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arXiv:2203.08156 · Physical Review Research

Symmetry-preserving boundary of (2+1)D fractional quantum Hall states

Ryohei Kobayashi

Published Aug 19, 2022• 11 citations
arXivHTMLPDF

Abstract

We investigate symmetry-preserving gapped boundary of $(2+1)$-dimensional $[(2+1)\mathrm{D}]$ topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)${}^{f}$-symmetric $(2+1)\mathrm{D}$ fermionic topological phases, which are beyond chiral central charge ${c}_{\ensuremath{-}}$ and electric Hall conductivity ${\ensuremath{\sigma}}_{H}$. These obstructions are given by a simple Gauss-Milgram-type formula valid for supermodular category and regarded as a higher version of ${c}_{\ensuremath{-}}$ and ${\ensuremath{\sigma}}_{H}$.

DOI
10.1103/physrevresearch.4.033137

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A (2+1)D topological ordered phase with U(1) symmetry may or may not have a symmetric gapped edge state, even if both thermal and electric Hall conductivity are vanishing. It is recently discovered that there are "higher" versions of Hall conductivity valid for fermionic fractional quantum Hall (FQH) states, which obstructs symmetry-preserving gapped edge state beyond thermal and electric Hall conductivity. In this paper, we show that one can extract higher Hall conductivity from a single wave function of an FQH state, by evaluating the expectation value of the "partial rotation" unitary which is a combination of partial spatial rotation and a U(1) phase rotation. This result is verified numerically with the fermionic Laughlin state with ν=1/3ν=1/3ν=1/3, 1/51/51/5, as well as the non-Abelian Moore-Read state. Together with topological entanglement entropy, we prove that the expectation values of the partial rotation completely determines if a bosonic/fermionic Abelian topological order with U(1) symmetry has a symmetry-preserving gappable edge state or not. We also show that thermal and electric Hall conductivity of Abelian topological order can be extracted by partial rotations. Even in non-Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type theorem constraining the low-energy spectrum of the bulk-boundary system. The generalization of higher Hall conductivity to the case with Lie group symmetry is also presented.
We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D(2+1)\mathrm{D}(2+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ is a central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity, ${(\ensuremath{-}1)}^{F}$, characterized by a nontrivial cohomology class $[{\ensuremath{\omega}}_{2}]\ensuremath{\in}{\mathcal{H}}^{2}({G}_{b},{\mathbb{Z}}_{2})$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the supermodular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in H3(Gb,K(C)){\mathcal{H}}^{3}({G}_{b},K(\mathcal{C}))H3(Gb​,K(C)), while the latter is valued in either ${\mathcal{H}}^{3}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$ or Z2(Gb,Z2){Z}^{2}({G}_{b},{\mathbb{Z}}_{2})Z2(Gb​,Z2​) depending on additional details of the theory. K(C)K(\mathcal{C})K(C) is the Abelian group of functions from anyons to U(1)\mathrm{U}(1)U(1) phases obeying the fusion rules, A\mathcal{A}A is the Abelian group defined by fusion of Abelian anyons, and $\ensuremath{\psi}$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over ${\mathcal{H}}^{2}({G}_{b},\mathcal{A}/{1,\ensuremath{\psi}})$. We study a number of examples in detail; in particular, we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid states of electrons.
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Abstract For a G-crossed braided extension of a unitary modular tensor category C\mathcal {C}C C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object A∈CA \in \mathcal {C}A∈C A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.