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Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems

Xiao-Gang Wen

TL;DR

This work develops a lattice-based framework for emergent and anomalous higher symmetries in condensed matter, connecting higher form symmetries to topological orders and to dynamical electromagnetic fields. It shows that EM condensed matter systems can host an anomalous $U(1)$-1-symmetry when monopoles are neglected, implying that any gapped phase must host nontrivial bosonic topological order. The paper provides concrete soluble lattice models realizing higher SPT phases and higher gauge theories, analyzes anomaly inflow and boundary realizations, and introduces criteria to detect anomalous 1-symmetries. It also discusses the topological robustness of emergent higher symmetries, their realizations in spacetime lattices, and systematic constructions of both hSPT and topologically ordered phases with higher symmetry, with explicit relevance to EM condensed matter systems and their phase structure.

Abstract

The usual condensed matter lattice theories do not include dynamical electromagnetic (EM) field and do not have higher symmetries naturally (unless we engineer fine-tuned toy models to realize higher symmetries). However, for gapped systems, (anomalous) higher symmetries can emerge from the usual condensed matter theories at low energies (usually in a spontaneously broken form). We pointed out that spontaneously broken emergent higher symmetries are nothing but a kind of topological orders. The emergent (anomalous) higher symmetries can be used to constrain possible phase transitions and possible phases induced by certain types of excitations in topological orders. (Anomalous) higher symmetry can also emerge in gapless systems if the gapless excitations contain gapless gauge fields. In particular, EM condensed matter systems that include the dynamical EM field have an anomalous $U(1)$ 1-symmetry if we ignore the magnetic monopoles. So EM condensed matter systems can realize some physical phenomena of anomalous higher symmetry. In particular, any gapped liquid phases of EM condensed matter systems (induced by arbitrary electric charge fluctuations and condensations) must have non-trivial bosonic topological orders.

Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems

TL;DR

This work develops a lattice-based framework for emergent and anomalous higher symmetries in condensed matter, connecting higher form symmetries to topological orders and to dynamical electromagnetic fields. It shows that EM condensed matter systems can host an anomalous -1-symmetry when monopoles are neglected, implying that any gapped phase must host nontrivial bosonic topological order. The paper provides concrete soluble lattice models realizing higher SPT phases and higher gauge theories, analyzes anomaly inflow and boundary realizations, and introduces criteria to detect anomalous 1-symmetries. It also discusses the topological robustness of emergent higher symmetries, their realizations in spacetime lattices, and systematic constructions of both hSPT and topologically ordered phases with higher symmetry, with explicit relevance to EM condensed matter systems and their phase structure.

Abstract

The usual condensed matter lattice theories do not include dynamical electromagnetic (EM) field and do not have higher symmetries naturally (unless we engineer fine-tuned toy models to realize higher symmetries). However, for gapped systems, (anomalous) higher symmetries can emerge from the usual condensed matter theories at low energies (usually in a spontaneously broken form). We pointed out that spontaneously broken emergent higher symmetries are nothing but a kind of topological orders. The emergent (anomalous) higher symmetries can be used to constrain possible phase transitions and possible phases induced by certain types of excitations in topological orders. (Anomalous) higher symmetry can also emerge in gapless systems if the gapless excitations contain gapless gauge fields. In particular, EM condensed matter systems that include the dynamical EM field have an anomalous 1-symmetry if we ignore the magnetic monopoles. So EM condensed matter systems can realize some physical phenomena of anomalous higher symmetry. In particular, any gapped liquid phases of EM condensed matter systems (induced by arbitrary electric charge fluctuations and condensations) must have non-trivial bosonic topological orders.

Paper Structure

This paper contains 43 sections, 211 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Symmetry and higher symmetry on lattice
  4. An example of anomaly-free (on-site) $Z_2$-1-symmetry
  5. The 3+1D bosonic lattice model
  6. $Z_2$-1-symmetry and unbreakable strings
  7. Emergence of generic higher symmetry in topological orders
  8. Generalized higher symmetry
  9. Spontaneous higher symmetry breaking and topological order
  10. The usefulness of higher symmetry in condensed matter
  11. An example of anomalous $Z_2$-1-symmetry
  12. The 2+1D bosonic lattice model
  13. On-site/non-on-site higher symmetry
  14. Simple lattice examples that realize higher symmetry protected topological phases
  15. A 3+1D model to realize a pure $Z_n$-1-SPT phase
  16. ...and 28 more sections

Figures (8)

  • Figure 1: ( Color online) A cubic lattice (black) and its dual cubic lattice (red). The index $\v i$ labels the faces of the cubic lattice and the links of the dual cubic lattice.
  • Figure 2: ( Color online) A qubit model, where qubits live on the links of a honeycomb lattice. $\v I$ labels the vertices and $\v i$ labels the links (the qubits) of the honeycomb lattice.
  • Figure 3: ( Color online) The string operators.
  • Figure 4: (Color online) The black lines describe a 2-dimensional spacetime complex $\cM^2$. The red lines describe the dual complex $\t \cM^2$.
  • Figure 5: (Color online) Two branched simplices with opposite orientations. (a) A branched simplex with positive orientation and (b) a branched simplex with negative orientation.
  • ...and 3 more figures