Quantum quenches in a spin-1 chain with tunable symmetry

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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

Imagine a long line of tiny, spinning tops (atoms) sitting next to each other on a table. In the world of quantum physics, these tops don't just spin; they can also "squash" or change shape in specific ways. This paper is about a team of scientists who decided to play a game with these tops to see how they behave when things suddenly change.

Here is the story of their experiment, broken down into simple concepts:

1. The Setup: A Line of Spinning Tops

Think of the scientists' system as a row of spin-1 atoms.

The "Spin": Imagine each atom is a spinning top. It can point Up, Down, or be flat (Zero).
The "Squash" (Quadrupole): Because these are special quantum tops, they can also be "squashed" like a pancake. This is called a quadrupolar state. It's like the top isn't spinning in a circle anymore but is wobbling in a specific shape.

The scientists built a model where they could control how much these tops "talk" to their neighbors. They had a special dial called $J_q$ .

Dial turned off ( $J_q = 0$ ): The tops only care about their spin direction (Up/Down). This is the "normal" world, which is chaotic and messy.
Dial turned all the way up ( $J_q = 1$ ): The tops start caring about their "squash" shape too. Suddenly, the rules of the game change completely. The system becomes integrable, which is a fancy way of saying it becomes perfectly organized and predictable, like a well-rehearsed dance troupe.

2. The Experiment: The "Quantum Quench"

A "quantum quench" is like hitting the reset button on a video game, but instead of starting over, you suddenly change the physics of the world.

The scientists prepared the line of tops in a specific pattern (like all pointing Up on the left, all Down on the right, or a mix of shapes). Then, they suddenly turned on the "squash" dial ( $J_q$ ) and watched what happened.

They asked: Do the tops eventually forget their starting pattern and settle into a random, hot mess (thermalization)? Or do they remember who they were?

3. The Big Discovery: The "Magic Rule"

When they turned the dial to the maximum setting ( $J_q = 1$ ), they found a new rule that nature suddenly obeyed.

The Old Rule: You can only count how many tops are pointing Up vs. Down (Total Magnetization).
The New Rule: You also have to count how many tops are "squashed" (Quadratic Magnetization).

The Analogy: Imagine a dance floor.

Normal World: Dancers can move anywhere as long as the total number of people stays the same. Eventually, they mix up completely, and you can't tell who started where.
The Magic World ( $J_q=1$ ): A new bouncer appears. He says, "You can move, but you must keep your specific dance partner and your specific dance style." Because of this strict rule, the dancers are trapped in a smaller room. They can't mix with everyone else. They get stuck in a loop, remembering their starting position forever.

This explains why some of their experiments showed the system "freezing" or "reviving" (going back to the start) instead of getting messy.

4. The Two Types of Starters

The scientists tried two different ways to start the line of tops:

A. The "Magnet" Starters (Z-Magnetization):
These were tops pointing Up, Down, or Flat.

What happened: When the "Magic Rule" kicked in, some patterns got stuck. For example, a pattern where every top was "Flat" (Nematic state) didn't move at all! It was like a statue. Another pattern with a single "Up" top in a sea of "Flats" would wiggle a bit but then bounce back to its original shape over and over again.
Why? The "Magic Rule" limited the number of places these tops could go. They ran out of room to explore, so they couldn't forget their past.

B. The "Phantom Helix" Starters:
These were tops arranged in a spiral, like a corkscrew.

What happened: In the normal world, these spirals are very stable and don't move much (they are "phantoms" because they carry momentum but no energy).
The Twist: When the scientists turned on the "squash" dial, these stable spirals melted faster than anything else! Even though the system became "integrable" (organized) at the max dial setting, the spiral patterns broke down quickly.
The Lesson: Sometimes, adding more rules (integrability) doesn't mean things stay calm. It depends entirely on how you started the game.

5. Why Does This Matter?

You might ask, "Who cares about spinning tops?"

This research is a roadmap for building future quantum computers and sensors.

Ultracold Atoms: Scientists can already create these spinning top lines in labs using lasers and super-cold atoms.
Controlling Chaos: By understanding these "Magic Rules," engineers can design materials that don't lose their information to heat. They can create quantum devices that stay stable and remember their data for a long time.
New Materials: It helps us understand strange materials (like certain metals) where atoms behave like these spinning tops.

Summary

The paper is about discovering that by tweaking a single knob, you can change a chaotic, messy quantum system into a highly organized one. However, this organization doesn't always mean "calm." Sometimes it means the system gets trapped in a loop, remembering its past forever, and other times it means it melts away instantly. It's a guide to understanding how to control the chaotic dance of the quantum world.

1. Problem Statement and Motivation

The paper investigates the non-equilibrium dynamics of interacting quantum systems, specifically focusing on how integrability and symmetry influence thermalization and relaxation processes.

Context: Recent advances in quantum simulators (e.g., ultracold atoms in optical lattices) allow for the realization of $SU(N)$-symmetric models with tunable $N$ . While equilibrium phases of $SU(N)$ models are well-studied, their far-from-equilibrium dynamics remain less understood.
Specific Gap: The authors aim to bridge the gap between non-integrable $SU(2)$ Heisenberg models and integrable $SU(3)$ Heisenberg models. They seek to understand how tuning the interaction strength affects the relaxation of various initial states, particularly looking for signatures of integrability (such as conserved quantities preventing thermalization) and the emergence of "phantom" states that defy standard thermalization.
Model: The study focuses on an anisotropic spin-1 Heisenberg chain where the symmetry can be tuned from $SU(2)$ to $SU(3)$ by adjusting a parameter $J_q$ controlling quadrupolar interactions.

2. Methodology

The authors employ a combination of analytical theory and numerical simulations:

Hamiltonian: The system is described by a Bilinear-Biquadratic (BB) model:
$H = J \sum_i \left[ S_i^z S_{i+1}^z + (S_i^x S_{i+1}^x + S_i^y S_{i+1}^y) + J_q \sum_i \mathbf{Q}_i \cdot \mathbf{Q}_{i+1} \right]$
where $\mathbf{Q}$ represents quadrupolar operators.
- Limits:
  - $J_q = 0$ : Reduces to the standard $SU(2)$ Heisenberg model (non-integrable for spin-1).
  - $J_q = 1$ : Corresponds to the $SU(3)$ symmetric Heisenberg model, which is integrable.
- Conservation Laws: Total magnetization $M = \sum S_i^z$ is conserved for all $J_q$ . Crucially, at $J_q = 1$ , a new conserved quantity emerges: quadratic magnetization $M^2 = \sum (S_i^z)^2$ .
Initial States: The study examines a wide range of experimentally accessible product states:
- Z-Magnetization Eigenstates: Domain walls (DW I, DW II), Antiferromagnetic states (AFM I, AFM II), Nematic (NM), and Nematic with Polarized Impurity (NPI).
- Phantom Helix States: Generalized spin-helix states that are approximate eigenstates of the $SU(2)$ limit but carry macroscopic momentum without energy.
Numerical Techniques:
- Time-Evolving Block Decimation (TEBD): Used to simulate time evolution for chains up to $L \approx 30-40$ sites. The wavefunction is represented as a Matrix Product State (MPS) with a bond dimension $\chi = 600$ .
- Exact Diagonalization (ED): Used for small chains ( $L \le 12$ ) to benchmark TEBD results and analyze finite-size effects.
- Observables: Local magnetization ( $\langle S_i^z \rangle$ ), entanglement entropy ( $S$ ), fidelity ( $F$ ), and various spin-spin and quadrupole-quadrupole correlation functions.

3. Key Contributions and Theoretical Framework

Discovery of a New Conserved Quantity: The authors rigorously prove that at the $SU(3)$ symmetric point ( $J_q = 1$ ), the quadratic magnetization $M^2$ becomes a conserved quantity.
Combinatorial Analysis of Hilbert Space: They develop a theoretical framework based on counting the number of accessible states ( $\Omega$ $Ω$ ) in the Hilbert subspace defined by fixed $M$ $M$ and $M^2$ $M^{2}$ .
- For $J_q < 1$ , only $M$ is conserved, leading to a large accessible subspace and rapid thermalization.
- For $J_q = 1$ , the constraint of fixed $M^2$ drastically reduces the number of accessible states. This reduction explains why certain states (like the Nematic state) become eigenstates (frozen dynamics) or exhibit fidelity revivals.
Correlation Freezing: They provide a combinatorial explanation for why specific correlation functions "freeze" at $J_q=1$ . If the initial state's symmetry sector only allows specific local alignments (e.g., only Ferroquadrupolar or only Antiferroquadrupolar), the corresponding correlation functions cannot evolve.

4. Key Results

A. Z-Magnetization Initial States

Dynamics Dependence on $J_q$ :
- General Trend: Increasing $J_q$ generally accelerates the decay of local magnetization due to enhanced quantum fluctuations from quadrupolar terms.
- Entanglement Entropy ( $S$ ): For most states, $S$ grows linearly before saturating. However, at $J_q=1$ , the saturation value is significantly lower for states with restricted $M^2$ (e.g., NPI state), confirming the reduction of accessible states.
- Fidelity: The Nematic (NM) state remains an eigenstate at $J_q=1$ (fidelity remains constant). The NPI state exhibits periodic fidelity revivals at $J_q=1$ , a hallmark of integrable dynamics, whereas it decays rapidly for $J_q < 1$ .
Correlation Freezing:
- At $J_q=1$ , correlation functions for states like NM and NPI freeze at their initial values because the conservation of $M^2$ restricts the system to a subspace where those local alignments are the only possibilities.
- Conversely, states like AFM II, which have a large number of accessible states even at $J_q=1$ , continue to thermalize rapidly.

B. Phantom Helix States

$J_q = 0$ Limit: These states are approximate eigenstates (phantom states) and show almost no dynamics (frozen observables).
$J_q > 0$ Behavior: Introducing quadrupolar interactions breaks the "phantom" condition.
- Thermalization: Unlike the expectation that integrability ( $J_q=1$ ) would preserve the state, the authors find that increasing $J_q$ leads to faster thermalization.
- Entanglement: The entanglement entropy grows rapidly, reaching the maximum numerical limit ( $S \approx 6.4$ ) for $J_q=1$ , indicating that the system fully explores its accessible Hilbert space despite the model being integrable. This suggests that the specific "phantom" structure is not robust against the $SU(3)$ symmetry.

5. Significance and Implications

Control of Non-Equilibrium Dynamics: The work demonstrates that the symmetry of a spin-1 chain can be used as a "knob" to control thermalization. By tuning $J_q$ , one can switch between regimes of rapid thermalization and regimes governed by integrability and conservation laws.
Experimental Relevance: The findings are directly applicable to current experimental platforms, such as ultracold alkaline-earth atoms in optical lattices (realizing $SU(N)$ models) and materials with $d^2$ or $d^8$ electronic configurations (e.g., $Y_2BaNiO_5$ ).
Theoretical Insight: The paper highlights that integrability does not guarantee the survival of specific non-equilibrium structures (like phantom helices) if the initial state does not align with the specific conserved quantities of the integrable point. It emphasizes the importance of the initial state's overlap with the conserved sectors (specifically $M^2$ ) in determining long-time dynamics.
New Conserved Quantity: The identification and utilization of $M^2$ conservation in the $SU(3)$ limit provides a powerful tool for predicting which initial states will thermalize and which will exhibit non-ergodic behavior (freezing or revivals).

In summary, the paper provides a comprehensive map of the non-equilibrium phase space of a spin-1 chain, revealing how the interplay between symmetry, integrability, and initial state preparation dictates the fate of quantum dynamics.