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

Abstract

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 ...

Level-spacing distribution of a fractal matrix

“…We shall consider the full spectrum of the Fibonacci matrix (eigenvalues and eigenvectors) and address questions related to Anderson localization and quantum chaos. We emphasise that we do not have in mind any particular physical system described by this matrix although such fractal matrices might be in general relevant for critical (between extended and localized) one-electron states in disordered or quasiperiodic potentials or possibly many-electrons states for interacting electrons in the presence of disorder [4]. We shall address the following questions concerning the Fibonacci matrix:…”

Coulomb repulsion versus Hubbard repulsion in a disordered chain

“…The convention in this work is that two particles at the same site cost an energy 2U (and not U as assumed in previous references [9][10][11][12]). An additional cost of energy U/p has to be paid by two particles separated by a distance p. The boundary conditions (BCs) are taken periodic (L/2 ≥ p ≥ 1).…”

“…Hubbard repulsion effectively couples a smaller density ρ eff 2 < ρ 2 , as explained in reference [9]. Let us review three consequences of this difference.…”

“…Additional information and references can be found in two recent conference proceedings [7,8]. For a "contact" interaction as Hubbard repulsion, the TIP system exhibits remarkable properties [9][10][11][12]. The mixing of the one body states inside a scale L 1 and the associated delocalization effect for sizes larger than L 1 is maximum for a Hubbard energy to kinetic energy ratio U/t ≈ U c , where U c is the fixed point of a duality transformation [10] mapping the weak U/t limit onto the large U/t limit.…”

“…the TIP eigenstates at U = 0), one finds that Coulomb repulsion mainly favors hopping terms between 2FP states nearby in energy, with energy separation of the order of the TIP level spacing ∆ 2 ∝ L −2 , where L denotes the size of the system. Hubbard repulsion mainly induces [10] hopping terms between 2FP states separated by a larger energy ∆ eff 2 > ∆ 2 , and the measure of the coupled 2FP states is multifractal [9].…”