Lindblad many-body scars

This paper introduces and characterizes "Lindblad many-body scars" as simultaneous eigenvectors of the Hamiltonian and dissipative parts of the Liouvillian in open quantum systems, demonstrating their existence in dissipative SYK models and XXZ chains with properties distinct from Hermitian scars, such as real eigenvalues, disorder-independent operator sizes, and unique entanglement features.

The Gist

Imagine a bustling, chaotic city where everyone is running in different directions, bumping into each other, and eventually, the whole system settles into a predictable, boring average state. In physics, this is called thermalization. If you drop a hot cup of coffee into a cold room, the heat spreads out until everything is lukewarm. In the quantum world, if you start a complex system with a specific pattern, that pattern usually gets scrambled and lost very quickly.

However, scientists have discovered "ghosts" in this chaos called Quantum Scars. These are special, stubborn states that refuse to get scrambled. They keep their shape and pattern, like a ripple in a pond that refuses to fade away, even though the water is supposed to be turbulent.

This paper explores what happens to these "ghosts" when the system isn't perfectly isolated but is instead leaking energy into its environment (like a cup of coffee sitting in a drafty room). The authors call these new ghosts Lindblad Many-Body Scars.

Here is a breakdown of their findings using simple analogies:

1. The Setup: The Leaky Bucket

Usually, physicists study these scars in "Hermitian" systems, which are like a perfectly sealed, frictionless box. But in the real world, systems are open; they interact with the environment.

• The Analogy: Imagine a drum being played. In a sealed room (Hermitian), the sound rings out clearly. In a windy, open field (the "Markovian bath" or environment), the sound gets distorted and fades.
• The Discovery: The authors asked: "Do the special, stubborn patterns (scars) survive in this windy, open field?" They found that yes, they do, but they change their nature. They don't just "ring out"; they become steady, decaying patterns that are mathematically predictable.

2. The Recipe for a Scar: The "Magic Ingredients"

To find these scars, you need two specific ingredients working together:

1. The Hamiltonian (The Music): The underlying rules of the system (like the notes on a drum).
2. The Jump Operators (The Wind): How the system talks to the environment.

The paper shows that if you choose the "wind" (the environment interaction) carefully, it can lock the system into a specific pattern.

• The Analogy: Think of a spinning top. Usually, friction makes it wobble and fall. But if you blow air on it in a very specific, rhythmic way (the "jump operators"), you can actually keep it spinning in a perfect, stable circle, defying the usual chaos. The authors found that for certain symmetries (like a specific type of rotational balance), these stable patterns are guaranteed to exist.

3. The "Size" of the Ghost

One of the most interesting things the authors measured is the "size" of these scars. In quantum physics, "size" refers to how many particles are involved in the pattern.

• The Analogy: Imagine a crowd of people.
  • Chaotic People (Non-scars): They are all jostling randomly. If you try to count how many people are in a specific group, the number fluctuates wildly. It's messy and unpredictable.
  • The Scar (The Ghost): These people are holding hands in a perfect, rigid line. The number of people in the line is exact and unchanging.
• The Finding: The authors found that for these Lindblad scars, the "size" is perfectly fixed. There is zero uncertainty. For the chaotic states, the size fluctuates. This is a perfect "fingerprint" to tell a scar apart from the chaos.

4. Entanglement: The "Social Network"

Quantum entanglement is like a super-strong social network where particles are linked no matter how far apart they are.

• The Finding: The scars are interesting because their "social network" depends entirely on how you look at them.
  • If you look at the system from one angle (partition), the scars might look like a tightly knit, low-entanglement group (easy to control).
  • If you look from another angle, they might look like a massive, highly entangled web.
• Why it matters: This is a double-edged sword for quantum computing. On one hand, low entanglement means these states are stable and good for storing information. On the other, the fact that they can be highly entangled means they are powerful resources for complex quantum tasks.

5. The "Rainbow" of Scars

The paper also looked at a specific model called the SYK model (a complex system of interacting particles).

• Majorana Fermions (The Simple Case): They found a few specific scars, like a small set of "VIPs" in the chaotic crowd.
• Complex Fermions (The Complex Case): They found a whole "rainbow" of scars. Because of a specific symmetry (U(1)), there are many more of these stable states, ranging from very simple to very complex.
• The Surprise: They also found a bunch of "mystery scars" in the middle of the spectrum that they couldn't fully explain yet. It's like finding a group of people in the crowd who are holding hands perfectly, but they aren't following any of the known rules. This suggests there might be hidden laws of physics we haven't discovered yet.

6. Why This Matters for the Future

• Quantum Computing: Quantum computers are notoriously fragile; they lose information quickly because of "noise" (the environment). These Lindblad scars are like shock absorbers. They show us how to design quantum systems that can resist noise and keep their information intact, even when they are leaking energy.
• Redefining Chaos: The paper suggests that the rules for how things "thermalize" (settle down) in open quantum systems are different from the rules we learned for closed systems. The "Gaussian" (bell curve) distribution we expect for chaos doesn't quite fit here; instead, we see "power-law" tails, meaning extreme events are more common than we thought.

Summary

In short, this paper discovers that even in a messy, leaking, chaotic quantum world, there are special, stable patterns that can be engineered. These "Lindblad Scars" are predictable, have fixed sizes, and can be tuned to be either simple or highly complex. They offer a promising new way to protect quantum information from the inevitable noise of the real world.

Technical Summary

1. Problem Statement

Quantum many-body scars (QMBS) are special eigenstates in quantum chaotic systems that violate the Eigenstate Thermalization Hypothesis (ETH), exhibiting non-thermal dynamics and potential utility for quantum information storage. Historically, QMBS research has been restricted to Hermitian (closed) systems. However, realistic quantum systems are often open, interacting with an environment (Markovian bath), leading to non-Hermitian dynamics described by Lindblad equations.

The paper addresses the following open questions:

• Do many-body scars exist in open quantum systems described by Lindblad dynamics?
• How are they defined and characterized in a non-Hermitian setting?
• What are their physical properties (operator size, entanglement entropy) compared to chaotic states?
• How does the presence of dissipation affect the ETH in these systems?

2. Methodology

The authors employ the vectorized Liouvillian formalism to study the dynamics of open quantum systems.

• Vectorization: The density matrix $\hat{\rho}$ is mapped to a state $|\rho\rangle$ in a doubled Hilbert space $\mathcal{H}^2 = \mathcal{H}_L \otimes \mathcal{H}_R$. The Lindblad equation $\partial_t \hat{\rho} = \mathcal{L}(\hat{\rho})$ becomes a Schrödinger-like equation $\partial_t |\rho\rangle = \mathcal{L} |\rho\rangle$, where $\mathcal{L} = -iH_0 + H_I$. Here, $H_0$ represents the Hamiltonian part ($H_L - H_R$) and $H_I$ represents the dissipative part.
• Definition of Lindblad Scars: The authors define a Lindblad many-body scar as a simultaneous eigenvector of the Hamiltonian part ($H_0$) and the dissipative part ($H_I$) of the vectorized Liouvillian. Crucially, the eigenvalue of such a state must be independent of the disorder realization (i.e., analytically determinable).
  • In operator language, an operator $\hat{O}$ corresponds to a scar if $[\hat{H}, \hat{O}] = 0$ and $\sum_\alpha (\hat{L}_\alpha \hat{O} \hat{L}_\alpha^\dagger - \frac{1}{2}\{\hat{L}_\alpha^\dagger \hat{L}_\alpha, \hat{O}\}) = \eta \hat{O}$.
• Models Studied:
  1. Dissipative Sachdev-Ye-Kitaev (SYK) Model: Both Majorana and complex fermion versions coupled to a Markovian bath via linear jump operators.
  2. Dissipative XXZ Spin Chain: A random spin-1/2 chain with dephasing noise.
• Characterization Tools:
  • Operator Size: A measure of the complexity of an operator (number of Majorana/Pauli strings).
  • Entanglement Entropy (EE): Calculated for two distinct bipartitions: Intersite (Left vs. Right copies) and Intrasite (spatial split within the copies).

3. Key Contributions and Results

A. Analytical Construction of Scars

The authors successfully constructed analytical Lindblad scars for both SYK and XXZ models:

• Majorana SYK: Identified 2 scars arising from parity symmetry ($\hat{P}$) and the Hamiltonian itself ($\hat{H}$), in addition to the identity (Thermofield Double state).
• Complex SYK: Identified $N/2 + 1$ scars due to $U(1)$ symmetry (particle number conservation). These are constructed using $p$-tuple operators $\hat{N}_p$. Additionally, they found 2 scars related to the Hamiltonian structure.
• XXZ Spin Chain: Identified $N/2 + 1$ scars corresponding to the $U(1)$ symmetry (total $Z$-spin), constructed using products of Pauli $Z$ operators.
• Numerical Discovery: In the complex SYK model, they numerically identified a large number of additional degenerate scars at the center of the spectrum ($\lambda = -N\mu/2$) that do not appear to be linked to the known symmetries, suggesting a richer structure.

B. Distinctive Physical Properties

1. Operator Size and Variance:

• Linear Relation: For both scars and non-scar states, the average operator size $\langle S \rangle$ is linearly related to the real part of the Liouvillian eigenvalue ($\text{Re}[\lambda]$).
• Variance as a Diagnostic: The key distinction is the variance of the operator size.
  • Scars: Since they are simultaneous eigenstates of the Liouvillian and the size operator, their size variance is zero.
  • Non-Scars (Chaotic): Exhibit a finite, positive variance.
• This provides a robust, disorder-independent method to identify scars in dissipative systems.

2. Entanglement Entropy (EE):

• Partition Dependence: The EE of Lindblad scars is highly sensitive to the choice of bipartition.
  • Intersite Partition (L vs. R): The Thermofield Double (TFD) scar has maximal entanglement (volume law), while other scars vary.
  • Intrasite Partition (Spatial split): The $U(1)$ scars in the complex SYK model exhibit low entanglement (area law or zero), indicating simple operator complexity.
• Tunability: Unlike Hermitian scars which often have fixed entanglement properties, Lindblad scars can exhibit a wide range of entanglement values depending on the partition and the specific symmetry involved.

3. Deviation from ETH:

• The authors analyzed the distribution of operator sizes for non-scar states.
• In Hermitian chaotic systems, ETH predicts a Gaussian distribution of observables.
• In these dissipative systems, the distribution of normalized operator sizes follows a power-law tail rather than a Gaussian. This suggests that the ETH in dissipative quantum chaos is qualitatively different from its Hermitian counterpart.

4. Significance

• Theoretical Framework: The paper establishes a rigorous definition and construction method for many-body scars in open quantum systems, bridging the gap between closed-system scars and realistic dissipative environments.
• Quantum Information: The identification of scars with low entanglement (in specific partitions) and purely real eigenvalues (implying decay rather than oscillation) offers new candidates for encoding quantum information in open systems, potentially overcoming thermalization issues.
• ETH in Open Systems: The discovery of power-law distributions for operator sizes challenges the standard Gaussian assumption of ETH in non-Hermitian settings, suggesting a need for a revised understanding of thermalization in open quantum chaos.
• Generality: The results are not limited to the SYK model (a holographic toy model) but extend to standard spin chains, indicating that Lindblad scars are a generic feature of dissipative quantum chaotic systems with specific symmetries.

5. Conclusion

The authors demonstrate that Lindblad many-body scars exist as simultaneous eigenstates of the Hamiltonian and dissipative parts of the Liouvillian. They are characterized by vanishing operator size variance and disorder-independent eigenvalues. While some scars (like the TFD) have high entanglement, others (like $U(1)$ scars) possess low entanglement in specific partitions, making them promising for quantum information tasks. Furthermore, the statistical properties of non-scar states in these systems deviate significantly from the standard ETH, exhibiting power-law distributions that require a new theoretical framework for dissipative quantum chaos.