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arXiv:1908.06643 · Physical review. B./Physical review. B

Taming two interacting particles with disorder

Diana Thongjaomayum, Alexei Andreanov, Thomas Engl, Sergej Flach A

Published Dec 16, 2019• 10 citations
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Abstract

We compute the scaling properties of the localization length ${\ensuremath{\xi}}_{2}$ of two interacting particles in a one-dimensional chain with diagonal disorder, and the connectivity properties of the Fock states. We analyze record large system sizes (up to $N=20\phantom{\rule{0.16em}{0ex}}000$) and disorder strengths (down to $W=0.5$). We vary the energy $E$ and the on-site interaction strength $u$. At a given disorder strength, the largest enhancement of ${\ensuremath{\xi}}_{2}$ occurs for $u$ of the order of the single-particle bandwidth and for two-particle states with energies at the center of the spectrum, $E=0$. We observe a crossover in the scaling of ${\ensuremath{\xi}}_{2}$ with the single-particle localization length ${\ensuremath{\xi}}_{1}$ into the asymptotic regime for ${\ensuremath{\xi}}_{1}>100$ ($W<1.0$). This happens due to the recovery of translational invariance and momentum conservation rules in the matrix elements of interconnected Fock eigenstates for $u=0$. The entrance into the asymptotic scaling is manifested through a nonlinear scaling function ${\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=F(u{\ensuremath{\xi}}_{1})$.

DOI
10.1103/physrevb.100.224203

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We compute the scaling properties of the localization length \ensuremathξ2{\ensuremath{\xi}}_{2}\ensuremathξ2​ of two interacting particles in a one-dimensional chain with diagonal disorder, and the connectivity properties of the Fock states. We analyze record large system sizes (up to N=20000N=20\phantom{\rule{0.16em}{0ex}}000N=20000) and disorder strengths (down to W=0.5W=0.5W=0.5). We vary the energy EEE and the on-site interaction strength uuu. At a given disorder strength, the largest enhancement of \ensuremathξ2{\ensuremath{\xi}}_{2}\ensuremathξ2​ occurs for uuu of the order of the single-particle bandwidth and for two-particle states with energies at the center of the spectrum, E=0E=0E=0. We observe a crossover in the scaling of \ensuremathξ2{\ensuremath{\xi}}_{2}\ensuremathξ2​ with the single-particle localization length \ensuremathξ1{\ensuremath{\xi}}_{1}\ensuremathξ1​ into the asymptotic regime for \ensuremathξ1>100{\ensuremath{\xi}}_{1}>100\ensuremathξ1​>100 (W<1.0W<1.0W<1.0). This happens due to the recovery of translational invariance and momentum conservation rules in the matrix elements of interconnected Fock eigenstates for u=0u=0u=0. The entrance into the asymptotic scaling is manifested through a nonlinear scaling function \ensuremathξ2/\ensuremathξ1=F(u\ensuremathξ1){\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=F(u{\ensuremath{\xi}}_{1})\ensuremathξ2​/\ensuremathξ1​=F(u\ensuremathξ1​).
We study the interaction-induced connectivity in the Fock space of two particles in a disordered one-dimensional potential. Recent computational studies showed that the largest localization length ${\ensuremath{\xi}}_{2}$ of two interacting particles in a weakly random tight-binding chain is increasing unexpectedly slow relative to the single-particle localization length ${\ensuremath{\xi}}_{1}$, questioning previous scaling estimates. We show this to be a consequence of the approximate restoring of momentum conservation of weakly localized single-particle eigenstates, and disorder-induced phase shifts for partially overlapping states. The leading resonant links appear among states which share the same energy and momentum. We substantiate our analytical approach by computational studies for up to ${\ensuremath{\xi}}_{1}=1000$. A potential nontrivial scaling regime sets in for ${\ensuremath{\xi}}_{1}\ensuremath{\approx}400$.
We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L, we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U. For intermediate U, the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t1 necessary to propagate over L1 , U de-favors the propagation. On the contrary, U favors a very slow delocalization after t1 , characterized by a log(t) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L1 and increase for L > L1 . The changes of the curvatures can be described by a conductance-like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L1.
We study the scaling of the localization length of two interacting bosons in a one-dimensional random lattice with the single particle localization length. We consider the short-range interaction assuming that the particles interact when located both on the same site. We discuss several regimes, among them one interesting weak Fock space disorder regime. In this regime we obtain a weak logarithmic scaling law. Numerical benchmark data support the absence of any strong enhancement of the two particle localization length.
We investigate the localization of two interacting particles in a one-dimensional random potential. Our definition of the two-particle localization length, \ensuremath{\xi}, is the same as that of von Oppen et al. [Phys. Rev. Lett. 76, 491 (1996)]. \ensuremath{\xi}'s for chains of finite lengths are calculated numerically using the recursive Green's function method for several values of the strength of the disorder, W,W,W, and the strength of interaction, U.U.U. When U=0,U=0,U=0, \ensuremath{\xi} approaches a value larger than half the single-particle localization length as the system size tends to infinity and behaves as $\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}{\ensuremath{\nu}}_{0}}$ for small WWW with ${\ensuremath{\nu}}_{0}=2.1\ifmmode\pm\else\textpm\fi{}0.1.$ When $U\ensuremath{\ne}0,$ we use the finite size scaling ansatz and find the relation $\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=2.9\ifmmode\pm\else\textpm\fi{}0.2.$ Moreover, data show the scaling behavior $\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}{\ensuremath{\nu}}_{0}}g(b|U|{/W}^{\ensuremath{\Delta}})$ with $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$
We study two interacting particles in a random potential chain by means of the transfer matrix method. The dependence of the two-particle localization length ${\ensuremath{\lambda}}_{2}$ on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of ${\ensuremath{\lambda}}_{2}$ as compared to the results for single particles is entirely due to the finite size of the systems considered. This is shown for a Hubbard-like on site interaction and also a long-range interaction.
Abstract We show that the crossover from the weak interaction limit towards the strong interaction limit may be accompanied by a delocalization effect in one dimensional disordered quantum models. The spin degrees of freedom are frozen and the spatial wave functions remain symmetric or antisymmetric when the strength U of a short range interaction is varied. The study concerns the excited states for two interacting particles and the ground state for a finite density of carriers. First, for two particles in a chain of length L , we establish a duality transformation mapping the behavior at weak U onto the behavior at strong U . For intermediate U , the mixing of the one body states and the interaction induced delocalization effect are maximum. Furthermore, if L ≈︂ L 1 (the one particle localization length), the system becomes weakly chaotic with critical spectral statistics. This weak chaos is related to the multifractality of the interaction matrix. For two particles starting close to each other, localization is reached in two steps. Before the time t 1 necessary to propagate over L 1 , U de‐favors the propagation. On the contrary, U favors a very slow delocalization after t 1 , characterized by a log( t ) spreading of the center of mass. Similarly, the curvatures of the energy levels with respect to an enclosed magnetic flux decrease as a function of U for L < L 1 and increase for L > L 1 . The changes of the curvatures can be described by a conductance‐like single scaling parameter. Second, using the density renormalization group algorithm, we have studied the ground state energy of a finite density of spinless fermions and its change under twisted boundary conditions. For a large disorder, a charge reorganization is induced by the interaction: When the system becomes instable between the inhomogeneous configuration driven by the random potential (Anderson insulator) and the homogeneous one driven by repulsive interactions (Mott insulator), the ground state sensitivity can be enhanced by orders of magnitude. In contrast, no enhancement occurs at weaker disorder, when there are many particles on a scale L 1 .