We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.

arXiv:1703.03418Physical review. B./Physical review. BJun 20, 201921 citations

Quantum field theoretic descriptions of topological phases in two and three dimensions

Zhenzhou Cheng

Jan 1, 20130 citations

Non-Abelian twist to integer quantum Hall states

Pedro L. S. Lopes, Victor L. Quito, Bo Han +1 more

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 ${E}_{8}$ exceptional Lie algebra. Through a theoretical coupled-wire construction, we provide an explicit microscopic model for this ${E}_{8}$ Abelian quantum Hall state, at filling $\ensuremath{\nu}=16$, and discuss how it is intimately related to topological paramagnets in $3+1$ dimensions. Still using coupled wires, we partition the ${E}_{8}$ 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 ${G}_{2}$ or ${F}_{4}$ 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 ${E}_{8}$ state that exchanges the ${G}_{2}$ and ${F}_{4}$ 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.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 ${E}_{8}$ exceptional Lie algebra. Through a theoretical coupled-wire construction, we provide an explicit microscopic model for this ${E}_{8}$ Abelian quantum Hall state, at filling $\ensuremath{\nu}=16$, and discuss how it is intimately related to topological paramagnets in $3+1$ dimensions. Still using coupled wires, we partition the ${E}_{8}$ 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 ${G}_{2}$ or ${F}_{4}$ 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 ${E}_{8}$ state that exchanges the ${G}_{2}$ and ${F}_{4}$ 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.

arXiv:1901.09043Physical review. B./Physical review. BAug 6, 201917 citations

Coupled-wire construction of non-Abelian higher-order topological phases

Jiaxin Pan, Longwen Zhou

Non-Abelian topological charges (NATCs), characterized by their noncommutative algebra, offer a framework for describing multigap topological phases beyond conventional Abelian invariants. While higher-order topological phases (HOTPs) host boundary states at corners or hinges, their characterization has largely relied on Abelian invariants such as winding and Chern numbers. Here, we propose a coupled-wire scheme of constructing non-Abelian HOTPs and analyze a non-Abelian second-order topological insulator as its minimal model. The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries and described by a topological vector that unites a non-Abelian quaternion charge with an Abelian winding number. Corner states emerge only when both invariants are nontrivial, whereas weak topological edge states of non-Abelian origins arise when the quaternion charge is nontrivial, enriching the bulk-edge-corner correspondence. The system further exhibits both non-Abelian and Abelian topological phase transitions, providing a unified platform that bridges these two distinct topological classes. Our work extends the understanding of HOTPs into non-Abelian regimes and suggests feasible experimental realizations in synthetic quantum systems, such as photonic or acoustic metamaterials.Non-Abelian topological charges (NATCs), characterized by their noncommutative algebra, offer a framework for describing multigap topological phases beyond conventional Abelian invariants. While higher-order topological phases (HOTPs) host boundary states at corners or hinges, their characterization has largely relied on Abelian invariants such as winding and Chern numbers. Here, we propose a coupled-wire scheme of constructing non-Abelian HOTPs and analyze a non-Abelian second-order topological insulator as its minimal model. The resulting Hamiltonian supports hybridized corner modes, protected by parity-time-reversal plus sublattice symmetries and described by a topological vector that unites a non-Abelian quaternion charge with an Abelian winding number. Corner states emerge only when both invariants are nontrivial, whereas weak topological edge states of non-Abelian origins arise when the quaternion charge is nontrivial, enriching the bulk-edge-corner correspondence. The system further exhibits both non-Abelian and Abelian topological phase transitions, providing a unified platform that bridges these two distinct topological classes. Our work extends the understanding of HOTPs into non-Abelian regimes and suggests feasible experimental realizations in synthetic quantum systems, such as photonic or acoustic metamaterials.

cond-mat.mes-hallCross-listedarXiv:2512.21179Dec 25, 2025

su{(2)}_{2}^{\phantom{\rule{0.16em}{0ex}}}

Moore-Read state, but the approach we outline here can be extended to general bosonic

su{(2)}_{k}^{\phantom{\rule{0.16em}{0ex}}}

topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.

{E}_{8}

exceptional Lie algebra. Through a theoretical coupled-wire construction, we provide an explicit microscopic model for this

{E}_{8}

Abelian quantum Hall state, at filling $\ensuremath{\nu}=16$, and discuss how it is intimately related to topological paramagnets in

3+1

dimensions. Still using coupled wires, we partition the

{E}_{8}

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

{G}_{2}

{F}_{4}

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

{E}_{8}

state that exchanges the

{G}_{2}

and

{F}_{4}

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.