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State-selective entanglement and point-gap topology in the unidirectional Bose-Hubbard chain

This paper investigates the unidirectional Bose-Hubbard model to reveal that while repulsive interactions suppress skin accumulation in the ground state, they induce a distinct, state-selective entanglement crossover and preserve point-gap topology in the steady state, characterized by specific spatial entropy maxima arising from competing subsystem configurations.

Original authors: Chuan-Fu Lin, Yu-Chin Tzeng, Po-Yao Chang

Published 2026-10-06

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CC BY 4.0

Original authors: Chuan-Fu Lin, Yu-Chin Tzeng, Po-Yao Chang

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

In the quantum world, particles do not always behave like the solid objects we see around us. Instead, they exist in a haze of possibilities, where a single particle can be in many places at once, and where the state of one particle is instantly linked to another, no matter how far apart they are. This deep connection is called entanglement, a phenomenon that serves as a fundamental measure of how complex a quantum system truly is. For decades, physicists have studied how these connections form in systems where particles move freely or interact in predictable ways. However, a newer and more puzzling area of research involves "non-Hermitian" systems. In these systems, the rules of movement are not perfectly balanced; particles might prefer to hop in one direction over another, creating a kind of one-way street for quantum matter. This imbalance can cause a strange effect where a vast number of particles pile up at the very edge of a material, a phenomenon known as the skin effect. While scientists knew that this edge accumulation changed the density of particles, they did not fully understand how it altered the intricate web of entanglement holding the system together, especially when the particles began to push against each other.

A team of researchers recently set out to untangle this mystery by studying a specific model of quantum particles called bosons moving along a one-way chain. They wanted to see what would happen if they turned up the repulsion between the particles, forcing them to avoid occupying the same space. In a standard scenario, one might expect that if the particles stop piling up at the edge, the complex entanglement would also disappear, returning the system to a simple, uniform state. The researchers tested this idea by simulating a chain of sixteen sites, a size large enough to reveal complex patterns but small enough to calculate with high precision. They compared two different ways of looking at the system: the ground state, which is the lowest energy configuration the system naturally settles into, and a "steady state," which represents the state that dominates the system after it has evolved for a long time under specific conditions.

The results revealed a surprising split in behavior. When the researchers looked at the ground state, their expectations held true. As they increased the repulsion between the particles, the pile-up at the edge vanished, and the particles spread out evenly across the chain. Correspondingly, the entanglement dropped significantly, fading away as the system approached a rigid, ordered state where particles were locked in place. This confirmed that for the lowest energy state, suppressing the edge accumulation also suppressed the quantum connections. However, the story was completely different for the steady state. Even when the repulsion was strong enough to flatten the density and remove the edge pile-up, the entanglement did not simply vanish. Instead, it underwent a strange, non-linear transformation. At intermediate levels of repulsion, the entanglement did not just rise or fall; it developed four distinct peaks at specific locations along the chain.

To understand why these peaks appeared, the researchers had to look deeper than just the number of particles in different sections of the chain. They found that simply counting the fluctuations in particle numbers could not explain the four peaks. Instead, the pattern emerged from a competition between different ways the particles could arrange themselves. At this intermediate level of repulsion, the system was dominated by a specific configuration where four pairs of particles sat together on single sites, while four sites were left completely empty. It was the interplay of these specific arrangements that created the four peaks in entanglement. When the researchers pushed the repulsion even higher, the pattern changed again. The four peaks disappeared, replaced by two strong peaks right at the very edges of the chain. This new pattern was explained by a different dominant configuration: a single pair of particles and a single empty site moving coherently across the system. The entanglement here was not caused by the particles being stuck in fixed positions, but by the uncertainty of where these specific pairs and holes were located as they moved.

Perhaps the most counterintuitive finding concerned the topology, or the global shape, of the system's energy spectrum. In these one-way systems, the energy levels can form a loop in a complex mathematical space. A property called a winding number measures how many times this loop wraps around a central point. Usually, if the skin effect is suppressed and the particles spread out, one might expect this loop to shrink and the winding number to vanish. Yet, the researchers found that even when the ground state showed no skin effect and the particles were evenly distributed, the winding number remained nonzero. They demonstrated this using a simplified two-site example, showing that the loop could become incredibly small without ever collapsing completely, as long as the interaction strength was finite. The loop only became undefined if the interaction became infinitely strong, a singular limit that is physically impossible to reach.

This work highlights that in interacting quantum systems, the behavior of the lowest energy state does not tell the whole story. The way a system settles over time can reveal a completely different landscape of entanglement and topology. The researchers showed that the suppression of particles at the edge does not automatically mean the suppression of quantum connections. By carefully distinguishing between the ground state and the long-term steady state, they uncovered a rich variety of behaviors where entanglement can persist, shift, and reorganize in ways that defy simple intuition. Their findings suggest that to truly understand these exotic quantum materials, scientists must look beyond the average density and examine the specific state of the system, as the path the system takes to get there can leave a lasting mark on its quantum structure.

Technical Summary: State-selective entanglement and point-gap topology in the unidirectional Bose–Hubbard chain

Problem Statement
This work investigates the interplay between repulsive interactions, non-Hermitian skin modes, and quantum entanglement in the unidirectional Bose–Hubbard model. While it is established that repulsive interactions can suppress skin accumulation in the ground state of this integrable model, the relationship between this suppression and the entanglement properties of many-body states remains unclear. Specifically, the paper addresses whether the suppression of skin density in the ground state implies a corresponding suppression of entanglement, or if entanglement behavior is "state-selective," varying significantly between the ground state and other dynamically relevant states (such as the steady state). Furthermore, the study examines how point-gap topology, characterized by the many-body winding number, behaves in the presence of strong interactions that suppress skin effects.

Methodology
The authors employ the exact Bethe ansatz solution for the unidirectional Bose–Hubbard chain to analyze the system's spectral properties and many-body wavefunctions.

Model: The system consists of $M$ bosons on an $N$ -site chain with unidirectional hopping ( $j+1 \to j$ ) and a weak closing link parameterized by $\epsilon$ . The Hamiltonian includes on-site repulsion $U$ .
Spectral Analysis: The authors utilize the algebraic Bethe ansatz to compute Bethe roots and many-body energies. They analyze the thermodynamic limit ( $N=M=250$ ) to identify the superfluid–Mott transition near $U/t \approx 5.83$ .
State Selection: Two distinct states are compared:
1. Ground State: The state minimizing the real part of the energy.
2. Steady State: Defined as the eigenstate with the largest imaginary energy in the zero-momentum sector ( $K=0$ ), which dominates the long-time limit of normalized nonunitary evolution.
Topological Characterization: The many-body point-gap winding number is calculated by inserting a flux through the closing link and tracking the evolution of the determinant of the Hamiltonian, rather than relying solely on the ground-state eigenvalue.
Entanglement Analysis: The study computes the second Rényi entropy ( $S_2$ ) for subsystems of varying lengths. To dissect the origin of entanglement features, the authors project the many-body wavefunction onto specific global onsite-pair manifolds (defined by the total number of boson pairs $D = \sum n_j(n_j-1)/2$ ) and compare the projected entropies with the full state entropy and subsystem particle-number variance.
System Sizes: Calculations are performed for $N=M=16$ (for detailed entanglement and density profiles) and $N=M=8$ (for complete spectral enumeration and winding number verification). A two-site exact solution is used to illustrate spectral loop behavior.

Key Contributions and Results

State-Selective Entanglement vs. Ground-State Suppression:
- Ground State: As interaction strength ( $U/t$ ) increases, the ground state undergoes a transition to a Mott-like state. The skin density accumulation is suppressed, the density becomes uniform, and the entanglement entropy decreases toward zero, consistent with a product-state Mott limit.
- Steady State: In contrast, the steady state (selected by the largest imaginary energy) exhibits a nonmonotonic entanglement crossover. Even at strong interactions where the ground state skin effect is suppressed, the steady state retains an asymmetric density profile and significant entanglement.
Spatial Maxima in Entanglement:
- At intermediate interaction ( $U/t = 10$ ), the second Rényi entropy of the steady state develops four distinct spatial maxima at subsystem sizes $\ell_A = 3, 6, 10, 13$ .
- The authors demonstrate that these maxima cannot be explained by the subsystem particle-number variance alone, which shows only a single broad peak.
- By resolving the reduced density matrix into subsystem-number sectors, the paper identifies that these four maxima arise from a competition between neighboring integer charges within a dominant configuration manifold. Specifically, the $D=4$ manifold (dominated by four doublons and four holons) reproduces the four-peak structure.
Strong Interaction Regime:
- At strong interaction ( $U/t = 50$ ), the steady state's entropy profile exhibits two maxima at the edges ( $\ell_A = 1, 15$ ).
- Projection onto the $D=1$ manifold (one doublon and one holon) closely reproduces the full entropy profile and edge peaks. This suggests the entanglement arises from the coherent spread of a single doublon-holon pair across the cut, rather than a fixed spatial separation.
Point-Gap Topology and Winding Number:
- The many-body winding number about the origin remains nonzero at strong interaction, even when the ground-state skin density is strongly suppressed.
- Using an exact two-site example, the authors show that the spectral loop can shrink arbitrarily small (vanishing as $U^{-1}$ ) while maintaining a nonzero winding number for any finite $U$ . The winding only becomes ill-defined in the singular limit $U/t \to \infty$ when the loop collapses onto the reference energy.
- Flux continuation of Bethe roots distinguishes between the open-arc regime (weak coupling) and the closed-loop regime (strong coupling), connecting root morphology to flux response without equating single-state behavior with the full determinant winding.

Significance and Claims
The paper claims to distinguish three distinct phenomena in interacting non-Hermitian systems: skin density accumulation, entanglement structure, and point-gap topology.

Decoupling of Phenomena: The results demonstrate that the suppression of skin density in the ground state does not imply the suppression of entanglement or the loss of point-gap topology. The entanglement properties are highly dependent on the specific many-body state selected (ground vs. steady).
Mechanism of Entanglement: The work provides a physical interpretation of entanglement in interacting bosonic systems by separating number fluctuations from configurational contributions. It shows that complex entanglement patterns (like the four maxima) are driven by the competition of specific configuration manifolds (e.g., doublon-holon sectors) rather than simple particle-number variance.
Robustness of Topology: The study highlights that point-gap topology, as measured by the many-body winding number, can persist even when the physical skin effect is suppressed by interactions, provided the interaction remains finite.

The authors conclude that a comprehensive understanding of non-Hermitian many-body systems requires separate examination of skin density, entanglement, and topology, as they do not necessarily evolve in tandem under interaction. The paper notes that while these findings are robust for the studied system sizes, future work is needed to determine if these crossovers persist under size scaling and in long-time evolution.

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