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Physical review. B, Condensed matter

Localization of multivibrational excitations

Marek S. Wartak, Lin Yang, Ching Yin Fong, Y. Ron Shen

Published Jun 15, 1988• 3 citations
Publisher

Abstract

We have studied the localization of multivibrational states in the one-dimensional boson system with on-site random anharmonicity. By introducing site basis states (local representation) we were able, in the limit of \ensuremath{\Delta}/\ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\ll}1, where \ensuremath{\Delta} is the coupling interaction between the nearest neighbors and \ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{} is the average on-site anharmonic energy, to reduce the problem to Anderson-like model for a one-dimensional electron system with diagonal disorder. Assuming a Cauchy distribution for on-site anharmonicity, we have calculated the degree of localization. It is shown that the degree of localization increases with the increase of the randomness.

DOI
10.1103/physrevb.37.10350

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We have studied the localization of multivibrational states in the one-dimensional boson system with on-site random anharmonicity. By introducing site basis states (local representation) we were able, in the limit of \ensuremath{\Delta}/\ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\ll}1, where \ensuremath{\Delta} is the coupling interaction between the nearest neighbors and \ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{} is the average on-site anharmonic energy, to reduce the problem to Anderson-like model for a one-dimensional electron system with diagonal disorder. Assuming a Cauchy distribution for on-site anharmonicity, we have calculated the degree of localization. It is shown that the degree of localization increases with the increase of the randomness.
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.
We study the localization problem in quantum stochastic mechanics. We start from the Edwards model for a particle in a bath of scattering centers and prove static localization of the ground state wavefunction of the particle in a one dimensional square well coupled to Dirac delta like scattering centers in arbitrary but fixed positions. We see how the localization increases for increasing coupling ggg and increasing number of scattering centers at constant density. Then we choose the scattering centers positions as pseudo random numbers with a uniform probability distribution and observe an increase in the localization of the average of the ground state over the many positions realizations. We discuss how this averaging procedure is consistent with a picture of a particle in a Bose-Einstein condensate of of non interacting boson scattering centers interacting with the particle with Dirac delta functions pair potential. We then study the dynamics of the ground state wave function. We conclude with a discussion of the affine quantization version of the Lax model which reduces to a system of contiguous square wells with walls in arbitrary positions independently of the coupling constant ggg.
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In one-dimensional quantum lattice models with open boundaries, we find and study localization at the lattice edge. We show that edge-localized eigenstates can be found in both bosonic and fermionic systems, specifically, in the Bose-Hubbard model with on-site interactions and in the spinless fermion model with nearest-neighbor interactions. We characterize the localization through spectral studies via numerical diagonalization and perturbation theory through considerations of the eigenfunctions and through the study of explicit time evolution. We concentrate on few-particle systems, showing how more complicated edge states appear as the number of particles is increased.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with NNN sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc{N}_{c}Nc​ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
The phase diagram of correlated disordered electron systems is calculated within dynamical mean-field theory for the Anderson-Falicov-Kimball model with nearest-neighbor and next-nearest-neighbor hoppings. The half-filled band is analyzed in terms of the chemical potential of the system using the geometric and arithmetic averages. We also introduce the on-site energies exhibiting a long-range correlated disorder, which generates a system with similar characteristics as the one created by a random independent variable distribution. A decrease in the correlated disorder reduces the extended phase.