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:1706.07113 · Journal of High Energy Physics

Effective theories for 2+1 dimensional non-Abelian topological spin liquids

Carlos A. Hernaski, Pedro R. S. Gomes

Published Oct 1, 2017• 5 citations
arXivHTMLPDF

Abstract

In this work we propose an effective low-energy theory for a large class of 2+1 dimensional non-Abelian topological spin liquids whose edge states are conformal degrees of freedom with central charges corresponding to the coset structure su(2) k ⊕ su(2) k ′ /su(2) k+k ′ . For particular values of k ′ it furnishes the series for unitary minimal and superconformal models. These gapped phases were recently suggested to be obtained from an array of one-dimensional coupled quantum wires. In doing so we provide an explicit relationship between two distinct approaches: quantum wires and Chern-Simons bulk theory. We firstly make a direct connection between the interacting quantum wires and the corresponding conformal field theory at the edges, which turns out to be given in terms of chiral gauged WZW models. Relying on the bulk-edge correspondence we are able to construct the underlying non-Abelian Chern-Simons effective field theory.

DOI
10.1007/jhep10(2017)021

Related papers

Non-Abelian topological spin liquids from arrays of quantum wires or spin chains

Po-Hao Huang, Jyong-Hao Chen, Pedro R. S. Gomes +3 more

In this paper, a family of non-Abelian topologically ordered states are built from arrays of quantum wires with fine-tuned electron-electron interactions. With the help of non-Abelian bosonization, the decoupled wires are described by conformal field theories. Electron-electron interactions are chosen to gap all conformal field theories except the one that describes the edge state. Hence, the construction makes it easy to engineer different topological ordered states. The authors prescribe, in particular, how to arrive at all the edge states described by the unitary conformal field theory minimal models using the spin degrees of freedom of the quantum wires alone. The construction provides a bridge between the field theory descriptions and the microscopic models of topological ordered states.

arXiv:1601.01094•Physical review. B./Physical review. B•May 13, 2016•52 citations

Chiral edge states in (2+1)-dimensional topological phases

Carlos A. Hernaski, Pedro R. S. Gomes

Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.

arXiv:1707.08048•International Journal of Modern Physics A•Oct 9, 2019•0 citations

Lattice spin models for non-Abelian chiral spin liquids

P. Lecheminant, Alexei M. Tsvelik

We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)${}_{n}$ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-$n/2$ ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)${}_{n}$ gapless edge excitations.

arXiv:1608.05977•Physical review. B./Physical review. B•Apr 26, 2017•39 citations

Bulk effective theory in coupled wire construction from generalized Wilson line

Yukihisa Imamura, Keisuke Totsuka

We reconsider the coupled wire construction, which is a useful method of obtaining two and three-dimensional topologically ordered states from an array of one-dimensional CFTs, and show that there is the hidden structure in the bulk of the constructed model. In order to uncover the structure hidden in the gapped bulk, we introduce a generalized Wilson line and show the necessity of introducing a new gauge field, which behaves like the Chern-Simons gauge field. From these examination, it is possible to show that in this formulation the bulk theory is the U(1) Chern-Simons gauge theory and the edge theory is the chiral Luttinger theory. We explicitly demonstrate our method for the Laughlin states and the chiral spin liquid state.

arXiv:1605.09235•May 30, 2016•1 citation

Dihedral twist liquid models from emergent Majorana fermions

Jeffrey C. Y. Teo, Yichen Hu

We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral Dk symmetry of SO(2n)1 , and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G twist liquids, where G=Z2 for C=U(1)l , SU(n)1 , and G=Z2 , Zk , Dk for C=SO(2n)1 . We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.

arXiv:2107.07242•Quantum•Mar 30, 2023•3 citations

Coupled spin- 12 ladders as microscopic models for non-Abelian chiral spin liquids

Po-Hao Huang, Jyong-Hao Chen, Adrian E. Feiguin +2 more

We construct a two-dimensional (2D) lattice model that is argued to realize a gapped chiral spin liquid with (Ising) non-Abelian topological order. The building blocks are spin-$\frac{1}{2}$ two-leg ladders with $\text{SU}(2)$-symmetric spin-spin interactions. The two-leg ladders are then arranged on rows and coupled through $\text{SU}(2)$-symmetric interactions between consecutive ladders. The intraladder interactions are tuned so as to realize $c=1/2$ Ising conformal field theory, a fact that we establish numerically via density matrix renormalization group studies. Time-reversal breaking interladder interactions are tuned so as to open a bulk gap in the 2D lattice system. This 2D system supports gapless chiral edge modes with Ising non-Abelian excitations but no charge excitations, in contrast to the Pfaffian non-Abelian fractional quantum Hall state.

arXiv:1611.02523•Physical review. B./Physical review. B•Apr 12, 2017•16 citations
In this work we propose an effective low-energy theory for a large class of 2+1 dimensional non-Abelian topological spin liquids whose edge states are conformal degrees of freedom with central charges corresponding to the coset structure su(2) k ⊕ su(2) k ′ /su(2) k+k ′ . For particular values of k ′ it furnishes the series for unitary minimal and superconformal models. These gapped phases were recently suggested to be obtained from an array of one-dimensional coupled quantum wires. In doing so we provide an explicit relationship between two distinct approaches: quantum wires and Chern-Simons bulk theory. We firstly make a direct connection between the interacting quantum wires and the corresponding conformal field theory at the edges, which turns out to be given in terms of chiral gauged WZW models. Relying on the bulk-edge correspondence we are able to construct the underlying non-Abelian Chern-Simons effective field theory.
In this paper, a family of non-Abelian topologically ordered states are built from arrays of quantum wires with fine-tuned electron-electron interactions. With the help of non-Abelian bosonization, the decoupled wires are described by conformal field theories. Electron-electron interactions are chosen to gap all conformal field theories except the one that describes the edge state. Hence, the construction makes it easy to engineer different topological ordered states. The authors prescribe, in particular, how to arrive at all the edge states described by the unitary conformal field theory minimal models using the spin degrees of freedom of the quantum wires alone. The construction provides a bridge between the field theory descriptions and the microscopic models of topological ordered states.
Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.
We suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)n{}_{n}n​ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-n/2n/2n/2 ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)n{}_{n}n​ gapless edge excitations.
We reconsider the coupled wire construction, which is a useful method of obtaining two and three-dimensional topologically ordered states from an array of one-dimensional CFTs, and show that there is the hidden structure in the bulk of the constructed model. In order to uncover the structure hidden in the gapped bulk, we introduce a generalized Wilson line and show the necessity of introducing a new gauge field, which behaves like the Chern-Simons gauge field. From these examination, it is possible to show that in this formulation the bulk theory is the U(1) Chern-Simons gauge theory and the edge theory is the chiral Luttinger theory. We explicitly demonstrate our method for the Laughlin states and the chiral spin liquid state.
We present a family of electron-based coupled-wire models of bosonic orbifold topological phases, referred to as twist liquids, in two spatial dimensions. All local fermion degrees of freedom are gapped and removed from the topological order by many-body interactions. Bosonic chiral spin liquids and anyonic superconductors are constructed on an array of interacting wires, each supports emergent massless Majorana fermions that are non-local (fractional) and constitute the SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral Dk symmetry of SO(2n)1 , and its promotion to a gauge symmetry by manipulating the locality of fermion pairs. Gauging the symmetry (sub)group generates the C/G twist liquids, where G=Z2 for C=U(1)l , SU(n)1 , and G=Z2 , Zk , Dk for C=SO(2n)1 . We construct exactly solvable models for all of these topological states. We prove the presence of a bulk excitation energy gap and demonstrate the appearance of edge orbifold conformal field theories corresponding to the twist liquid topological orders. We analyze the statistical properties of the anyon excitations, including the non-Abelian metaplectic anyons and a new class of quasiparticles referred to as Ising-fluxons. We show an eight-fold periodic gauging pattern in SO(2n)1/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.
We construct a two-dimensional (2D) lattice model that is argued to realize a gapped chiral spin liquid with (Ising) non-Abelian topological order. The building blocks are spin-12\frac{1}{2}21​ two-leg ladders with SU(2)\text{SU}(2)SU(2)-symmetric spin-spin interactions. The two-leg ladders are then arranged on rows and coupled through SU(2)\text{SU}(2)SU(2)-symmetric interactions between consecutive ladders. The intraladder interactions are tuned so as to realize c=1/2c=1/2c=1/2 Ising conformal field theory, a fact that we establish numerically via density matrix renormalization group studies. Time-reversal breaking interladder interactions are tuned so as to open a bulk gap in the 2D lattice system. This 2D system supports gapless chiral edge modes with Ising non-Abelian excitations but no charge excitations, in contrast to the Pfaffian non-Abelian fractional quantum Hall state.