A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases

This paper introduces a "symmetry breaking long exact sequence" (SBLES) of invertible field theories to classify symmetry breaking phases by relating the 't Hooft anomalies of localized defect excitations to the anomalies of the broken symmetry, thereby providing a new computational framework for understanding defect anomalies, higher Berry phases, and symmetry protected topological phases.

The Gist

Imagine you are a detective trying to solve a mystery about the hidden rules of the universe. In the world of quantum physics, particles often follow strict "symmetry" rules—like a dance where everyone must move in perfect unison. Sometimes, these rules are broken (like when a dancer decides to spin the other way), creating a new phase of matter.

This paper is about what happens at the fault lines where these rules break. Specifically, it looks at the "scars" or "defects" (like cracks in ice, whirlpools in water, or domain walls between magnets) that form when symmetry is broken. The authors discovered a powerful mathematical tool to predict exactly what kind of weird, gapless (energy-less) particles must live inside these scars, and why they can't just disappear.

Here is the breakdown using everyday analogies:

1. The Mystery: The "Anomaly"

Think of a Symmetry as a rulebook for a game. Sometimes, the rulebook has a hidden glitch called an Anomaly.

• The Glitch: Imagine a game where the rules say "You must always have an even number of players," but the physics of the game forces you to have an odd number. You can't fix this by just changing the players; the glitch is built into the game's DNA.
• The Consequence: In physics, if a system has this glitch (an anomaly), it cannot be "gapped" (made boring and quiet). It must have some active, noisy, gapless particles on its surface or edges. It's like a car that refuses to turn off its engine no matter how much you try to silence it.

2. The Setting: Breaking the Rules (Symmetry Breaking)

Now, imagine you introduce a "symmetry-breaking field." This is like telling the dancers, "Okay, stop dancing in unison; everyone pick your own partner."

• When the symmetry breaks, the system usually settles down into a quiet, gapped state (a calm sea).
• The Problem: Sometimes, even after you break the rules, you still find a "whirlpool" (a defect) in the middle of the calm sea. Inside this whirlpool, the particles are still noisy and gapless.
• The Question: Why is this whirlpool still noisy? Is it because the original glitch (anomaly) forced it to be? Or is there a new reason?

3. The Detective's Tool: The "Long Exact Sequence" (SBLES)

The authors built a mathematical machine called the Symmetry Breaking Long Exact Sequence (SBLES). Think of this as a three-step translation machine that connects three different worlds:

Step A: The "Residual" Check (The Obstruction)

• The Analogy: Imagine you are trying to pack a suitcase (the system) to make it perfectly flat (gapped). You try to break the symmetry to flatten it.
• The Check: The machine asks: "Even if you break the symmetry, is there still a hidden lump in the suitcase?"
• The Result: If the answer is "Yes," the suitcase cannot be flattened. There is a Residual Family Anomaly. This is a "No Entry" sign. It tells you that no matter how you try to break the symmetry, you cannot get rid of the noise. The defect is unavoidable.

Step B: The "Defect" Reconstruction (The Smith Map)

• The Analogy: Imagine you found a noisy whirlpool in a calm lake. You want to know what caused the whole lake to be weird in the first place.
• The Check: The machine looks at the noise inside the whirlpool and works backward. It says, "Ah! The noise inside this whirlpool is so specific that it proves the entire lake must have had a hidden glitch."
• The Result: This is the Defect Anomaly Map. It allows physicists to look at a small defect and deduce the properties of the entire bulk material. It's like looking at a single drop of water to know the chemical composition of the whole ocean.

Step C: The "Index" (The Ambiguity)

• The Analogy: Sometimes, the machine finds that two different types of whirlpools could explain the same lake glitch. Or, it finds a whirlpool that exists even when the lake is perfectly normal (no glitch).
• The Check: This is the Index Map. It measures the "twist" or "winding number" of the symmetry breaking. Think of it like winding a rubber band around a finger. You can wind it once, twice, or three times. The "Index" counts how many times you wound it.
• The Result: This explains the ambiguity. It tells us that there might be multiple ways to create a defect, or that a defect can exist even without a bulk glitch, provided it has a specific "topological twist" (like a Berry phase). It's the mathematical way of saying, "There's a hidden twist in the fabric of space here."

4. The "Higher Berry Phase" (The Invisible Thread)

The paper also talks about Higher Berry Phases.

• The Analogy: Imagine you are walking in a circle. If you walk around a magnetic field, you might feel a "kick" (a phase shift) when you return to the start, even if you didn't touch the field. This is a Berry phase.
• The Twist: The authors found that when you have a whole family of these systems (changing parameters like temperature or magnetic field), the "kick" isn't just a number; it's a higher-dimensional shape.
• The Connection: The "Index Map" in their sequence is essentially counting these kicks. It's like counting how many times a rubber band wraps around a finger as you twist the whole system.

Summary: Why is this important?

Before this paper, physicists knew that defects (like domain walls) often had weird particles, but they didn't have a universal rulebook to predict exactly what those particles would be or when they would appear.

• The "Long Exact Sequence" is the rulebook. It connects the "Glitch" of the whole universe to the "Noise" of the defect.
• It solves the "Obstruction" problem: It tells you when a defect is impossible to create (because the system is too "glitchy" to be quiet).
• It solves the "Ambiguity" problem: It tells you when there are multiple ways to create a defect, or when a defect can exist without a bulk glitch.

In a nutshell:
The authors found a mathematical "Rosetta Stone" that translates the language of Global Symmetry Glitches into the language of Local Defect Noise. They showed that you can't just look at the defect in isolation; you have to look at the whole "family" of symmetry-breaking possibilities to understand why the noise is there. It's a powerful new tool for classifying the exotic phases of matter that make up our universe, from superconductors to topological insulators.

Technical Summary

1. Problem Statement

The paper addresses the classification and constraints of symmetry breaking phases in quantum many-body systems, specifically focusing on the localized gapless excitations (defects) that arise at the cores of topological defects such as domain walls, vortices, and hedgehogs.

While the phenomenon of anomaly matching (relating bulk anomalies to boundary or defect anomalies) is well-established, previous work (e.g., [HKT20a]) left open two critical issues:

1. Obstruction: Not all symmetry-breaking patterns allow for the existence of a local, gapped defect. The conditions under which a symmetry-breaking phase can support a local defect were not fully characterized.
2. Ambiguity: Even when a defect exists, its anomaly is not uniquely determined by the bulk anomaly alone. There is an ambiguity arising from the specific pattern of symmetry breaking (the "family" of ground states).

The authors aim to provide a complete mathematical framework to:

• Determine the obstruction to the existence of local defects in symmetry-breaking phases.
• Classify the anomalies of these defects.
• Resolve the ambiguity in defect anomalies by relating them to the topology of the order parameter space.

2. Methodology

The authors employ a combination of invertible field theory (IFT) and cobordism theory to construct a unified mathematical structure.

• Invertible Field Theories (IFTs): They treat 't Hooft anomalies as obstructions to gauging, which are classified by the group of deformation classes of invertible field theories in $D+1$ dimensions ($\Omega^{D+1}_{G,s,\eta}$).
• Higher Berry Phases: They utilize the concept of family anomalies, where the theory depends on a parameter space (the order parameter space). This relates to higher Berry phases and the bulk-boundary correspondence in parameterized systems.
• The Symmetry Breaking Long Exact Sequence (SBLES): The core methodological innovation is the derivation of a long exact sequence of abelian groups. This sequence connects three distinct maps:
  1. Residual Family Anomaly ($Res_\rho$): Maps a bulk anomaly to an anomaly in the family of symmetry-broken theories.
  2. Defect Anomaly Map ($Def_\rho$): Reconstructs the bulk anomaly from the anomaly of a localized defect.
  3. Index Map ($Ind_\rho$): Computes the anomaly of a defect within an anomaly-free family, acting as a generalized index theorem.

The mathematical backbone relies on the SPT-cobordism conjecture, identifying anomaly groups with Anderson duals of Thom spectra (cobordism groups).

3. Key Contributions

A. The Symmetry Breaking Long Exact Sequence (SBLES)

The paper establishes the following exact sequence for a symmetry group $G$ and a representation $\rho$ (the order parameter):
$$\cdots \to \Omega^{D+1-k}_{G,s,\eta+\rho} \xrightarrow{Def_\rho} \Omega^{D+1}_{G,s,\eta} \xrightarrow{Res_\rho} \Omega^{D+1}_{G,s,\eta}(S(\rho)) \xrightarrow{Ind_\rho} \Omega^{D-k+1}_{G,s,\eta+\rho} \to \cdots$$
Where:

• $\Omega^{D+1}_{G,s,\eta}$: Anomalies of $G$-symmetric theories in $D$ dimensions.
• $\Omega^{D+1}_{G,s,\eta}(S(\rho))$: Anomalies of $G$-equivariant families parametrized by the unit sphere $S(\rho)$ of the order parameter.
• $\Omega^{D+1-k}_{G,s,\eta+\rho}$: Anomalies of the defect theory (codimension $k$).

B. The Three Maps

1. Residual Family Anomaly ($Res_\rho$):

   • Definition: Measures the obstruction to "gapping" the theory uniformly over the order parameter space $S(\rho)$.
   • Physical Meaning: If $Res_\rho(\omega) \neq 0$, the symmetry-breaking phase cannot support a local, non-degenerate gapped defect. The system must have a "diabolical locus" (gap closing) somewhere in the parameter space.
   • Result: This generalizes the concept of residual anomalies of unbroken subgroups to include topological obstructions from the parameter space itself (higher Berry phases).

2. Defect Anomaly Map ($Def_\rho$):

   • Definition: Reconstructs the bulk anomaly $\omega$ from the anomaly $\alpha$ of a localized defect.
   • Physical Meaning: If a local defect exists (i.e., $Res_\rho(\omega)=0$), the bulk anomaly is the image of the defect anomaly under this map. This generalizes the Smith homomorphism to arbitrary groups and representations.
   • Result: It provides a formula to calculate bulk SPT phases from the properties of vortices or domain walls.

3. Index Map ($Ind_\rho$):

   • Definition: Computes the anomaly of a defect inside an anomaly-free bulk family.
   • Physical Meaning: It quantifies the ambiguity in the defect anomaly. Even if the bulk is trivial, a defect can carry an anomaly if the symmetry-breaking family is topologically non-trivial.
   • Result: This is a generalization of the Callias index theorem to interacting systems and higher dimensions, linking the winding of the order parameter to the zero-mode anomaly.

C. Group Cohomology Formulation

The authors show that in the context of group cohomology (SPTs classified by $H^{D+1}(G, U(1))$), the SBLES is equivalent to the Thom-Gysin sequence associated with the fibration $S(\rho) \to S(\rho)//G \to BG$. This provides explicit formulas for the maps using Euler classes and integration over fibers.

4. Key Results and Examples

The paper applies the SBLES to numerous concrete examples, demonstrating its computational power:

• 2+1D Majorana Fermions (Time Reversal):
  • Shows that a single Majorana fermion with $T^2=(-1)^F$ is not gappable by a pair of T-odd operators (residual family anomaly is non-zero), even though a single mass term gaps it. This explains the necessity of a gapless mode or degeneracy in certain parameter configurations.
• Adjoint QCD (SU(2)):
  • Analyzes chiral symmetry breaking in 3+1D SU(2) QCD with adjoint fermions. The residual family anomaly is identified as a non-trivial class in $H^5$, predicting specific degeneracies in the phase diagram at $\theta = \pi$.
• 3+1D Dirac and Weyl Fermions:
  • Derives the anomaly matching for vortices in Dirac/Weyl semimetals. The defect anomaly map correctly reconstructs the bulk chiral anomaly from the 1+1D Weyl fermion living on the vortex core.
• p + ip Superfluids:
  • Computes the index map for vortices in a p+ip superfluid. The map correctly predicts that a vortex binds a Majorana zero mode (gravitational anomaly $\mathbb{Z}_2$), resolving the ambiguity of the defect anomaly.
• Higher Symmetry Breaking ($Z/3, Z/4, SU(2)$):
  • The sequence is used to compute anomaly classifications for $Z/3$ and $Z/4$ symmetry breaking, revealing complex structures involving torsion groups (e.g., $\mathbb{Z}_9$, $\mathbb{Z}_{32}$) that are difficult to derive via standard spectral sequences.
  • For $SU(2)$ breaking, it confirms that skyrmions in chiral symmetry breaking are fermions (Witten's anomaly).

5. Significance

• Unified Framework: The SBLES unifies previously disparate concepts: anomaly matching, obstruction to gapping, and higher Berry phases, into a single algebraic structure.
• Computational Tool: It offers a more efficient method for classifying anomalies than traditional spectral sequence calculations. By combining different symmetry breaking patterns, one can bootstrap the classification of anomalies for a given symmetry group.
• New Physical Predictions: The framework predicts new constraints on phase diagrams (e.g., where gaps must close) and clarifies the nature of defect modes in interacting systems, extending the Callias index theorem beyond free fermions.
• Connection to LSM Theorems: The paper suggests a deep connection between defect anomaly matching and the Lieb-Schultz-Mattis (LSM) theorems, proposing that the reconstruction of bulk anomalies from defects is a form of point-group LSM theorem.
• Mathematical Rigor: It provides a rigorous cobordism-theoretic foundation for the "Smith isomorphism" and defect anomaly matching, moving beyond heuristic arguments.

In summary, this paper provides a comprehensive "dictionary" for translating between bulk symmetries, order parameter topology, and the anomalies of defects, fundamentally advancing the understanding of symmetry breaking in topological phases of matter.