betekintés - Scientific Computing - # Non-Hermitian Many-Body Localization

Spread Complexity as a Diagnostic Tool for Non-Hermitian Many-Body Localization Transitions

Q: How does the dimensionality of the system affect the behavior of spread complexity in the context of non-Hermitian many-body localization?

The provided text primarily focuses on one-dimensional (1D) systems and doesn't directly address the impact of dimensionality on spread complexity in non-Hermitian many-body localization (MBL). However, we can extrapolate some insights based on general principles and known effects of dimensionality on MBL: Higher dimensions generally disfavor localization: In higher dimensions, particles have more paths to delocalize, making it harder for disorder to localize all eigenstates. This suggests that the MBL transition, if it exists in higher dimensions, would occur at stronger disorder strengths compared to 1D. Impact on spread complexity: Since spread complexity dynamics are sensitive to the ergodic/localized nature of the system, we can expect dimensionality to influence it: Ergodic phase: In the ergodic phase, the peak height of singular value spread complexity might decrease with increasing dimensionality. This is because higher dimensionality leads to a larger Hilbert space, potentially slowing down the initial growth of complexity. MBL phase: In the MBL phase, the suppression of complexity growth observed in 1D might be less pronounced or even absent in higher dimensions, reflecting the weaker localization. TFD and CDW complexity: Similar trends might be observed for TFD and CDW spread complexities. For instance, the saturation value of CDW complexity, which is lower in the MBL phase for 1D systems with TRS, might increase with dimensionality. Direct investigation in higher dimensions is crucial to confirm these hypotheses. Numerical studies of spread complexity in 2D or 3D non-Hermitian disordered systems would be valuable to understand the interplay of dimensionality, non-Hermiticity, and localization on complexity dynamics.

Q: Could the observed difference in CDW spread complexity between models with and without TRS be attributed to the existence of different types of localization in these systems?

Yes, the distinct CDW spread complexity behaviors between models with and without time-reversal symmetry (TRS) likely stem from different localization mechanisms at play: Models with TRS: These systems can exhibit a conventional MBL phase characterized by localized eigenstates and a complete set of local integrals of motion (LIOMs). In this regime, the CDW state, being an ordered state, struggles to explore the Hilbert space due to the constraints imposed by LIOMs, resulting in lower complexity saturation. Models without TRS: The absence of TRS allows for more complex non-Hermitian terms, potentially leading to non-Hermitian localization. This type of localization might not necessarily imply a complete set of LIOMs or strictly localized eigenstates. Instead, it can arise from the non-trivial structure of eigenstates in the complex energy plane, even when they have some degree of spatial delocalization. This could explain the higher complexity saturation observed for CDW in the MBL phase of models without TRS. The CDW state, even in the presence of disorder, might still be able to explore a larger portion of the Hilbert space compared to the TRS case due to the absence of strong LIOM constraints. Further investigation is needed to confirm the precise nature of localization in the model without TRS. Analyzing the eigenspectrum, entanglement properties, and the structure of eigenstates in the complex energy plane would provide valuable insights into the localization mechanism and its connection to the observed CDW complexity behavior.

Q: Can the insights gained from studying spread complexity in non-Hermitian systems be applied to understand other non-equilibrium phenomena in quantum many-body systems?

Yes, the insights from studying spread complexity in non-Hermitian systems hold promising potential for understanding various non-equilibrium phenomena in quantum many-body systems: Open quantum systems: Non-Hermitian Hamiltonians naturally describe open quantum systems interacting with an environment. Spread complexity can provide insights into the dynamics of entanglement, information scrambling, and thermalization in such systems, complementing traditional approaches based on master equations. Driven-dissipative systems: Many experimental platforms involve driving and dissipation, leading to non-equilibrium steady states. Spread complexity can help characterize these states, distinguish different dynamical phases, and understand the interplay of driving, dissipation, and interactions. Quantum information scrambling: The growth of spread complexity, particularly the presence or absence of a peak, can signal the efficiency of information scrambling in non-Hermitian systems. This is relevant for understanding the dynamics of quantum information in noisy environments and designing robust quantum devices. Topological phases in non-Hermitian systems: Recent studies have explored the interplay of topology and non-Hermiticity. Spread complexity might offer a new perspective on characterizing and distinguishing different non-Hermitian topological phases, especially in the context of dynamical probes. Beyond these specific examples, the key takeaway is that spread complexity provides a versatile tool for probing the dynamics of quantum information and entanglement in non-equilibrium settings. Its application to non-Hermitian systems opens up exciting avenues for exploring a broader class of phenomena relevant to various fields, including condensed matter physics, quantum information science, and quantum simulation.

Alapfogalmak

This study investigates the effectiveness of spread complexity, a measure of quantum chaos, in characterizing non-Hermitian many-body localization transitions, revealing its ability to distinguish between ergodic and localized phases and detect complex-real eigenvalue transitions.

Kivonat

Bibliographic Information:

Ganguli, M. (2024). Spread Complexity in Non-Hermitian Many-Body Localization Transition. arXiv preprint arXiv:2411.11347v1.

Research Objective:

This study aims to explore the utility of spread complexity, a concept rooted in quantum information theory, as a tool to understand and characterize the transition from a chaotic (ergodic) phase to a localized (many-body localized or MBL) phase in non-Hermitian quantum systems.

Methodology:

The research employs numerical simulations of one-dimensional disordered interacting hard-core boson models exhibiting non-Hermitian many-body localization transitions. Two specific models are considered: one with time-reversal symmetry (TRS) featuring asymmetric hopping and another without TRS incorporating onsite particle loss and gain. The study investigates three types of spread complexity: singular value spread complexity, thermofield double (TFD) state spread complexity, and charge density wave (CDW) state spread complexity.

Key Findings:

The peak height in the singular value spread complexity serves as an effective order parameter for distinguishing between ergodic and MBL phases in both models, irrespective of TRS.
The saturation value of TFD spread complexity is sensitive to the presence of complex eigenvalues in the system, with lower saturation values indicating a higher fraction of complex eigenvalues. This behavior allows for the detection of complex-real eigenvalue transitions in systems with TRS.
The CDW spread complexity exhibits distinct behavior in models with and without TRS. In the TRS model, the MBL phase is characterized by a lower saturation value compared to the ergodic phase. However, the model without TRS shows the opposite trend, with higher saturation in the MBL phase.

Main Conclusions:

The study demonstrates the efficacy of spread complexity as a valuable tool for analyzing non-Hermitian many-body localization transitions. Specifically, singular value spread complexity effectively differentiates between ergodic and localized phases, while TFD spread complexity reveals complex-real eigenvalue transitions. The distinct behavior of CDW spread complexity in models with and without TRS highlights the role of TRS in influencing localization transitions.

Significance:

This research contributes significantly to the understanding of non-Hermitian quantum systems, particularly in the context of many-body localization. The findings provide valuable insights into the relationship between quantum chaos, localization, and eigenvalue properties in these systems.

Limitations and Future Research:

The study primarily focuses on one-dimensional models. Further research is needed to explore the applicability of spread complexity in higher-dimensional non-Hermitian systems. Additionally, a more in-depth analytical understanding of the observed behavior, particularly the contrasting trends in CDW spread complexity for models with and without TRS, is crucial for a comprehensive understanding of these transitions.

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arxiv.org

Statisztikák

The study uses system sizes of L=12 and L=14 for analyzing singular value spread complexity.
The analysis includes simulations with 250 disorder realizations for both the models and random matrix ensembles.
The parameters used for the models are J=1, U=2, g=0.1, and γ=0.1.

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Főbb Kivonatok

Spread Complexity in Non-Hermitian Many-Body Localization Transition

by Maitri Gangu... : arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11347.pdf

Spread Complexity in Non-Hermitian Many-Body Localization Transition

Mélyebb kérdések

How does the dimensionality of the system affect the behavior of spread complexity in the context of non-Hermitian many-body localization?

The provided text primarily focuses on one-dimensional (1D) systems and doesn't directly address the impact of dimensionality on spread complexity in non-Hermitian many-body localization (MBL). However, we can extrapolate some insights based on general principles and known effects of dimensionality on MBL:

Higher dimensions generally disfavor localization: In higher dimensions, particles have more paths to delocalize, making it harder for disorder to localize all eigenstates. This suggests that the MBL transition, if it exists in higher dimensions, would occur at stronger disorder strengths compared to 1D.
Impact on spread complexity:  Since spread complexity dynamics are sensitive to the ergodic/localized nature of the system, we can expect dimensionality to influence it:

Ergodic phase: In the ergodic phase, the peak height of singular value spread complexity might decrease with increasing dimensionality. This is because higher dimensionality leads to a larger Hilbert space, potentially slowing down the initial growth of complexity.
MBL phase: In the MBL phase, the suppression of complexity growth observed in 1D might be less pronounced or even absent in higher dimensions, reflecting the weaker localization.


TFD and CDW complexity: Similar trends might be observed for TFD and CDW spread complexities. For instance, the saturation value of CDW complexity, which is lower in the MBL phase for 1D systems with TRS, might increase with dimensionality.
Direct investigation in higher dimensions is crucial to confirm these hypotheses. Numerical studies of spread complexity in 2D or 3D non-Hermitian disordered systems would be valuable to understand the interplay of dimensionality, non-Hermiticity, and localization on complexity dynamics.

Could the observed difference in CDW spread complexity between models with and without TRS be attributed to the existence of different types of localization in these systems?

Yes, the distinct CDW spread complexity behaviors between models with and without time-reversal symmetry (TRS) likely stem from different localization mechanisms at play:

Models with TRS: These systems can exhibit a conventional MBL phase characterized by localized eigenstates and a complete set of local integrals of motion (LIOMs). In this regime, the CDW state, being an ordered state, struggles to explore the Hilbert space due to the constraints imposed by LIOMs, resulting in lower complexity saturation.
Models without TRS: The absence of TRS allows for more complex non-Hermitian terms, potentially leading to non-Hermitian localization. This type of localization might not necessarily imply a complete set of LIOMs or strictly localized eigenstates. Instead, it can arise from the non-trivial structure of eigenstates in the complex energy plane, even when they have some degree of spatial delocalization. This could explain the higher complexity saturation observed for CDW in the MBL phase of models without TRS. The CDW state, even in the presence of disorder, might still be able to explore a larger portion of the Hilbert space compared to the TRS case due to the absence of strong LIOM constraints.
Further investigation is needed to confirm the precise nature of localization in the model without TRS. Analyzing the eigenspectrum, entanglement properties, and the structure of eigenstates in the complex energy plane would provide valuable insights into the localization mechanism and its connection to the observed CDW complexity behavior.

Can the insights gained from studying spread complexity in non-Hermitian systems be applied to understand other non-equilibrium phenomena in quantum many-body systems?

Yes, the insights from studying spread complexity in non-Hermitian systems hold promising potential for understanding various non-equilibrium phenomena in quantum many-body systems:

Open quantum systems: Non-Hermitian Hamiltonians naturally describe open quantum systems interacting with an environment. Spread complexity can provide insights into the dynamics of entanglement, information scrambling, and thermalization in such systems, complementing traditional approaches based on master equations.
Driven-dissipative systems: Many experimental platforms involve driving and dissipation, leading to non-equilibrium steady states. Spread complexity can help characterize these states, distinguish different dynamical phases, and understand the interplay of driving, dissipation, and interactions.
Quantum information scrambling: The growth of spread complexity, particularly the presence or absence of a peak, can signal the efficiency of information scrambling in non-Hermitian systems. This is relevant for understanding the dynamics of quantum information in noisy environments and designing robust quantum devices.
Topological phases in non-Hermitian systems: Recent studies have explored the interplay of topology and non-Hermiticity. Spread complexity might offer a new perspective on characterizing and distinguishing different non-Hermitian topological phases, especially in the context of dynamical probes.
Beyond these specific examples, the key takeaway is that spread complexity provides a versatile tool for probing the dynamics of quantum information and entanglement in non-equilibrium settings. Its application to non-Hermitian systems opens up exciting avenues for exploring a broader class of phenomena relevant to various fields, including condensed matter physics, quantum information science, and quantum simulation.