Lieb-Schultz-Mattis theorem from gauge constraints

This paper establishes a novel Lieb-Schultz-Mattis theorem for a one-dimensional $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory coupled to matter, demonstrating that kinematic Gauss law constraints generate a U(1) symmetry which forbids trivial gapped ground states and necessitates either spontaneous symmetry breaking or gapless excitations, the latter exhibiting free Dirac fermion behavior with specific correlation decay.

The Gist

The Big Picture: A Rulebook for Quantum Chains

Imagine you have a long, circular necklace made of beads. In the world of quantum physics, these beads aren't just plastic; they are tiny magnets (spins) that can point in different directions. Usually, physicists try to figure out what happens to these beads when they get cold. Do they freeze into a perfect, quiet pattern? Or do they keep jittering around forever?

This paper introduces a new set of rules for a specific type of necklace. The authors built a model where the beads are connected by invisible "gauge" strings. The most important rule of this model is the Gauss Law. Think of the Gauss Law as a strict bouncer at a club: it says, "No two neighbors can wear the same outfit." If a bead is wearing a "Red" shirt, the string connecting it to the next bead must be "Blue" or "Green," never Red.

The Main Discovery: The "No Quiet Zone" Theorem

The authors discovered a powerful mathematical rule (a variation of the famous Lieb-Schultz-Mattis or LSM theorem) that applies to this specific necklace.

The Analogy:
Imagine trying to arrange a line of dancers so that everyone is perfectly still and happy (a "gapped" ground state). In many physics systems, you can do this. But in this specific model, the authors proved it is impossible to have a perfectly still, simple arrangement.

Why? Because of a conflict between two types of symmetry:

1. Translation: If you slide the whole necklace one step to the right, the rules look the same.
2. Reflection: If you look at the necklace in a mirror, the rules look the same.

The authors found that the "bouncer" (the Gauss Law) creates a hidden "U(1) symmetry"—a kind of internal clock or rhythm for the system. This clock ticks in a way that is friendly to sliding (translation) but hates looking in the mirror (reflection). It's like a clock that runs forward when you walk left, but runs backward when you walk right.

The Result:
Because of this conflict, the system cannot settle down into a boring, frozen state. It is forced to do one of two things:

• Break the symmetry: The dancers spontaneously decide to break the mirror rule (e.g., everyone leans left instead of right).
• Stay jittery: The dancers never stop moving; the system remains "gapless" (fluid and active) even at absolute zero.

The paper proves that you cannot have a trivial, frozen, gapped state in this system. The "bouncer" (Gauss Law) forces the system to be interesting.

Finding the "Sweet Spot" (The Gapless Point)

The authors didn't just prove the system can't be frozen; they also found a specific setting (a specific "gapless point") where they could solve the math exactly.

The Analogy:
At this specific setting, the complex necklace of beads and strings transforms into a simpler system: a line of free-floating fermions (think of them as ghostly, non-interacting particles). However, there's a catch: the total number of these ghosts must follow a strict rule (a constraint on the total number).

At this point, the system behaves like a river flowing smoothly. The authors calculated how disturbances (ripples) in this river behave. They found that if you poke the system at one point, the effect of that poke fades away as you move further away, but it does so in a very specific, wavy pattern:

• It oscillates (like a wave: up, down, up, down).
• It gets weaker very slowly (following a specific mathematical power law).

This behavior is described by "free Dirac fermions," which is a fancy way of saying the system acts like a perfect, massless fluid of quantum particles.

Why This Matters (According to the Paper)

1. New Source of Rules: Usually, theorems like LSM come from the internal properties of the particles (like their spin). This paper shows that constraints (the Gauss Law) alone can create these powerful rules. It's like saying the shape of the room forces the furniture to be arranged in a specific way, even if the furniture itself has no opinion.
2. A New Playground: This model provides a perfect test bed for studying "topological defects." Imagine a knot in the necklace that can't be untied. The authors suggest this model is a great place to study how these knots behave when the system is in different phases.
3. Verification: They used powerful computer simulations (DMRG) to confirm that the system behaves exactly as their math predicted, showing it has a "central charge" of 1, which confirms it acts like a single channel of free-moving quantum particles.

Summary in One Sentence

The authors built a quantum necklace with a strict "no neighbors alike" rule, proving that this rule forces the system to either break symmetry or stay fluid, and they found a specific setting where the system acts like a perfect, flowing river of quantum particles.

Technical Summary

Technical Summary: Lieb-Schultz-Mattis Theorem from Gauge Constraints

Problem Statement
Lieb-Schultz-Mattis (LSM) theorems impose nontrivial constraints on the ground-state properties of quantum many-body systems with spatial and internal symmetries, typically prohibiting a trivial gapped ground state. While recent work has linked LSM theorems to quantum anomalies and established them in pure $U(1)$ gauge theories, a key open question remains: Can an LSM-type theorem arise in a theory of matter coupled to gauge fields with a discrete gauge group? Specifically, can the kinematic structure (Gauss law constraints) of such a gauge theory lead to nontrivial constraints on the low-energy physics within the Gauss law subspace?

Methodology
The authors construct a one-dimensional $Z_2 \times Z_2$ gauge theory coupled to matter on a periodic chain. The model features a three-dimensional Hilbert space on each site (matter) and each link (gauge). The Hamiltonian is defined as:
$$H = \sum_{j, \alpha=x,y,z} \left( t_\alpha S^\alpha_{j\sigma} S^\alpha_{j\tau} S^\alpha_{j+1\sigma} - K_\alpha (S^\alpha_{j\tau})^2 \right)$$
where $S^\alpha$ are spin-1 operators. The system possesses local gauge symmetries generated by operators $A^\alpha_j = \Sigma^\alpha_{j-1\tau}\Sigma^\alpha_{j\sigma}\Sigma^\alpha_{j\tau}$, where $\Sigma^\alpha = \exp(i\pi S^\alpha)$.

The analysis proceeds by restricting the system to the Gauss law subspace $\mathcal{V}_G$, defined by the condition $A^\alpha_j |\psi\rangle = |\psi\rangle$ for all sites and generators. Within this subspace, the authors:

1. Analyze Symmetries: Investigate the commutation relations of the Hamiltonian with lattice translation ($T$) and reflection ($R$) operators.
2. Identify Hidden Symmetry: Construct a conserved quantity $M$ (a $U(1)$ generator) derived from the kinematic constraints of the Gauss law.
3. Prove LSM Theorem: Demonstrate that the generator $M$ commutes with $T$ but anticommutes with $R$, leading to a proof that a trivial gapped ground state is impossible.
4. Map to Fermions: Perform a non-local mapping from the constrained Hilbert space to a chain of qubits, and subsequently via a Jordan-Wigner transformation to free fermions, to identify specific points in the parameter space.
5. Calculate Correlations: Utilize the theory of Toeplitz determinants and the Fisher-Hartwig conjecture to determine the asymptotic behavior of correlation functions at the identified critical point.

Key Contributions and Results

1. LSM Theorem from Kinematic Constraints: The primary result is the demonstration that imposing the Gauss law in a $Z_2 \times Z_2$ gauge theory coupled to matter generates an emergent $U(1)$ symmetry. The generator $M$ of this symmetry satisfies $[M, T] = 0$ and $\{M, R\} = 0$. This algebraic structure forces the ground state to either spontaneously break a symmetry or be gapless, ruling out a trivial gapped phase for any translation- and reflection-invariant Hamiltonian in this subspace. This establishes a novel mechanism where the LSM theorem originates from the kinematic constraints of a gauge theory rather than a global symmetry of the matter sector alone.

2. Gapless Point and Dirac Fermions: The authors identify a specific point in the parameter space ($t_x=t_y=t_z=1, K_x=K_y=K_z=0$) where the system is gapless. Through a non-local mapping, the Hamiltonian at this point is shown to be equivalent to a system of free Dirac fermions subject to a constraint on the total fermion number: $\exp(i\frac{2\pi}{3}(L+N)) = 1$. The central charge of this critical theory is determined to be $c=1$, verified numerically via Density Matrix Renormalization Group (DMRG) calculations of the bipartite entanglement entropy.

3. Correlation Function Asymptotics: At the gapless point, the asymptotic behavior of the two-point correlation function for the local gauge-invariant quantity $(S^\alpha_{j\tau})^2$ is derived. The result is:
   $$\langle (S^\alpha_{j\tau})^2 (S^\beta_{j+r\tau})^2 \rangle - \frac{4}{9} \propto \frac{\cos(\pi r)}{r^{2/9}}$$
   This power-law decay with an oscillatory term confirms the critical nature of the point.

4. Spin-1/2 Variant: The paper briefly analyzes a spin-1/2 version of the model where the local symmetry operators do not all commute. It is shown that this leads to a projective representation of the symmetry group, resulting in an extensive degeneracy (at least $2^L$) for every energy eigenvalue.

Significance
The paper claims to answer the question of whether discrete gauge theories coupled to matter can support LSM-type theorems in the affirmative. The significance lies in uncovering a mechanism where the symmetry responsible for the LSM theorem (the emergent $U(1)$) originates directly from the kinematic structure (Gauss law) of the gauge theory. This provides a new perspective on how constraints in gauge theories can dictate low-energy physics. Furthermore, the model serves as a natural platform for studying topological defects within families of spontaneously symmetry-broken (SSB) phases and the nature of singularities in the phase diagram at the identified gapless point. The work also connects the study of constrained Hilbert spaces to phenomena such as Hilbert space fragmentation and quantum many-body scars.