🔬 condensed matter

Ergodicity breaking in matrix-product-state effective Hamiltonians

This paper demonstrates that the DMRG effective Hamiltonian, typically used for ground-state approximations, serves as a powerful spectral probe capable of characterizing ergodicity breaking, many-body localization, and quantum many-body scars in large interacting quantum systems beyond the reach of exact diagonalization.

Original authors: Andrew Hallam, Jared Jeyaretnam, Zlatko Papić

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Hallam, Jared Jeyaretnam, Zlatko Papić

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: Finding the "Ghost" in the Machine

Imagine you are trying to understand how a massive, chaotic crowd of people behaves. In physics, this "crowd" is a quantum system made of billions of tiny particles (like atoms). Usually, when these particles interact, they eventually settle down into a state of thermal equilibrium—like a cup of hot coffee cooling down to room temperature. They forget their initial state and become a uniform, messy soup. This is called thermalization.

However, sometimes this doesn't happen. Sometimes, the crowd gets "stuck" in a specific pattern, or a few special people in the crowd keep dancing in a loop while everyone else stops. Physicists call these exceptions Ergodicity Breaking.

Many-Body Localization (MBL): The crowd gets stuck in a frozen, disordered state because of "noise" or disorder (like a room full of random obstacles).
Quantum Many-Body Scars (QMBS): The crowd is chaotic, but a few special "ghosts" (scar states) keep performing a perfect dance routine, refusing to join the chaos.

The Problem: To study these phenomena, you usually need to look at the "middle" of the crowd's energy spectrum. But for large systems, the math is so complex that even the world's fastest supercomputers can't solve it. It's like trying to count every grain of sand on a beach while the tide is coming in.

The Solution: This paper introduces a clever trick using a tool called DMRG (Density Matrix Renormalization Group). Usually, DMRG is used to find the ground state (the calmest, lowest energy state) of a system. The authors realized that the "effective Hamiltonian" (a mathematical object DMRG builds to do its job) actually contains a hidden map of the entire system's behavior, not just the calmest part.

The Analogy: The "Local Neighborhood Map"

Think of the quantum system as a giant city.

Exact Diagonalization (The Old Way): To understand the traffic patterns of the whole city, you try to map every single street, every car, and every driver at once. This works for a small town, but for a metropolis, the map is too big to draw.
The DMRG Effective Hamiltonian (The New Way): Instead of mapping the whole city, you pick one specific house (a small subsystem) and ask: "If I stand here, what does the rest of the city look like to me?"

The authors built a "Local Neighborhood Map" (the Effective Hamiltonian). Even though this map is only based on one small spot, it turns out to be incredibly powerful. By looking at the "spectrum" (the list of possible energy levels) of this local map, they can deduce what is happening in the whole city.

The Two Main Discoveries

The paper tested this "Local Neighborhood Map" on two different types of quantum cities:

1. The Frozen City (Many-Body Localization)

Imagine a city where the roads are full of random potholes and barriers (disorder).

What happens: In a normal city, traffic flows freely (thermalizes). In this "Frozen City," the cars get stuck in local pockets. They can't move past the potholes.
The Test: The authors used their Local Map to look at the traffic flow. They found that the map perfectly predicted the transition from "free-flowing traffic" to "frozen traffic."
The Result: The map could even spot "Ergodic Bubbles." Imagine a small, sunny patch in the frozen city where the potholes are smooth. The cars there can move freely, acting like a tiny thermal island. The Local Map could see this island and measure how far its "warmth" spreads into the frozen city. This helps scientists understand if the whole city will eventually melt (avalanche instability) or stay frozen forever.

2. The Dancing Ghosts (Quantum Many-Body Scars)

Now imagine a city that is chaotic and noisy, but there are a few special dancers who refuse to stop dancing.

What happens: Most people in the city are chaotic, but these "Scar" states are like a perfect, repeating loop in a song that never gets lost in the noise.
The Test: The authors targeted one of these "dancers" and built their Local Map.
The Result: The map didn't just show that one dancer; it revealed the entire family of dancers. It showed a "tower" of special states, all connected like a ladder. Even though the map was built on just one state, it somehow "remembered" the structure of the other special states nearby. It's like looking at one brick and realizing the whole wall is built in a specific, repeating pattern.

Why This Matters

This is a game-changer for two reasons:

Size Matters: Previously, we could only study these weird quantum behaviors in very small systems (like a 10-person crowd). With this new method, we can study systems with 50, 100, or even more particles—sizes that were previously impossible to calculate. It's like being able to predict the weather for a whole continent instead of just a single backyard.
Versatility: The same tool that finds the "frozen" states also finds the "dancing ghosts." It's a universal key for unlocking the secrets of quantum chaos.

The Takeaway

The authors discovered that the "side effects" of a standard computer algorithm (DMRG) are actually a treasure trove of information. By treating the algorithm's internal "local view" as a diagnostic tool, they created a powerful new microscope. This microscope allows us to see how quantum systems break the rules of thermodynamics, helping us understand everything from why some materials never conduct heat to how quantum computers might stay stable in the future.

In short: They found a way to look at a small piece of a quantum puzzle and realize that piece contains the instructions for the whole picture.

1. Problem Statement

Interacting quantum many-body systems exhibit rich collective behaviors, including thermalization and its breakdown (ergodicity breaking). Key phenomena such as Many-Body Localization (MBL) and Quantum Many-Body Scars (QMBS) are governed by mid-spectrum eigenstates.

The Challenge: Characterizing these states is difficult because they typically obey the Eigenstate Thermalization Hypothesis (ETH) and possess high entanglement (volume-law scaling). This restricts numerical studies to small system sizes accessible via Exact Diagonalization (ED).
The Limitation of MPS: Matrix Product State (MPS) methods, specifically the Density Matrix Renormalization Group (DMRG), are highly efficient for low-entanglement states (ground states or MBL states) but are traditionally considered unsuitable for the highly entangled thermal phase.
The Gap: While DMRG is routinely used to find ground states, the DMRG effective Hamiltonian ( $H_{\text{eff}}$ )—a byproduct of the algorithm used to update local tensors—has not been fully explored as a probe for the spectral properties of the system far from equilibrium, particularly regarding ergodicity breaking in large systems.

2. Methodology

The authors propose using the DMRG effective Hamiltonian ( $H_{\text{eff}}$ ) as a versatile spectral probe for large systems ( $N \sim 50$ ), bypassing the limitations of ED.

Construction of $H_{\text{eff}}$ :
- An MPS $|\psi(A)\rangle$ is prepared, typically targeting an approximate eigenstate of the full Hamiltonian $H$ (using DMRG-S for excited states).
- The Hamiltonian $H$ is represented as a Matrix Product Operator (MPO).
- By contracting the MPO with the MPS tensors everywhere except for a small subsystem of $M$ sites (usually $M=1$ ), an effective Hamiltonian $H_{\text{eff}}$ is constructed.
- $H_{\text{eff}}$ acts on the local tensor space at the center of the chain, with dimension $\chi^2 d^M$ (where $\chi$ is the bond dimension and $d$ is the physical dimension).
Spectral Analysis:
- Instead of just finding the ground state of $H_{\text{eff}}$ , the authors exactly diagonalize the full spectrum of $H_{\text{eff}}$ .
- The eigenvectors of $H_{\text{eff}}$ are reshaped into MPS tensors to reconstruct states $|\psi(A, v_j)\rangle$ in the full Hilbert space.
- Selection Criteria: Since $|\psi(A, v_j)\rangle$ are not exact eigenstates of $H$ , the authors filter them based on energy variance ( $\sigma_H^2 = \langle H^2 \rangle - \langle H \rangle^2$ ) to select states that are good approximations to true eigenstates.
Diagnostics:
- Level Spacing Ratio ( $\langle r \rangle$ ): To distinguish between Poisson statistics (MBL) and Gaussian Orthogonal Ensemble (GOE) statistics (thermal).
- Spectral Form Factor (SFF): To probe long-range spectral correlations and extract the Thouless time.
- Entanglement Entropy ( $S_E$ ): To distinguish between area-law (MBL) and volume-law (thermal) scaling.
- Spatial Resolution: By varying the position of the $M$ -site window, they perform spatially resolved probes (e.g., detecting "ergodic bubbles").

3. Key Contributions

Reinterpretation of $H_{\text{eff}}$ : The paper establishes that the DMRG effective Hamiltonian, traditionally a tool for ground state optimization, encodes detailed spectral information about the mid-spectrum and far-from-equilibrium dynamics.
Scalability: The method allows for the study of ergodicity breaking in system sizes ( $N=50$ ) and bond dimensions ( $\chi$ up to 128) that are inaccessible to ED, while maintaining high accuracy in diagnosing phase transitions.
Unified Framework: It demonstrates a single methodology capable of capturing both strong ergodicity breaking (MBL) and weak ergodicity breaking (QMBS).

4. Key Results

A. Many-Body Localization (Random-Field XXZ Chain)

Transition Detection: In the random-field XXZ model, the spectrum of $H_{\text{eff}}$ successfully captures the transition from the thermal (GOE) to the MBL (Poisson) regime.
Level Statistics: The mean level spacing ratio $\langle r \rangle$ shows a clear crossover from $\approx 0.53$ (GOE) to $\approx 0.39$ (Poisson) as disorder $W$ increases.
Spectral Form Factor: The SFF exhibits the characteristic "dip-ramp-plateau" structure of chaotic systems at weak disorder, which is suppressed in the MBL regime. The Thouless time extracted matches previous ED estimates.
Critical Exponents: Finite-size scaling of the entanglement entropy variance yields a critical disorder $W_c \approx 4.4$ and critical exponents $\nu \approx 0.89$ , consistent with ED studies.
Ergodic Bubbles: The authors simulate a "rare region" (low disorder) embedded in a high-disorder bulk. $H_{\text{eff}}$ reveals that at intermediate disorder, the thermal grain hybridizes nearby degrees of freedom (enhancing entanglement), whereas at strong disorder, the influence is spatially confined, confirming stability against avalanches.

B. Quantum Many-Body Scars (PXP Model)

Scars in Chaotic Systems: Applied to the PXP model (Rydberg atom arrays), which is chaotic but hosts QMBS.
Tower Reconstruction: By targeting a specific energy $E_{\text{target}}$ within a scar tower using DMRG-S, the resulting $H_{\text{eff}}$ spectrum reproduces the entire tower of scar states.
Asymmetry: The method accurately captures lower-energy scar towers ( $n' < n$ ) relative to the target, while higher-energy towers are less captured, attributed to the approximate $su(2)$ algebraic structure of the scars.
Identification: Scar states are identified by their anomalously low entanglement entropy and high overlap with the $|Z_2\rangle$ state, distinct from the thermal bulk.

5. Significance and Implications

Beyond Exact Diagonalization: This work provides a robust numerical tool to explore quantum thermalization and its breakdown in the thermodynamic limit (or large finite limits) where ED fails.
Local vs. Global: The authors clarify that $H_{\text{eff}}$ does not reconstruct the global spectrum of $H$ (which is impossible for volume-law states at fixed $\chi$ ) but rather provides the optimal local representation of the Hamiltonian. Despite this, the local correlations are sufficient to diagnose global phase transitions and ergodicity breaking.
Future Directions: The method opens avenues for studying:
- Interplay between MBL and QMBS.
- Disorder-free localization and Hilbert space fragmentation.
- Integration with machine learning for phase detection.
- Application to quantum processors for variational excited state preparation.

In conclusion, the paper demonstrates that the DMRG effective Hamiltonian is a powerful, versatile probe for non-equilibrium quantum physics, capable of resolving the intricate spectral signatures of both MBL and QMBS in large systems.