Anderson localization in an interacting fermionic system

Abstract: In the present article, we discuss the role played by the interaction in the Anderson localization problem, for a system of interacting fermions in a one-dimensional disordered lattice, described by the Fermi Hubbard Hamiltonian, in presence of an on-site random potential. We show that, given the proper identification of the elementary excitations of the system described in terms of doublons and unpaired particles, the Anderson localization picture survives. Ensuing a "global quench", we show that the system exhibits a rich localization scenario, which can be ascribed to the nearly-free dynamics of the elementary excitations of the Hubbard Hamiltonian.

• The paper reveals that Anderson localization persists in the presence of two-body interactions by distinguishing doublon and unpaired particle dynamics.
• The paper employs TEBD simulations on a 40-atom lattice to show exponential localization with distinct lengths for doublons and unpaired particles.
• The paper highlights potential applications in quantum systems design by elucidating how disorder and interactions govern localization phenomena.

Anderson Localization in an Interacting Fermionic System: A Detailed Analysis

Introduction

The paper "Anderson Localization in an Interacting Fermionic System" by Francesco Massel explores the nuanced dynamics of Anderson Localization (AL) in systems characterized by interactions between fermions. Specifically, it explores a setup involving a one-dimensional disordered lattice using the Fermi-Hubbard Hamiltonian, which introduces an on-site random potential amidst interacting fermions.

Summary of Key Findings

The research delineates the survival of the Anderson localization picture even in the presence of two-body interactions within a many-body context. Notably, the study pioneers a description based on identifying elementary excitations in terms of doublons (bound pairs) and unpaired particles. This identification stands crucial, as it enables the preservation of AL for these distinct excitations even amid complex interactions and disorder.

By employing a global quantum quench to initiate the expansion of the interacting fermionic cloud, the study observes rich localization dynamics driven predominantly by the nearly-free propagation dynamics of these elementary excitations. The numerical simulations, performed through a time-evolving block decimation (TEBD) algorithm, robustly support the theoretical framework, demonstrating the exponential localization behavior anticipated by the Anderson model.

Detailed Numerical Results

The analyzed system comprises 40 fermionic atoms in a lattice of 256 sites, initially forming a band insulator in the lattice's central region. Post quench, the particles are observed to either remain localized due to disorder or exhibit expansion characterized by two separate localization lengths:

1. Unpaired Particles: They follow the standard AL localization length.
2. Doublons: Demonstrate a distinct localization behavior explained by an effective hopping parameter altered by interactions.

Simulations with $U=0$ show conventional Anderson localization for non-interacting particles, with expansion behavior starkly contrasting against cases of $U \neq 0$. For an assorted set of conditions ($U=0$, $V_0/J=5$ compared to $U/J=-5$, $V_0/J=1$), the localization dynamics showcase similar lengths, reaffirming the proposed two-fluid description.

Theoretical Implications

This research crucially underscores the significance of dissecting many-body systems into their elementary excitations for a thorough understanding of disorder and interaction dynamics. The theoretical implications extend beyond fermionic systems; such frameworks could be instrumental in examining bosonic or mixed quantum systems, offering insight into the robustness of AL in interacting scenarios.

Moreover, the discussion presents intriguing theoretical considerations, such as the independence of the localization length with respect to the interaction sign. Such findings stimulate deeper theoretical examinations of particle-hole symmetry and its ramifications on localization characteristics beyond mean-field and perturbative approaches.

Practical Implications and Future Directions

Practically, the implications portend advancements in designing and controlling quantum systems embedded in disorder, particularly within cold atomic setups where control parameters like dimensionality and interaction strengths are tunable. Explorations down this path may elucidate new mechanisms for localization exploiting interaction-disorder interplay.

Going forward, further investigations could expand this analysis to encompass higher-dimensional lattices, differing disorder distributions, and study their exact impacts on localization phenomena. Additionally, analyzing the impact of non-equilibrium dynamics such as time-dependent potentials or interactions could unveil new quantum behaviors pertinent to quantum computing and information storage technologies.

In summary, Massel's work provides a clarifying lens through which the intertwined effects of disorder and interactions in quantum systems can be understood, with definitive insights underpinning both current applications and future inquiries.