⚛️ quantum physics

Thermalization of Quantum Many-Body Scars in Kinetically Constrained Systems

This paper resolves the tension between quantum many-body scars and thermalization by embedding kinetically constrained systems into a Lindblad framework and reformulating the eigenstate thermalization hypothesis to show that both scar and thermal states ultimately obey grand canonical statistics.

Original authors: Jia-wei Wang, Xiang-Fa Zhou, Guang-Can Guo, Zheng-Wei Zhou

Published 2026-03-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jia-wei Wang, Xiang-Fa Zhou, Guang-Can Guo, Zheng-Wei Zhou

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 "Party" That Never Ends

Imagine a massive, chaotic party in a giant ballroom (this is the Quantum System). Usually, when you throw a party, everyone eventually mingles, drinks, and the room reaches a state of "thermal equilibrium." Everyone is just a blur of activity, and if you look at any small corner of the room, it looks exactly like the statistical average of the whole room. In physics, this is called Thermalization, and the rule that explains it is the Eigenstate Thermalization Hypothesis (ETH).

The Problem:
Sometimes, the party doesn't work that way. In certain systems (called Kinetically Constrained Systems), there are strict rules. For example, "You can only dance if your neighbor is standing still." These rules create a traffic jam in the dance floor.

In these jammed systems, most people eventually mingle (thermalize), but a special group of dancers (called Quantum Many-Body Scars) refuses to mix. They keep doing the exact same dance move over and over again, perfectly synchronized, forever. They break the rules of the party. They are the "non-thermal" outliers.

The New Discovery: The "Open Window" Experiment

The authors of this paper asked a tricky question: If these special dancers are breaking the rules of the closed room, what happens if we open a window?

They didn't just watch the closed room; they imagined the room leaking energy and information to the outside world (an Open System). They built a mathematical model (a Master Equation) that simulates this "leakage."

The Analogy of the Leaky Bucket:
Imagine the dancers are in a bucket with a hole in the bottom.

The Thermal Dancers: They are heavy, clumsy, and fall out of the bucket very quickly.
The Scar Dancers: They are surprisingly light and aerodynamic. They fall out of the bucket much slower than the others.

The authors discovered that by measuring how fast these dancers fall out of the bucket (their decay rate), they could tell them apart. The "Scar" dancers have a unique, slow decay signature.

The Solution: The "Grand Canonical" Rulebook

Once they realized these "Scar" dancers were just behaving differently because of the "leakage," the authors rewrote the rulebook for the party.

Old Rulebook (ETH):
"Everyone in the room is the same. If you know the total energy, you know everything about any specific person."

Result: This rulebook failed to predict the behavior of the Scar dancers. They looked like glitches.

New Rulebook (Grand Canonical Ensemble):
The authors realized the Scar dancers aren't glitches; they are just following a more complex rule. They introduced a new variable: "Quasi-Particle Number."

Think of the "Quasi-Particle" as a ticket or a token that a dancer holds.

In the old rulebook, you only cared about the Energy (how loud the music is).
In the new rulebook, you care about Energy AND the Number of Tokens (how many "blockades" or constraints are active).

By adding this second variable (the token count), the authors found that both the chaotic thermal dancers and the stubborn Scar dancers fit perfectly into a single, unified statistical model called the Grand Canonical Ensemble.

The "Aha!" Moment

The paper essentially says:

The Scar isn't a bug; it's a feature. The "non-thermal" behavior is just a specific type of thermal behavior that happens when you account for the system's constraints (the tokens).
The "Leak" explains the "Lock." By studying how the system leaks energy (dissipation), they found that the Scar states are simply the ones that are "sticky" and don't leak as fast.
Unification. They unified the two worlds. The chaotic thermal states and the orderly Scar states are now both described by the same Grand Canonical statistics. It's like realizing that a marathon runner and a sprinter are both "athletes," just governed by slightly different energy budgets.

Summary in One Sentence

The authors found that the stubborn, non-mixing "Quantum Scars" aren't breaking the laws of physics; they are just following a more detailed version of the thermalization rules that accounts for the system's internal "traffic jams," effectively unifying the chaotic and the orderly under one mathematical umbrella.

Why This Matters

This is a big deal because it solves a fundamental tension in physics. For years, scientists thought "Thermalization" (everything mixing) and "Scars" (everything staying separate) were opposites. This paper shows they are actually two sides of the same coin, governed by a unified thermodynamic law. It gives us a new way to understand and predict how complex quantum systems behave, which is crucial for building future quantum computers.

1. Problem Statement

The Eigenstate Thermalization Hypothesis (ETH) posits that individual eigenstates of non-integrable quantum many-body systems act as thermal ensembles, where local observables match microcanonical averages. However, Quantum Many-Body Scars (QMBS) represent a class of non-thermal eigenstates that violate ETH, exhibiting persistent oscillations and weak ergodicity breaking.

While QMBS in exactly solvable models (e.g., AKLT) are understood via algebraic structures (Krylov subspaces), the mechanism behind QMBS in kinetically constrained systems (e.g., the PXP model) remains elusive. These systems lack rigorous algebraic structures, making it difficult to explain:

Why scar states fail to thermalize while surrounding thermal states do not.
What statistical ensemble governs the thermalization of these constrained systems.
How to unify the description of scar states and thermal states under a single thermodynamic framework.

2. Methodology

The authors propose a novel framework that maps constrained unitary dynamics to an open quantum system described by a Lindblad-like master equation.

A. Embedding Constraints in Dissipative Dynamics

Construction: They define a general constrained system via a Hamiltonian $H$ acting on a subspace $\mathcal{H}$ , enforced by blockade projectors $\pi_k$ .
Non-Hermitian Mapping: They construct a non-Hermitian Hamiltonian $H_N$ and engineered dissipation channels ( $L_{k,\sigma}$ ) such that the constrained subspace $\mathcal{H}$ becomes a Jump-Free Subspace (JFS).
Master Equation: The system evolves under $\partial_t \rho = \mathcal{L}(\rho) = -i[H_N, \rho] + \mathcal{J}(\rho)$ $\partial_{t} ρ = L (ρ) = - i [H_{N}, ρ] + J (ρ)$ .
- Within $\mathcal{H}$ , the "jumping" terms $\mathcal{J}(\rho)$ cancel out, recovering the unitary dynamics of $H$ .
- Outside $\mathcal{H}$ , the system experiences rapid decay (Zeno effect) back into the constrained subspace or leakage out of it.
Approximation: By tuning a parameter $c \to \infty$ , they derive a master equation with only positive dissipation rates ( $\mathcal{L}_+$ ), which approximates the constrained unitary dynamics while allowing for the study of decay rates.

B. Identifying the "Quasi-Particle" Operator

By analyzing the decay of eigenstates $|E_i\rangle$ under the dissipative process, they observe that the decay rate $\alpha_i$ is proportional to the expectation value of a specific operator $\hat{N}$ .
They define $\hat{N} = \sum_k L_{k,2}^\dagger L_{k,2}$ as a quasi-particle counter. Unlike conserved charges in traditional integrable systems, $\hat{N}$ is not conserved ( $[H, \hat{N}] \neq 0$ ) but serves as a dynamical variable characterizing the state's proximity to the blockade constraints.

C. Reformulating the ETH

The authors propose that the diagonal matrix elements of local observables $\hat{O}$ in eigenstates $|E_i\rangle$ are not solely functions of energy $E$ , but of both energy and the quasi-particle number $N = \langle \hat{N} \rangle_i$ :
$\langle E_i | \hat{O} | E_i \rangle \approx O(E, N)$
This leads to a Grand Canonical Ensemble description:
$\rho(\beta, \mu) = \frac{\sum_i e^{-\beta(E_i - \mu \langle \hat{N} \rangle_i)} |E_i\rangle\langle E_i|}{\mathcal{Z}}$
where $\beta$ is the inverse temperature and $\mu$ is the chemical potential associated with the quasi-particle number.

3. Key Results

A. Slow Decay of Scar States

Numerical simulations on 1D spin chains (spin-1 and spin-3/2) demonstrate that scar eigenstates exhibit significantly slower decay rates ( $\alpha_i$ ) compared to thermal eigenstates within the same energy window.
The decay rate is linearly proportional to the expectation value of the quasi-particle number: $\alpha_i \approx -\gamma_2 \langle \hat{N} \rangle_i$ .
Scar states have lower $\langle \hat{N} \rangle_i$ , explaining their stability (slow decay) in the dissipative picture.

B. Unification via Grand Canonical Statistics

Canonical Failure: Standard ETH (Canonical Ensemble) fails to predict the local properties of scar states, showing large deviations.
Grand Canonical Success: The proposed Grand Canonical Ensemble (incorporating $\mu$ and $N$ ) accurately predicts the expectation values of local observables for both scar and thermal eigenstates.
Schatten Distance: The reduced density matrices of eigenstates are shown to be extremely close to the Grand Canonical ensemble predictions, whereas the Canonical ensemble shows large deviations for scar states.

C. Generalizability

The framework was tested on:
1. Spin-1 chains (equivalent to the PXP model).
2. Spin-3/2 chains.
3. High-dimensional PXP models with different blockade constraints.
In all cases, the Grand Canonical description successfully unified the thermalization behavior of scar and thermal states.

4. Key Contributions

Dissipative Perspective on QMBS: The paper provides a new physical mechanism for QMBS: scar states are eigenstates that decay slowly in a corresponding engineered dissipative process due to their low "quasi-particle" content.
Revised ETH: It extends the Eigenstate Thermalization Hypothesis to include a constraint-dependent variable (quasi-particle number), transforming the thermalization paradigm from a Canonical to a Grand Canonical framework.
Unified Thermodynamic Rule: It resolves the tension between non-ergodicity (scars) and thermalization by showing that both are governed by the same statistical rule, provided the correct ensemble (Grand Canonical) is used.
Experimental Feasibility: The proposed Lindblad-like master equations with positive dissipation rates ( $\mathcal{L}'_+$ ) can be approximated experimentally using open quantum systems, offering a pathway to observe these phenomena.

5. Significance

This work fundamentally shifts the understanding of thermalization in kinetically constrained systems. Instead of viewing scars as "exceptions" to thermalization, the authors demonstrate that they are thermalized states governed by a broader statistical ensemble.

Theoretical Impact: It bridges the gap between open quantum systems and closed-system thermalization, suggesting that "constraints" can be effectively modeled as "chemical potentials" for emergent quasi-particles.
Broader Applicability: The methodology offers a unified route to analyze generalized thermalization in complex quantum systems, potentially applicable to other non-ergodic phases like Many-Body Localization (MBL), though the authors note this remains an open question for future research.

In summary, the paper establishes that kinetically constrained quantum systems thermalize according to Grand Canonical statistics, where the "scar" phenomenon arises naturally from the specific values of the quasi-particle number in the eigenstates.