arXiv:2104.07015 · Nuclear Physics B

Non-abelian bosonization in two and three spatial dimensions and applications

Yen-Ta Huang, D. H. Lee

Published Sep 27, 2021• 9 citations

Abstract

In this paper, we generalize Witten's non-abelian bosonization in (1+1)-D to two and three spatial dimensions. Our theory applies to fermions with relativistic dispersion. The bosonized theories are non-linear sigma models with level-1 Wess-Zumino-Witten terms. We apply the bosonization results to the SU(2) gauge theory of the π-flux phase, critical spin liquids in 1,2,3 spatial dimensions, and twisted bilayer graphene.

DOI: 10.1016/j.nuclphysb.2021.115565

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arXiv:2104.07015 · Nuclear Physics B

Non-abelian bosonization in two and three spatial dimensions and applications

Yen-Ta Huang, D. H. Lee

Published Sep 27, 2021• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.nuclphysb.2021.115565

Related papers

Non-abelian bosonization of fermion symmetry-protected topological states

Yen-Ta Huang, Dung‐Hai Lee

Applying the results of Ref.[1], we carry out the non-abelian bosonization for a class of free fermion symmetry-protected topological states (SPTs). The resulting boson theories are non-linear sigma models with topological $θ$ term, thus unifying the field theory description of the (free) fermionic and bosonic SPTs. Tuning $θ$ from $0$ to $2π$ triggers the topological trivial to non-trivial phase transitions. In addition, applying the same idea to the symmetry-protected critical spin liquids, we obtain the NL$\s$ models for spin SPTs living in one spatial dimension higher.Applying the results of Ref.[1], we carry out the non-abelian bosonization for a class of free fermion symmetry-protected topological states (SPTs). The resulting boson theories are non-linear sigma models with topological $θ$ term, thus unifying the field theory description of the (free) fermionic and bosonic SPTs. Tuning $θ$ from $0$ to $2π$ triggers the topological trivial to non-trivial phase transitions. In addition, applying the same idea to the symmetry-protected critical spin liquids, we obtain the NL$\s$ models for spin SPTs living in one spatial dimension higher.

arXiv:2206.09393Jun 19, 20220 citations

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arXiv:2509.18966Sep 23, 20250 citations

Non-Abelian bosonization of topological insulators and superconductors

Yen-Ta Huang, D. H. Lee

Applying the method of [Nucl. Phys. B 972, 115565 (2021)], which bosonizes massless relativistic free fermions, we derive the (non-Abelian) bosonized theory for free fermion topological insulators and superconductors that have, in addition to the U(1) charge, time reversal and charge conjugation symmetries and flavor symmetries. For the case we consider, the flavor symmetries render the topological classification $\mathbb{Z}$ . The results are nonlinear $\ensuremath{\sigma}$ models with the topological $\ensuremath{\theta}$ term. In addition, we present the theory of a class of bosonic symmetry-protected topological states, whose boundaries are critical spin liquids.Applying the method of [Nucl. Phys. B 972, 115565 (2021)], which bosonizes massless relativistic free fermions, we derive the (non-Abelian) bosonized theory for free fermion topological insulators and superconductors that have, in addition to the U(1) charge, time reversal and charge conjugation symmetries and flavor symmetries. For the case we consider, the flavor symmetries render the topological classification $\mathbb{Z}$. The results are nonlinear $\ensuremath{\sigma}$ models with the topological $\ensuremath{\theta}$ term. In addition, we present the theory of a class of bosonic symmetry-protected topological states, whose boundaries are critical spin liquids.

Physical review. B./Physical review. BDec 19, 20220 citations

θ

term, thus unifying the field theory description of the (free) fermionic and bosonic SPTs. Tuning

θ

from

0

2π

triggers the topological trivial to non-trivial phase transitions. In addition, applying the same idea to the symmetry-protected critical spin liquids, we obtain the NL$\s$ models for spin SPTs living in one spatial dimension higher.

Dual topological nonlinear sigma models of

\text{QED}

theory by dimensional reduction and monopole operators

Nonlinear

σ

models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or

\text{QED}

theory. In this paper, we propose a dimensional reduction scheme to derive the

\text{U}(1)

gauged topological NLSM in

n

-dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the

N_f=4

\text{QED}_3

with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold

\frac{\text{U(4)}}{\text{U(2)}\times \text{U(2)}}

coupled with a dynamical

\text{U(1)}

gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the

\text{U(1)}

monopole operators and the corresponding quantum number.

I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space

\mathrm{St}_{N,N-D-1} \cong \mathrm{SO}(N)/\mathrm{SO}(D+1)

, with a particular focus on

D = 3

. I investigate in particular whether such a theory admits IR-stable fixed points of the renormalization group flow. Such fixed points have been suggested to describe conformal phases of matter that do not have a known dual (super-)renormalizable Lagrangian for

N \geq 7

D = 3

. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The

D

-dimensional expressions enable the computation, by analytic computation, of beta functions in

D = 2 + ε

, at least to first non-trivial order. In

D = 2

, a stable fixed point is found, serving a generalization of the famed

\mathrm{SU}(2)_k

Wess-Zumino-Witten conformal field theory; it annihilates in

D = 2 + ε

with an unstable fixed point which splits off from the Gaussian one for

ε> 0

. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for

N \geqslant 6

, the annihilation appears to occur only for

ε> 1

, suggesting the existence of a stable phase in

D = 3

. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for

N = 6

, which is dual to QED

_3

with

N_\mathrm{f} = 4

fermion flavors. The predictions for the

N = 7

Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.

\mathbb{Z}

. The results are nonlinear $\ensuremath{\sigma}$ models with the topological $\ensuremath{\theta}$ term. In addition, we present the theory of a class of bosonic symmetry-protected topological states, whose boundaries are critical spin liquids.