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:1612.02831 · Physical review. B./Physical review. B

Edge-entanglement correspondence for a gapped topological phase with symmetry

Maciej Koch-Janusz, Kusum Dhochak, Erez Berg

Published May 8, 2017• 7 citations
arXivHTMLPDF

Abstract

The correspondence between the edge theory and the entanglement spectrum is firmly established for the chiral topological phases. We study a gapped, topologically ordered, nonchiral state with a conserved $\text{U}(1)$ charge and show that the entanglement Hamiltonian contains not only the information about topologically distinct edges such a phase may admit, but also which of them will be realized in the presence of symmetry-breaking/-conserving perturbations. We introduce an exactly solvable, charge-conserving lattice model of a ${\mathbb{Z}}_{2}$ spin liquid and derive its edge theory and the entanglement Hamiltonian, also in the presence of perturbations. We construct a field theory of the edge and study its renormalization group flow. We show the precise extent of the correspondence between the information contained in the entanglement Hamiltonian and the edge theory.

DOI
10.1103/physrevb.95.205110

Related papers

Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality

Wen Wei Ho, Łukasz Cincio, Heidar Moradi +2 more

For chiral topological phases there exist analytical proofs of a one-to-one correspondence between the low lying spectrum of edge states and that of the entanglement spectrum. Whether or not such an edge-entanglement spectrum correspondence could apply to non-chiral phases has been unclear. Here it is shown that in the Wen-plaquette model, a non-chiral ${Z}_{2}$ topological phase, such a correspondence is exact in the absence of perturbations. While no such correspondence exists in the case of a general perturbation, the correspondence can be shown to apply for perturbations restricted to be translation invariant along the edge/entanglement cut.

arXiv:1411.6932•Physical Review B•Mar 11, 2015•43 citations

Symmetry Enriched Topological Phases and Their Edge Theories

Christopher Phillip Heinrich

In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.

•Jan 1, 2017•0 citations

Gapless chirality liquid with symmetry-protected edge spins

Shunsuke C. Furuya, Katsuhiro Morita

We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, $\mathbb Z_3\times\mathbb Z_2^T$, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local $\mathbb{Z}_3$ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the $\mathbb{Z}_3$ on-site symmetry, the $\mathbb{Z}_2^T$ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.

arXiv:2207.02485•Jul 6, 2022•0 citations

Topological Order and Universal Properties of Gapped Quantum Systems

Heidar Moradi

Phases of gapped quantum liquids are topologically ordered and have very interesting physical features that are completely robust against any local perturbation that do not close the bulk energy gap. These universal properties are hidden in the ground states of these systems, as different patterns of many-body long-range entanglement. In this thesis we study the universal properties of gapped quantum liquids from various perspectives. We propose the notion of Universal Wavefunction Overlap as a way of extracting almost complete information about the underlying entanglement structure in a system with topological order. We propose an efficient numerical methods to use these universal wavefunction overlaps as topological order parameters and demonstrate their usefulness with concrete numerical computations. In 2 + 1D these overlaps correspond to known quantities and contain information about anyonic particle excitations. We show that in 3 + 1D, these overlaps contain information about linked multi-string braiding processes, in particular three-string braiding. \n \nIn the second part of this thesis, we study boundary physics of systems with topological order. We investigate the correspondence between edge and entanglement spectra for non-chiral topological systems in general and with the presence of extra symmetries and dualities. We also show that by local deformations of the fixed-point wavefunction on non- chiral topological orders, all possible edge theories can be extracted from its entanglement Hamiltonian. Finally we introduce the notion of fermionic gapped boundaries and see how the phase diagram of the simplest topological orders get significantly enriched.

•Nov 15, 2018•0 citations

Universal edge information from wave-function deformation

Wen Wei Ho, Łukasz Cincio, Heidar Moradi +1 more

It is well known that the bulk physics of a topological phase constrains its possible edge physics through the bulk-edge correspondence. Therefore, the different types of edge theories that a topological phase can host constitute a universal piece of data which can be used to characterize topological order. In this paper, we argue that, beginning from only the fixed-point wave function (FPW) of a nonchiral topological phase and by locally deforming it, all possible edge theories can be extracted from its entanglement Hamiltonian (EH). We give a general argument, and concretely illustrate our claim by deforming the FPW of the Wen-plaquette model, the quantum double of ${\mathbb{Z}}_{2}$. In that case, we show that the possible EHs of the deformed FPW reflect the known possible types of edge theories, which are generically gapped but gapless if translational symmetry is preserved. We stress that our results do not require an underlying Hamiltonian---thus, this lends support to the notion that a topological phase is indeed characterized by only a set of quantum states and can be studied through its FPWs.

arXiv:1510.02982•Physical review. B./Physical review. B•Jun 6, 2017•5 citations

SU(2)1 chiral edge modes of a critical spin liquid

Didier Poilblanc, Norbert Schuch, Ian K. Affleck

Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-$\frac{1}{2}$ quantum spin liquids with ${\mathbb{Z}}_{2}$ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)${}_{1}$), as is the case for the $\ensuremath{\nu}$=$\frac{1}{2}$ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.

arXiv:1602.05969•Physical review. B./Physical review. B•May 18, 2016•43 citations
The correspondence between the edge theory and the entanglement spectrum is firmly established for the chiral topological phases. We study a gapped, topologically ordered, nonchiral state with a conserved U(1)\text{U}(1)U(1) charge and show that the entanglement Hamiltonian contains not only the information about topologically distinct edges such a phase may admit, but also which of them will be realized in the presence of symmetry-breaking/-conserving perturbations. We introduce an exactly solvable, charge-conserving lattice model of a Z2{\mathbb{Z}}_{2}Z2​ spin liquid and derive its edge theory and the entanglement Hamiltonian, also in the presence of perturbations. We construct a field theory of the edge and study its renormalization group flow. We show the precise extent of the correspondence between the information contained in the entanglement Hamiltonian and the edge theory.
For chiral topological phases there exist analytical proofs of a one-to-one correspondence between the low lying spectrum of edge states and that of the entanglement spectrum. Whether or not such an edge-entanglement spectrum correspondence could apply to non-chiral phases has been unclear. Here it is shown that in the Wen-plaquette model, a non-chiral Z2{Z}_{2}Z2​ topological phase, such a correspondence is exact in the absence of perturbations. While no such correspondence exists in the case of a general perturbation, the correspondence can be shown to apply for perturbations restricted to be translation invariant along the edge/entanglement cut.
In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.
We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, Z3×Z2T\mathbb Z_3\times\mathbb Z_2^TZ3​×Z2T​, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local Z3\mathbb{Z}_3Z3​ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the Z3\mathbb{Z}_3Z3​ on-site symmetry, the Z2T\mathbb{Z}_2^TZ2T​ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.
Phases of gapped quantum liquids are topologically ordered and have very interesting physical features that are completely robust against any local perturbation that do not close the bulk energy gap. These universal properties are hidden in the ground states of these systems, as different patterns of many-body long-range entanglement. In this thesis we study the universal properties of gapped quantum liquids from various perspectives. We propose the notion of Universal Wavefunction Overlap as a way of extracting almost complete information about the underlying entanglement structure in a system with topological order. We propose an efficient numerical methods to use these universal wavefunction overlaps as topological order parameters and demonstrate their usefulness with concrete numerical computations. In 2 + 1D these overlaps correspond to known quantities and contain information about anyonic particle excitations. We show that in 3 + 1D, these overlaps contain information about linked multi-string braiding processes, in particular three-string braiding. \n \nIn the second part of this thesis, we study boundary physics of systems with topological order. We investigate the correspondence between edge and entanglement spectra for non-chiral topological systems in general and with the presence of extra symmetries and dualities. We also show that by local deformations of the fixed-point wavefunction on non- chiral topological orders, all possible edge theories can be extracted from its entanglement Hamiltonian. Finally we introduce the notion of fermionic gapped boundaries and see how the phase diagram of the simplest topological orders get significantly enriched.
It is well known that the bulk physics of a topological phase constrains its possible edge physics through the bulk-edge correspondence. Therefore, the different types of edge theories that a topological phase can host constitute a universal piece of data which can be used to characterize topological order. In this paper, we argue that, beginning from only the fixed-point wave function (FPW) of a nonchiral topological phase and by locally deforming it, all possible edge theories can be extracted from its entanglement Hamiltonian (EH). We give a general argument, and concretely illustrate our claim by deforming the FPW of the Wen-plaquette model, the quantum double of Z2{\mathbb{Z}}_{2}Z2​. In that case, we show that the possible EHs of the deformed FPW reflect the known possible types of edge theories, which are generically gapped but gapless if translational symmetry is preserved. We stress that our results do not require an underlying Hamiltonian---thus, this lends support to the notion that a topological phase is indeed characterized by only a set of quantum states and can be studied through its FPWs.
Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)1{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.