Towers of Quantum Many-body Scars from Integrable Boundary States

This paper constructs models exhibiting towers of quantum many-body scars with restricted spectrum generating algebra structures and sub-volume law entanglement by utilizing tilted Néel integrable boundary states, a method extendable to two-dimensional systems.

The Gist

Imagine you have a giant, chaotic dance floor filled with thousands of dancers (quantum particles). Usually, when you start a dance, everyone eventually gets tired, mixes up, and settles into a uniform, boring "thermal" state where no one remembers the original choreography. This is called thermalization, and it's the rule for almost everything in the universe.

However, sometimes, a few dancers refuse to mix. They keep dancing in a perfect, repeating loop, ignoring the chaos around them. In physics, these special dancers are called Quantum Many-Body Scars (QMBS). They are like "ghosts" of the original order that refuse to fade away.

This paper is about a new way to build a whole tower of these special dancers, rather than just finding one or two by accident. Here is the breakdown using simple analogies:

1. The Problem: Finding a Needle in a Haystack

For a long time, scientists knew these "scar" states existed, but finding them was like looking for a needle in a haystack. They were rare, isolated, and didn't do much interesting. The authors wanted to build a whole family of them that could dance together in a synchronized, repeating pattern.

2. The Secret Ingredient: The "Integrable Boundary State" (IBS)

The authors used a special mathematical tool called an Integrable Boundary State (IBS).

• The Analogy: Imagine a perfectly symmetrical, frozen statue made of ice. In the world of quantum physics, this "statue" is a special starting point that has a hidden connection to a perfectly ordered, solvable world (called an "integrable model").
• The Twist: The authors realized that if you take this "frozen statue" (specifically a "tilted Néel state," which is just a fancy way of saying spins pointing up and down but slightly tilted) and place it inside a chaotic, messy system, it doesn't melt. Instead, it acts as a parent that can birth many children.

3. Building the "Tower"

In previous work, this method only produced one or two scar states. In this paper, the authors showed that because the "tilt" of the statue can be adjusted, they can generate a whole tower of scar states.

• The Analogy: Think of a piano. Usually, a chaotic room (the quantum system) makes noise. But the authors found a way to tune the room so that specific keys (energy levels) ring out perfectly.
• The "Tower": They didn't just find one key; they found a whole scale of keys that are perfectly spaced apart (like steps on a ladder). This structure is called a Restricted Spectrum Generating Algebra (RSGA). It means the energy levels are equally spaced, allowing the system to bounce back and forth in a perfect rhythm.

4. The Dance: Periodic Revivals

The most exciting part is what happens when you start the system.

• The Analogy: If you drop a ball in a chaotic room, it bounces randomly and stops. But if you drop a ball in this special "scar" system, it bounces up, comes down, and returns to your hand exactly as it was, over and over again, forever.
• The Result: The authors showed that if you start with a specific pattern (like a checkerboard of spins), the system will "remember" that pattern and return to it perfectly every few seconds. It never forgets its past. This is called periodic revival.

5. Low Entanglement: The "Quiet" States

Usually, when quantum particles interact, they get "entangled," meaning they become a messy, inseparable web of information. This is why things thermalize.

• The Analogy: Imagine a crowded party where everyone is talking to everyone else. That's high entanglement. The scar states are like a group of people standing in a corner, talking only to each other, ignoring the rest of the party. They stay "clean" and simple.
• The Finding: The authors proved mathematically that these scar states have very low "entanglement entropy." They are simple, ordered, and refuse to get messy, even though they are surrounded by chaos.

6. Going 2D: From a Line to a Grid

So far, this was all done in a 1D line (like beads on a string). The authors then asked: "Can we do this on a 2D grid (like a chessboard)?"

• The Breakthrough: Yes! They showed that you can treat a 2D grid as a collection of many 1D lines running through it. By using the same "tilted statue" trick on these lines, they created scar states that work on a 2D surface. This is a huge deal because most real-world materials are 2D or 3D, not 1D.

Summary

This paper is like an architect who discovered a new blueprint.

1. Old Way: We knew a few "magic rooms" existed where chaos didn't happen, but we couldn't build more.
2. New Way: The authors found a "master key" (the tilted Néel state) that unlocks a whole tower of magic rooms.
3. The Result: These rooms allow quantum systems to remember their past perfectly, bouncing in a loop forever without getting messy.
4. The Future: They showed this works not just in a line, but on a whole grid, opening the door to building quantum computers or simulators that can stay stable and ordered for a long time, defying the usual rules of chaos.

In short: They found a way to build a quantum echo chamber where the sound never fades, using a clever mathematical trick involving "tilted" starting points.

Technical Summary

1. Problem Statement

Quantum thermalization in isolated many-body systems is typically governed by the Eigenstate Thermalization Hypothesis (ETH), which posits that individual energy eigenstates behave thermally. However, Quantum Many-Body Scars (QMBS) are a class of non-thermal eigenstates that violate ETH, often exhibiting low entanglement entropy and leading to persistent coherent dynamics (periodic revivals) when the system is quenched from specific initial states.

While QMBS have been observed in various systems (e.g., Rydberg atom chains), a systematic method for constructing models with multiple QMBS forming a "tower" (a sequence of equally spaced energy eigenstates) remains a challenge. Previous work by the authors using Integrable Boundary States (IBS) could only generate isolated QMBS (one or two states) that lacked non-trivial dynamics. The central problem addressed in this paper is: How can one construct non-integrable models hosting a tower of exact QMBS using IBS, and what are their dynamical and algebraic properties?

2. Methodology

The authors employ a constructive approach based on Integrable Boundary States (IBS). The methodology proceeds as follows:

• Starting Point: They utilize tilted Néel states ($|\psi_{1,2}(\alpha)\rangle$), which are known IBS for spin-1/2 Heisenberg and XYZ chains. These states are annihilated by all parity-odd conserved charges ($Q_{2k+1}$) of the underlying integrable Hamiltonian.
• Hamiltonian Construction: They construct non-integrable Hamiltonians of the form:
  $$H = H_{NI} + \sum t_k Q_{2k+1}$$
  where $H_{NI}$ is a non-integrable perturbation term designed such that the tilted Néel states are its zero-energy eigenstates. The conserved charges $Q_{2k+1}$ are added to ensure the tilted Néel states remain exact eigenstates of the full Hamiltonian.
• Generating the Tower: By projecting the parameterized tilted Néel states onto subspaces with definite magnetization (using a lowering operator $O^-_\pi$), they generate a sequence of states $|\Psi_n\rangle$.
• Symmetry Breaking: They extend the construction from models with U(1) symmetry (spin rotation about the y-axis) to completely anisotropic models (XYZ model) where this symmetry is explicitly broken.
• Higher Dimensions: The 1D construction is generalized to 2D square lattices and arbitrary bipartite graphs by decomposing the 2D Hamiltonian into an array of 1D paths.

3. Key Contributions

A. Construction of QMBS Towers

The paper demonstrates that tilted Néel states serve as "parent states" for a tower of exact QMBS. Unlike previous IBS-based constructions that yielded isolated scars, this method generates a sequence of states $|\Psi_n\rangle$ ($n=0, 1, \dots, L$) that are exact eigenstates of non-integrable Hamiltonians.

B. Restricted Spectrum Generating Algebra (RSGA)

The authors prove that the constructed tower of QMBS possesses an RSGA structure.

• The Hamiltonian $H$, the reference state $|\Psi_0\rangle$, and the lowering operator $O^-$ satisfy specific commutation relations (e.g., $[H, O^-]|\Psi_0\rangle = E O^-|\Psi_0\rangle$ and higher-order commutators vanishing).
• Significance: This algebraic structure guarantees that the energy levels of the scar states are equally spaced (equidistant), which is a prerequisite for perfect periodic revivals.

C. Extension to Anisotropic and Higher-Dimensional Models

• XYZ Model: The method is generalized to the fully anisotropic XYZ model. Even when U(1) symmetry is broken ($J_x \neq J_z$), the tilted Néel states remain IBS, and the tower of scars persists. In this case, the RSGA order increases from 2 to 4.
• 2D Lattices: The authors construct a 2D model on a square lattice where the Hamiltonian is a sum of 1D interactions along specific paths. The 2D tilted Néel states are shown to be exact scars with low entanglement. This approach is further generalized to any bipartite graph (e.g., honeycomb lattice).

4. Key Results

• Non-Integrability: Numerical analysis of level-spacing statistics (Wigner-Dyson distribution) confirms that the constructed Hamiltonians are non-integrable and not many-body localized, ensuring the scars exist within a chaotic spectrum.
• Low Entanglement (Sub-Volume Law): The authors calculate the half-chain entanglement entropy (EE) for the scar states.
  • The EE scales as $S_A \sim \frac{1}{2} \ln L$ (in 1D) and $\sim \frac{1}{2} \ln N$ (in 2D), which is significantly lower than the volume-law scaling of thermal states.
  • This confirms the states are exact QMBS and violate ETH.
• Periodic Revival Dynamics:
  • When the system is quenched from a Néel state ($|Z_2\rangle$), which is a superposition of the scar tower, the fidelity $F(t)$ exhibits perfect periodic revivals with period $T = \pi/h_y$.
  • Analytical derivation shows $F(t) = \cos^L(h_y t)$.
  • In contrast, a period-3 state ($|Z_3\rangle$), which is not a superposition of the scar tower, thermalizes rapidly with fidelity decaying to zero.
• Analytical Correlation Functions: The spin-spin correlation function $C_r^x(t)$ for the Néel state shows perfect periodic oscillations, analytically derived as $(-1)^r \sin^2(2h_y t)$.

5. Significance and Implications

• Systematic Construction: This work provides a robust, systematic framework for generating models with multiple QMBS, moving beyond isolated examples. It bridges the gap between integrable models (via IBS) and non-integrable scar models.
• Algebraic Insight: The identification of the RSGA structure in these non-integrable models offers a deeper algebraic understanding of why QMBS towers exist and why they are equidistant in energy.
• Dimensional Scalability: The successful extension to 2D and arbitrary bipartite graphs suggests that QMBS are not limited to 1D chains but can be engineered in higher-dimensional quantum simulators (e.g., Rydberg atom arrays or superconducting qubits).
• Connection to Integrability: The results illuminate a profound connection between the non-thermal behavior of scars and the integrability of boundary states, suggesting that "partial solvability" via IBS is a powerful tool for designing quantum materials with tailored dynamical properties.

In conclusion, Sanada et al. have established a powerful method to engineer "scarred" quantum matter, demonstrating that integrable boundary states can serve as the foundation for rich, non-thermal dynamics in complex, non-integrable many-body systems.