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Journal of Physics Conference Series

Complexity Measures in the Tight-Binding Model

Boris Atenas, Sergio Curilef

Published Sep 1, 2024• 1 citation
Publisher

Abstract

Abstract The deformation of a wave packet is a significant topic in classical and quantum mechanics. Understanding this phenomenon is relevant in the study of various physical systems. In this work, we characterize the evolution of a highly localized wave packet in a tight-binding lattice. We investigate the behavior of the probability distribution associated with the wave packet and the accompanying complexity measures. We take information entropy, disequilibrium, disorder, and complexity measures to describe the localization-delocalization process from a highly localized initial pulse, showing the particles moving in a lattice. The main result is obtained from the entropy definition (Logarithmic and Linear) and the inverse of the participant ratio to describe the expected localization-delocalization process, evoking two definitions of Complexity: CLMC and CSDL .

DOI
10.1088/1742-6596/2839/1/012010

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Abstract The deformation of a wave packet is a significant topic in classical and quantum mechanics. Understanding this phenomenon is relevant in the study of various physical systems. In this work, we characterize the evolution of a highly localized wave packet in a tight-binding lattice. We investigate the behavior of the probability distribution associated with the wave packet and the accompanying complexity measures. We take information entropy, disequilibrium, disorder, and complexity measures to describe the localization-delocalization process from a highly localized initial pulse, showing the particles moving in a lattice. The main result is obtained from the entropy definition (Logarithmic and Linear) and the inverse of the participant ratio to describe the expected localization-delocalization process, evoking two definitions of Complexity: CLMC and CSDL .
Understanding wave-packet deformation and spreading is central to the description of transport and localization phenomena in both classical and quantum systems. In this work, we introduce information-theoretic and complexity-based indicators. We analyze the evolution of the probability distribution using Shannon and linear entropies, together with the inverse participation ratio, to quantify localization–delocalization processes and their associated fluctuations. These measures naturally motivate the use of statistical complexity definitions, including the López–Mancini–Calbet (CLMC) and Shiner–Davison–Landsberg (CSDL) complexities. In addition, we compute the probability current for both single-particle and two-particle configurations, providing a dynamical description of transport in terms of particle motion. The influence of an external field on entropy production, complexity, and probability currents is also examined, highlighting the interplay between information spreading, localization, and transport in lattice systems.
Spread complexity and localization in PT\mathcal{PT}PT-symmetric systems
We present a framework for investigating wave function spreading in PT\mathcal{PT}PT-symmetric quantum systems using spread complexity and spread entropy. We consider a tight-binding chain with complex on-site potentials at the boundary sites. In the PT\mathcal{PT}PT-unbroken phase, the wave function is delocalized. We find that in the PT\mathcal{PT}PT-broken phase, it becomes localized on one edge of the tight-binding lattice. This localization is a realization of the non-Hermitian skin effect. Localization in the PT\mathcal{PT}PT-broken phase is observed both in the lattice chain basis and the Krylov basis. Spread entropy, entropic complexity, and a further measure that we term the Krylov inverse participation ratio probe the dynamics of wave function spreading and quantify the strength of localization probed in the Krylov basis. The number of Krylov basis vectors required to store the information of the state reduces with the strength of localization. Our results demonstrate how measures in Krylov space can be used to characterize the non-hermitian skin effect and its localization phase transition.
Characterizing complexity and criticality in quantum systems requires diagnostics that are both computationally tractable and physically insightful. We apply a measure of quantum state complexity for n-qubit systems, defined as the divergence between the Shannon or von Neumann entropy of the computational basis distribution and the second-order Renyi entropy. This quantity has already been used earlier termed as structural entropy and it is particularly powerful as the Renyi entropy is directly related to state purity, linear entropy, and the inverse participation ratio, providing a clear physical grounding. While other Renyi orders could be used, the second order offers a deep and established connection to these key physical quantities. We first validate the measure in canonical noise channels, showing it peaks at the boundary between quantum and classical regimes. We then demonstrate its power in many-body physics. For systems exhibiting a many-body localization transition - including deformed random matrix ensembles and a disordered Heisenberg spin chain - the complexity measure reliably signals the crossover from integrable/localized to quantum-chaotic/ergodic phases. Crucially, the maximum complexity occurs in the non-ergodic yet extended states at the transition, precisely capturing the critical region where the system is neither fully localized nor thermalized. Furthermore, within the chaotic phase, the measure correlates with the survival probability of local excitations, revealing a spectrum of thermalization properties. Our results establish that the entropic complexity is a simple, versatile, and effective probe for identifying nontrivial quantum regimes and transitions giving a new and alternative insight into such systems.
The wave function temporal evolution on the one-dimensional (1D) lattice is considered in the tight-binding approximation.The lattice consists of N equal sites and one impurity site (donor).The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor.The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2.The wave packet width grows with time ∼ t 1/3 and its amplitude decreases ∼ t -1/3 .The wave packet reflects multiply from the lattice ends.Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.
Quantum lattices are pivotal in the burgeoning fields of quantum materials and information science. Novel experimental techniques allow the preparation and monitoring of wave packet dynamics on quantum lattices with high spatiotemporal resolution. We present an analytical study of wave packet diffusivity and diffusion length on tight-binding quantum lattices subject to stochastic noise. Our analysis reveals the crucial role of spatial coherence and predicts a set of novel phenomena: (1) noise can enhance the transient diffusivity and diffusion length of spatially extended initial states; (2) standing or traveling initial states, with large momentum, spread faster than a localized initial state and exhibit a noise-induced peak in the transient diffusivity; (3) the differences in the diffusivity or diffusion length of extended and localized initial states have a universal dependence on initial width. These predictions suggest the possibility of controlling the wave packet dynamics by spatial manipulations, which will have implications for materials science and quantum technologies.
The wave function temporal evolution on the one-dimensional(1D) lattice is considered in the tight-binding approximation. The lattice consists of N equal sites and one impurity site(donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time ～ t1/3and its amplitude decreases ～ t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.