Probing the Global Delocalization Transition in the de Moura-Lyra Model with the Kernel Polynomial Method

Abstract

In this paper, we report numerical calculations of the localization length in a non-interacting one-dimensional tight-binding model at zero tem¬perature, holding a correlated disorder model with an algebraic power-spectrum (de Moura-Lyra model). Our calculations were based on a Kernel Polynomial implementation of the Thouless formula for the inverse localization length of a general nearest-neighbor 1D tight-binding model with open boundaries. Our results confirm the delocalization of all eigenstates in de Moura-Lyra model for α > 1 and a localization length which diverges as ξ ∝ (1 – α)–1 for α → 1–, at all energies in the weak disorder limit (as previously seen in [12]).

Probing band-center anomaly with the Kernel polynomial method

“…The kernel polynomial method (KPM) [22] is a polynomial expansion-based technique and can efficiently compute the spectral function of a large disordered quantum system. It has been successfully applied to condensed matter problems [23][24][25]27], for underpinning the Anderson transitions in non-interacting disordered systems. In the KPM technique, the Hamiltonian and all energy scales need to be normalized into the standard domain of orthogonality of the Chebyshev polynomials -1 1 (] [).…”

Anomalous correlation-induced dynamical phase transitions

“…With regard to the phase transition, Pires et al 41 demonstrated that the delocalization phase transition may occur at without a mobility edge in the perturbative regime. It was found that the localization length diverge as in limit in the thermodynamic limit, confirmed by the analytical perturbative calculations 41 , 42 .…”

“…The randomness in the potential is demonstrated as a long-range spatially correlated disorder under spectral density, with being the strength of correlation of the spectral density that controls the roughness of the potential landscapes. The disordered potential amplitude , is given by 39 – 42 , 45 , 47 , where is a normalization constant, imposing unit variance of the local potential with zero mean, and are the N /2 independent random phases which are uniformly distributed in the interval [ . It is very important to emphasize that the disorder distribution takes the following sinusoidal form of wavelength N with a vanishing noise,

Figure 1 (Color online) A schematic representation of quantum quench process under correlated Anderson model.

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“…The main focus is to study the quench dynamics under correlated model in different regimes 41 , 42 . In the case of the initial eigenstate of the prequench Hamiltonian is a plane-wave state with eigenenergy , where, a represents the lattice spacing.…”

“…We assign this structure to be one of the reasons for the emergence of delocalization transition in the system. Further, using the generalized Thouless formula 50 , the localization length of the correlated Anderson model for can be analytically calculated as 41 , 42 ,

Figure 7 (Color online) Log-linear scale: The time-evolution of the Loschmidt echo for various prequench correlation exponent with system size and sample averaged over 2048 realizations of disorder. The Loschmidt echoes are well fitted (green dashed-dotted curves) by Eq.

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