Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics

TL;DR

The paper introduces the loop self-statistics $\mu$ as a lattice invariant for 3+1D ${\mathbb Z}_2$ gauge theories with fermionic charges, distinguishing ordinary FcBl (fermionic charges, bosonic loops) from the anomalous FcFl (both fermionic) phase. It provides a rigorous, universal definition of $\mu$ via a 36-step membrane-process at a tetrahedral geometry, showing $\mu$ is well-defined only when the gauge charge is a fermion and equals $+1$ for FcBl while $-1$ occurs for FcFl. To substantiate the FcFl phase, the authors construct an explicit 4+1D Walker–Wang-like model with a boundary hosting FcFl, prove the bulk is invertible (two copies disentangle to a product state), and show the FcFl boundary matches all-fermion QED in anomaly content. The work connects to braided fusion 2-category classifications and cobordism predictions, implying a nontrivial ${\mathbb Z}_2$-classified 4+1D invertible phase with action $S=\frac{1}{2}\int w_2 w_3$. Overall, the paper provides both a concrete solvable lattice realization of anomalous 3+1D fermionic toric code and a robust diagnostic for gravitational anomalies in higher-dimensional boundary theories, with broad implications for invertible phases and topological order classifications.

Abstract

Quasiparticle excitations in $3+1$ dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in $3+1$ dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics, that distinguishes two bosonic topological orders that both superficially resemble $3+1$ d ${\mathbb{Z}}_2$ gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic ${\mathbb{Z}}_2$ gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl), and, as we argue, can only exist at the boundary of a non-trivial 4+1d invertible bosonic phase, stable without any symmetries, i.e. it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable $4+1$ d Walker-Wang model and computing the loop self-statistics in the fermionic ${\mathbb{Z}}_2$ gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2-categories by Johnson-Freyd, and with the cobordism prediction of a non-trivial ${\mathbb{Z}}_2$ classified $4+1$ d invertible phase with action $S=\frac{1}{2} \int w_2 w_3$.

Figures (16)

• Figure 2.1: T-junction process used to measure statistics of identical particles
• Figure 2.2: All $33$ loop configurations. The length scale is much longer than the correlation length. Whenever two of these configurations look identical in some local region, we require that their reduced density matrices in this local region be identical. This in particular means that there are $6 = {4 \choose 2}$ different reduced density matrices in the neighborhood of each vertex. We have colored the interior edges blue and the outside edges black for clarity. The bottom three configurations appear twice in the sequence defining our invariant; all other configurations appear exactly once.
• Figure 2.3: A construction of an operator $M_{ij}$ satisfying the requisite conditions, described in \ref{['subsec:data']}. In this construction, $M_{ij}$ is obtained as the composition of a face operator (blue), which nucleates a loop of gauge flux in the interior of the plaquette, with controlled shallow circuits, which splice this loop into the perimeter of the plaquette. The edge operators (red) are controlled by the occupation numbers of the corresponding edges, and the vertex operators (purple) are controlled by the local configurations near the corresponding vertices. All distances in the figure are much greater than the correlation length. This process is designed in such a way that \ref{['eq:Mcondition']} is satisfied, as can be checked by comparing the local reduced density matrices on both sides of this \ref{['eq:Mcondition']}.
• Figure 2.4: Sequence of moves used in defining the loop self-statistics. We start with the configuration $|{\mathfrak{c}}_1\rangle$ in the upper left, and, at each step, apply the operator below the configuration to obtain the next configuration (reading left to right and up to down). This sequence of operators is written as a product in \ref{['eq:defhI']}. Note that the loop comes back to itself at the end, but the orientation along it reverses. At the top we illustrate our labeling scheme for the vertices of the tetrahedron.
• Figure 2.5: There is a unique triple of distinct $j,i,k$, such that the only non-trivial commutation relation is $S_{0j}$ with $M_{ik}$. In the case shown in the figure, this non-trivial commutation relation is between $S_{03}$ and $M_{24}$.

Theorems & Definitions (27)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition C.1
• Lemma C.2
• proof
• Definition E.1
• Proposition E.2
• proof