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arXiv:2410.01094 · Physical review. B./Physical review. B

Topological orders with classical Lie group symmetries from coupling electron wires

Pak Kau Lim, Michael Mulligan, Jeffrey C. Y. Teo

Published Apr 21, 2025• 1 citation
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Abstract

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 ${E}_{8}$ quantum Hall state. We use exactly solvable models of coupled electron wires to construct states with $\mathrm{SU}{(m)}_{n}, \mathrm{SO}{(m)}_{n}$, or $\mathrm{Sp}{(m)}_{n}$ topological order for various levels $n$. 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.

DOI
10.1103/physrevb.111.165140

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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.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
At a fixed magnetic filling fraction, fractional quantum Hall states may display a plethora of interaction-induced competing phases. Effective Chern-Simons theories have suggested the existence of multiple interaction-induced short-range entangled phases also at integer quantum Hall plateaus. Among these, a bosonic phase has been proposed with edge modes carrying representations based on the E8{E}_{8}E8​ exceptional Lie algebra. Through a theoretical coupled-wire construction, we provide an explicit microscopic model for this E8{E}_{8}E8​ Abelian quantum Hall state, at filling $\ensuremath{\nu}=16$, and discuss how it is intimately related to topological paramagnets in 3+13+13+1 dimensions. Still using coupled wires, we partition the E8{E}_{8}E8​ state into a pair of non-Abelian, long-range entangled states. These two states occur at filling $\ensuremath{\nu}=8$, demonstrating that even topological order may also exist at integer Hall plateaus. These phases are bosonic, carry chiral edge theories with either G2{G}_{2}G2​ or F4{F}_{4}F4​ internal symmetries, and host Fibonacci anyonic excitations in the bulk. This suggests that the $\ensuremath{\nu}=8$ quantum Hall plateau may provide an unexpected platform to realize decoherence-free quantum computation by anyon braiding. We also find that these topological ordered phases are related by a notion of particle-hole conjugation based on the E8{E}_{8}E8​ state that exchanges the G2{G}_{2}G2​ and F4{F}_{4}F4​ Fibonacci states. We argue that these phases can be tracked down by their electric and thermal Hall transport satisfying a distinctive Wiedemann-Franz law $\left({\ensuremath{\kappa}}_{xy}/{\ensuremath{\sigma}}_{xy}\right)/\left[\left({\ensuremath{\pi}}^{2}{k}_{B}^{2}T\right)/3{e}^{2}\right]<1$, even at integer magnetic filling factors.
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.
Non-Abelian excitations are sought after because of the promise they hold for topological quantum computation. However, in spite of many proposals of systems in which they can occur intrinsically, observation has proved to be elusive. In this thesis, we study ways in which we can engineer such excitations by combining two parent Abelian theories, and as a result allow for defects which host non-Abelian zero modes. We first formulate the symmetries of the anyonic excitations of Abelian topological states. Then, we study properties of semiclassical point defects, called twist defects, which permute orbiting anyons by their symmetries. We see how to realize these defects at domain walls on the gapped edges of non-chiral fractional quantum Hall states. We have distinct gapped edge phases in one to one correspondence with the anyonic symmetries. Non-Abelian twist defects appear between different gapped phases. We study the twist defects for a variety of bosonic ADE fractional quantum spin Hall systems for which the anyonic symmetries turn out to be given by their Dynkin diagrams. We also look at superconducting heterostructures with fermionic quantum Hall states and show the presence of anyonic symmetries and twist defects which physically correspond to a fermion parity flip. This can result in Majorana bound states at such defects instead of more exotic parafermions as was believed before. This is because the presence of the superconductor necessitates the consideration of a larger bosonic topological order. We then investigate the twist defects further by promoting the anyonic symmetries from global to local. This deconfines the defects and results in non-Abelian topological states called twist liquids. Finally, we study the emergence of new topological order from two constituent systems driven by the condensation of a composite boson. This allows us to study a variety of systems with non-Abelian excitations.