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Boundary topological orders of (4+1)d fermionic $\mathbb{Z}_{2N}^{\mathrm{F}}$ SPT states

Meng Cheng, Juven Wang, Xinping Yang

TL;DR

This work analyzes (3+1)d fermionic systems with anomalous $\\mathbb{Z}_{2N}^{\\mathrm{F}}$ symmetry by embedding them as boundaries of (4+1)d fermionic SPTs. Using a crystalline correspondence to map to a $C_N\\times\\mathbb{Z}_2^{\\mathrm{F}}$ symmetry and a block-state construction, the authors explicitly construct symmetric gapped boundaries for the bulk SPT when the anomaly index satisfies certain divisibility conditions. They show that for the Majorana-chain decoration ($\\nu=N$) the minimal symmetric boundary is a (3+1)d $\\mathbb{Z}_4$ gauge theory, while for the $p_x+ ip_y$ decoration ($\\nu=N/2$) the boundary is non-TQFT and highly anisotropic; for other cases, a symmetric gapped boundary does not arise in their framework, in line with Cordova–Ohmori no-go results. The results illuminate how different fermionic SPT decorations saturate anomalies and offer perspectives on Higgsing continuous symmetries, potential links to the Standard Model anomaly structure, and future exploration of higher-dimensional boundary topological orders.

Abstract

We investigate (3+1)d topological orders in fermionic systems with an anomalous $\mathbb{Z}_{2N}^{\mathrm{F}}$ symmetry, where its $\mathbb{Z}_2^{\mathrm{F}}$ subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of $ν$ copies of Weyl fermions of the same chirality. Inspired by the crystalline correspondence principle, we deform the anomalous $\mathbb{Z}_{2N}^\mathrm{F}$ symmetry of (3+1)d Weyl fermion to the anomalous $C_N \times \mathbb{Z}_2^\mathrm{F}$ symmetry. Then we microscopically construct symmetry-preserving gapped boundary states of the closely-related (4+1)d $C_N\times \mathbb{Z}_2^{\mathrm{F}}$ symmetry-protected topological (SPT) state (with $C_N$ being the $N$-fold rotation), whenever it is possible. In particular, for $ν=N$, we show that the (3+1)d symmetric gapped state admits a topological $\mathbb{Z}_4$ gauge theory description at low energy, and propose that a similar theory saturates the corresponding $\mathbb{Z}_{2N}^\mathrm{F}$ anomaly. For $N \nmid ν$, our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for $ν=N/2$, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological order inhomogeneously. For other values of $ν$, no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

Boundary topological orders of (4+1)d fermionic $\mathbb{Z}_{2N}^{\mathrm{F}}$ SPT states

TL;DR

This work analyzes (3+1)d fermionic systems with anomalous symmetry by embedding them as boundaries of (4+1)d fermionic SPTs. Using a crystalline correspondence to map to a symmetry and a block-state construction, the authors explicitly construct symmetric gapped boundaries for the bulk SPT when the anomaly index satisfies certain divisibility conditions. They show that for the Majorana-chain decoration () the minimal symmetric boundary is a (3+1)d gauge theory, while for the decoration () the boundary is non-TQFT and highly anisotropic; for other cases, a symmetric gapped boundary does not arise in their framework, in line with Cordova–Ohmori no-go results. The results illuminate how different fermionic SPT decorations saturate anomalies and offer perspectives on Higgsing continuous symmetries, potential links to the Standard Model anomaly structure, and future exploration of higher-dimensional boundary topological orders.

Abstract

We investigate (3+1)d topological orders in fermionic systems with an anomalous symmetry, where its subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of copies of Weyl fermions of the same chirality. Inspired by the crystalline correspondence principle, we deform the anomalous symmetry of (3+1)d Weyl fermion to the anomalous symmetry. Then we microscopically construct symmetry-preserving gapped boundary states of the closely-related (4+1)d symmetry-protected topological (SPT) state (with being the -fold rotation), whenever it is possible. In particular, for , we show that the (3+1)d symmetric gapped state admits a topological gauge theory description at low energy, and propose that a similar theory saturates the corresponding anomaly. For , our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for , we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological order inhomogeneously. For other values of , no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

Paper Structure

This paper contains 12 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Z2N symmetry and the nonperturbative global anomaly
  3. CN symmetry
  4. Construction of symmetric gapped boundary
  5. General considerations
  6. nu: non-TQFT gapped state
  7. Z4 gauge theory
  8. N=2p
  9. N=0 mod 8
  10. Conclusion and discussions
  11. (4+1)d SPT with CN symmetry
  12. Z16 class anomaly of Z4 symmetry in the Standard Model

Figures (2)

  • Figure 1: Block construction for a (3+1)d symmetric gapped state on the boundary of (4+1)d bulk $C_N \times \mathbb{Z}_2^{\rm F}$ FSPT state. Here we demonstrate the $N=4$ case.
  • Figure 2: The left figure demonstrates the "ungauged" description of (3+1)d $N=4$ symmetric gapped boundary construction: $p_x - \space\mathrm{i}\space p_y$ superconductors are added to cancel the chiral central charge, then we further decorate the (2+1)d planes with $\mathbb{Z}_4 \times \mathbb{Z}_2^{\rm F}$ SPTs to gap out everything. The right figure shows the actual TQFT with a static network of gauged FSPT defects after we gauge the $\mathbb{Z}_4$ symmetry.