The Impact of Disorder on Scale-Free Localization in Non-Hermitian Systems: A Size-Dependent Transition to Anderson Localization

Core Concepts

Introducing disorder to a non-Hermitian system with scale-free localized (SFL) states leads to a size-dependent Anderson transition, where SFL states transform into Anderson-localized states at a critical disorder strength that decreases with increasing system size.

Abstract

Bibliographic Information:

Yılmaz, B., Yuce, C., & Bulutay, C. (2024). From scale-free to Anderson localization: a size-dependent transition. arXiv preprint arXiv:2411.00389.

Research Objective:

This study investigates the impact of disorder on a one-dimensional Hermitian lattice with a single non-Hermitian impurity, focusing on the transition from scale-free localization to Anderson localization. The authors aim to determine if and how the critical disorder strength required for this transition depends on the system size.

Methodology:

The researchers employ a theoretical model of a one-dimensional lattice with a non-Hermitian impurity and random on-site energies representing disorder. They numerically solve the Schrödinger equation for this system under both periodic and open boundary conditions to analyze the eigenstate localization properties. The critical disorder strength is identified as the point where the system's sensitivity to boundary conditions disappears, indicating complete Anderson localization. The inverse participation ratio (IPR) is used to quantify the degree of localization for different disorder strengths and system sizes.

Key Findings:

• Introducing disorder to the system gradually converts both extended and scale-free localized (SFL) states into Anderson-localized states.
• The critical disorder strength (Wc) required for complete Anderson localization is size-dependent and decreases as the system size increases, following a power law relationship Wc ∝ N^α, where α is a negative exponent.
• The average IPR values at Wc decrease with increasing system size, indicating a stronger size dependence for smaller systems.

Main Conclusions:

The study demonstrates that the presence of a single non-Hermitian impurity in an otherwise Hermitian lattice leads to a size-dependent Anderson transition. This behavior contrasts with the Hermitian Anderson model and the Hatano-Nelson model, where the transition point is either zero or independent of system size. The findings highlight the unique interplay between scale-free localization and Anderson localization in non-Hermitian systems.

Significance:

This research contributes to the understanding of localization phenomena in non-Hermitian systems, which are gaining increasing attention due to their relevance to open and dissipative systems. The size-dependent Anderson transition observed in this study has implications for the design and control of transport properties in such systems.

Limitations and Future Research:

The study focuses on a one-dimensional lattice model. Investigating higher-dimensional systems and different types of non-Hermitian impurities could reveal further insights into the interplay between scale-free and Anderson localization. Additionally, exploring the role of interactions in these systems would be an interesting avenue for future research.

Stats

The scaling exponent α is found to vary between -0.36 and -0.41 depending on the parameters γ and δ when averaged over 10,000 realizations.
For a completely delocalized state, the IPR approaches 1/N.

Quotes

"The phenomenon of scale-free localization challenges conventional understanding of localization, as a single non-Hermitian impurity in an otherwise Hermitian lattice can generate a significant number of scale-free localized (SFL) modes."
"In contrast, our model shows that both extended and SFL states transition to Anderson-localized states, leading to a size-dependent, non-zero transition point due to the size-dependent nature of SFL states."

Key Insights Distilled From

From scale-free to Anderson localization: a size-dependent transition

by Burc... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00389.pdf

Deeper Inquiries

How would the presence of multiple non-Hermitian impurities influence the Anderson transition in this system?

Introducing multiple non-Hermitian impurities into the Hermitian Anderson model would create a more complex interplay of scale-free localization and Anderson localization, significantly impacting the Anderson transition. Here's how:

Increased Competition between Localization Mechanisms: Each impurity would act as a source of scale-free localization, potentially with different strengths and ranges depending on their individual non-Hermitian coupling parameters (δ and γ). This would compete with the disorder-induced Anderson localization, making the overall localization behavior dependent on the spatial distribution and strengths of the impurities.

Modification of the Critical Disorder Strength: The critical disorder strength (Wc) would likely increase with the number of impurities. Each impurity contributes to a tendency towards scale-free localization, requiring stronger disorder to induce the Anderson transition. The exact relationship between Wc and the number of impurities would depend on their arrangement and coupling strengths.

Emergence of Complex Localization Landscapes: The presence of multiple impurities could lead to the formation of complex localization landscapes. Regions closer to impurities might exhibit stronger scale-free localization, while regions farther away could be dominated by Anderson localization. This could result in a heterogeneous distribution of localization lengths within the system.

Potential for Novel Phenomena: The interplay of multiple non-Hermitian impurities could give rise to novel phenomena not observed with a single impurity. For instance, depending on the relative phases of the impurities, constructive or destructive interference between the scale-free localized states could occur, potentially leading to enhanced or suppressed localization.
Investigating these aspects would require extensive numerical simulations and analytical modeling, considering factors like impurity density, spatial correlations between impurities, and the specific non-Hermitian coupling parameters of each impurity.

Could the size-dependent Anderson transition observed in this study be experimentally verified in real-world systems, such as photonic lattices or cold atom systems?

Yes, the size-dependent Anderson transition predicted in this study could potentially be experimentally verified in real-world systems like photonic lattices or cold atom systems. Here's how:
Photonic Lattices:

Implementation: Photonic lattices offer a versatile platform to emulate the tight-binding model described in the paper. The non-Hermitian impurity can be realized using techniques like controlled losses (e.g., using absorbing elements) or gain-loss structures that break PT-symmetry. Disorder can be introduced by randomly varying the refractive index or position of the lattice sites.

Measurement: The Anderson transition can be probed by injecting light into the lattice and observing the transmission spectrum. In the localized regime, the transmission would be suppressed, while in the delocalized regime, light would propagate through the lattice. By varying the system size (number of lattice sites) and disorder strength, the size-dependent shift in the critical disorder strength (Wc) could be observed.
Cold Atom Systems:

Implementation: Cold atoms trapped in optical lattices provide another suitable platform. The non-Hermitian impurity can be engineered using techniques like spatially dependent loss (e.g., using a focused laser beam to remove atoms) or by coupling the atoms to a dissipative auxiliary system. Disorder can be introduced using speckle potentials or by superimposing incommensurate lattices.

Measurement: The Anderson transition can be studied by monitoring the expansion dynamics of the atomic cloud. In the localized regime, the expansion would be suppressed, while in the delocalized regime, the atoms would spread out. By varying the lattice size and disorder strength, the size-dependent Anderson transition could be observed.
Challenges:
While these experimental platforms offer promising avenues, some challenges need to be addressed:

Control of Disorder: Precise control over the disorder realization is crucial for accurate measurements. Techniques for generating well-characterized and reproducible disorder are essential.

Finite Size Effects: Real-world systems are inherently finite. Careful analysis is required to distinguish the size-dependent Anderson transition from finite-size effects.

Dissipation and Decoherence: In experimental systems, dissipation and decoherence can play a significant role, potentially obscuring the effects of the non-Hermitian impurity. Minimizing these effects is crucial for observing the predicted phenomena.
Despite these challenges, the experimental verification of the size-dependent Anderson transition in these systems would be a significant achievement, providing valuable insights into the interplay of disorder and non-Hermiticity in condensed matter physics.

If we consider the concept of entropy in the context of information flow, how does the transition from scale-free to Anderson localization relate to the system's ability to process and store information?

The transition from scale-free to Anderson localization, viewed through the lens of entropy and information flow, reveals an intriguing connection to a system's capacity to process and store information.

Scale-Free Localization and Information Transport: In the scale-free localized regime, the eigenstates extend across a significant portion of the system, albeit with a distinct spatial profile. This suggests a potential for long-range information transport. The system can be viewed as having channels for information flow, facilitated by the extended nature of the scale-free localized states. The entropy, in this case, would be relatively high, reflecting a degree of delocalization and the potential for information spreading.

Anderson Localization and Information Confinement: As the system transitions to Anderson localization, the eigenstates become exponentially localized, confining information to smaller spatial regions. The system's ability to transport information over long distances is significantly reduced. The localized states act as traps for information, hindering its flow. Consequently, the entropy decreases, reflecting the increased order and reduced uncertainty in the system's state due to information confinement.

Implications for Information Processing: This transition has significant implications for information processing tasks. In the scale-free localized regime, the system might be suitable for tasks requiring long-range correlations or information transfer. However, the presence of disorder-induced Anderson localization could limit the efficiency of such processes. On the other hand, the Anderson localized regime, with its confined states, might be advantageous for information storage. The localized states could act as robust memory elements, preserving information within their localized regions.

Entropy as a Measure of Information Capacity: The change in entropy during the transition can be interpreted as a change in the system's capacity to store or process information. A higher entropy in the scale-free localized regime suggests a larger effective space for information to be distributed, potentially enabling more complex information processing. Conversely, the lower entropy in the Anderson localized regime indicates a reduced information capacity but potentially enhanced information storage capabilities due to the localized nature of the states.
In summary, the transition from scale-free to Anderson localization, viewed from an information-theoretic perspective, represents a shift from a regime potentially suitable for information transport to one more suited for information storage. The entropy serves as a valuable measure of this changing information capacity, providing insights into the system's ability to manipulate and retain information.