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Theoretical and Mathematical Physics

Computation of localization degree in the sense of the Anderson criterion for a one-dimensional diagonally disordered system

G. G. Kozlov

Published Feb 1, 2010• 5 citations
Publisher
DOI
10.1007/s11232-010-0019-1

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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.
Using locally the projector postulate (wave-function collapse) we consider a one-dimensional disordered system under local (one-site) iterative measurements. We found that Anderson localization is broken due to these decoherence effects. Explicitly, the probability of return falls exponentially to zero and the delocalization time is computed. For infinite size systems the delocalization time remains finite. Quantum Zeno effect emerges in the corresponding limit of a rapid sequence of measurements including off-diagonal disorder.
We present the study of the structure of many-body eigenfunctions in a one-dimensional disordered spin chain.We discuss the choice of an appropriate basis in the Hilbert space, where the problem can be seen as an Anderson model defined on a high-dimensional non-trivial graph, determined by the many-body Hamiltonian.The comparison with the usual behavior of wave-functions in finite dimensional Anderson localization allows us to put in light the main differences of the many-body case.At high disorder, the typical eigenfunctions do not seem to localize though they occupy a infinitesimal portion of the Hilbert space in the thermodynamic limit.We perform a detailed analysis of the distribution of the wave-function coefficients and their peculiar scaling in the small and large disorder phase.We propose a criterion to identify the position of the transition by looking at the long tails of these distributions.The results coming from exact diagonalization show signs of breaking of ergodicity when the disorder reaches a critical value that agrees with the estimation of the many-body localization transition in the same model.