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arXiv:cond-mat/9705081 · Physical review. B, Condensed matter

Localization of two interacting particles in a one-dimensional random potential

Pil Hun Song, Doochul Kim

Published Nov 15, 1997• 20 citations
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Abstract

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, $W,$ and the strength of interaction, $U.$ When $U=0,$ \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 $W$ 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 $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$

DOI
10.1103/physrevb.56.12217

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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 \ensuremathξ\ensuremath∼W\ensuremath−\ensuremathν0\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}{\ensuremath{\nu}}_{0}}\ensuremathξ\ensuremath∼W\ensuremath−\ensuremathν0​ for small WWW with \ensuremathν0=2.1\ifmmode±\else\textpm\fi0.1.{\ensuremath{\nu}}_{0}=2.1\ifmmode\pm\else\textpm\fi{}0.1.\ensuremathν0​=2.1\ifmmode±\else\textpm\fi0.1. When U\ensuremath≠0,U\ensuremath{\ne}0,U\ensuremath=0, we use the finite size scaling ansatz and find the relation \ensuremathξ\ensuremath∼W\ensuremath−\ensuremathν\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}\ensuremath{\nu}}\ensuremathξ\ensuremath∼W\ensuremath−\ensuremathν with \ensuremathν=2.9\ifmmode±\else\textpm\fi0.2.\ensuremath{\nu}=2.9\ifmmode\pm\else\textpm\fi{}0.2.\ensuremathν=2.9\ifmmode±\else\textpm\fi0.2. Moreover, data show the scaling behavior \ensuremathξ\ensuremath∼W\ensuremath−\ensuremathν0g(b∣U∣/W\ensuremathΔ)\ensuremath{\xi}\ensuremath{\sim}{W}^{\ensuremath{-}{\ensuremath{\nu}}_{0}}g(b|U|{/W}^{\ensuremath{\Delta}})\ensuremathξ\ensuremath∼W\ensuremath−\ensuremathν0​g(b∣U∣/W\ensuremathΔ) with \ensuremathΔ=4.0\ifmmode±\else\textpm\fi0.5.\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.\ensuremathΔ=4.0\ifmmode±\else\textpm\fi0.5.
In a recent letter [Phys. Rev. Lett. 78, 515 (1997); cond-mat/9612034] Roemer and Schreiber report on numerical calculations that led them to conclude that the previously observed enhancement of the localization length L2L_2L2​ of two interacting particles (TIP) vanishes in the thermodynamic limit. We point out that such a claim (i) is in conflict with the scaling theory of localization, (ii) ignores a consistent picture from a wealth of published numerical data and analytical investigations, and (iii) directly contradicts new numerical results obtained with a Green function method that is well adapted to the problem under study. The claim of Roemer and Schreiber is based on extrapolating from L<360 to infinity. Our data clearly shows the two particle localization length L2L_2L2​ to be significantly enhanced and independent of system size between L=100 and L=1000, as expected on physical grounds.
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 λ2λ_2λ2​ on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of λ2λ_2λ2​ 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 onsite interaction and also a long-range interaction.
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.
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=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 ${\ensuremath{\xi}}_{2}$ 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 ${\ensuremath{\xi}}_{2}$ with the single-particle localization length ${\ensuremath{\xi}}_{1}$ into the asymptotic regime for ${\ensuremath{\xi}}_{1}>100$ (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 ${\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=F(u{\ensuremath{\xi}}_{1})$.
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 theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, ${V}_{ij}\ensuremath{\propto}{\left|i\ensuremath{-}j\right|}^{\ensuremath{-}\ensuremath{\alpha}}$, where the exponent of the interaction range $\ensuremath{\alpha}$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\ensuremath{\alpha}$, the critical exponent experiences a sharp increase at about ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.2$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with an ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.