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Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries

Rongge Xu, Zhi-Hao Zhang

TL;DR

This work develops an enriched fusion category framework to describe macroscopic observables in 1+1D gapped phases with Abelian onsite symmetry $G$, showing that spacetime observables organize into a braided–fusion structure governed by the Drinfeld center relation with the state category. Through a concrete 1+1D lattice model, it demonstrates that the full set of topological sectors of states and operators realizes $\text{Z}_1(\text{Rep}(G))$, with symmetry breaking and SPT orders captured by the enriched categories ${}^{\text{Z}_1(\text{Rep}(G))}\text{Rep}(G)$, ${}^{\text{Z}_1(\text{Rep}(G))}\text{Vec}_G$, and their variants ${}_{F_H}\text{Rep}(G)_{F_H}$. The authors develop equivariantization as a physical tool to obtain $G$-symmetric sectors, and they connect bulk anyon condensation via Lagrangian algebras to 1d gapped phases through holographic duality and topological Wick rotation, providing explicit lattice realizations and boundary/domain-wall descriptions. The results offer a unified, holography-based view of gapped quantum phases, dualities, and boundaries, with potential extensions to nonabelian and fermionic settings and broader applications to condensed matter and high-energy contexts.

Abstract

In this work, we analyze the macroscopic observables in the 1+1D gapped phases with Abelian onsite symmetries and show that the spacetime observables for each gapped phase form a clear structure that can be mathematically described by enriched fusion categories, which uncovers the behavior of nonlocal excitations that were blurry in traditional Landau paradigm. These categorical descriptions not only generate the known classification results for symmetry preserving and breaking phases, but also unifies lattice dualities in a broader picture. After analyzing the general lattice model together with their boundaries, we give explicit examples including nontrivial SPT phase, where nontrivial boundaries can be given directly through our classification. Using enriched categorical descriptions, the lattice dualities and their gapped phases are unified under a holographic duality between an 2d. topological order with gapped 1d boundaries and 1+1D gapped quantum liquids with a categorical symmetry, which shed light on a unified definition of all quantum phases.

Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries

TL;DR

This work develops an enriched fusion category framework to describe macroscopic observables in 1+1D gapped phases with Abelian onsite symmetry , showing that spacetime observables organize into a braided–fusion structure governed by the Drinfeld center relation with the state category. Through a concrete 1+1D lattice model, it demonstrates that the full set of topological sectors of states and operators realizes , with symmetry breaking and SPT orders captured by the enriched categories , , and their variants . The authors develop equivariantization as a physical tool to obtain -symmetric sectors, and they connect bulk anyon condensation via Lagrangian algebras to 1d gapped phases through holographic duality and topological Wick rotation, providing explicit lattice realizations and boundary/domain-wall descriptions. The results offer a unified, holography-based view of gapped quantum phases, dualities, and boundaries, with potential extensions to nonabelian and fermionic settings and broader applications to condensed matter and high-energy contexts.

Abstract

In this work, we analyze the macroscopic observables in the 1+1D gapped phases with Abelian onsite symmetries and show that the spacetime observables for each gapped phase form a clear structure that can be mathematically described by enriched fusion categories, which uncovers the behavior of nonlocal excitations that were blurry in traditional Landau paradigm. These categorical descriptions not only generate the known classification results for symmetry preserving and breaking phases, but also unifies lattice dualities in a broader picture. After analyzing the general lattice model together with their boundaries, we give explicit examples including nontrivial SPT phase, where nontrivial boundaries can be given directly through our classification. Using enriched categorical descriptions, the lattice dualities and their gapped phases are unified under a holographic duality between an 2d. topological order with gapped 1d boundaries and 1+1D gapped quantum liquids with a categorical symmetry, which shed light on a unified definition of all quantum phases.
Paper Structure (37 sections, 8 theorems, 165 equations, 19 figures, 2 tables)

This paper contains 37 sections, 8 theorems, 165 equations, 19 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Holographic duality and topological Wick rotation
  3. Classification of gapped phases and anyon condensation
  4. Topological sectors of states with onsite G-symmetry
  5. The category of topological sectors of states and group actions
  6. Equivariantization and G-symmetric topological sectors of states
  7. General analysis for lattice model with abelian onsite G-symmetry
  8. The lattice model and ground states
  9. Topological sectors of operators
  10. The topological skeletons
  11. Symmetry preserving case (the trivial SPT)
  12. Symmetry completely broken case
  13. General cases (symmetry partially broken)
  14. 0+1D boundaries and domain walls with abelian onsite symmetry
  15. Boundaries
  16. ...and 22 more sections

Key Result

Proposition 3.7

Let $G$ be a finite group and $H \subseteq G$ be a subgroup. Then the fusion category ${}_{F_H} \mathrm{Rep}(G)_{F_H}$ is monoidally equivalent to the equivariantization $\mathop{\mathrm{Fun}}\nolimits(\mathrm{Vec}_{G/H},\mathrm{Vec}_{G/H})^G$.

Figures (19)

  • Figure 1: The left hand side depicts an ($n$+1)d topological order with an $n$d gapped boundary. After the topological Wick rotation, we get its holographic dual as depicted in the right hand side, which is an anomaly-free $n+1$D quantum liquid phase that can be described by ${ \hbox{$\scaleobj{0.7}{\EuScript{B}}$} {\EuScript{S}}}$.
  • Figure 2: (a) depicts a normal 2d anyon condensation process from a $2$d topological order $\EuScript{B}$ to vacuum $\mathrm{Vec}$, which generates a boundary phase described by $\EuScript{S} \simeq \EuScript{B}_A$. After topological Wick rotation, we get (b), an anomaly-free 1+1D phase described by ${ \hbox{$\scaleobj{0.7}{\EuScript{B}}$} {\EuScript{S}}} \simeq { \hbox{$\scaleobj{0.7}{\text{Z}_1(\EuScript{S})}$} {\EuScript{S}}}$. The original boundary phase $\EuScript{S}$ now becomes the topological sector of sates, and the original 2d bulk phase $\EuScript{B}$ becomes the categorical symmetry that acts on those states.
  • Figure 3: A picturesque explanation of the holographic duality between 2d topological orders with gapped boundaries and 1d gapped quantum phases. For the left part of (a), we have 2d toric code mode with the smooth boundary $\mathrm{Rep}(\undefined{Z}_2)$ and rough boundary $\mathrm{Vec}_{\undefined{Z}_2}$. After topological Wick rotation, these two gapped boundaries correspond to the two gapped phases of transverse Ising chain, namely, they become the $\undefined{Z}_2$ SPT phase ${}^{\text{Z}_1(\mathrm{Rep}(\undefined{Z}_2))}\mathrm{Rep}(\undefined{Z}_2)$ and the symmetry-breaking phase ${}^{\text{Z}_1(\mathrm{Rep}(\undefined{Z}_2))}\mathrm{Vec}_{\undefined{Z}_2}$. More generally, we have (b), in which the gapped boundaries of 2d quantum double model classified by Lagrangian algebra $A(H)$, one to one correspond to $1$d anomaly-free gapped phases ${ \hbox{$\scaleobj{0.7}{\text{Z}_1(\mathrm{Rep}(G))}$} {\text{Z}_1(\mathrm{Rep}(G))_{A_{(H)}}}}$ with onsite symmetries after topological Wick rotation.
  • Figure 4: The fusion of topological defect in a 1+1D topological order $\mathsf{C}$.
  • Figure 5: A topological sector of states without symmetry $\mathcal{T}_{g,h}$ can be generated by the state $\lvert\psi_{g,h,i}\rangle$, which can be interpreted as a 0+1D domain wall on 1+1D lattice model.
  • ...and 14 more figures

Theorems & Definitions (67)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Example 1.5
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.4
  • ...and 57 more