Symmetry Spans and Enforced Gaplessness

This paper introduces a new mechanism called "symmetry spans," where a global symmetry embedded into two larger symmetries enforces infrared gaplessness when no compatible gapped phase exists, offering a powerful tool to establish symmetry-enforced gaplessness even with discrete and non-anomalous continuous symmetries.

The Gist

The Big Picture: Why Can't Everything Be Still?

Imagine you are trying to build a machine that sits perfectly still (a "gapped" state). In the quantum world, "still" means the system has a clear, calm ground state with no jittery energy fluctuations. Usually, if you have a system with certain rules (symmetries), you can build a stable, still machine.

However, physicists have long known that sometimes, the rules of the game make it impossible for the system to be still. The system is forced to be "gapless"—meaning it must always be buzzing, vibrating, or flowing like a river. This is called Symmetry-Enforced Gaplessness.

For a long time, we thought this only happened if the rules were "broken" or "anomalously" weird (like a rule that contradicts itself). But this paper introduces a new, clever trick. It shows that you can force a system to be restless even if all the individual rules look perfectly normal and consistent on their own.

The Core Idea: The "Symmetry Span"

The authors introduce a concept called a Symmetry Span. Think of it like a bridge or a tension cable.

Imagine a small, simple rule (let's call it Rule E).
Now, imagine two larger, more complex rulebooks: Rule C and Rule D.

The "Span" happens when Rule E is secretly embedded inside both Rule C and Rule D at the same time.

• Rule C says: "If you follow me, you must be a specific type of still machine."
• Rule D says: "If you follow me, you must be a different type of still machine."

Here is the catch: Rule E is the common ground. If your system follows Rule E, it must be able to satisfy the requirements of both Rule C and Rule D simultaneously.

The Analogy of the Impossible Suit:
Imagine you are a tailor trying to make a suit (the quantum state).

1. Client C (Rule C) says: "I need a suit that fits perfectly if I wear a red hat."
2. Client D (Rule D) says: "I need a suit that fits perfectly if I wear a blue hat."
3. The Constraint (Rule E): You are only allowed to use the fabric patterns that work for both clients.

If Client C requires a suit that is tight and stiff, but Client D requires a suit that is loose and flowing, and there is no single suit design that can be both tight/stiff AND loose/flowing at the same time, then no suit can be made.

In physics terms: If there is no "gapped" (still) state that satisfies both Rule C and Rule D, the system is forced to be gapless (restless/vibrating). It has no choice but to flow.

Why This Is a Big Deal

Previously, to prove a system must be restless, physicists had to find a "broken" rule (an anomaly) that made the system unstable. It was like saying, "The suit is impossible because the fabric is cursed."

This paper says: "No, the fabric isn't cursed. The fabric is fine. The problem is that you are trying to wear two different hats at once, and the shape of the suit required for Hat A is mathematically incompatible with the shape required for Hat B."

This is powerful because:

1. It works with normal rules: You don't need "cursed" or anomalous continuous symmetries in the starting material (the UV scale). You can start with perfectly normal, discrete rules (like flipping a switch) and normal continuous rules (like rotating a dial).
2. It's buildable: Because the starting rules are normal, we can actually build these systems on a computer or in a lab (on a "lattice").

The Examples: How They Did It

The authors built specific examples in 1+1 dimensions (a line of atoms, like a chain).

Example 1: The Tambara-Yamagami (TY) Span

• The Setup: They took a simple symmetry (flipping a coin, $Z_N$) and embedded it into two larger worlds.
  • World 1: A continuous rotation symmetry (like a dial).
  • World 2: A "non-invertible" symmetry. This is a fancy way of saying a rule that acts like a "duality" or a mirror. If you apply it, you don't just get a new state; you get a state that is a mix of possibilities.
• The Conflict: The continuous world says, "If you are still, you must be in a specific topological state (SPT)." The duality world says, "If you are still, you must be in a different topological state."
• The Result: Since a single state cannot be both, the system is forced to be gapless. It flows like a liquid.

Example 2: The Rep(D8) Span

• They used a symmetry group called $D_8$ (related to the symmetries of a square/dihedral group).
• They showed that if you try to fit this symmetry into a continuous rotation world, the "still" states required by the rotation world clash with the "still" states allowed by the $D_8$ world.
• Again, the system is forced to be restless.

The "Lattice" Connection

One of the hardest things in physics is taking a beautiful, abstract mathematical theory and building it with real atoms on a grid (a lattice). Usually, "anomalous" continuous symmetries are impossible to build on a grid without breaking them.

This paper solves that. They showed how to build these "Symmetry Spans" using only normal, discrete symmetries (like flipping spins) and normal continuous symmetries (like rotating spins) on a grid.

• They wrote down specific "Hamiltonians" (the energy recipes for the atoms).
• They proved that if you build a chain of atoms with these specific rules, the atoms cannot settle down. They must vibrate.
• Even though the atoms start with "normal" rules, as you zoom out to look at the big picture (the infrared limit), the system looks like it has a weird, anomalous continuous symmetry. The anomaly emerges from the clash of the two normal rules.

Summary in a Nutshell

1. The Problem: We want to know when a quantum system is forced to be restless (gapless).
2. The Old Way: Look for "broken" or "cursed" rules (anomalies).
3. The New Way (This Paper): Look for a Symmetry Span. This is when a simple rule is squeezed between two larger rules that demand contradictory "still" states.
4. The Result: If the demands contradict, the system has no choice but to be restless.
5. The Bonus: We can build these systems in the lab using standard, non-cursed rules. The "weirdness" (anomaly) appears naturally as a result of the system trying to satisfy conflicting demands.

It's like trying to balance a pencil on its tip while two people push it from opposite sides with different forces. The pencil (the quantum system) can't stand still; it must fall and move. The paper gives us the blueprint for exactly how to set up those two pushes.

Technical Summary

1. Problem Statement

The central challenge in theoretical physics is determining the infrared (IR) fate of a quantum system based on its ultraviolet (UV) data. While 't Hooft anomaly matching has been the primary tool for establishing symmetry-enforced gaplessness (the phenomenon where global symmetry forces a system to be gapless), this mechanism has historically relied heavily on anomalous continuous symmetries.

• The Limitation: Realizing anomalous continuous symmetries directly on a lattice is difficult and often incomplete. Most lattice models naturally possess discrete symmetries or non-anomalous continuous symmetries.
• The Question: Can one enforce gaplessness using only discrete symmetries and non-anomalous continuous symmetries in the UV, without relying on pre-existing continuous anomalies?

2. Methodology: The "Symmetry Span" Mechanism

The authors introduce a new mechanism called a Symmetry Span. This approach relies on the incompatibility of constraints arising from multiple embeddings of a single symmetry into larger symmetry structures.

• Definition of a Symmetry Span: A configuration where a global symmetry $E$ is simultaneously embedded into two larger symmetries, $C$ and $D$, forming a commutative diagram:
  $$E \xrightarrow{i_C} C \quad \text{and} \quad E \xrightarrow{i_D} D$$
  where $C$ and $D$ are sub-categories of a larger faithful symmetry $S_{faith}$.
• The Compatibility Constraint:
  • Let $\text{TQFT}(X)$ denote the category of gapped phases (Topological Quantum Field Theories) with symmetry $X$.
  • Any gapped phase with the full symmetry $S_{faith}$ must restrict to a valid gapped phase for both $C$ and $D$.
  • Therefore, the restriction of the phase to the common sub-symmetry $E$ must lie in the intersection of the pullbacks of the gapped phases of $C$ and $D$:
    $$i_C^* \text{TQFT}(C) \cap i_D^* \text{TQFT}(D) \neq \{0\}$$
• Enforced Gaplessness: If this intersection is empty (i.e., no gapped phase exists that is consistent with both embeddings), the system is forced to be gapless. This occurs even if $E$, $C$, and $D$ individually admit gapped phases or are non-anomalous in isolation.

3. Key Contributions

A. Theoretical Framework

• Generalization of Anomaly Matching: The paper generalizes the concept of anomaly matching to scenarios involving non-invertible symmetries (described by fusion categories) and continuous symmetries.
• Pullback of Module Categories: The authors utilize the mathematical framework of module categories over fusion categories. They demonstrate how embedding functors (tensor functors) pull back gapped phases. Crucially, they show that the pullback map is often affine rather than linear (due to SPT phases), which is key to creating the incompatibility required for gaplessness.
• Independence from Continuous Anomalies: The mechanism works even if the continuous symmetry $G$ in the span is non-anomalous in the UV. The gaplessness arises from the structural conflict between the discrete non-invertible symmetry and the continuous symmetry, rather than a pre-existing continuous anomaly.

B. Specific Examples in 1+1 Dimensions

The authors construct explicit examples where the intersection condition fails:

1. $Z_N$ embedded in Tambara-Yamagami ($TY(Z_N)$) and $U(1)$:
   • $TY(Z_N)$ is a non-invertible symmetry that forbids a unique gapped vacuum preserving the $Z_N$ subgroup (it forces spontaneous symmetry breaking or degeneracy).
   • Embedding $Z_N$ into a connected continuous group $G$ (like $U(1)$) forces any gapped phase to be a direct sum of SPTs (since continuous symmetries cannot be spontaneously broken in gapped 1+1D phases).
   • The pullback of the $TY(Z_N)$ phases does not contain the specific SPTs required by the $U(1)$ embedding, leading to a contradiction and enforced gaplessness.
2. $Z_2 \times Z_2$ embedded in $Rep(D_8)$ and $U(1) \times U(1)$:
   • $Rep(D_8)$ (representation category of the dihedral group) has specific SPT phases that, under a specific embedding, pull back to a non-trivial $Z_2 \times Z_2$ SPT.
   • However, embedding $Z_2 \times Z_2$ into $U(1) \times U(1)$ (with trivial SPTs) pulls back to the trivial SPT.
   • The mismatch between the required SPTs enforces gaplessness.
3. Type III Anomaly: The paper connects the $Rep(D_8)$ case to a $Z_2^3$ symmetry with a Type III anomaly, showing they are related by gauging/duality transformations (Kennedy-Tasaki).

C. Continuum and Lattice Realizations

• Continuum Examples: The authors identify known Conformal Field Theories (CFTs) that realize these spans, such as:
  • $U(1)_{2k}$ WZW CFT (realizing the $TY(Z_k)$ span).
  • $Spin(4)_1$ WZW CFT (realizing the $Rep(D_8)$ and Type III anomaly spans).
• Lattice Hamiltonians: A major contribution is the construction of explicit lattice Hamiltonians that realize these symmetry spans in the UV.
  • They construct spin chains with $Z_N$ and $TY(Z_N)$ symmetries, and chains with $Z_2 \times Z_2$ and $Rep(D_8)$ symmetries.
  • These models utilize Onsager algebras and Jordan-Wigner transformations to demonstrate that the Hamiltonians are free (or effectively free) and gapless.
  • Crucially, these lattice models possess non-anomalous continuous symmetries in the UV, yet flow to gapless IR theories with emergent anomalous continuous symmetries.

4. Results

• Proof of Gaplessness: The paper rigorously proves that for specific symmetry spans, the intersection of allowed gapped phases is empty, thereby enforcing gaplessness.
• Lattice Feasibility: The authors demonstrate that symmetry-enforced gaplessness can be realized in lattice models without requiring anomalous continuous symmetries at the microscopic level. This provides a concrete path to studying such phenomena in condensed matter systems.
• Duality Connections: The work clarifies the relationship between non-invertible symmetries (like $TY$ and $Rep(D_8)$) and standard anomalies (Type III) via gauging and duality transformations (e.g., Kennedy-Tasaki).

5. Significance

• New Mechanism for Gaplessness: This paper establishes a robust mechanism for enforcing gaplessness that does not rely on the traditional "anomalous continuous symmetry" route. It expands the toolkit for identifying gapless phases in quantum many-body systems.
• Bridging Discrete and Continuous: It successfully bridges the gap between discrete non-invertible symmetries and continuous symmetries, showing how their interplay can constrain IR physics.
• Lattice Realization: By providing explicit lattice Hamiltonians, the authors move the concept of symmetry-enforced gaplessness from abstract continuum field theory to concrete, potentially experimentally relevant lattice models. This is particularly significant given the historical difficulty of realizing anomalous continuous symmetries on the lattice.
• Future Directions: The work opens avenues for exploring symmetry spans in higher dimensions ($d > 2$) and understanding the full symmetry algebras (beyond Onsager algebras) generated by these spans.

In summary, Ando and Ohmori introduce the Symmetry Span as a powerful new criterion for gaplessness, demonstrating that the incompatibility of gapped phases under multiple symmetry embeddings can force a system to be gapless, even when all constituent symmetries are individually well-behaved in the UV.