Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries

TL;DR

The paper develops a comprehensive, category-theoretic classification of 2+1D gapped quantum liquids with finite on-site symmetry, unifying bosonic and fermionic cases through $\mathrm{UMTC}_{/\mathcal{E}}$ and their modular extensions. It introduces a triple $(\mathcal{C},\mathcal{M},c)$ to label bulk excitations, symmetry gauging, and edge central charge, and proves (where possible) a correspondence between modular extensions and bosonic/fermionic SPT states via $H^3(G,U(1))$ and its fermionic analogs. A second, data-driven description using fusion/spin data $(\tilde{N}^{ab}_{c},\tilde{s}_a; N^{ij}_k,s_i; \mathcal{N}^{IJ}_K,\mathcal{S}_I;c)$ is offered to facilitate numerical tabulation of GQLs, SETs, and SPTs; the authors discuss practical computation via condensable algebras, $L_{\mathcal{E}}$, and stacking rules $\boxtimes_{\mathcal{E}}$. The work provides extensive examples, enumerating modular extensions for bosonic/fermionic symmetry groups (including $Z_2$, $Z_3$, $Z_5$, $S_3$, and their fermionic counterparts), and explains how SPT phases emerge as $c=0$ sectors while nontrivial SET orders require nontrivial modular extensions. Overall, the framework enables systematic construction and counting of 2+1D GQLs with symmetry, with implications for realizing and classifying SETs and SPTs in interacting systems.

Abstract

Gapped quantum liquids (GQL) include both topologically ordered states (with long range entanglement) and symmetry protected topological (SPT) states (with short range entanglement). In this paper, we propose a classification of 2+1D GQL for both bosonic and fermionic systems: 2+1D bosonic/fermionic GQLs with finite on-site symmetry are classified by non-degenerate unitary braided fusion categories over a symmetric fusion category (SFC) $\cal E$, abbreviated as $\text{UMTC}_{/\cal E}$, together with their modular extensions and total chiral central charges. The SFC $\cal E$ is $\text{Rep}(G)$ for bosonic symmetry $G$, or $\text{sRep}(G^f)$ for fermionic symmetry $G^f$. As a special case of the above result, we find that the modular extensions of $\text{Rep}(G)$ classify the 2+1D bosonic SPT states of symmetry $G$, while the $c=0$ modular extensions of $\text{sRep}(G^f)$ classify the 2+1D fermionic SPT states of symmetry $G^f$. Many fermionic SPT states are studied based on the constructions from free-fermion models. But it is not clear if free-fermion constructions can produce all fermionic SPT states. Our classification does not have such a drawback. We show that, for interacting 2+1D fermionic systems, there are exactly 16 superconducting phases with no symmetry and no fractional excitations (up to $E_8$ bosonic quantum Hall states). Also, there are exactly 8 $Z_2\times Z_2^f$-SPT phases, 2 $Z_8^f$-SPT phases, and so on. Besides, we show that two topological orders with identical bulk excitations and central charge always differ by the stacking of the SPT states of the same symmetry.

Figures (2)

• Figure 1: Consider a physical situation in which the excitations in the $2+1$D bulk are given by a modular extension $\cM$ of $\Rp(G)$, and those on the gapped boundary by the UFC $\cM_A$. Consider a simple particle $e\in \Rp(G)$ in the bulk moving toward the boundary. The bulk-to-boundary map is given by the central functor $-\otimes A: \cM \to \cM_A$, which restricted to $\Rp(G)$ is nothing but the forgetful functor $F:\Rp(G) \to \mathrm{Vec}$. Let $x$ be a simple excitation in $\cM_A$ sitting next to $F(e)$. We move $F(e)$ along the semicircle $\gamma_1$ (defined by the half-braiding), then move along the semicircle $\gamma_2$ (defined by the symmetric braiding in the trivial phase $\mathrm{Vec}$).
• Figure 2: (a) A torus with a flat $G$-connection (described by a symmetry twist along the dashed loop). The thin solid loop is a braiding path. (b) A handle is deformed into a very thin one. (c) A very thin handle can be viewed as two defects, and each defect corresponds to the added particle in the modular extension.

Theorems & Definitions (45)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Conjecture 1
• Definition 5
• Definition 6
• Remark 1
• Conjecture 2
• Corollary 1