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Network construction of non-Abelian topological phases

Hernán Guillermo Bueno Xavier

Published May 5, 2023• 0 citations
OpenAlex

Abstract

In this thesis we use a network of chiral junctions to realize non-Abelian topological phases in two dimensions. Our construction is based on boundary fixed points of low-energy field theories describing Y junctions of critical spin-chain models. To set the stage, we first study a single junction of gapless spin-1 chains belonging to the SU(2)2 universality class, whose spectrum includes fractional excitations such as Majorana fermions. We find that chiral fixed points appear as special points on a transition line that separates two regimes described by open boundary conditions. Remarkably, along this transition the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. We then construct a family of topological chiral spin liquids departing from a honeycomb network made of critical spin-S chains. The chiral spin liquid phase harbors SU(2)k=2S anyons, which stem from the underlying WessZumino-Witten models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2)2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our work provides a controllable analytical framework to study non-Abelian topological phases, sheding new light on the stability of such phases in artificial quantum materials.

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In this thesis we use a network of chiral junctions to realize non-Abelian topological phases in two dimensions. Our construction is based on boundary fixed points of low-energy field theories describing Y junctions of critical spin-chain models. To set the stage, we first study a single junction of gapless spin-1 chains belonging to the SU(2)2 universality class, whose spectrum includes fractional excitations such as Majorana fermions. We find that chiral fixed points appear as special points on a transition line that separates two regimes described by open boundary conditions. Remarkably, along this transition the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. We then construct a family of topological chiral spin liquids departing from a honeycomb network made of critical spin-S chains. The chiral spin liquid phase harbors SU(2)k=2S anyons, which stem from the underlying WessZumino-Witten models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2)2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our work provides a controllable analytical framework to study non-Abelian topological phases, sheding new light on the stability of such phases in artificial quantum materials.
We use a network of chiral junctions to construct a family of topological chiral spin liquids in two spatial dimensions. The chiral spin liquid phase harbors SU(2) _k k anyons, which stem from the underlying SU(2) _k k WZW models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2) _2 2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our results shed new light on the stability of non-Abelian topological phases in artificial quantum materials.
Junctions of one-dimensional systems are of great interest to the development of synthetic materials that harbor topological phases. We study a junction of three gapless spin-1 chains described by the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Wittten model and coupled by exchange and chiral three-spin interactions. We show that a chiral fixed point appears as a special point on the transition line separating two regimes described by open boundary conditions, corresponding to decoupled chains and the formation of a boundary spin singlet state. Along this transition line, the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. Since the spectrum of the junction contains fractional excitations such as Majorana fermions, in this paper, we set the stage for network constructions of non-Abelian chiral spin liquids.
We show that a honeycomb lattice of Heisenberg spin-1/2 chains with three-spin junction interactions allows for controlled analytical studies of chiral spin liquids (CSLs). Tuning these interactions to a chiral fixed point, we find a Kalmeyer-Laughlin CSL phase which here is connected to the critical point of a boundary conformal field theory. Our construction directly yields a quantized spin Hall conductance and localized spinons with semionic statistics as elementary excitations. We also outline the phase diagram away from the chiral point where spinons may condense. Generalizations of our approach can provide microscopic realizations for many other CSLs.
We use junctions of critical spin-1 chains as the basic elements to construct a honeycomb network that harbors a gapless chiral spin liquid phase. The low-energy modes are described by spin-1 Majorana fermions that form a two-dimensional Fermi surface when the interactions at the junctions are tuned to the vicinity of chiral fixed points with staggered chirality. We discuss the physical properties and the stability of this chiral spin liquid phase against perturbations from the point of view of the effective field theory for the network. We find clear connections with the excitation spectrum obtained in parton constructions on the kagome lattice.
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 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.