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

Lukasz Fidkowski, Ashvin Vishwanath, Max A. Metlitski

TL;DR

The work identifies a new obstruction for surface topological orders on 3+1D intrinsically fermionic SPTs with onsite symmetry, captured by a cohomology class $[\rho] \in H^3(G, \mathbb{Z}_2)$. It provides an explicit example for $G=\mathbb{Z}_2 \times \mathbb{Z}_4$ with a $\mathbb{Z}_4$ gauge STO whose symmetry action permutes anyons and carries fractionalization, yielding a nontrivial $[\rho]$ that matches the bulk invariant. A crystalline cousin is analyzed, and a general anomaly-matching framework is developed to unite bosonic and fermionic $H^3$ anomalies via symmetry actions and fractionalization on anyons, along with a systematic obstruction theory for extending symmetry actions from a subcategory to the full topological order. The paper also explores the $n=2$ bosonic limit, demonstrating how the anomaly can vanish in the doubled case and connecting to known bosonic SPT surface physics. Overall, the results offer a concrete path to classify and realize STOs for fermionic SPTs and clarify how surface anomalies reflect bulk fermionic topology.

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 $G = Z_2 \times Z_4$. 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.

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

TL;DR

The work identifies a new obstruction for surface topological orders on 3+1D intrinsically fermionic SPTs with onsite symmetry, captured by a cohomology class . It provides an explicit example for with a gauge STO whose symmetry action permutes anyons and carries fractionalization, yielding a nontrivial that matches the bulk invariant. A crystalline cousin is analyzed, and a general anomaly-matching framework is developed to unite bosonic and fermionic anomalies via symmetry actions and fractionalization on anyons, along with a systematic obstruction theory for extending symmetry actions from a subcategory to the full topological order. The paper also explores the bosonic limit, demonstrating how the anomaly can vanish in the doubled case and connecting to known bosonic SPT surface physics. Overall, the results offer a concrete path to classify and realize STOs for fermionic SPTs and clarify how surface anomalies reflect bulk fermionic topology.

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 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 . Using a network model of the surface we derive a candidate surface topological order given by a gauge theory. We relate our findings to anomalies valued in with various coefficients introduced previously in both bosonic and fermionic settings, and describe a general framework that unifies these various anomalies.

Paper Structure

This paper contains 18 sections, 73 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction.
  2. ${\mathbb Z}_2 \times {\mathbb Z}_4 \times {\mathbb Z}^f_2$ 3+1D fermion SPT and its crystalline cousin.
  3. Surface topological order for ${\mathbb Z}_2 \times {\mathbb Z}_4 \times {\mathbb Z}^f_2$ SPT.
  4. Surface topological order for the $\mathbb{Z}_2 \times \mathbb{Z} \times \mathbb{Z}^f_2$ crystalline fermion SPT.
  5. General anomaly matching condition for super-cohomology fermionic SPTs with onsite symmetries.
  6. Bulk fermionic SPT order
  7. Surface topological orders and symmetry
  8. Example: surface topological order for $\mathbb{Z}_2 \times \mathbb{Z}_4 \times \mathbb{Z}_2^f$ SPT.
  9. Braided auto-equivalence action $\phi$
  10. Fractionalization class $[\omega]$
  11. Relation to the topological order constructed for the crystalline $\mathbb{Z}_2 \times \mathbb{Z} \times \mathbb{Z}_2^f$ SPT
  12. $n = 2$ STO
  13. General $H^3$ obstruction
  14. Special case
  15. Discussion
  16. ...and 3 more sections

Figures (5)

  • Figure 1: A 3+1D crystalline SPT with $\mathbb{Z}_2 \times \mathbb{Z} \times \mathbb{Z}^f_2$ symmetry constructed by stacking layers of 2+1D $\nu =2$$\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SPTs along the $y$ direction. Brown lines denote gapless edge modes.
  • Figure 2: A dislocation in the crystalline SPT of fig. \ref{['fig:stack']} hosts a $\nu=2$$\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SPT edge mode. The brown circles denote these edge modes propagating perpendicular to the plane $yz$ plane.
  • Figure 3: Top: Constructing a topological order for the $z = 0$ surface of the crystaline SPT in Fig. \ref{['fig:stack']}. The surface area between neighboring $\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SPT edges is decorated with a strip of $\mathbb{Z}_4$ topological order (orange). These strips are then "stitched" together to produce a macroscopic $\mathbb{Z}_4$ topological order and gap out the modes associated with the $\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SPT edges. Bottom: A thought experiment illuminating why the stitching procedure works: a $\mathbb{Z}_4$ gauge theory stacked with a $\nu = 2$$\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SPT (left) is, in fact, in the same $\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SET phase as the $\mathbb{Z}_4$ gauge theory without the additional SPT (right).
  • Figure 4: Expanded top view of the surface decoration in figure \ref{['fig:stackdec']} (top). $\nu^i_R$ (red, $\mathbb{Z}_2$ charged) and $\nu^i_L$ (blue, $\mathbb{Z}_2$ neutral) are edge modes of $\nu = 2$$\mathbb{Z}_2 \times \mathbb{Z}^f_2$ SPTs in the stack. $\phi^i$, $\theta^i$, $\bar{\phi}^i$, $\bar{\theta}^i$ are the edge states associated with strips of $\mathbb{Z}_4$ topological order. The terms in Eq. (\ref{['gap3']}) gap out all edge modes without breaking the symmetry and sew together the $\mathbb{Z}_4$ topological orders.
  • Figure 5: $U(1)$ monopole-antimonopole pair in the bulk of the crystalline SPT with $\mathbb{Z}_2 \times \mathbb{Z} \times U(1)$ symmetry. The $\mathbb{Z}_2$ charge carried by this configuration is equal to the number of $\nu =2$$\mathbb{Z}_2 \times U(1)$ SPT layers between the monopoles, i.e. the generators of $\mathbb{Z}_2$ and $\mathbb{Z}$ anti-commute on the monopole.