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arXiv:cond-mat/9702084 · Physical Review Letters

Comment on “No Enhancement of the Localization Length for Two Interacting Particles in a Random Potential”

Klaus M. Frahm, Axel Müller–Groeling, Jean‐Louis Pichard, Dietmar Weinmann A

Published Jun 23, 1997• 17 citations
arXivHTMLPDF

Abstract

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 $L_2$ 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 $L_2$ to be significantly enhanced and independent of system size between L=100 and L=1000, as expected on physical grounds.

DOI
10.1103/physrevlett.78.4889

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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 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 $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$
This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a ddd-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].
We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. This leads to a much weaker enhancement than proposed earlier. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by these data.
Abstract 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 λ 1 with disorder strength W is observed, λ 1 ∝ 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 λ 2 ∝ W —4 and can be much larger than <$>\mitlambda λ 1 . For three and four particles, delocalization is even stronger. Based on analytical arguments, an upper bound for the n ‐particle localization length λ n is derived and shown to be in agreement with the numerical data, λ n ∝ λ 1 . Although the localization length increases superexponentially with particle number and can become arbitrarily large for small disorder, it does not diverge for finite λ 1 and n . Hence, no extended states exist in one dimension, at least for spinless fermions.
Finite-size effects arise when a sample of particles is not sufficient to provide a statistically satisfactory description of the bulk environment of a physical system. As a consequence, a reliable estimate of finite-size effects in many-particle systems is key to judge the validity of a theoretical model or the accuracy of a numerical simulation. In this context, we propose the use of a theorem on the free-energy cost for separating a system into smaller independent subsystems [J. Stat. Mech.: Theory Exp. (2017) 083201; Lett. Math. Phys. 112, 97 (2022)] to estimate the relevance of finite-size effects in thermodynamic quantities from computer simulations. The key aspect of this study is that for two-body potentials, as mostly occurring in physics, the method requires only two-body distribution functions and the particle number density. The calculation of the involved physical quantities can be done numerically on a three-dimensional grid. In some cases even analytical estimates are possible and as an example the uniform interacting electron gas in the ground state is considered; we derive an approximating scaling law for the finite-size effects.
With the help of von Neumann entropy, we study numerically the localization properties of two interacting particles (TIP) with on-site interactions in one-dimensional disordered, quasiperiodic, and slowly varying potential systems, respectively. We find that for TIP in disordered and slowly varying potential systems, the spectrum-averaged von Neumann entropy ⟨Ev⟩⟨{E}_{v}⟩⟨Ev​⟩ first increases with interaction UUU until its peak, then decreases as UUU gets larger. For TIP in the Harper model [S. N. Evangelou and D. E. Katsanos, Phys. Rev. B 56, 12797 (1997)], the functions of ⟨Ev⟩⟨{E}_{v}⟩⟨Ev​⟩ versus UUU are different for particles in extended and localized regimes. Our numerical results indicate that for these two-particle systems, the von Neumann entropy is a suitable quantity to characterize the localization properties of particle states. Moreover, our studies propose a consistent interpretation of the discrepancies between previous numerical results.