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arXiv:1806.06584 · Journal of Physics Condensed Matter

Spectral functions of one-dimensional systems with correlated disorder

Niaz Ali Khan, João M. Viana Parente Lopes, J. P. Santos Pires, J. M. B. Lopes dos Santos

Published Jan 31, 2019• 10 citations
arXivHTMLPDF

Abstract

We investigate the spectral function of Bloch states in a one-dimensional tight-binding non-interacting chain with two different models of static correlated disorder, at zero temperature. We report numerical calculations of the single-particle spectral function based on the Kernel polynomial method, which has an [Formula: see text] computational complexity. These results are then confirmed by analytical calculations, where precise conditions were obtained for the appearance of a classical limit in a single-band lattice system. Spatial correlations in the disordered potential give rise to non-perturbative spectral functions shaped as the probability distribution of the random on-site energies, even at low disorder strengths. In the case of disordered potentials with an algebraic power-spectrum, [Formula: see text] [Formula: see text], we show that the spectral function is not self-averaging for [Formula: see text].

DOI
10.1088/1361-648x/ab03ad

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We investigate the spectral function of Bloch states in a one-dimensional tight-binding non-interacting chain with two different models of static correlated disorder, at zero temperature. We report numerical calculations of the single-particle spectral function based on the Kernel polynomial method, which has an [Formula: see text] computational complexity. These results are then confirmed by analytical calculations, where precise conditions were obtained for the appearance of a classical limit in a single-band lattice system. Spatial correlations in the disordered potential give rise to non-perturbative spectral functions shaped as the probability distribution of the random on-site energies, even at low disorder strengths. In the case of disordered potentials with an algebraic power-spectrum, [Formula: see text] [Formula: see text], we show that the spectral function is not self-averaging for [Formula: see text].
The decay of (disorder-averaged) static spin correlation functions at T = 0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling J ∆ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At ∆ = 0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N = 100 sites. At ∆ ̸ = 0 Lanczos diagonalizations are carried out for chains with up to N = 22 sites. The longitudinal correlation function 〈Sz 0Sz r 〉 is found to exhibit a power-law decay with an exponent that varies with ∆ and, for nonzero ∆, also with the width of the λi-distribution. The results for the transverse correlation function 〈S x 0 Sx r 〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on. 1 Typeset using REVTEXThe combination of randomness and quantum fluctuations is well-known to be a fertile ground for interesting physical phenomena including Anderson localization. In onedimensional (1D) tight-binding systems the rule is that disorder always leads to localization. However, if the randomness is purely off-diagonal, the localization length diverges at the band center, 1 which is bound to affect the decay law of correlation functions. One particular off-diagonally disordered fermion model, the 1D half-filled tight-binding model with random hopping, is equivalent to the special XX case ( ∆ = 0) of the 1D s = 1/2 XXZ model with random transverse exchange coupling, described by the Hamiltonian H = J ∑
We investigate a one-dimensional tight-binding model in which onsite potentials [Formula: see text] exhibit power-law spatial correlations (with exponent α) and the hopping amplitudes decay as [Formula: see text]. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the [Formula: see text] plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio r, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behavior across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the [Formula: see text] parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.
Results for the single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ($t\ensuremath{-}J$ model) obtained by numerical and analytical methods are discussed. The numerical calculation of the spectral function is performed for small systems of up to 18 sites using the Lanczos method. A detailed analysis of the spectral functions for the $t\ensuremath{-}J$ model in one and two dimensions is given. We also investigate the effect of the next-nearest-neighbor hopping terms, which appear in order t2U\frac{{t}^{2}}{U}Ut2​ and are omitted in the standard $t\ensuremath{-}J$ model. The role of quantum fluctuations on the shape of the spectra and the fundamental difference between one and two dimensions is elucidated. In 2D we find evidence for a coherent quasiparticle band of width 2J2J2J with dispersion $^{\ensuremath{\omega}}\mathrm{k}\ensuremath{\sim}(\frac{J}{2}){(cos{k}_{x}+cos{k}_{y})}^{2}$ and spin S=12S=\frac{1}{2}S=21​, which emerges at the low-energy edge of the incoherent spectrum. This form of dispersion is due to effective next-nearest-neighbor processes related to spin flips and agrees with our extension of the approach of Brinkman and Rice to finite JJJ.
The spectral function of a single hole in the two-dimensional Hubbard model is studied numerically through exact diagonalization for a nine-site cluster over a wide range of interaction strengths U. In the weak-interaction limit, the spectral function consists of sharp quasiparticle peaks corresponding to the allowed single-particle excitations of the noninteracting system. As the interaction strength increases, the spectral weight begins to spread over a large range of energies, and satellite structures appear at higher excitation energy. The band-narrowing effect and the quasiparticle weight are investigated. The results are discussed in connection with other analytical and numerical studies on the spectral function.
Abstract We investigate two one-dimensional tight-binding models with disorder that have extended states at zero energy. We use exact and partial diagonalisation of the Hamiltonian to obtain the eigenmodes and the associated participation ratios, and the transfer-matrix method to determine the localisation length. The first model has no on-site disorder, but random couplings. While the participation ratio remains finite at zero energy, the localisation length diverges logarithmically as the energy goes to zero. We provide an intuitive derivation of this logarithmic divergence based on the weak coupling of the two sublattices. The second model has a conserved quantity as the row sums of the Hamiltonian are zero. This model can be represented as a harmonic chain with random couplings, or as a diffusion model on a lattice with random links. We find, in agreement with existing analytical calculations, that the number of system-spanning eigenmodes increases proportionally to the square root of the system size, and we related this power law to other power laws that characterise the scaling behaviour of the eigenmodes, the participation ratio, the localisation length, and their dependence on energy and system size. When disorder is so strong that the smallest hopping terms can be arbitrarily close to zero, all these power laws change, and we show a crossover between the two scaling regimes. All these results are explained by intuitive arguments based on scaling.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size LLL in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.