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Higher-form entanglement asymmetry. Part I. The limits of symmetry breaking

Francesco Benini, Eduardo García-Valdecasas, Stathis Vitouladitis

TL;DR

This work extends entanglement asymmetry to higher-form symmetries, providing an entropic diagnostic of symmetry breaking for 0-form and p-form cases. By formulating the symmetry action through topological defects and employing path-integral/replica methods, the authors derive an entropic Coleman–Mermin–Wagner theorem, with ΔS scaling logarithmically with system size when breaking is allowed (d>p+2) and vanishing in the forbidden regime (d≤p+2). They also establish subregion theorems, showing how symmetry breaking can be detected within spatial subregions via Rényi entropies and theta-function expressions tied to winding sectors and instanton data. The results connect universal logarithmic entanglement contributions to the number of broken generators and set the stage for extensions to non-Abelian, higher-group, and noninvertible symmetries, as well as to torsion and more general geometric settings. Overall, the paper provides a rigorous entropic framework for diagnosing and quantifying higher-form symmetry breaking across scales.

Abstract

Entanglement asymmetry is a relative entropy that faithfully diagnoses symmetry breaking in quantum states, possibly within a spatial subregion. In this work, we extend such framework to higher-form symmetries and compute entanglement asymmetry in theories with spontaneously-broken continuous zero- and higher-form symmetries. One of our central results is an entropic Coleman--Mermin--Wagner theorem, for $0$- and $p$-form symmetries, valid also on subregions, which forbids spontaneous breaking of continuous $p$-form symmetries in spacetime dimensions $d\leq p+2$. Our theorem not only qualifies symmetry breaking, it also quantifies it: spontaneous breaking triggers a nonvanishing entanglement asymmetry that grows monotonically towards the infrared, and counts the number of Goldstone fields. Along the way, we derive standalone results concerning the entanglement entropy of Goldstone bosons and gauge fields.

Higher-form entanglement asymmetry. Part I. The limits of symmetry breaking

TL;DR

This work extends entanglement asymmetry to higher-form symmetries, providing an entropic diagnostic of symmetry breaking for 0-form and p-form cases. By formulating the symmetry action through topological defects and employing path-integral/replica methods, the authors derive an entropic Coleman–Mermin–Wagner theorem, with ΔS scaling logarithmically with system size when breaking is allowed (d>p+2) and vanishing in the forbidden regime (d≤p+2). They also establish subregion theorems, showing how symmetry breaking can be detected within spatial subregions via Rényi entropies and theta-function expressions tied to winding sectors and instanton data. The results connect universal logarithmic entanglement contributions to the number of broken generators and set the stage for extensions to non-Abelian, higher-group, and noninvertible symmetries, as well as to torsion and more general geometric settings. Overall, the paper provides a rigorous entropic framework for diagnosing and quantifying higher-form symmetry breaking across scales.

Abstract

Entanglement asymmetry is a relative entropy that faithfully diagnoses symmetry breaking in quantum states, possibly within a spatial subregion. In this work, we extend such framework to higher-form symmetries and compute entanglement asymmetry in theories with spontaneously-broken continuous zero- and higher-form symmetries. One of our central results is an entropic Coleman--Mermin--Wagner theorem, for - and -form symmetries, valid also on subregions, which forbids spontaneous breaking of continuous -form symmetries in spacetime dimensions . Our theorem not only qualifies symmetry breaking, it also quantifies it: spontaneous breaking triggers a nonvanishing entanglement asymmetry that grows monotonically towards the infrared, and counts the number of Goldstone fields. Along the way, we derive standalone results concerning the entanglement entropy of Goldstone bosons and gauge fields.

Paper Structure

This paper contains 35 sections, 232 equations, 4 figures.

Table of Contents

  1. Introduction and summary
  2. Entanglement asymmetry for higher-form symmetries
  3. Path integral methods
  4. An entropic Coleman--Mermin--Wagner theorem
  5. Quantisation of p-form gauge theory
  6. Asymmetry in p-form gauge theory
  7. Various physical limits
  8. Very hot and very cold.
  9. Very large and very small (target).
  10. The Coleman--Mermin--Wagner theorem
  11. Non-Abelian asymmetry
  12. Complete symmetry breaking
  13. Partial symmetry breaking
  14. Subregion theorem for U(1) 0-form symmetry
  15. Partition functions and winding sectors
  16. ...and 20 more sections

Figures (4)

  • Figure 1: A cartoon of the replica manifold $X_\mathsf{n}$. The coloured discs denote the location of the branch cuts $\mathbb{B}_R^{d-1}$, which are identified cyclically.
  • Figure 2: Entanglement asymmetry for a compact scalar in $d=3$. We show the Rényi asymmetries $\Delta\mathcal{S}_\mathsf{n}$ for $\mathsf{n}=2,3,4,5$, the numerical extrapolation to $\mathsf{n} \rightarrow1$, and the asymptotic limits that the $\mathsf{n} \rightarrow 1$ must reproduce.
  • Figure 3: Entanglement asymmetry for a compact scalar in $d=4,5,6$. We show the numerical extrapolation to $\mathsf{n}=1$ and the asymptotic limits. We have fitted the additive constant in \ref{['eq:EntasBigmur']}.
  • Figure 4: $X_\mathsf{n}$ drawn as a $d$-sphere with $\mathsf{n}$ boundaries, as explained in \ref{['App:TopRepMan']}. The boundaries, denoted by crossed circles get mapped to the conductors. Of these, $B_\mathsf{n}$ is grounded. The relevant basis, $\qty{\gamma_i}$ of relative homology cycles is drawn in red.