⚛️ quantum physics

Non-zero Momentum Implies Long-Range Entanglement When Translation Symmetry is Broken in 1D

This paper demonstrates that in one-dimensional systems with broken translation symmetry, the magnitude of the expectation value of the translation operator serves as a reliable proxy for long-range entanglement in delocalized states, effectively acting as a momentum-space analog to Resta's formula for localization length.

Original authors: Amanda Gatto Lamas, Taylor L. Hughes

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amanda Gatto Lamas, Taylor L. Hughes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Mood Ring" of Quantum Matter

Imagine you have a crowd of people (electrons) in a room. In physics, we often want to know: Are these people acting as a chaotic, independent mob, or are they all secretly holding hands in a giant, invisible chain?

Short-Range Entangled (SRE): The people are just standing around. If you look at one person, they aren't connected to anyone far away. They are "localized."
Long-Range Entangled (LRE): The people are all holding hands in a massive, invisible chain that spans the whole room. If you pull one person, someone on the other side feels it. They are "delocalized."

For a long time, physicists had a special trick to tell these two groups apart, but it only worked if the room was perfectly symmetrical (like a dance floor where everyone moves in perfect unison). This paper asks: What if the room is messy? What if the floor is uneven, or the people are moving randomly? Can we still tell if they are holding hands?

The authors say: Yes! And they found a new way to look at it using "momentum" (how fast and in what direction things are moving) instead of just looking at where they are standing.

The Old Trick: The "Perfect Dance Floor"

In a perfect, symmetrical system, physicists used to check the total momentum of the group.

If the group was "holding hands" (LRE), their total momentum would be weird and changeable.
If they were just standing around (SRE), their momentum would be boring and fixed.

But this only worked if the "dance floor" (the lattice) was perfect. If the floor was broken or uneven (broken translation symmetry), this old trick stopped working.

The New Discovery: The "Translation Meter"

The authors developed a new tool called the Translation Operator Expectation Value (let's call it $z_t$ ).

Think of $z_t$ as a "Shake Meter."

Imagine you gently shake the entire room.
If the people are holding hands (Delocalized/LRE): The whole chain moves together. The "Shake Meter" reads 1.0 (Maximum). The system is rigid and connected.
If the people are standing alone (Localized/SRE): When you shake the room, everyone just jiggles in place or gets stuck. The "Shake Meter" reads 0. The system is loose and disconnected.

The Key Insight: Even if the floor is messy and the people aren't moving in a perfect pattern, if you shake the room and the "Shake Meter" reads high, you know they are holding hands (entangled). If it reads zero, they are just standing alone.

The Analogy of the "Blurry Photo"

To understand why this works, imagine taking a photo of the crowd.

The "Position" Photo (Where they are):
- If the crowd is localized (stuck in one spot), the photo is sharp and clear. You can see exactly where everyone is.
- If the crowd is delocalized (spread out), the photo is a blur. You can't tell where anyone is specifically.
The "Momentum" Photo (How they are moving):
- Physics has a rule (Heisenberg's Uncertainty Principle) that says: If you know exactly where they are, you know nothing about how they are moving, and vice versa.
- Localized Crowd: The position photo is sharp, so the "movement photo" is a blurry mess. It looks like static noise. The "Shake Meter" ( $z_t$ ) is 0.
- Delocalized Crowd: The position photo is a blur, so the "movement photo" is sharp and focused. Everyone is moving in a coordinated way. The "Shake Meter" ( $z_t$ ) is 1.

The Paper's Breakthrough: The authors proved that for messy, broken-symmetry systems, you can look at the "movement photo" (momentum distribution). If it's sharp (high $z_t$ ), the system is entangled. If it's a blur (low $z_t$ ), it's not.

The "Flux" Test: The Magic Twist

The paper also discusses a cool experiment called "Flux Insertion." Imagine twisting the room like a towel.

For a Delocalized (Entangled) System: When you twist the room, the "movement photo" shifts. It's like turning a dial; the whole pattern moves. This proves the system is sensitive to the twist.
For a Localized (Non-Entangled) System: When you twist the room, the "movement photo" is so blurry (flat) that you can't tell if it moved or not. It looks exactly the same. The system is "blind" to the twist.

This confirms that even in messy systems, entanglement makes the system sensitive to changes in the environment, while non-entangled systems are oblivious.

Why Does This Matter?

It works on messy systems: Real-world materials (like disordered alloys or complex crystals) are rarely perfect. This new method works even when the "floor" is broken.
It's a new diagnostic tool: Physicists can now look at the "momentum" of a system to instantly know if it has "long-range entanglement" (a key feature of exotic quantum states) without needing to do impossible calculations.
It bridges two worlds: It connects the idea of "localization" (where particles are stuck) with "entanglement" (quantum connections), showing they are two sides of the same coin in 1D systems.

Summary in One Sentence

Just as a perfectly synchronized dance troupe moves as one unit, a quantum system with "long-range entanglement" will show a sharp, coordinated pattern in its momentum, even if the system itself is messy and broken; this paper gives us the mathematical ruler to measure that coordination.

1. Problem Statement

The paper addresses a fundamental question in condensed matter physics: Can the momentum properties of a quantum state directly encode its entanglement nature (specifically, whether it is Short-Range Entangled (SRE) or Long-Range Entangled (LRE)) when the system lacks translation symmetry?

Context: A previous result by Gioia and Wang (2022) established that for translation-symmetric states, a non-trivial momentum eigenvalue implies LRE. However, this relies on the state being an eigenstate of the translation operator, which is not true for states that break translation symmetry (e.g., due to disorder, quasiperiodicity, or inhomogeneous potentials).
Challenge: For generic states breaking translation symmetry, it is often impractical to find a finite-depth quantum circuit connecting them to a translation-symmetric state to determine their entanglement class. There is a need for a direct measure based on the state's own properties to distinguish between localized (SRE) and delocalized (LRE) phases in 1D periodic systems.

2. Methodology

The authors develop a theoretical framework based on characteristic functions and uncertainty relations for unitary operators, drawing an analogy to Resta's formula for the localization length in position space.

Key Operators:
- Modular Position Operator ( $U$ ): $U = e^{i \frac{2\pi}{L} \hat{X}}$ , where $\hat{X}$ is the many-body position operator. Its expectation value magnitude $z_x = |\langle U \rangle|$ relates to position-space localization (Resta's formula).
- Translation Operator ( $T$ ): $T = e^{ia \hat{K}}$ , where $\hat{K}$ is the many-body momentum operator and $a$ is the lattice spacing. The authors focus on the magnitude $z_t = |\langle T \rangle|$ .
Theoretical Derivation:
- The authors utilize the uncertainty relation between $U$ and $T$ . When the filling is incommensurate ( $a N_e / L \notin \mathbb{Z}$ ), these operators do not commute, imposing a trade-off: a state localized in position space must be delocalized in momentum space, and vice versa.
- They derive an expression for the second cumulant (spread) of the many-body momentum distribution, $C(K)_2$ , in terms of $z_t$ . In the continuum limit ( $a \to 0$ ), the relationship is:
  $C(K)_2 \approx -\frac{2}{a^2} \ln(z_t)$
- Hypothesis:
  - Localized (SRE) states: Have a flat/uniform momentum distribution ( $C(K)_2 \to \infty$ ), implying $z_t \to 0$ .
  - Delocalized (LRE) states: Have a sharply peaked momentum distribution ( $C(K)_2 \to 0$ ), implying $z_t \to 1$ .
Numerical Validation: The authors test this hypothesis on several 1D lattice models:
1. Deterministic Dimer Model (DDM): A tunable model with well-defined continuum and thermodynamic limits.
2. Simple Self-Dual (SSD) Model: A model where position and momentum space translations are symmetric.
3. Random Dimer Model (RDM): A disordered model lacking a clear continuum limit.
4. Aubry-Andre (AA) Model: A quasiperiodic model lacking a clear continuum limit.

3. Key Contributions

Momentum-Space Analogue of Resta's Formula: The paper establishes that $z_t = |\langle T \rangle|$ serves as a direct probe for delocalization (and thus LRE) in 1D systems, even when translation symmetry is broken. This is the momentum-space counterpart to Resta's formula for position-space localization.
Role of Limits: The authors rigorously distinguish between the thermodynamic limit ( $L \to \infty$ $L \to \infty$ ) and the continuum limit ( $a \to 0$ $a \to 0$ ).
- They show that $z_t \to 1$ for delocalized states is an exact result only in the continuum limit.
- Conversely, $z_x \to 1$ for localized states is exact in the thermodynamic limit.
- This clarifies why $z_t$ is a superior probe for delocalization in the continuum limit, while $z_x$ is better for localization in the thermodynamic limit.
Flux Sensitivity in Broken Symmetry: The paper investigates the phase of the translation operator, $\arg \langle T \rangle$ . They demonstrate that while translation-symmetric LRE states shift their momentum under flux insertion, translation-broken LRE states also exhibit flux sensitivity in their momentum distribution. However, for SRE (localized) states, the momentum distribution becomes flat ( $z_t \to 0$ ), rendering the phase ill-defined and the distribution insensitive to flux shifts.
Single-Particle vs. Many-Body Probes: For models like the DDM, the authors show that the many-body $z_t$ can be inferred from the product of single-particle $z_t$ values, provided the single-particle states are localized in momentum space.

4. Results

Deterministic Dimer Model (DDM):
- In the delocalized phase (small potential strength), $z_t \to 1$ and $z_x \to 0$ .
- In the localized phase, $z_t \to 0$ and $z_x \to 1$ (or partial localization).
- The continuum limit sharpens the transition, making $z_t$ approach 1 more distinctly for delocalized states.
Simple Self-Dual Model (SSD):
- At the self-dual point, the thermodynamic and continuum limits coincide.
- $z_t$ and $z_x$ exhibit symmetric behavior around the transition point, confirming the duality of the measures.
Models without Continuum Limits (RDM and AA):
- In the Random Dimer Model, $z_t$ decays rapidly before the actual localization transition, making it a less sharp indicator than $z_x$ . However, the derivative of the momentum localization length ( $\lambda_k$ ) derived from $z_t$ still accurately pinpoints the transition.
- In the Aubry-Andre model, the quasiperiodicity creates a multi-peaked momentum distribution even in the delocalized phase, causing $z_t < 1$ before the transition. Again, $z_t$ is not a high-contrast indicator, but the momentum localization length $\lambda_k$ remains sensitive to the transition.
Flux Insertion:
- Numerical simulations of the RDM confirm that in the delocalized phase, the momentum distribution shifts visibly with flux insertion.
- In the localized phase, the distribution is flat; flux insertion shifts a flat distribution to another flat distribution, making the shift undetectable. This confirms that flux sensitivity (via the momentum distribution) can distinguish SRE from LRE even without translation symmetry.

5. Significance

New Diagnostic Tool: The paper provides a practical, computable metric ( $z_t$ ) to identify Long-Range Entanglement in 1D systems where traditional methods (like finding a circuit to a symmetric state) are infeasible.
Refinement of Localization Theory: It clarifies the distinct roles of the thermodynamic and continuum limits in localization measures, correcting potential misconceptions about when $z_t$ or $z_x$ are exact.
Universality: While the paper focuses on 1D, the authors suggest that the expectation value of the translation operator may generalize to higher dimensions and topologically ordered systems, offering a pathway to detect LRE in more complex scenarios.
Pedagogical Value: The introduction of the Deterministic Dimer and Simple Self-Dual models provides clean theoretical testbeds for studying localization transitions and the interplay between position and momentum space entanglement.

In summary, the authors successfully extend the connection between momentum and entanglement to symmetry-broken systems, proving that the magnitude of the translation operator expectation value is a robust proxy for delocalization (and thus LRE) in 1D, particularly when analyzed within the appropriate continuum limit.