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Topological phases from higher gauge symmetry in 3+1D

Alex Bullivant, Marcos Calçada, Zoltán Kádár, Paul Martin, João Faria Martins

TL;DR

The paper constructs an exactly solvable Hamiltonian for 3+1D topological phases with finite 2-group symmetry (crossed module), showing its ground-state projector equals the Yetter homotopy 2-type TQFT amplitude on $M^3\times[0,1]$ and extending the framework to arbitrary dimensions. It establishes a comprehensive higher lattice gauge theory (HLGT) formalism with 1- and 2-holonomies, fake-flatness, and 2-flatness constraints, and connects the model to ordinary lattice gauge theories and to Walker-Wang models via a duality with the symmetric braided fusion category $ ext{M}( ext{E})$. The work provides explicit ground-state degeneracy calculations for select manifolds, clarifies the relation to 4D TQFT state-sums, and offers a path to incorporate 4-cocycle phases and boundary theories. Altogether, the results illuminate the structure of 3+1D topological order with higher gauge symmetry and expose a tight link between HLGT, Yetter TQFT, and Walker-Wang constructions with potential physical realizations in topological insulators and related systems.

Abstract

We propose an exactly solvable Hamiltonian for topological phases in $3+1$ dimensions utilising ideas from higher lattice gauge theory, where the gauge symmetry is given by a finite 2-group. We explicitly show that the model is a Hamiltonian realisation of Yetter's homotopy 2-type topological quantum field theory whereby the groundstate projector of the model defined on the manifold $M^3$ is given by the partition function of the underlying topological quantum field theory for $M^3\times [0,1]$. We show that this result holds in any dimension and illustrate it by computing the ground state degeneracy for a selection of spatial manifolds and 2-groups. As an application we show that a subset of our model is dual to a class of Abelian Walker-Wang models describing $3+1$ dimensional topological insulators.

Topological phases from higher gauge symmetry in 3+1D

TL;DR

The paper constructs an exactly solvable Hamiltonian for 3+1D topological phases with finite 2-group symmetry (crossed module), showing its ground-state projector equals the Yetter homotopy 2-type TQFT amplitude on and extending the framework to arbitrary dimensions. It establishes a comprehensive higher lattice gauge theory (HLGT) formalism with 1- and 2-holonomies, fake-flatness, and 2-flatness constraints, and connects the model to ordinary lattice gauge theories and to Walker-Wang models via a duality with the symmetric braided fusion category . The work provides explicit ground-state degeneracy calculations for select manifolds, clarifies the relation to 4D TQFT state-sums, and offers a path to incorporate 4-cocycle phases and boundary theories. Altogether, the results illuminate the structure of 3+1D topological order with higher gauge symmetry and expose a tight link between HLGT, Yetter TQFT, and Walker-Wang constructions with potential physical realizations in topological insulators and related systems.

Abstract

We propose an exactly solvable Hamiltonian for topological phases in dimensions utilising ideas from higher lattice gauge theory, where the gauge symmetry is given by a finite 2-group. We explicitly show that the model is a Hamiltonian realisation of Yetter's homotopy 2-type topological quantum field theory whereby the groundstate projector of the model defined on the manifold is given by the partition function of the underlying topological quantum field theory for . We show that this result holds in any dimension and illustrate it by computing the ground state degeneracy for a selection of spatial manifolds and 2-groups. As an application we show that a subset of our model is dual to a class of Abelian Walker-Wang models describing dimensional topological insulators.

Paper Structure

This paper contains 28 sections, 2 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Higher lattice gauge theory
  3. Crossed modules
  4. Lattices and lattice paths
  5. Gauge fields
  6. The Hamiltonian model
  7. Gauge transformations
  8. Covariance and 2-flatness constraints
  9. The Hamiltonian
  10. Relation to the 4D Yetter TQFT
  11. Ground state projection as a 4D state sum
  12. Ordinary Pure Lattice gauge theory based on a finite group
  13. 2+1D lattice gauge theory with a finite group
  14. Higher lattice gauge theory based on a finite crossed module
  15. Hamiltonians corresponding to lattice gauge theories
  16. ...and 13 more sections

Key Result

Theorem 4.1

Let $M^2$ be a 2-manifold with lattice $L$ and let $\Delta$ be the 3-dimensional lattice of $M^2\times [0,1]$ that restricts to $L_0\simeq L_1\simeq L$ at the boundaries $M^2\times \{0\}$ and $M^2\times \{1\}$. Let the internal edge set be $\{v\times [0,1]\}_{v\in L^0}$. Let $L^i_j$ refer to $L_j$ a

Figures (4)

  • Figure 1: Resolution of 6-valent vertex to a trivalent vertex.
  • Figure 2: Trivalent plaquette with oriented edges for Walker-Wang model.
  • Figure 3: Examples of the dual of a cubic lattice. The edges of the original lattice are black and the dual edges blue.
  • Figure 4: The left figure shows one of the internal faces connecting corresponding boundary edges of $L_j$, the top edge label $g_i^1$ is determined by fake flatness of the rectangle. By our choice of total order on $\Delta^0$, the basepoint of the rectangle is $s(i)$ and the fake flatness constraint reads as the composition (\ref{['fifo']}). The right figure shows a blob bounded by the "bucket" consisting of the green pentagon of $L_0$ with label $e_p^0$ and the rectangles with labels $e_i\in E,i\in bd(p_0)$. The 2-holonomy of the bucket is $\tilde{e}_p^0$ the "lid pentagon" is coloured by $e_p^1$. By 2-flatness of the blob $\tilde{e}_p^0=e_p^1$.

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 4.2