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

Abstract

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-1/2 two-leg ladders with SU (2)-symmetric spin-spin interactions. The two-leg ladders are then arranged on rows and coupled through SU (2)-symmetric interactions between consecutive ladders. The intraladder interactions are tuned so as to realize the c = 1/2 Ising conformal field theory, a fact that we establish numerically via Density Matrix Renormalization Group (DMRG) studies. Time-reversal breaking inter-ladder 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.That point particles may obey non-Abelian braiding statistics in (2+1)-dimensional spacetime has been known in quantum-field theory since the 1980's [1][2][3][4][5]. Moore and Read showed in 1991 that certain Pfaffian wave functions support quasi-particles with non-Abelian braiding statistics [6]. This discovery opened the possibility that non-Abelian braiding statistics could be found in certain fractional quantum Hall plateaus [6][7][8][9].A second physical platform to realize braiding statistics that is neither bosonic nor fermionic is provided by quasitwo-dimensional quantum spin magnets with a gapped chiral spin-liquid ground state [10, 11]. Quasi-twodimensional arrays of quantum spin chains also have the potential for realizing gapped spin liquid ground states with quasi-particles obeying Abelian or non-Abelian braiding statistics [12][13][14][15].In this paper, we construct a two-dimensional (2D) lattice model, depicted in Fig. 1, that is argued to realize a non-Abelian chiral spin liquid. This 2D model consists of an array of coupled one-dimensional (1D) two-leg quantum spin-1/2 ladders. The inter-ladder coupling leads to a bulk gap, while gapless modes remain at the boundaries. The chiral edge states correspond to the Ising conformal field theory (CFT) with central charge c = 1/2, similarly to the Moore-Read Pfaffian state. However, in contrast to the Pfaffian quantum Hall state, there is no additional c = 1 chiral bosonic charge-carrying edge mode. By the bulk-edge correspondence, the bulk of the coupled spin-ladder model is a gapped chiral spin liquid supporting Ising non-Abelian topological order [16,17].To obtain this result, we argue that the aforementioned lattice model is a regularization of one of the interacting quantum-field theories presented in Ref. 14, one that supports chiral non-Abelian topological order. We start from coupled two-leg ladders (called bundles in Ref. 14), on which sites quantum spin-1/2 degrees of freedom are localized. Two ingredients are needed. First, the interactions within the two-leg ladders should be fine-tuned so as to realize the Ising universality class in (1+1)-dimensional spacetime, the Ising criticality in short. Second, the...

Model of chiral spin liquids with Abelian and non-Abelian topological phases

“…This was also the case in Ref. 19. However, instead of relying on two-body spin-1/2 interactions which are reduced to marginal current-current interactions at low energies, as was the case in Ref.…”

“…However, instead of relying on two-body spin-1/2 interactions which are reduced to marginal current-current interactions at low energies, as was the case in Ref. 19, we shall rely in this paper on four-body spin-1/2 interactions which are reduced to a mass term for three out of the four gapless Majorana fields that encode the critical theory of two decoupled antiferromagnetic quantum spin-1/2 chains. Once a single spin-1/2 ladder is tuned to the Ising critical point, we couple the ladders as was done in Ref.…”

“…Once a single spin-1/2 ladder is tuned to the Ising critical point, we couple the ladders as was done in Ref. 19. The resulting lattice model is depicted in Fig.…”

Model of spin liquids with and without time-reversal symmetry

“…The phase boundary between the CD phase and the RS phase is a continuous phase transition belonging to the two-dimensional Ising universality class. The numerical evidence for this Ising transition is supported by the finite-size scaling of the leg-dimer order parameter 14,27…”

“…with H and H deduced from H and H by the substitution S i,m → S i,m . The low energy limit of Hamiltonian H inter-ladder was obtained using bosonization in Ref 14. and 15 (see also Ref.…”