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Physical review. E

Localization-delocalization transition in discrete-time quantum walks with long-range correlated disorder

C.V.C. Mendes, G. M. A. Almeida, M. L. Lyra, F. A. B. F. de Moura

Published Feb 12, 2019• 19 citations
Publisher

Abstract

We study the effects of spatially long-range correlated phase disorder on the Hadamard quantum walk on a line. The shift operator is built to exhibit an intrinsic disorder distribution featuring long-range correlations. To impose such, we resort to fractional Brownian motion with power-law spectrum 1/k^{2α} with α≥0 being the exponent that controls the degree of correlations. We discuss the scaling behavior of the walker's wave packet and report a localization-delocalization transition controlled by α. We unveil two intermediate dynamical regimes between exponential localization and full delocalization.

DOI
10.1103/physreve.99.022117

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We investigate the evolution dynamics of inhomogeneous discrete-time one-dimensional quantum walks displaying long-range correlations in both space and time. The associated quantum coin operators of internal states are built to exhibit random inhomogeneity distribution of long-range correlations embedded in the time evolution protocol through a fractional Brownian motion with spectrum following a power-law behavior, S(k)∼1/k^{ν}. From extensive numerical simulations with averages over a large number of independent realizations of the phases of quantum coins, the power-law correlated disorder encoded in the coin phases is shown to give rise to a wide variety of spreading patterns of the qubit states, from localized to subdiffusive, diffusive, and superdiffusive (including ballistic) behavior, depending on the relative strength of the parameters driving the correlation degree. Dispersion control is possible in one-dimensional discrete-time quantum walks by tuning the long-range correlation properties assigned to the inhomogeneous quantum coin operator.
We study the one-dimensional quantum Heisenberg ferromagnet with exchange couplings exhibiting long-range correlated disorder with power spectrum proportional to ${1/k}^{\ensuremath{\alpha}},$ where k is the wave vector of the modulations on the random coupling landscape. By using the renormalization group, integration of the equations of motion, and exact diagonalization, we compute the spin-wave localization length and the mean-square displacement of the wave packet. We find that, associated with the emergence of extended spin waves in the low-energy region for \ensuremath{\alpha}>1, the wave-packet mean-square displacement changes from a long-time super diffusive behavior for \ensuremath{\alpha}1 to a long-time ballistic behavior for \ensuremath{\alpha}>1. At the vicinity of \ensuremath{\alpha}=1, the mobility edge separating the extended and localized phases is shown to scale with the degree of correlation as ${E}_{c}\ensuremath{\propto}(\ensuremath{\alpha}\ensuremath{-}{1)}^{1/3}.$
We investigate time-independent disorder on several two-dimensional discrete-time quantum walks. We find numerically that, contrary to claims in the literature, random onsite phase disorder, spin-dependent or otherwise, cannot localize the Hadamard quantum walk; rather, it induces diffusive spreading of the walker. In contrast, split-step quantum walks are generically localized by phase disorder. We explain this difference by showing that the Hadamard walk is a special case of the split-step quantum walk, with parameters tuned to a critical point at a topological phase transition. We show that the topological phase transition can also be reached by introducing strong disorder in the rotation angles. We determine the critical exponent for the divergence of the localization length at the topological phase transition, and find $\ensuremath{\nu}=2.6$, in both cases. This places the two-dimensional split-step quantum walk in the universality class of the quantum Hall effect.
Systems with purely off-diagonal disorder may have peculiar features such as the localization-delocalization transition and long-range correlations in their wave functions. To motivate possible experimental studies of the physics of off-diagonal disorder (e.g., in systems with random nearest-neighbor hopping), we study in detail disordered discrete-time quantum walk in a finite chain. Starting from a transfer matrix approach, we show, both theoretically and computationally, that the dynamics of the quantum walk with disorder manifests all the main features of systems with off-diagonal disorder. We also propose how to prepare a remarkable delocalized zero mode from a localized and easy-to-prepare initial zero mode using an adiabatic protocol that changes the disorder strength slowly. Numerical experiments are also performed with encouraging results.
A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.
We describe the critical behavior of the Anderson-like transition predicted to occur in the 2D tight-binding model with isotropic scale-free long-range correlated disorder characterized by a power-law spectral density. We explore the scale invariance of the participation function relative fluctuation at the critical point to locate the mobility edge as a function of the power-law spectral density exponent α 2D . The states near the band center, which exhibit power-law localization for uncorrelated disorder, become delocalized for α 2D >2. In addition, we consider the finite-size scaling hypothesis to estimate the correlation length critical exponent. We find that the critical exponent ν depends on α 2D , thus indicating that correlations in the disorder distribution are indeed relevant in this regime, in agreement with the extended Harris criterion.