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Hamiltonian models for topological phases of matter in three spatial dimensions

Dominic J. Williamson, Zhenghan Wang

TL;DR

The paper presents commuting projector Hamiltonians for a broad class of (3+1)D topological phases derived from unitary G-crossed braided fusion categories (UGxBFCs), unifying lattice realizations of state-sum TQFTs with CYWW, 2-group gauge theories, and Kashaev's TQFT. It develops two complementary construction routes—tensor-network/PEPO-based and graphical-calculus-based—for UGxBFC-based TQFTs, and provides explicit lattice realizations on BCC/3-torus geometries, including detailed analysis of ground-state degeneracies and excitations. It demonstrates that Kashaev's TQFTs are closely linked to CYWW models (invertible for odd N, premodular for even N) and shows how UGxBFC data recovers known 3+1D constructions while extending them to defect sectors and higher consistency conditions. The work situates these models within a broader hierarchy of state-sum and higher-category theories, discusses potential anomaly considerations (H^3/H^4), and outlines boundary and higher-categorical generalizations, offering a versatile framework for realizing and analyzing (3+1)D topological orders on the lattice.

Abstract

We present commuting projector Hamiltonian realizations of a large class of (3+1)D topological models based on mathematical objects called unitary G-crossed braided fusion categories. This construction comes with a wealth of examples from the literature of symmetry-enriched topological phases. The spacetime counterparts to our Hamiltonians are unitary state sum topological quantum fields theories (TQFTs) that appear to capture all known constructions in the literature, including the Crane-Yetter-Walker-Wang and 2-Group gauge theory models. We also present Hamiltonian realizations of a state sum TQFT recently constructed by Kashaev whose relation to existing models was previously unknown. We argue that this TQFT is captured as a special case of the Crane-Yetter-Walker-Wang model, with a premodular input category in some instances.

Hamiltonian models for topological phases of matter in three spatial dimensions

TL;DR

The paper presents commuting projector Hamiltonians for a broad class of (3+1)D topological phases derived from unitary G-crossed braided fusion categories (UGxBFCs), unifying lattice realizations of state-sum TQFTs with CYWW, 2-group gauge theories, and Kashaev's TQFT. It develops two complementary construction routes—tensor-network/PEPO-based and graphical-calculus-based—for UGxBFC-based TQFTs, and provides explicit lattice realizations on BCC/3-torus geometries, including detailed analysis of ground-state degeneracies and excitations. It demonstrates that Kashaev's TQFTs are closely linked to CYWW models (invertible for odd N, premodular for even N) and shows how UGxBFC data recovers known 3+1D constructions while extending them to defect sectors and higher consistency conditions. The work situates these models within a broader hierarchy of state-sum and higher-category theories, discusses potential anomaly considerations (H^3/H^4), and outlines boundary and higher-categorical generalizations, offering a versatile framework for realizing and analyzing (3+1)D topological orders on the lattice.

Abstract

We present commuting projector Hamiltonian realizations of a large class of (3+1)D topological models based on mathematical objects called unitary G-crossed braided fusion categories. This construction comes with a wealth of examples from the literature of symmetry-enriched topological phases. The spacetime counterparts to our Hamiltonians are unitary state sum topological quantum fields theories (TQFTs) that appear to capture all known constructions in the literature, including the Crane-Yetter-Walker-Wang and 2-Group gauge theory models. We also present Hamiltonian realizations of a state sum TQFT recently constructed by Kashaev whose relation to existing models was previously unknown. We argue that this TQFT is captured as a special case of the Crane-Yetter-Walker-Wang model, with a premodular input category in some instances.

Paper Structure

This paper contains 23 sections, 2 theorems, 72 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. State Sum TQFTs
  4. Hamiltonians, Tensor Network Ground States and PEPO Symmetries
  5. Review of the UGxBFC TQFT
  6. UGxBFC
  7. Example: $\mathbb{Z}_3$ Tambara-Yamagami Category
  8. State Sum from UGxBFC
  9. Review of Kashaev's TQFT
  10. Lattice Model for Kashaev's TQFT
  11. The Hamiltonian
  12. Degeneracy, Statistics, and the Ground State Wave Function
  13. Example: $\mathbb{Z}_2$ Case
  14. Back to the general case
  15. Lattice Model for UGxBFC TQFT
  16. ...and 8 more sections

Key Result

Lemma 1

For the commuting, projector, real-space blocking RG fixed point Hamiltonian term $h_v$ we have $\dim V[{T^3}]=Z[{T^4}]= \text{Tr}[{\pi Z[{v'*{\mathop{\mathrm{st}}\nolimits}_{v}}]\,}] = \text{Tr}[{\pi (\mathds{1} - h_v)}]$ for the permutation $\pi$ given below.

Figures (16)

  • Figure 1: Summary of the results in Ref.higherdto.
  • Figure 2: The pentagon equation.
  • Figure 3: Consistency of the global symmetry action and projective phases.
  • Figure 4: Consistency of fusion with: $\eta$ (left), and $U$ (right).
  • Figure 5: Yang-Baxter relation in a UGxBFC.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Proposition 1
  • proof