Two interacting particles in a disordered chain III: Dynamical aspects of the interplay disorder-interaction

Arias, S. de Toro; Waintal, Xavier; Pichard, Jean‐Louis

doi:10.1007/s100510050838

Cited by 22 publications

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“…In a second paper [15], a study of the TIP spectral fluctuations will be presented, showing that statistics is critical (as for the one body spectrum at a mobility edge) if U is large enough, accompanied by multifractal wavefunctions in the TIP eigenbasis for U = 0. In a third paper [16], a study of the dynamics of a TIP wave packet will be presented, showing that the center of mass exhibits anomalous diffusion between L 1 and L 2 . These three studies provide consistent and complementary observations supporting our claim: multifractality and criticality are relevant concepts for a TIP system with on site interaction in one dimension.…”

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Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

Waintal

Pichard

1998

Eur. Phys. J. B

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Abstract. For N interacting particles in a one dimensional random potential, we study the structure of the corresponding network in Hilbert space. The states without interaction play the role of the "sites". The hopping terms are induced by the interaction. When the one body states are localized, we numerically find that the set of directly connected "sites" is multifractal. For the case of two interacting particles, the fractal dimension associated to the second moment of the hopping term is shown to characterize the Golden rule decay of the non interacting states and the enhancement factor of the localization length. The wave functions of one particle in a random potential have been extensively studied. In two dimensions within the localization domains [1] the large fluctuations of their amplitudes have a multifractal character. In one dimension, the elastic mean free path l and the localization length L 1 coincide, preventing a single one particle wave function to be multifractal over a significant range of scales. The description of the correlations existing between the localized eigenstates is more difficult. This is quite unfortunate, since a local two-body interaction reorganizes the non interacting electron gas in a way which depends on the spatial overlap of (four) different one particle states. When one writes the N -body Hamiltonian in the basis built out from the one particle states (eigenbasis without interaction), this overlap determines the interaction matrix elements, i.e. the hopping terms of the corresponding network in Hilbert space. In this work, we numerically study the distribution of the hopping terms in one dimension, when the one body states are localized. It has been observed [2] that this distribution is broad and non Gaussian. We give here numerical evidence that this distribution is multifractal. Moreover, since the obtained Rényi dimensions do not depend on L 1 , simple power laws describe how the moments scale with the characteristic length L 1 of the one body problem. Since the main applications we consider (Golden rule decay of the non interacting states, enhancement factor of the localization length for two interacting particles) depend on the square of the hopping terms, we are mainly interested by the scaling of the second moment. For a size L ≈ L 1 , we show that, contrary to previous assumptions, the N -body eigenstates without interaction directly coupled by the square of the hopping terms have not a density of the order of PACS. The dimension f (α(q = 2)) (≈ 1.75 for hopping terms involving four different one body states) characterizes the fractal set of N -body eigenstates without interaction which are directly coupled by the square of the hopping terms. We consider N electrons described by an Hamiltonian including the kinetic energy and a random potential, plus a two-body interaction:The operators d + ασ (d ασ ) create (destroy) an electron in a one body eigenstate |α > of spin σ. Noting Ψ α (n) the amplitude on site n of the state |α > with energy ǫ α , the interaction ...

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Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

Waintal

Pichard

1998

Eur. Phys. J. B

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“…In this article, the localization behaviour of two hardcore bosons has been studied from the spread of the particles over a finite two-dimensional disordered lattice from an initial occupation at adjacent sites. The case of such localization of two interacting particles in one-dimensional lattices has been investigated thoroughly before [11][12][13][14][15][16][17] since the 1990s. These studies were motivated by the observation of persistent currents in 1D wires [18][19][20].…”

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Localization Parameters for Two Interacting Particles in Disordered Two-Dimensional Finite Lattices

Chattaraj

2018

Condensed Matter

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I study spreading of two interacting hardcore bosons in disordered two-dimensional finite lattices from an initial occupation of two adjacent sites. The parameters related to the spreading of the particles provide an insight on the effect of interaction. I find that the presence of interaction makes the particles less localized than the non-interacting ones within the range of disorder strength W ≤ 4 and interaction strength V ≤ 4 . If the interaction strength is higher, then particles localize more. A transition with changes in the character of dominant correlations is found at critical disorder strengths for each chosen strength of interaction. The nature of correlations between the particles as nearest neighbours becomes dominant beyond these disorder strengths.

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Quantum diffusion and scaling of localized interacting electrons

Halfpap

2001

Annalen der Physik

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A new numerical method is introduced that enables a reliable study of disorder‐induced localization of interacting particles. It is based on a quantum mechanical time evolution calculation combined with a finite size scaling analysis. The time evolution of up to four particles in one dimension is studied and localization lengths are defined via the long‐time saturation values of the mean radius, the inverse participation ratio and the center of mass extension. A systematic study of finite size effects using the finite size scaling method is performed in order to extract the localization lengths in the limit of an infinite system size. For a single particle, the well‐known scaling of the localization length λ1 with disorder strength W is observed, λ1 ∝ W—2. For two particles, an interaction‐induced delocalization is found, confirming previous results obtained by numerically calculating matrix elements of the two‐particle Green's function: in the limit of small disorder, the localization length increases with decreasing disorder as λ2 ∝ W—4 and can be much larger than <$>\mitlambda λ1. For three and four particles, delocalization is even stronger. Based on analytical arguments, an upper bound for the n‐particle localization length λn is derived and shown to be in agreement with the numerical data, λn ∝ λ1. Although the localization length increases superexponentially with particle number and can become arbitrarily large for small disorder, it does not diverge for finite λ1 and n. Hence, no extendedstates exist in one dimension, at least for spinless fermions.

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Two interacting particles in a disordered chain III: Dynamical aspects of the interplay disorder-interaction

Cited by 22 publications

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Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

Localization Parameters for Two Interacting Particles in Disordered Two-Dimensional Finite Lattices

Quantum diffusion and scaling of localized interacting electrons

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