One dimensional gapped quantum phases and enriched fusion categories

TL;DR

This work validates the proposal that enriched fusion categories yield a unified categorical description of all 1d gapped/gapless quantum liquids, including symmetry-breaking, SPT/SET orders, and certain gapless phases, by analyzing Ising and Kitaev chains. It demonstrates that spacetime observables in each gapped phase form a fusion category enriched in a braided fusion category with trivial center, and that boundary theories satisfy the boundary-bulk relation, thereby connecting lattice models to enriched-categorical structures. The authors develop and apply the topological Wick rotation framework to extract the topological skeletons and local quantum-symmetry data, and they provide a concrete classification of all 1d gapped phases with finite onsite symmetry (bosonic and fermionic), via centers Z1(Rep(G)) and associated Lagrangian algebras, linking to established classifications. The results unify symmetry-protected, symmetry-breaking, and Morita-equivalent phases within a single enriched-category paradigm and pave the way for systematic lattice-model realizations of all 1d gapped phases in terms of enriched fusion categories.

Abstract

In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these models. In the end, we provide a classification of and the categorical descriptions of all 1-dimensional (the spatial dimension) gapped quantum phases with a finite onsite symmetry.

Figures (5)

• Figure 1: This picture depicts the physical meaning of the classification theorem of $n$d SPT/SET orders given in Theorem \ref{['thm:classification_SET']}. There are two ways to interpret this picture. One was provided in KLWZZ20a, where $\text{Z}_1(\EuScript{R})$ is regarded as the 1-dimensional higher bulk of the SPT/SET order and the vertical direction is the ($n+$1)-th spatial direction. The other one was provided in KZ20b, where the vertical direction is the time direction and $\text{Z}_1(\EuScript{R})$ is viewed as the background category of an enriched $n$-category ${}^{\text{Z}_1(\EuScript{R})}\EuScript{R}$ or ${}^{\text{Z}_1(\EuScript{R})}\EuScript{S}$ (see Appendix \ref{['sec:enriched-categories']}), the hom spaces of which encode the spacetime observables of the SPT/SET orders.
• Figure 2: This picture depicts all observables in LWLL in a 1+1D CFT. In particular, $V$ denotes the local quantum symmetry, and $M_{a,b}$ is a space of defects fields, and $M_{a,a}$ defines a topological defect line (TDL). All $M_{a,b}$, together with the labels $a,b,c,\cdots \in\EuScript{S}$, form an enriched category ${}^{\text{Z}_1(\EuScript{S})}\EuScript{S}$ with $\hom_{{}^{\text{Z}_1(\EuScript{S})}\EuScript{S}}(a,b)=M_{a,b}$.
• Figure 3: These pictures illustrate two gapped boundaries of the trivial 1d $\mathbb{Z}_2$ SPT order in two different ways.
• Figure 4: These pictures illustrate two gapped boundaries of the $\mathbb{Z}_2$-symmetry broken phase in two different ways.
• Figure 5: This picture depicts a 2d spatial configuration that can realize two 1+1D gapped phases appeared in 1d Kitaev chain and their boundaries via topological Wick rotation. We use $\EuScript{Y}_{m \leftrightarrow e}$ to denote the invertible domain wall associated to the braided auto-equivalence $\text{Z}_1(\mathrm{sVec}) \to \text{Z}_1(\mathrm{sVec})$ defined by $m\leftrightarrow e$. By KZ18, $\EuScript{Y}_{m \leftrightarrow e}$ can be mathematically described by the category $\mathop{\mathrm{Fun}}\nolimits_{\mathrm{sVec}|\mathrm{sVec}}(\mathrm{Vec}, \mathrm{Vec})$ of $\mathrm{sVec}$-$\mathrm{sVec}$-bimodule functors.

Theorems & Definitions (36)

• Remark 1.2
• Example 1.3
• Remark 1.4
• Example 1.5
• Remark 1.6
• Example 1.7
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4