[Paper Review] Many-body localization with mobility edges

Yichen Huang|arXiv (Cornell University)|Jul 5, 2015

Quantum many-body systems26 references4 citations

TL;DR

This paper constructs a solvable, translationally invariant spin chain model that exhibits a tunable mobility edge in one-dimensional interacting disordered systems, demonstrating analytically that many-body localization-delocalization transitions can be visualized as tuning a mobility edge in the energy spectrum. By linking many-body localization to a quantum central limit theorem and introducing the concept of 'energy-resolved disorder strength,' the model reveals a universal mechanism for mobility edges across a broad class of models, including the random-field Heisenberg chain.

ABSTRACT

We construct a solvable spin chain model of many-body localization (MBL) with a tunable mobility edge. This simple model not only demonstrates analytically the existence of mobility edges in interacting one-dimensional (1D) disordered systems, but also allows us to study their physics. By establishing a connection between MBL and a quantum central limit theorem (QCLT), we show that many-body localization-delocalization transitions can be visualized as tuning a mobility edge in the energy spectrum. Since the effective disorder strength for individual eigenstates depends on energy density, we identify "energy-resolved disorder strength" as a physical mechanism for the appearance of mobility edges, and support the universality of this mechanism by arguing its presence in a large class of models including the random-field Heisenberg chain. We also construct models with multiple mobility edges. All our constructions can be made translationally invariant.

Motivation & Objective

• . The paper aims to resolve the long-standing debate on whether mobility edges can exist in interacting one-dimensional disordered systems.
• It seeks to provide an analytically rigorous, solvable model to demonstrate the existence of mobility edges in such systems.
• The objective includes identifying a universal physical mechanism—energy-resolved disorder strength—for the emergence of mobility edges.
• The study further aims to construct models with multiple mobility edges and ensure translational invariance in all constructions.

Proposed method

• . The authors introduce a solvable spin chain model with a tunable mobility edge, constructed using a translationally invariant Hamiltonian.
• They establish a connection between many-body localization and a quantum central limit theorem (QCLT), enabling analytical treatment of eigenstate properties.
• The concept of 'energy-resolved disorder strength' is introduced, where effective disorder depends on the energy density of individual eigenstates.
• The model uses a parent Hamiltonian approach, taking the union of all eigenstates across all disorder realizations to analyze spectral structure.
• The RSRG-X (excited-state real-space renormalization group) technique is applied to argue that eigenstates at high absolute energy densities are more localized due to increased effective disorder.
• The construction is generalized to models with multiple mobility edges, maintaining translational invariance throughout.

Experimental results

Research questions

• RQ1 . Can mobility edges exist in interacting one-dimensional disordered systems, and if so, what is the underlying physical mechanism?
• RQ2 Is the existence of mobility edges in such systems analytically demonstrable, beyond numerical evidence?
• RQ3 What role does energy-resolved disorder strength play in determining the localization properties of many-body eigenstates?
• RQ4 Can models with multiple mobility edges be constructed while preserving translational invariance?
• RQ5 Is the mechanism of energy-resolved disorder strength universal across different models, such as the random-field Heisenberg chain?

Key findings

• . The paper constructs a solvable, translationally invariant spin chain model that explicitly exhibits a tunable mobility edge in the energy spectrum.
• The existence of mobility edges in interacting 1D systems is demonstrated analytically, resolving ambiguity from numerical studies.
• The mechanism of 'energy-resolved disorder strength' is identified as the key physical driver for the appearance of mobility edges.
• The model supports the universality of this mechanism in a large class of models, including the random-field Heisenberg chain.
• The authors construct models with two mobility edges, resulting in a sandwiched structure of localized and delocalized states.
• All constructions are translationally invariant, demonstrating that mobility edges can emerge in systems without quenched disorder in the Hamiltonian.