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arXiv:1601.01094 · Physical review. B./Physical review. B

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

Po-Hao Huang, Jyong-Hao Chen, Pedro R. S. Gomes, Titus Neupert, Claudio Chamon, Christopher Mudry

Published May 13, 2016• 52 citations
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Abstract

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.

DOI
10.1103/physrevb.93.205123

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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.
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.
Starting from an array of interacting fermionic quantum wires, we construct a family of non-Abelian topologically ordered states of matter in three spatial dimensions (3D). These states of matter inherit their non-Abelian topological properties from the su(2)k su(2)^{\,}_{k}su(2)k​ conformal field theories that characterize the constituent interacting quantum wires in the decoupled limit. Thus, the resulting topological phases can be viewed as 3D generalizations of the (bosonic) su(2)k su(2)^{\,}_{k}su(2)k​ Read-Rezayi sequence of fractional quantum Hall states. Focusing in detail on the su(2)2 su(2)^{\,}_{2}su(2)2​ case, we first review how to determine the nature of the non-Abelian topological order (in particular, the topological degeneracy on the torus) in the two-dimensional (2D) case, before generalizing this approach to the 3D case. We also investigate the 2D boundary of the 3D phases, and show for the su(2)2 su(2)^{\,}_{2}su(2)2​ case that there are anomalous gapless surface states protected by an analog of time-reversal symmetry, similar to the massless Dirac surface states of the noninteracting 3D topological insulator.
We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non Abelian states. In the present context, the coset construction arises when the original chiral modes are fractionalized into coset sectors, and the different sectors acquire energy gaps due to coupling in "different directions". The coupled wire construction can also can be used to describe anisotropic lattice systems, and provides a starting point for models of fractional and non-Abelian Chern insulators. This paper also includes an extended introduction to the coupled wire construction for Abelian quantum Hall states, which was introduced earlier.
We study the topological order that arises from chiral states with SU(N) or SO(N) edge-state symmetry. This extends our previous study of topological orders that descend from the bosonic E8{E}_{8}E8​ quantum Hall state. We use exactly solvable models of coupled electron wires to construct states with SU(m)n,SO(m)n\mathrm{SU}{(m)}_{n}, \mathrm{SO}{(m)}_{n}SU(m)n​,SO(m)n​, or Sp(m)n\mathrm{Sp}{(m)}_{n}Sp(m)n​ topological order for various levels nnn. We use our constructions to write down string operators for various non-Abelian anyons. We thereby provide a systematic, model derivation of quantum Hall states, topological superconductors, and spin liquids with emergent non-Abelian quasiparticle excitations, including those of Ising, metaplectic, and Fibonacci type.
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.
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)SO(N)SO(N) Kac-Moody Wess-Zumino-Witten algebra at level 1. We focus on the dihedral DkD_kDk​ symmetry of SO(2n)1SO(2n)_1SO(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\mathcal{C}/GC/G twist liquids, where G=Z2G=\mathbb{Z}_2G=Z2​ for C=U(1)l\mathcal{C}=U(1)_lC=U(1)l​, SU(n)1SU(n)_1SU(n)1​, and G=Z2G=\mathbb{Z}_2G=Z2​, Zk\mathbb{Z}_kZk​, DkD_kDk​ for C=SO(2n)1\mathcal{C}=SO(2n)_1C=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/GSO(2n)_1/GSO(2n)1​/G by identifying the non-chiral components of the twist liquids with discrete gauge theories.