🔬 mesoscale physics

Scale-free localization in pseudo-Hermitian quantum walk

This paper demonstrates that a single non-Hermitian defect in one- and two-dimensional quantum walks induces scale-free localization, a phenomenon intrinsically linked to pseudo-Hermitian phase transitions and the emergence of bound states.

Original authors: Haiting Zhang, Lei Xiao, Cui-Xian Guo, Quan Lin, Peng Xue

Published 2026-10-23

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CC BY 4.0

Original authors: Haiting Zhang, Lei Xiao, Cui-Xian Guo, Quan Lin, Peng Xue

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 as waves of probability, spreading out and interfering with one another. For decades, physicists have studied these systems using a mathematical framework called Hermitian quantum mechanics, which guarantees that energy is conserved and that the total probability of finding a particle always adds up to one. However, the real world is rarely so perfect. Systems often lose energy to their surroundings or gain energy from external sources, a situation described by non-Hermitian physics. In these open systems, strange and counterintuitive behaviors emerge. One such behavior is the "skin effect," where particles pile up at the edges of a container, and another is the "exceptional point," a special condition where different states of a system merge into one. Understanding how these systems respond to imperfections, such as a single flaw or defect in an otherwise perfect structure, is crucial for designing future quantum technologies, from ultra-sensitive sensors to new types of computers.

A team of researchers has now discovered a new way that a single flaw can reshape the behavior of a quantum particle. They studied a model known as a quantum walk, which is essentially a mathematical simulation of a particle hopping from one spot to another on a grid, much like a person taking steps on a sidewalk. In their study, they introduced a single point of gain and loss—a non-Hermitian defect—into a one-dimensional line of these steps. What they found was a phenomenon they call scale-free localization. Usually, when a particle gets trapped or localized near a defect, the size of the trap depends on the strength of the defect. But in this specific setup, the researchers observed that the particle's probability cloud, when normalized by the system size, maintained a shape that remained exactly the same whether the system had a hundred steps or a thousand. The spatial profile of these states is invariant with respect to the system size, meaning the distribution scales perfectly with the boundaries.

The researchers mapped out the behavior of this system as they increased the strength of the defect. They identified two distinct moments where the system's nature changed abruptly. The first transition occurred when the system lost a specific type of symmetry, causing some of the particle's possible states to become complex numbers rather than simple real numbers. At this precise moment, the system began to host the scale-free localized states. As they increased the defect strength even further, a second transition happened. This time, the system developed tightly bound states, where the particle became stuck very close to the defect, behaving differently from the scale-free states. By carefully measuring how the particle's probability was distributed across the grid, the team could pinpoint exactly where these two transitions occurred. They found that the scale-free states had a unique signature: their localization length grew linearly with the system size, whereas the tightly bound states remained fixed in size, independent of the system's boundaries.

To ensure this was not just a quirk of a one-dimensional line, the team extended their investigation to a two-dimensional grid. They introduced the same single defect into this larger, flat landscape. The results were strikingly similar. The defect induced the same scale-free localization, proving that this phenomenon is universal and not limited to a single line of steps. The researchers showed that the spatial profile of the particle's wave function, when normalized, remained invariant across different system sizes, confirming that the effect holds true in higher dimensions as well. This work suggests that a single imperfection can act as a powerful tool to control quantum states in ways that were previously unexplored. By understanding how these scale-free states form and how they differ from traditional bound states, scientists gain a new perspective on defect physics in non-Hermitian systems. This insight could pave the way for experimental investigations in photonics and other platforms, offering a fresh avenue for manipulating quantum matter in the presence of gain and loss.

Technical Summary: Scale-free Localization in Pseudo-Hermitian Quantum Walks

Problem Statement
Non-Hermitian systems exhibit phenomena distinct from Hermitian counterparts, such as the non-Hermitian skin effect and exceptional points. While global gain/loss mechanisms have been extensively studied, the impact of a single non-Hermitian defect on system properties remains less explored. Specifically, the phenomenon of "scale-free localization," where specific eigenstates possess localization lengths scaling linearly with system size, has been identified theoretically but its relationship to multiple phase transitions and its behavior in higher dimensions (beyond 1D) are largely unexplored. This work addresses the mechanism by which a single non-Hermitian defect induces scale-free localization in quantum walks (QWs) and investigates the associated phase transitions and universality in two dimensions.

Methodology
The authors construct a split-step quantum walk model on a 1D lattice under periodic boundary conditions, introducing a single non-Hermitian defect via on-site gain and loss terms. The evolution operator is defined as $U = D^{-1}SR(\theta)DSR(\theta)$ , where $D$ represents the defect with non-Hermitian strength $\gamma$ , $S$ is the conditional shift operator, and $R(\theta)$ is the coin operator. An effective Hamiltonian $H = i \ln U$ is derived to analyze the energy spectrum.

The study employs the following analytical and numerical tools:

Spectral Analysis: Examining the eigenvalues of $H$ to identify Pseudo-Hermitian (PH) symmetry breaking (transition from real to complex-conjugate eigenvalues) and the emergence of bound states.
Order Parameters: Using the ratio of complex eigenvalues ( $\rho_{com}$ ) and the maximal imaginary eigenvalue ( $(ImE)_{max}$ ) to map phase diagrams.
Localization Metrics: Calculating the Inverse Participation Ratio (IPR) and Normalized Participation Ratio (NPR) to characterize spatial profiles. The scaling of IPR with system size ( $L$ ) is used to distinguish between scale-free localized states (LSs) and bound states.
Dimensional Extension: Generalizing the model to a 2D split-step QW with a single defect at the origin to test the universality of the phenomenon.

Key Contributions and Results

Induction of Scale-Free Localization: The study demonstrates that a single non-Hermitian defect in a 1D QW induces scale-free localization. Upon breaking PH symmetry, certain extended states transition into scale-free LSs. These states are characterized by spatial profiles that remain invariant with respect to system size when normalized, and their IPR scales linearly with $1/L$ .
Two Distinct Phase Transitions: The system undergoes two sequential phase transitions as the non-Hermitian strength $\gamma$ $γ$ increases:
1. PH Symmetry Breaking: Occurs at a critical point $\gamma_c$ . Here, eigenvalues become complex, and scale-free LSs emerge. The critical point is analytically derived as $\gamma_c = \ln(\tan(\theta/2))$ for specific coin angles.
2. Bound State Emergence: Occurs at a second critical point $\gamma_b > \gamma_c$ . This transition is marked by the appearance of two pairs of bound states. These states exhibit spatial profiles that vary with system size and an IPR that remains constant (independent of $L$ ).
Identification of Critical Points: The authors utilize the averaged IPR ( $\langle IPR \rangle$ ) and averaged NPR ( $\langle NPR \rangle$ ) as observables. Singularities in $\ln\langle IPR \rangle$ and $\ln\langle NPR \rangle$ precisely identify the critical points $\gamma_c$ and $\gamma_b$ , corroborating the spectral analysis.
Universality in 2D: The investigation is extended to 2D QWs. A single non-Hermitian defect in the 2D lattice also induces scale-free localization. The IPR of these states scales linearly with $1/L^2$ , and their normalized spatial profiles overlap across different system sizes, confirming the universality of the phenomenon.

Significance
The paper claims to provide a valuable perspective for exploring defect physics in non-Hermitian quantum systems. By linking scale-free localization to PH symmetry breaking and identifying a secondary transition to bound states, the work clarifies the interplay between single defects and global system properties. The demonstration of this phenomenon in 2D QWs suggests that scale-free localization is a robust feature of non-Hermitian defects, offering a new avenue for studying non-Hermitian defect physics and novel localized states. The authors note that these findings may facilitate future experimental investigations, particularly in photonic platforms where gain/loss and dimensional extension are naturally accessible.

Technical Summary: Scale-free Localization in Pseudo-Hermitian Quantum Walks

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