⚛️ quantum physics

Many-Body Mobility Edge and Non-Hermitian Skin Effect in an Interacting Quasi-Periodic Spin Chain

This paper investigates an interacting non-Hermitian quasi-periodic spin chain to reveal a "D-shaped" many-body mobility edge that unifies the interplay of interactions, non-Hermiticity, and localization, thereby delineating distinct regimes of many-body localization and the many-body skin effect characterized by multifractal scaling and anomalous boundary drift.

Original authors: Lavoisier Wah, Ayan Banerjee, Flore K. Kunst

Published 2026-08-10

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CC BY 4.0

Original authors: Lavoisier Wah, Ayan Banerjee, Flore K. Kunst

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 world where the rules of physics are slightly tweaked, allowing particles to behave in ways that seem impossible in our everyday reality. This is the realm of quantum many-body physics, a field that studies how huge crowds of tiny particles, like electrons or atoms, interact with one another. Usually, scientists look at two main behaviors: "thermalization," where particles mix up and forget their past (like a drop of ink spreading in water), and "localization," where particles get stuck in place and remember exactly where they started (like a drop of ink refusing to spread). A key concept here is the "mobility edge," which acts like a border line in the energy of the system; on one side of the line, particles flow freely, and on the other, they are frozen.

Now, imagine adding a twist: what if the universe wasn't perfectly balanced? In "non-Hermitian" physics, systems can lose or gain energy, or particles can move more easily in one direction than the other, creating a kind of cosmic one-way street. When you mix this one-way traffic with the "stuck" behavior of localization and the "mixing" of interactions, things get incredibly complicated. Scientists care about this because understanding these strange states could help us build better quantum computers or new types of materials that control light and electricity in novel ways. But until now, figuring out exactly how these three ingredients—non-Hermitian effects, interactions, and disorder—dance together has been a massive puzzle.

In this new study, researchers Lavoisier Wah, Ayan Banerjee, and Flore K. Kunst decided to build a digital playground to solve this puzzle. They created a model of a chain of quantum spins (think of them as tiny magnets) that interact with each other, move in a one-way direction, and face a bumpy, repeating landscape of disorder. By running detailed computer simulations, they discovered something surprising and beautifully structured: a "D-shaped" boundary.

Imagine a map of a country where the weather changes. Usually, you might expect a straight line separating a sunny region from a rainy one. But in this quantum world, the researchers found a shape that looks like the letter "D." Inside the curve of the "D," the particles are free to roam and mix (extended states). Outside the curve, they are stuck in place (localized states). What makes this "D" special is that it separates these two worlds not just by how strong the disorder is, but also by the energy of the particles. This means that in the exact same system, with the same settings, some particles can be flowing freely while others right next to them are frozen solid.

The study also uncovered a phenomenon called the "many-body skin effect." Picture a crowd of people in a hallway. In a normal hallway, people spread out evenly. But in this quantum hallway, the rules are rigged so that everyone gets pushed toward one specific wall, piling up there. The researchers found that when the particles are in the "flowing" part of their "D-shaped" world, they exhibit this skin effect, crowding against the edge of the chain. However, as they cross the "D-shaped" border into the "frozen" zone, this crowding disappears, and the particles get stuck in their original spots, refusing to move or forget their past.

Using various digital tools to measure how "spread out" the particles were, the team mapped out this entire landscape. They confirmed that the "D-shaped" edge is a real, robust feature of this quantum system, appearing consistently whether they looked at the energy levels or watched how the particles moved over time. While the paper suggests this is a fundamental feature of such open quantum systems, it notes that proving these exact behaviors in real-world experiments (like with cold atoms or light) is a next step for future research. For now, this work provides a clear, unified picture of how these three chaotic forces interact, revealing a hidden order in the chaos of the quantum world.

Technical Summary: Many-Body Mobility Edge and Non-Hermitian Skin Effect in an Interacting Quasi-Periodic Spin Chain

Problem Statement
The paper addresses the unexplored collective behavior of non-Hermitian (NH) many-body systems where topology, localization, and boundary effects intertwine. While single-particle mobility edges (energy-dependent transitions between localized and extended states) are known in specific Hermitian and quasi-periodic settings, their existence and nature in interacting NH systems remain debated. Specifically, the interplay between the Many-Body Skin Effect (MBSE)—where eigenstates accumulate at boundaries under open boundary conditions (OBCs)—and Many-Body Localization (MBL) in the presence of quasi-periodic disorder is largely uncharacterized. The authors aim to determine if a Many-Body Mobility Edge (MBME) exists in such systems and how it is shaped by the competition between NH asymmetry, interactions, and disorder.

Methodology
The authors study a one-dimensional interacting non-Hermitian spin chain of length $L$ subject to a quasi-periodic longitudinal field. The system is governed by the Hamiltonian:
$H = \sum_{i=1}^L \left[ t((1+\gamma)\sigma^-_i \sigma^+_{i+1} + (1-\gamma)\sigma^+_i \sigma^-_{i+1}) + J\sigma^z_i \sigma^z_{i+1} + g\sigma^x_i + h_i \sigma^z_i \right]$
where $t$ is the hopping amplitude, $\gamma$ controls the non-Hermitian asymmetry, $J$ is the nearest-neighbor Ising coupling, $g$ is the transverse field, and $h_i = h \cos(2\pi\alpha i + \phi)$ is the quasi-periodic (Aubry-André) disorder potential.

The study employs exact diagonalization (using Jax) for system sizes up to $L=14$ . The authors analyze the system under both Periodic Boundary Conditions (PBCs) and OBCs. To characterize the phases, they utilize a suite of diagnostics:

Multifractal Dimension ( $D_q$ ): Specifically the second moment $D_2$ , calculated from the right eigenstates to quantify localization in the many-body Hilbert space.
Complex Level-Spacing Statistics: The mean gap ratio $\bar{r}$ and angular statistics $-\cos\theta$ of complex eigenvalues to distinguish between Ginibre (delocalized) and Poisson (localized) statistics.
Many-Body Inverse Participation Ratio (MIPR): To measure real-space particle distribution.
Complex Eigenenergy Fraction ( $f$ ): The ratio of eigenvalues with non-zero imaginary parts to the total number of eigenvalues.
Dynamical Observables: Time evolution of particle density, charge imbalance $c(\tau)$ , and entanglement entropy $S_E(\tau)$ following a Néel state quench.

Key Contributions and Results

Discovery of a "D-Shaped" Many-Body Mobility Edge (MBME):
The central finding is the identification of a robust, "D-shaped" region in the phase diagram (spanning disorder strength $h$ and energy density $\epsilon$ ) where eigenstates are extended, surrounded by a background of localized states.
- Structure: Within this D-shaped island, eigenstates near the spectral center are extended ( $D_2 \to 1$ ), while those at the spectral edges are localized ( $D_2 \to 0$ ).
- Coexistence: This structure signifies that localized and extended eigenstates coexist within the same Hamiltonian at the same disorder strength, separated by a critical energy density.
- Robustness: This D-shaped feature persists under both PBCs and OBCs, confirming it as a bulk property rather than a boundary artifact.
Characterization of the Many-Body Skin Effect (MBSE):
The authors demonstrate that the MBSE induces multifractal scaling in the eigenstates.
- Under OBCs, the non-Hermitian asymmetry causes an accumulation of wavefunction amplitudes at one boundary.
- This manifests as a significant suppression of the multifractal dimension $\langle D_2 \rangle$ under OBCs compared to PBCs in the delocalized regime.
- The gap between OBC and PBC $\langle D_2 \rangle$ serves as a Hilbert-space fingerprint of the real-space boundary accumulation.
Phase Transitions and Spectral Properties:
- Extended-to-MBL Transition: As the disorder strength $h$ increases, the system transitions from an extended phase (dominated by MBSE under OBCs) to an MBL phase. This transition is mediated by the broad D-shaped MBME region.
- Complex-to-Real Transition: The system exhibits a spectral transition from complex eigenvalues (Ginibre ensemble statistics) to real eigenvalues (Poisson statistics) as disorder increases. The critical point for this spectral transition ( $h_c \approx 7.1$ ) is slightly higher than the point where Hilbert-space localization ( $D_2$ ) sets in ( $h_c \approx 6.5$ ), suggesting these are distinct but related boundaries.
- Role of Interactions: Increasing the interaction strength $J$ promotes localization, shifting the MBME toward lower disorder strengths.
Dynamical Signatures:
- MBSE: In the weak disorder regime, time evolution shows a drift of particle density toward the boundary, confirming the skin effect.
- MBL: In the strong disorder regime, the system retains memory of the initial Néel state, with charge imbalance saturating to a finite value and entanglement entropy growing slowly (logarithmically), characteristic of MBL.
- MBME: In the intermediate regime, dynamical observables show a mixed behavior where delocalization occurs in part of the chain while localization persists in others, consistent with the energy-resolved nature of the MBME.

Significance and Claims
The paper claims to provide a unified framework for understanding the interplay between non-Hermiticity, interactions, and disorder.

Defining Signature: The authors identify the "D-shaped" MBME as a clear, defining signature of the transition in open quantum many-body systems, distinguishing it from simple delocalization or full localization.
Multifractality as a Diagnostic: They establish that multifractal scaling, particularly the difference between OBC and PBC $\langle D_2 \rangle$ , is a robust indicator of the MBSE and the critical nature of the MBME.
Universality: The results suggest that the D-shaped mobility edge, previously established in Hermitian systems, extends to interacting non-Hermitian chains, shaping the physics of open quantum systems.
Future Outlook: The authors note that while they have identified these phases, the construction of local integrals of motion (LIOMs) for the NH MBL phase and the persistence of the D-shaped edge under random disorder remain open theoretical questions. They suggest that the identified dynamical observables could be probed in ultracold atomic gases, photonic lattices, and digital quantum processors.

Technical Summary: Many-Body Mobility Edge and Non-Hermitian Skin Effect in an Interacting Quasi-Periodic Spin Chain

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