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

This paper extends the framework of entanglement asymmetry to higher-form symmetries to establish an entropic Coleman-Mermin-Wagner theorem that forbids spontaneous breaking of continuous $p$-form symmetries in dimensions $d \leq p+2$, while demonstrating that such breaking generates a monotonically growing asymmetry that quantifies the number of Goldstone fields.

The Gist

Imagine you are trying to understand a complex, crowded party. In physics, this "party" is the quantum world, and the "guests" are particles and fields. For a long time, physicists have used a tool called Symmetry to understand how these parties behave. Symmetry is like a rule that says, "If we swap two guests or rotate the room, the party looks exactly the same."

Sometimes, a party breaks these rules. Maybe everyone decides to stand on one side of the room, or everyone starts dancing in a specific direction. This is called Symmetry Breaking.

For decades, physicists had a strict rule (the Coleman-Mermin-Wagner Theorem) about when this breaking can happen. The rule was: "If the room is too small (in terms of dimensions), the guests can't agree on a direction. They will always be too jittery and chaotic to pick a side. You need a big enough room for order to emerge."

The New Tool: The "Entanglement Asymmetry" Meter

This paper introduces a new, super-sensitive tool to measure this chaos. Instead of just asking, "Did the guests pick a side?" (which is hard to see directly), the authors invented a meter called Entanglement Asymmetry.

Think of Entanglement as a secret handshake between two groups of guests. If the guests in the "Red Zone" are deeply connected to the guests in the "Blue Zone," they are entangled.

Entanglement Asymmetry measures how "unfair" or "lopsided" this connection is when you try to force the Red Zone to look like the Blue Zone.

• If the symmetry is unbroken: The Red and Blue zones are perfectly balanced. The meter reads Zero.
• If the symmetry is broken: The zones are out of sync. The meter reads a positive number. The bigger the number, the more the symmetry is broken.

The Big Discovery: "Higher-Form" Parties

The authors took this meter and applied it to a new kind of party: Higher-Form Symmetries.

• Normal Symmetry (0-form): Imagine a rule about individual guests (points).
• Higher-Form Symmetry (p-form): Imagine a rule about lines of guests, sheets of guests, or even volumes of guests.

Think of it like this:

• 0-form: A rule about how individual people stand.
• 1-form: A rule about how a rope of people holds hands.
• 2-form: A rule about how a blanket of people covers a table.

The paper asks: "Can these ropes or blankets break symmetry? Can they decide to all point North?"

The New "Room Size" Rule

The authors discovered a new version of the "Room Size" rule for these ropes and blankets.

• The Old Rule: You need a room with at least 3 dimensions (length, width, height) for normal guests to break symmetry.
• The New Rule: For a "rope" (1-form), you need a room with at least 4 dimensions. For a "sheet" (2-form), you need 5 dimensions.

The Formula: You need $d > p + 2$ dimensions to break symmetry.

• If your room is too small (e.g., a 2D video game world), a "rope" of guests can never agree on a direction. They will always be too jittery.
• If the room is big enough, they can lock into place, and the "Entanglement Asymmetry" meter will start ticking up.

The "Subregion" Surprise

Here is the most fascinating part. Usually, if a symmetry is broken, it's broken everywhere. But this paper shows that you can look at just a small slice of the party (a subregion).

• In the Ultraviolet (Tiny Slice): If you look at a tiny speck of the party, the meter reads Zero. The symmetry looks restored. The jitteriness of the tiny speck is too high to let order form.
• In the Infrared (Huge Slice): As you zoom out and look at a larger and larger chunk of the party, the meter starts to rise. The symmetry breaking becomes visible.

The Analogy: Imagine trying to organize a line of people.

• If you look at just two people, they might be fidgeting and can't agree on a line.
• If you look at two thousand people, they can easily form a straight line.
• The "Entanglement Asymmetry" meter tells you exactly how many people you need to look at before the line becomes visible. It grows logarithmically (slowly but steadily) as you get bigger.

Why This Matters

1. It's a Universal Ruler: This meter works for both simple particles and complex "ropes" of particles. It gives a precise number for how much symmetry is broken, not just a "yes or no."
2. It Counts the Goldstones: In physics, when symmetry breaks, new massless particles called "Goldstone bosons" appear (like ripples in a pond). This meter counts exactly how many of these ripples exist.
3. It Works in Small Spaces: It proves that even if you are trapped in a small part of the universe (like inside a black hole or a tiny lab), you can still detect if the universe as a whole has broken symmetry, provided you look at the right scale.

Summary in One Sentence

The authors built a new "symmetry detector" that proves you need a sufficiently large, multi-dimensional room for complex "ropes" and "sheets" of quantum particles to organize themselves, and this detector can measure exactly how organized they are, even if you only look at a small piece of the room.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the characterization of spontaneous symmetry breaking (SSB) in quantum field theories (QFTs), specifically extending the concept of entanglement asymmetry to higher-form symmetries.

• Context: Traditional diagnostics for SSB rely on local order parameters (Landau paradigm). However, "beyond Landau" phases (e.g., topological orders, phases with higher-form symmetries) lack local order parameters.
• Entanglement Asymmetry: Originally introduced for 0-form (standard) symmetries, this quantity measures the relative entropy between a state $\rho$ and its symmetrized version $\rho_S$. It serves as an entropic order parameter: $\Delta S = 0$ if the symmetry is preserved, and $\Delta S > 0$ if broken.
• Gap: While entanglement asymmetry was known for 0-form symmetries, its application to continuous $p$-form symmetries (where symmetry operators are supported on $(d-p-1)$-dimensional manifolds) was unexplored. Furthermore, the behavior of these quantities in spatial subregions (local SSB) versus global states needed clarification, particularly regarding the Coleman–Mermin–Wagner (CMW) theorem.

2. Methodology

The authors employ a combination of algebraic QFT definitions, path integral techniques, and replica tricks to compute entanglement asymmetry.

A. Definition of Higher-Form Entanglement Asymmetry

• Symmetrization: For a $p$-form symmetry group $G$, the symmetrized density matrix $\rho_S$ is defined by averaging over topological defects (operators) labeled by cycles in the homology group $H_{d-p-1}(\Sigma; G)$.
• Formula:
  $$\Delta S_{G[p]}(\rho) = S(\rho_S) - S(\rho)$$
  where $S$ is the von Neumann entropy.
• Replica Trick: To compute this, the authors utilize Rényi asymmetries ($\Delta S_n$) defined via the ratio of traces:
  $$\Delta S_{n, G[p]} = \frac{1}{1-n} \log \frac{\text{tr}(\rho_S^n)}{\text{tr}(\rho^n)}$$
  The physical entanglement asymmetry is recovered via the limit $n \to 1$.

B. Path Integral Construction

• Replica Manifold: The calculation involves a Euclidean path integral on an $n$-fold branched cover of spacetime ($X_n$), constructed by cyclically gluing $n$ copies of the manifold along a subregion $A$.
• Factorization: To handle the reduced density matrix $\rho_A$, the entangling surface is regulated (thickened) to allow for a factorized Hilbert space.
• Decomposition: The partition function $Z$ is decomposed into oscillator (Gaussian fluctuations) and instanton (winding/topological) sectors.
  • Crucially, the oscillator contribution cancels out in the asymmetry calculation, leaving the instanton sector as the sole contributor. This isolates the universal, topological data related to symmetry breaking.

C. Specific Models Analyzed

1. Abelian $p$-form Gauge Theory: Modeled as a free $p$-form Maxwell theory (effective theory of broken $p$-form symmetry).
2. Non-Abelian 0-form Symmetry: Modeled as a Non-Linear Sigma Model (NLSM) with target space $G$ or $G/H$.
3. Geometry: Calculations are performed on spatial slices of the form $S^p_r \times S^{d-p-1}_R$ (for global states) and spherical subregions $B^{d-1}_R$ (for subregion theorems).

3. Key Contributions and Results

A. Entropic Coleman–Mermin–Wagner (CMW) Theorem

The authors derive a rigorous entropic version of the CMW theorem for both 0-form and $p$-form symmetries.

• Global Result: For a continuous $p$-form symmetry in $d$ spacetime dimensions:
  • If $d \leq p + 2$: The entanglement asymmetry vanishes ($\Delta S = 0$). The symmetry is unbroken, and the ground state is unique.
  • If $d > p + 2$: The entanglement asymmetry grows logarithmically with the system size $R$:
    $$\Delta S \sim \frac{N(d - p - 2)}{2} \log(\mu R)$$
    Here, $N$ is the number of broken generators. This logarithmic divergence signals spontaneous symmetry breaking.
• Significance: This confirms that continuous $p$-form symmetries cannot break spontaneously in low dimensions, providing an entropic proof of the generalized CMW theorem.

B. Subregion Theorems

The paper establishes that entanglement asymmetry can detect SSB within a spatial subregion $A$ of size $R$.

• Behavior:
  • UV Limit ($\mu R \ll 1$): The asymmetry vanishes. The symmetry appears restored at short distances.
  • IR Limit ($\mu R \gg 1$): The asymmetry approaches the global result, growing logarithmically.
• Monotonicity: The asymmetry is a monotonic function of the subregion size, acting as an RG monotone. This implies that symmetry breaking is a low-energy phenomenon that emerges as one integrates out high-energy modes.

C. Quantitative Results for Goldstone Modes

• Conjecture Verification: The authors confirm a conjecture by Metlitski and Grover regarding the entanglement entropy of Goldstone fields. They show that the entanglement asymmetry isolates the universal logarithmic term in the entanglement entropy.
• Coefficient: The coefficient of the logarithm is precisely determined by the number of broken generators ($N$) and the dimensionality, not by the number of polarizations.
  $$b_{p\text{-form}} = \frac{N(d - p - 2)}{2}$$
• Closed-Form Expressions: The authors derive exact closed-form expressions for the Rényi asymmetries of a compact scalar ($p=0$) and $p$-form gauge fields in $d=3$ and $d=4$ dimensions. These involve Siegel theta functions and specific determinants of "instanton matrices" derived from electrostatic analogies.

D. Non-Abelian Symmetries

The framework is extended to non-Abelian symmetries ($G \to H$).

• The results generalize naturally: the coefficient of the logarithmic growth depends on $\dim(G/H)$, the dimension of the coset space (number of broken generators).
• The analysis holds for both complete and partial symmetry breaking.

4. Technical Details and Derivations

• Instanton Matrix ($M_{n,d}$): A central technical achievement is the computation of the instanton matrix for the replica manifold. The authors map this problem to an electrostatic capacitance problem on a sphere with $n$ punctures.
• Electrostatics Analogy: The matrix elements are derived by solving the Laplace equation with specific monodromy conditions around the entangling surface.
  • For $d=3$, the authors recover known results involving $\tan(\pi k/n)$.
  • For $d=4$, they propose a new analytic form: $C_4(k/n) \propto k/n(1-k/n)$.
• Theta Functions: The partition functions are expressed in terms of Siegel theta functions $\Theta(M)$, where $M$ is the instanton matrix. The asymptotic behavior of these functions in the large-volume limit yields the logarithmic scaling.

5. Significance and Implications

1. Unified Diagnostic: Entanglement asymmetry is established as a robust, universal diagnostic for SSB that works for both 0-form and higher-form symmetries, and for both Abelian and non-Abelian groups.
2. Beyond Landau: It provides a tool to study symmetry breaking in phases where no local order parameter exists (e.g., topological phases, deconfined phases).
3. RG Flow Interpretation: The monotonicity of asymmetry with respect to subregion size reinforces the view of entanglement measures as probes of Renormalization Group flows. It demonstrates that symmetry restoration is a UV phenomenon, while breaking is an IR phenomenon.
4. Quantitative Precision: The paper moves beyond qualitative statements to provide exact quantitative formulas for the asymmetry in specific dimensions, linking the coefficient of the divergence directly to the number of broken generators.
5. Future Directions: The work opens avenues for studying non-invertible symmetries, higher-group symmetries, and the role of torsion in homology within the context of entanglement asymmetry.

In summary, this paper successfully generalizes the concept of entanglement asymmetry to higher-form symmetries, proving a quantitative entropic version of the Coleman–Mermin–Wagner theorem and providing exact analytical results for the asymmetry in various dimensions and geometries.