[Paper Review] Surface Topological Order and a new 't Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs

Lukasz Fidkowski, Ashvin Vishwanath+1 more|arXiv (Cornell University)|Apr 23, 2018

Parallel Computing and Optimization Techniques19 citations

TL;DR

This paper identifies a new 't Hooft anomaly in 3+1D fermionic symmetry-protected topological (SPT) phases protected by unitary symmetries, where the surface exhibits topological order with anyon permutations that are anomalous and classified by $H^3(G, \mathbb{Z}_2)$. The authors construct a surface topological order via a $\mathbb{Z}_4$ gauge theory and demonstrate that the symmetry action on anyons—specifically, the nontrivial braiding and fusion statistics—cannot be realized in isolation, requiring the bulk SPT to cancel the anomaly.

ABSTRACT

Symmetry protected topological (SPT) phases are well understood in the context of free fermions and in the context of interacting but essentially bosonic models. Recently it has been realized that intrinsically fermionic SPTs exist which only appear in interacting models. Here we show that the 3+1 dimensional realizations of these phases have surface states characterized by a new 't Hooft anomaly, captured by a $H^3(G, Z_2)$ class. This is encoded in the anomalous action of symmetry on the surface states with topological order, which must necessarily permute the anyons. We discuss in detail an example with symmetry group $G = Z_2 imes Z_4$. Using a network model of the surface we derive a candidate surface topological order given by a $Z_4$ gauge theory. We relate our findings to anomalies valued in $H^3$ with various coefficients introduced previously in both bosonic and fermionic settings, and describe a general framework that unifies these various anomalies.

Motivation & Objective

• To understand the surface topological order (STO) of 3+1D fermionic SPTs protected by unitary symmetries, particularly when no free-fermion or bosonic analogs exist.
• To identify and characterize a new type of 't Hooft anomaly in fermionic SPTs, distinct from known $H^4(G,U(1))$ anomalies in bosonic systems.
• To establish a framework linking the anomalous symmetry action on surface anyons to the bulk SPT, using the $\mathbb{Z}_2 \times \mathbb{Z}_4$ symmetry group as a concrete example.
• To unify fermionic and bosonic anomalies by showing that the $H^3(G,A)$ anomaly for anyon permutation is a universal feature across both settings.

Proposed method

• Construct a surface network model realizing a $\mathbb{Z}_4$ gauge theory as the candidate surface topological order.
• Analyze the symmetry action on anyons, particularly the nontrivial permutation of anyons under the $\mathbb{Z}_2 \times \mathbb{Z}_4$ group, which leads to an anomalous symmetry realization.
• Use the $S$-matrix and Verlinde formula to derive characters of the anyon fusion algebra and identify Abelian anyons $b$ such that $\omega_a = M^*_{ab}$, ensuring consistency with topological order.
• Prove that the anomaly is captured by a class in $H^3(G, \mathbb{Z}_2)$, where $G = \mathbb{Z}_2 \times \mathbb{Z}_4$, by showing that the symmetry fractionalization on anyons cannot be realized in 2+1D without the bulk.
• Relate the fermionic anomaly to the bosonic $H^4(G,U(1))$ classification via a generalized framework that includes both $H^3(G,A)$ and $H^4(G,U(1))$ anomalies.
• Demonstrate that the monopole-antimonopole tunneling process in the bulk induces a nontrivial $\mathbb{Z}_2$ charge change on the surface, confirming the anomaly through a physical realization of the anyon braiding statistics.

Experimental results

Research questions

• RQ1 What type of surface topological order do 3+1D fermionic SPTs with purely unitary symmetries realize?
• RQ2 How is the symmetry action on surface anyons anomalous, and what cohomology class captures this anomaly?
• RQ3 Can the $H^3(G, \mathbb{Z}_2)$ anomaly for anyon permutation be consistently realized in a 2+1D surface theory, and how does it relate to the bulk SPT?
• RQ4 How do fermionic SPTs with $\sigma \in H^2(G, \mathbb{Z}_2)$ and $\rho \in H^3(G, \mathbb{Z}_2)$ differ from bosonic SPTs in terms of surface anomalies?
• RQ5 What is the unified framework that connects $H^3(G,A)$ anomalies in fermionic systems with $H^4(G,U(1))$ anomalies in bosonic systems?

Key findings

• The surface of a 3+1D fermionic SPT with $G = \mathbb{Z}_2 \times \mathbb{Z}_4$ symmetry realizes a $\mathbb{Z}_4$ gauge theory as its topological order, with anyons $1, e, m, \mu$ and nontrivial anyon braiding.
• The symmetry action permutes the anyons in a way that is anomalous and cannot be realized in isolation, with the anomaly classified by a nontrivial class in $H^3(G, \mathbb{Z}_2)$.
• The anomaly arises because the generators of $\mathbb{Z}_2$ and $\mathbb{Z}$ symmetries anti-commute on the anyon $m^2$, mimicking the bulk monopole tunneling process.
• The authors prove that a set of phases $\omega_a$ on anyons satisfying $\omega_a \omega_b = \omega_c$ for $N^c_{ab} \neq 0$ and $\omega_f = 1$ can be written as $\omega_a = M^*_{ab}$ for some Abelian anyon $b$, which is unique up to fusion with the fermion anyon $f$.
• The $H^3(G, \mathbb{Z}_2)$ anomaly for anyon permutation is a universal feature in fermionic SPTs and generalizes the known $H^4(G,U(1))$ anomaly in bosonic SPTs.
• The framework unifies fermionic and bosonic anomalies by showing that $H^3(G,A)$ anomalies for anyon fractionalization are a necessary condition for the existence of nontrivial fermionic SPTs when $\sigma \neq 0$.