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:1812.10523 · Physical review. B./Physical review. B

Corner modes and ground-state degeneracy in models with gaugelike subsystem symmetries

Julian May-Mann, Taylor L. Hughes

Published Oct 4, 2019• 4 citations
arXivHTMLPDF

Abstract

Subsystem symmetries are intermediate between global and gauge symmetries. One can treat these symmetries either like global symmetries that act on subregions of a system, or gauge symmetries that act on the regions transverse to the regions acted upon by the symmetry. We show that this latter interpretation can lead to an understanding of global, topology-dependent features in systems with subsystem symmetries. We demonstrate this with an exactly solvable lattice model constructed from a two-dimensional system of bosons coupled to a vector field with a one-dimensional subsystem symmetry. The model is shown to host a robust ground-state degeneracy that depends on the spatial topology of the underlying manifold, and localized zero-energy modes on corners of the system. A continuum field theory description of these phenomena is derived in terms of an anisotropic, modified version of the Abelian $K$-matrix Chern-Simons field theory. We show that this continuum description can lead to geometric-type effects such as corner states and edge states whose character depends on the orientation of the edge.

DOI
10.1103/physrevb.100.165108

Related papers

Subsystem symmetry protected topological order

Yizhi You, Trithep Devakul, F. J. Burnell +1 more

In this work, we introduce a new type of topological order that is protected by subsystem symmetries that act on lower-dimensional subsets of lattice many-body system, e.g., along lines or planes in a three-dimensional system. The symmetry groups for such systems exhibit a macroscopic number of generators in the infinite-volume limit. We construct a set of exactly solvable models in $2D$ and $3D$, which exhibit such subsystem SPT (SSPT) phases with one-dimensional subsystem symmetries. These phases exhibit analogs of phenomena seen in SPTs protected by global symmetries: gapless edge modes, projective realizations of the symmetries at the edge, and nonlocal order parameters. Such SSPT phases are proximate, in theory space, to previously studied phases that break the subsystem symmetries and phases with fracton order, which result upon gauging them.

arXiv:1803.02369•Physical review. B./Physical review. B•Jul 10, 2018•191 citations

Rotational symmetry protected edge and corner states in Abelian topological phases

Naren Manjunath, Abhinav Prem, Yuan-Ming Lu

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with nontrivial strong topological invariants (protected by internal symmetries) with additional ``weak'' topological indices. In this paper, we study the edge physics of systems with a nontrivial shift invariant, which is protected by either a continuous $\text{U}{(1)}_{r}$ or discrete ${\text{C}}_{n}$ rotation symmetry, along with internal $\text{U}{(1)}_{c}$ charge conservation. Specifically, we construct an interface between two systems that have the same Chern number but are distinguished by their Wen-Zee shift, and, through analytic arguments supported by numerics, we show that the interface hosts counterpropagating gapless edge modes that cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian subalgebra framework for gapped boundaries, while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and we compute them in the case in which the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in strongly interacting symmetry enriched topological phases.

arXiv:2303.08148•Physical review. B./Physical review. B•May 16, 2023•8 citations

TOPOLOGICAL ORDERS IN RIGID STATES

Xiao-Gang Wen

We study a new kind of ordering — topological order — in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2 + 1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a non-Abelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2 + 1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations. The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states. We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90° rotation.

International Journal of Modern Physics B•Feb 1, 1990•1,115 citations

Fractonlike phases from subsystem symmetries

J. P. Ibieta Jimenez, L. N. Queiroz Xavier, M. Petrucci +1 more

We study models with fractonlike order based on ${\mathbb{Z}}_{2}$ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy ${S}_{A}$ of these models in a subregion $A$ of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to $A$.

arXiv:1908.07601•Physical review. B./Physical review. B•Jul 2, 2020•11 citations

Dyonic zero-energy modes

Morten I. K. Munk, Asbjørn Rasmussen, Michele Burrello

One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a twofold ground-state degeneracy related to the global ${\mathbb{Z}}_{2}$ symmetry associated with fermionic parity. By extending the symmetry to the ${\mathbb{Z}}_{N}$ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group $G$. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of $G$. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.

arXiv:1807.09286•Physical review. B./Physical review. B•Dec 21, 2018•11 citations

Fractons with twisted boundary conditions and their symmetries

Tom Rudelius, Nathan Seiberg B, Shu-Heng Shao

We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the $xy$ plane. The ground-state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground-state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in $2+1$ and $3+1$ dimensional systems. It originates from the underlying subsystem symmetry but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus $\ensuremath{\tau}={\ensuremath{\tau}}_{1}+i{\ensuremath{\tau}}_{2}$ has rational ${\ensuremath{\tau}}_{1}=k/m$. Then, in systems based on $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized ${\mathbb{Z}}_{m}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{m}$, which leads to an $m$-fold ground state degeneracy. In systems based on ${\mathbb{Z}}_{N}$ symmetries, like the X-cube model, each of these two ${\mathbb{Z}}_{m}$ factors is replaced by ${\mathbb{Z}}_{gcd(N,m)}$.

arXiv:2012.11592•Physical review. B./Physical review. B•May 7, 2021•42 citations
Subsystem symmetries are intermediate between global and gauge symmetries. One can treat these symmetries either like global symmetries that act on subregions of a system, or gauge symmetries that act on the regions transverse to the regions acted upon by the symmetry. We show that this latter interpretation can lead to an understanding of global, topology-dependent features in systems with subsystem symmetries. We demonstrate this with an exactly solvable lattice model constructed from a two-dimensional system of bosons coupled to a vector field with a one-dimensional subsystem symmetry. The model is shown to host a robust ground-state degeneracy that depends on the spatial topology of the underlying manifold, and localized zero-energy modes on corners of the system. A continuum field theory description of these phenomena is derived in terms of an anisotropic, modified version of the Abelian KKK-matrix Chern-Simons field theory. We show that this continuum description can lead to geometric-type effects such as corner states and edge states whose character depends on the orientation of the edge.
In this work, we introduce a new type of topological order that is protected by subsystem symmetries that act on lower-dimensional subsets of lattice many-body system, e.g., along lines or planes in a three-dimensional system. The symmetry groups for such systems exhibit a macroscopic number of generators in the infinite-volume limit. We construct a set of exactly solvable models in 2D2D2D and 3D3D3D, which exhibit such subsystem SPT (SSPT) phases with one-dimensional subsystem symmetries. These phases exhibit analogs of phenomena seen in SPTs protected by global symmetries: gapless edge modes, projective realizations of the symmetries at the edge, and nonlocal order parameters. Such SSPT phases are proximate, in theory space, to previously studied phases that break the subsystem symmetries and phases with fracton order, which result upon gauging them.
Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with nontrivial strong topological invariants (protected by internal symmetries) with additional ``weak'' topological indices. In this paper, we study the edge physics of systems with a nontrivial shift invariant, which is protected by either a continuous U(1)r\text{U}{(1)}_{r}U(1)r​ or discrete Cn{\text{C}}_{n}Cn​ rotation symmetry, along with internal U(1)c\text{U}{(1)}_{c}U(1)c​ charge conservation. Specifically, we construct an interface between two systems that have the same Chern number but are distinguished by their Wen-Zee shift, and, through analytic arguments supported by numerics, we show that the interface hosts counterpropagating gapless edge modes that cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian subalgebra framework for gapped boundaries, while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and we compute them in the case in which the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in strongly interacting symmetry enriched topological phases.
We study a new kind of ordering — topological order — in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2 + 1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a non-Abelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2 + 1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations. The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states. We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90° rotation.
We study models with fractonlike order based on Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy SA{S}_{A}SA​ of these models in a subregion AAA of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to AAA.
One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a twofold ground-state degeneracy related to the global Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with fermionic parity. By extending the symmetry to the ZN{\mathbb{Z}}_{N}ZN​ group, it is possible to engineer systems hosting topological parafermionic modes. In this work, we address one-dimensional systems with a generic discrete symmetry group GGG. We define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, we map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of GGG. We show that the so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero modes appear when suitable position-dependent couplings are considered.
We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the xyxyxy plane. The ground-state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground-state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in 2+12+12+1 and 3+13+13+1 dimensional systems. It originates from the underlying subsystem symmetry but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus $\ensuremath{\tau}={\ensuremath{\tau}}_{1}+i{\ensuremath{\tau}}_{2}$ has rational ${\ensuremath{\tau}}_{1}=k/m$. Then, in systems based on $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized ${\mathbb{Z}}_{m}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{m}$, which leads to an mmm-fold ground state degeneracy. In systems based on ZN{\mathbb{Z}}_{N}ZN​ symmetries, like the X-cube model, each of these two Zm{\mathbb{Z}}_{m}Zm​ factors is replaced by Zgcd(N,m){\mathbb{Z}}_{gcd(N,m)}Zgcd(N,m)​.