⚛️ quantum physics

Ground state and persistent oscillations in the quantum East model

This paper demonstrates that in the 1D quantum East model with open boundaries, the ground state and a specific low-entanglement excited state are well-approximated by spin-coherent product states, leading to size-independent energy gaps and persistent coherent oscillations driven by boundary physics rather than many-body scars.

Original authors: Adway Kumar Das, Achilleas Lazarides

Published 2026-03-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adway Kumar Das, Achilleas Lazarides

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

Imagine a long line of people standing in a row, each holding a spinning top that can point either Up or Down. This is our "quantum East model."

In the real world, if you wanted to flip one person's top, you'd just grab it and turn it. But in this specific quantum world, there's a strict rule: You can only flip a person's top if the person standing immediately to their left is already pointing "Up."

This rule creates a "traffic jam" of sorts. If everyone is pointing Down, no one can move. If the person on the far left is Up, they can flip their neighbor, who can then flip the next one, and so on. This is a "kinetically constrained" system—it's not that the people can't move, it's that the rules make it very hard to get things started.

The Two Main Discoveries

The researchers studied what happens when the "rules" of this game are set to be extremely strict (a parameter called $s$ going to negative infinity). They found two surprising things:

1. The "Perfectly Ordered" Ground State

Usually, in complex quantum systems, the lowest energy state (the "ground state") is a messy, chaotic soup where every particle is entangled with every other particle. It's like a crowded dance floor where everyone is holding hands with everyone else in a giant, tangled knot.

However, in this specific model, the researchers found that the ground state is surprisingly simple. It's like a military formation. Every single person in the line is standing at the exact same angle, perfectly synchronized.

The Analogy: Imagine a choir where every singer is holding the exact same note, perfectly in tune, with no one getting distracted.
Why it matters: This state has almost no "entanglement" (no messy knots). It's a "spin-coherent state," meaning the whole system acts like one giant, simple wave rather than a chaotic mess.

2. The "Edge Mode" and the Eternal Swing

Next, they looked at what happens if you take that perfectly ordered line and flip only the very last person on the right end.

The Analogy: Imagine that military formation again. Everyone is still in perfect sync, except the very last soldier on the right is doing a 180-degree turn.
The Surprise: In most quantum systems, if you poke the edge of a system, the disturbance ripples through the whole line and gets lost in the chaos. But here, that "flipped last soldier" creates a special state that is almost identical to the ground state, just with one twist.

Because this "flipped edge" state is so special, it doesn't just sit there. It starts oscillating (swinging back and forth) with the ground state.

The Analogy: Think of a pendulum. If you push a normal quantum system, it quickly loses energy and stops (thermalizes). But this specific setup is like a perfectly frictionless pendulum. Once you push it, it swings back and forth forever, never stopping, never getting tired.
The Result: Even if you wait for a very long time, the system remembers exactly how it started. It never settles down into a random, "thermal" state.

Why is this a Big Deal?

Usually, when you have a closed quantum system (one that doesn't talk to the outside world), it eventually forgets its past. It "thermalizes." If you start with a specific pattern, that pattern eventually dissolves into a random mess, just like a drop of ink spreading in water.

This paper shows a way to break that rule without needing disorder or randomness (which is usually how scientists explain systems that don't thermalize).

The "Scar" vs. "Edge" Distinction: Recently, physicists discovered "Quantum Scars," which are special states that allow systems to remember their past. However, those usually happen in the "middle" of the energy spectrum.
The New Mechanism: The oscillations in this paper come from the edge of the system. It's a "boundary effect." The physics of the very last spin creates a special door that lets the system keep swinging. It's distinct from the "Scar" phenomenon and offers a new way to keep quantum information alive.

The Takeaway

The researchers discovered that by tuning the rules of a simple line of quantum spins, they can create a system where:

The ground state is surprisingly simple and ordered (like a synchronized choir).
A tiny change at the very end of the line creates a special "edge mode."
This edge mode causes the whole system to swing back and forth forever, refusing to forget its initial state.

This is like finding a door in a locked room that leads to a hallway where time seems to stand still. It suggests that we might be able to design quantum computers or sensors that are much more stable and resistant to "forgetting" their data, simply by engineering the edges of the system.

1. Problem Statement

The paper investigates the thermalization properties and ground state structure of the 1D Quantum East Model with open boundary conditions. This model is a kinetically constrained spin chain without quenched disorder, known for exhibiting a quantum phase transition (QPT) at a parameter $s=0$ .

Context: While ergodic systems typically thermalize (local observables relax to Gibbs ensembles), and Many-Body Localized (MBL) systems fail to thermalize due to disorder, the authors explore whether non-ergodic behavior or "weak ergodicity breaking" can occur in clean, translationally invariant systems.
Specific Focus: The study focuses on the limit $s \to -\infty$ and the region $-\infty < s < 0$ . The goal is to characterize the ground state ( $|GS\rangle$ ) and identify special excited eigenstates that might evade thermalization, potentially leading to persistent coherent oscillations.

2. Methodology

The authors employ a combination of analytical approximations, numerical diagonalization, and dynamical analysis:

Hamiltonian Analysis: They analyze the irreducible Hamiltonian of the East model:
$\hat{H}_{East} = \frac{\hat{I} + \hat{N}}{2} - e^{-s} \left( \hat{\sigma}_1^x + \sum_{j=1}^{L-1} \hat{n}_j \hat{\sigma}_{j+1}^x + \hat{n}_L \right)$
where $\hat{n}_j = (1-\hat{\sigma}_j^z)/2$ .
Ansatz Construction: In the $s \to -\infty$ limit, they propose spin-coherent product states as approximations for the ground state and specific excited states. These states are defined as tensor products of single-spin states $|\theta_j\rangle = \cos(\theta_j/2)|1\rangle + \sin(\theta_j/2)|0\rangle$ .
Fractal Dimension Analysis: To characterize the delocalization properties in Fock space, they calculate the fractal dimensions $D_q$ $D_{q}$ of the wavefunctions in both the $Z$ $Z$ -basis (computational basis) and $X$ $X$ -basis.
- $D_q = 1$ : Ergodic/Extended.
- $D_q = 0$ : Localized.
- $0 < D_q < 1$ : Multifractal.
Entanglement Entropy: They compute the bipartite von Neumann entanglement entropy ( $S_{vN}$ ) to distinguish between area-law (non-ergodic) and volume-law (ergodic) scaling.
Dynamical Probes:
- Survival Probability: $R(t) = |\langle \Psi(0) | \Psi(t) \rangle|^2$ to test for non-decay.
- Green's Functions: Retarded Green's functions and spectral functions to identify edge modes.
- Level Statistics: Analysis of level spacing ratios and number variance to determine ergodicity in the bulk spectrum.

3. Key Contributions and Results

A. The Ground State at $s \to -\infty$

Spin-Coherent Approximation: The authors demonstrate that the ground state $|GS\rangle$ is accurately approximated by a uniform spin-coherent state $|\Theta\rangle = |\pi/3\rangle^{\otimes L}$ .
Multifractality vs. Real Space: While the state is multifractal in the Fock space (specifically in the $Z$ -basis, with $D_z^\infty \approx 0.47$ ), it exhibits low entanglement entropy ( $S_{vN} \approx 0.047$ ) and is spatially homogeneous in real space. This coexistence of multifractality and low entanglement is a unique feature.
Chiral Symmetry: The Hamiltonian at $s \to -\infty$ possesses a chiral symmetry that pairs eigenstates. Applying the parity operator to the ground state ansatz yields an approximation for the "anti-ground state."

B. The Edge Mode ( $|ES_1\rangle$ )

Identification of a Special Excited State: The authors identify a low-entanglement excited eigenstate $|ES_1\rangle$ that lies above the ground state.
Structure: $|ES_1\rangle$ $∣ E S_{1} ⟩$ differs from the ground state only by a $\pi$ -rotation of the rightmost boundary spin.
- Ansatz: $|ES_1\rangle \approx |\pi/3\rangle^{\otimes (L-1)} \otimes |4\pi/3\rangle \equiv |\bullet_L\rangle$ .
Edge Mode Characterization:
- It has low entanglement entropy ( $S_{vN} \approx 0.109$ ), independent of system size.
- It behaves as an edge mode: The Green's function analysis shows a sharp peak in the spectral function at a constant energy $\omega_0 \approx 0.8767$ with a finite lifetime, independent of system size $L$ .
- The bulk spectrum remains ergodic (volume law entanglement, GOE level statistics), but this specific state is pinned to the boundary.

C. Persistent Oscillations for $-\infty < s < 0$

Mechanism: As $s$ increases from $-\infty$ towards $0$, the edge-coherent state $|\bullet_L\rangle$ (initial state) develops significant overlap with two distinct eigenstates ( $|GS\rangle$ and $|ES_1\rangle$ ).
Energy Gap: The energy gap between these two states remains size-independent (constant) in the thermodynamic limit.
Dynamics: Because the initial state is a superposition of two eigenstates with a constant energy gap, the system exhibits persistent coherent oscillations in both global observables (survival probability) and local observables (magnetization).
Distinction from Scars: The authors explicitly distinguish this mechanism from:
- Quantum Many-Body Scars (QMBs): Unlike the PXP model, these oscillations do not arise from a hypercube-like Fock space structure or weak ergodicity breaking of the bulk.
- MBL: The bulk remains ergodic; the non-thermal behavior is strictly a boundary phenomenon.

D. The Quantum Phase Transition at $s=0$

At $s=0$ (Rokhsar-Kivelson point), the ground state becomes $|+\rangle^{\otimes L}$ (fully extended in $X$ , localized in $Z$ ).
For $s > 0$ , the ground state becomes exponentially localized in real space near the boundary.
The transition at $s=0$ is marked by a divergence in the localization length and a change in the fractal dimensions of the ground state.

4. Significance

Boundary-Induced Non-Ergodicity: The paper establishes a new mechanism for persistent oscillations in closed quantum systems driven purely by boundary physics in a translationally invariant model. This challenges the assumption that non-ergodicity requires disorder (MBL) or specific bulk kinetic constraints (Scars).
Edge Modes in Kinetically Constrained Models: It identifies a robust edge mode in the East model that survives in the thermodynamic limit, distinct from topological edge modes, arising from the specific kinetic constraints of the model.
Multifractality in Ground States: The finding that the ground state is multifractal in Fock space yet has low entanglement and is spatially uniform provides a nuanced example of non-ergodic behavior that does not fit standard MBL or scar paradigms.
Theoretical Framework: The work suggests that kinetically constrained models may host a broader class of "scar-like" phenomena rooted in boundary modes, opening new avenues for understanding thermalization breakdown in clean systems.

In summary, the authors show that the 1D Quantum East model hosts a unique boundary-localized excited state that, when superposed with the ground state, prevents thermalization and leads to persistent oscillations, offering a distinct pathway to non-ergodicity in clean quantum systems.