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arXiv:1503.05051 · Physical Review B

Coupled-wire construction of chiral spin liquids

Tobias Meng, Titus Neupert, Martin Greiter, Ronny Thomale

Published Jun 17, 2015• 97 citations
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Abstract

We develop a coupled-wire construction of chiral spin liquids. The starting points are individual wires of electrons in the Mott regime that are subject to a Zeeman field and Rashba spin-orbit coupling. Suitable spin-flip couplings between the wires yield an Abelian chiral spin liquid state which supports spinon excitations above a bulk gap and chiral edge states. The approach generalizes to non-Abelian chiral spin liquids at level $k$ with parafermionic edge states.

DOI
10.1103/physrevb.91.241106

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We develop a coupled-wire construction of chiral spin liquids. The starting points are individual wires of electrons in the Mott regime that are subject to a Zeeman field and Rashba spin-orbit coupling. Suitable spin-flip couplings between the wires yield an Abelian chiral spin liquid state which supports spinon excitations above a bulk gap and chiral edge states. The approach generalizes to non-Abelian chiral spin liquids at level kkk with parafermionic edge states.
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 describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground state degeneracy on the torus signalling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.
We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground-state degeneracy on the torus signaling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field-theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.
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.