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Twisted Injectivity in PEPS and the Classification of Quantum Phases

Oliver Buerschaper

TL;DR

This work advances the 2D gapped phase classification by introducing a twisted PEPS framework based on a finite group G and a 3-cocycle ω, leading to (G, ω)-injective and (G, ω)-isometric tensors. It constructs a family of gapped, frustration-free Hamiltonians whose ground-state structure on a torus is governed by c^ω-regular pair conjugacy classes and connects the isometric form to Dijkgraaf-Witten TQFT, with universality determined by the cohomology class of ω. Through quasi-adiabatic evolution, the authors show that almost twisted isometric PEPS remain in the same universality class as their isometric point, providing a local normal form that captures intrinsic 2D topological order. The results point toward a broader framework linking tensor-network states, DW-type topological order, and higher-categorical constructions (e.g., Turaev-Viro/Levin-Wen) for classifying 2D quantum phases.

Abstract

We introduce a class of projected entangled pair states (PEPS) which is based on a group symmetry twisted by a 3-cocycle of the group. This twisted symmetry gives rise to a new standard form for PEPS from which we construct a family of local Hamiltonians which are gapped, frustration-free and include fixed points of the renormalization group flow. Moreover, we advance the classification of 2D gapped quantum spin systems by showing how this new standard form for PEPS determines the emergent topological order of these local Hamiltonians. Specifically, we identify their universality class as Dijkgraaf-Witten topological quantum field theory (TQFT).

Twisted Injectivity in PEPS and the Classification of Quantum Phases

TL;DR

This work advances the 2D gapped phase classification by introducing a twisted PEPS framework based on a finite group G and a 3-cocycle ω, leading to (G, ω)-injective and (G, ω)-isometric tensors. It constructs a family of gapped, frustration-free Hamiltonians whose ground-state structure on a torus is governed by c^ω-regular pair conjugacy classes and connects the isometric form to Dijkgraaf-Witten TQFT, with universality determined by the cohomology class of ω. Through quasi-adiabatic evolution, the authors show that almost twisted isometric PEPS remain in the same universality class as their isometric point, providing a local normal form that captures intrinsic 2D topological order. The results point toward a broader framework linking tensor-network states, DW-type topological order, and higher-categorical constructions (e.g., Turaev-Viro/Levin-Wen) for classifying 2D quantum phases.

Abstract

We introduce a class of projected entangled pair states (PEPS) which is based on a group symmetry twisted by a 3-cocycle of the group. This twisted symmetry gives rise to a new standard form for PEPS from which we construct a family of local Hamiltonians which are gapped, frustration-free and include fixed points of the renormalization group flow. Moreover, we advance the classification of 2D gapped quantum spin systems by showing how this new standard form for PEPS determines the emergent topological order of these local Hamiltonians. Specifically, we identify their universality class as Dijkgraaf-Witten topological quantum field theory (TQFT).

Paper Structure

This paper contains 21 sections, 20 theorems, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Structure
  5. Twisted Injectivity
  6. Branched Tensors
  7. Twisted Symmetry as Matrix Product Operator
  8. Closure on a Torus
  9. Parent Hamiltonians
  10. Twisted Isometry
  11. Standard Form
  12. Dijkgraaf-Witten Form
  13. Cohomology Classes
  14. Parent Hamiltonians
  15. Gapped Paths and Almost Twisted Isometry
  16. ...and 6 more sections

Key Result

Proposition 1

The MPOs $\{V^\omega(g)\mid g\in G\}$ form a unitary representation of $G$.

Figures (4)

  • Figure 1: The closure tensor elements $\braket{\alpha_1,\dots,\alpha_4}{M}$, $\braket{\alpha_1,\dots,\alpha_4}{P}$ and $\braket{\alpha_1,\dots,\alpha_4}{Q}$.
  • Figure 2: The closure tensors $M_0(g,h)$ and $M_1(g,h)$.
  • Figure 3: The $(G,\omega)$-action on the closure state $\lvert M_0(g,h)\rangle$.
  • Figure 4: Intersecting regions $R_1$ and $R_2$ corresponding to $(G,\omega)$-injective tensors $(A_iB_j)$ and $(B_jC_k)$ respectively. Each such region $R_v$ is naturally associated with a vertex $v$ of the cellulation.

Theorems & Definitions (53)

  • Definition 1: Branched polygon
  • Definition 2: Branched tensor
  • Definition 3: Twisted symmetry MPO
  • Remark 1
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 4: Virtual symmetry
  • Lemma 1: Concatenation of twisted symmetry
  • proof
  • ...and 43 more