction of a U ( 1 ) -symmetric toric code (TC) model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the TC Hamiltonian. This “term-dropping” protocol introduces global symmetries that go beyond the alternative framework of “ungauging” topological order in symmetry-deformed models and gives rise to models such as the U ( 1 ) TC or X Y TC . These models inherit (emergent) subsystem symmetries (from the original 1-form symmetry of the TC) that can give rise to (subextensive) ground-state degeneracies, which can still be organized by the eigenvalues of Wilson loop operators. However, we demonstrate that these models do support topological or fracton order (as has been conjectured in the literature) due to the loss of (emergent) gauge symmetry. An extreme deformation of the TC is the quantum dimer model (QDM), which we discuss along the family of symmetry-deformed models from the perspective of subsystem symmetries, sublattice modulation, and quantum order-by-disorder mechanisms resulting in rich phase diagrams. For the QDM, this allows us to identify an emergent SO(2) symmetry for what appears to be a gapless ground state (by numerical standards) that is unstable to the formation of a plaquette valence bond solid upon sublattice modulation.

arXiv:2506.00114•Physical Review Research•Dec 8, 2025•2 citations
This paper demonstrates that the ground state subspace of systems with topological order - such as the toric code and systems with fracton order - can stay real under a robust set of non-Hermitian perturbations. This preservation of the reality of eigenvalues is sensitive to the size of the system, the conditions for which can be formulated both algebraically and geometrically.
In this paper, based on a non-Hermitian toric-code model we found the fragileness of intrinsic topological orders under non-Hermitian perturbations. The effect of non-Hermitian avalanche for a designed toric-code model is uncovered: for the designed toric-code model with special external fields, a tiny non-Hermitian perturbation (local imaginary state-selective dissipation) leads to anomalous topological degeneracy and breaks down the bulk-degeneracy correspondence (a correspondence between bulk quasi-particles and topologically protected degenerate ground states). To support our theoretical predictions, we calculated the ground-state degeneracy and fidelity susceptibility of the ground states based on the non-Hermitian toric-code model on lattice and that on lattice. The numerical results are consistent with theoretical prediction.
Many-body localization (MBL) has been proposed to enable and protect topological order in all eigenstates, vastly expanding the traditional ground-state setting. However, for the most intriguing case of two-dimensional (2D) systems with anyons and topology-dependent degeneracies, the dense many-body spectrum challenges studying this MBL protection numerically. Here we use large-scale full-spectrum variational to demonstrate MBL-protected topological order in the disordered 2D toric code perturbed by magnetic fields. We show that the system has topological local integrals of motion (tLIOMs) for magnetic field strengths below h c ≈ 0.1 times the toric-code coupling scale. Combining tLIOMs with exact diagonalization, we also identify high-energy topological multiplets in the dense many-body spectrum. The phase diagram we find is consistent with toric-code and trivial MBL phases being separated by an intervening thermal phase. Published by the American Physical Society 2024
Abstract Topologically ordered matter is famously robust to small perturbations. If a Hamiltonian has a topologically ordered (TQFT) ground state with a gap, then adding a small pertrubation (on the scale of the gap) to the Hamiltonian cannot change its TQFT properties. We explore this robustness with the example of the toric code. We show that the robustness is guaranteed by the fact that the toric code has a protected code space. We further discuss how the properties of the quasiparticles are also unchanged under perturbations of the Hamiltonian. These properties might be unsurprising, given the rigidity of the properties of TQFTs. We define the notion of topological order and give a definition of a topological phase of matter.
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.
Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.
Motivated by the recent introduction of a U ( 1 ) -symmetric toric code (TC) model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the TC Hamiltonian. This “term-dropping” protocol introduces global symmetries that go beyond the alternative framework of “ungauging” topological order in symmetry-deformed models and gives rise to models such as the U ( 1 ) TC or X Y TC . These models inherit (emergent) subsystem symmetries (from the original 1-form symmetry of the TC) that can give rise to (subextensive) ground-state degeneracies, which can still be organized by the eigenvalues of Wilson loop operators. However, we demonstrate that these models do support topological or fracton order (as has been conjectured in the literature) due to the loss of (emergent) gauge symmetry. An extreme deformation of the TC is the quantum dimer model (QDM), which we discuss along the family of symmetry-deformed models from the perspective of subsystem symmetries, sublattice modulation, and quantum order-by-disorder mechanisms resulting in rich phase diagrams. For the QDM, this allows us to identify an emergent SO(2) symmetry for what appears to be a gapless ground state (by numerical standards) that is unstable to the formation of a plaquette valence bond solid upon sublattice modulation.