Physical Review Letters

No Enhancement of the Localization Length for Two Interacting Particles in a Random Potential

Rudolf A. Römer, Michael Schreiber

Published Jan 20, 1997• 66 citations

Abstract

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.

DOI: 10.1103/physrevlett.78.515

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Physical Review Letters

No Enhancement of the Localization Length for Two Interacting Particles in a Random Potential

Rudolf A. Römer, Michael Schreiber

Published Jan 20, 1997• 66 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.78.515

Related papers

No enhancement of the localization length for two interacting particles in a random potential

Rudolf A. Römer, Michael Schreiber

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 $λ_2$ on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of $λ_2$ 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 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 $λ_2$ on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of $λ_2$ 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 onsite interaction and also a long-range interaction.

arXiv:cond-mat/9612034Dec 3, 19960 citations

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Pil Hun Song, Doochul Kim

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 $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$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 $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$

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Diana Thongjaomayum, Alexei Andreanov, Thomas Engl +1 more

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Ole Halfpap

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 <$>\mitlambda λ 1 . For three and four particles, delocalization is even stronger. Based on analytical arguments, an upper bound for the n ‐particle localization length λ n is derived and shown to be in agreement with the numerical data, λ n ∝ λ 1 . Although the localization length increases superexponentially with particle number and can become arbitrarily large for small disorder, it does not diverge for finite λ 1 and n . Hence, no extended states exist in one dimension, at least for spinless fermions.Abstract A new numerical method is introduced that enables a reliable study of disorder‐induced localization of interacting particles. It is based on a quantum mechanical time evolution calculation combined with a finite size scaling analysis. The time evolution of up to four particles in one dimension is studied and localization lengths are defined via the long‐time saturation values of the mean radius, the inverse participation ratio and the center of mass extension. A systematic study of finite size effects using the finite size scaling method is performed in order to extract the localization lengths in the limit of an infinite system size. For a single particle, the well‐known scaling of the localization length λ 1 with disorder strength W is observed, λ 1 ∝ W —2 . For two particles, an interaction‐induced delocalization is found, confirming previous results obtained by numerically calculating matrix elements of the two‐particle Green's function: in the limit of small disorder, the localization length increases with decreasing disorder as λ 2 ∝ W —4 and can be much larger than <$>\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.

Annalen der PhysikJul 6, 20011 citation

Mobility edge of two interacting particles in three-dimensional random potentials

Filippo Stellin, Giuliano Orso

We investigate Anderson transitions for a system of two particles moving in a three-dimensional disordered lattice and subject to on-site (Hubbard) interactions of strength $U$ . The two-body problem is exactly mapped into an effective single-particle equation for the center-of-mass motion, whose localization properties are studied numerically. We show that, for zero total energy of the pair, the transition occurs in a regime where all single-particle states are localized. In particular, the critical disorder strength exhibits a nonmonotonic behavior as a function of $|U|$ , increasing sharply for weak interactions and converging to a finite value in the strong-coupling limit. Within our numerical accuracy, short-range interactions do not affect the universality class of the transition.We investigate Anderson transitions for a system of two particles moving in a three-dimensional disordered lattice and subject to on-site (Hubbard) interactions of strength $U$. The two-body problem is exactly mapped into an effective single-particle equation for the center-of-mass motion, whose localization properties are studied numerically. We show that, for zero total energy of the pair, the transition occurs in a regime where all single-particle states are localized. In particular, the critical disorder strength exhibits a nonmonotonic behavior as a function of $|U|$, increasing sharply for weak interactions and converging to a finite value in the strong-coupling limit. Within our numerical accuracy, short-range interactions do not affect the universality class of the transition.

arXiv:1903.06214Physical review. B./Physical review. BJun 28, 20195 citations

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 study two interacting particles in a random potential chain by means of the transfer matrix method. The dependence of the two-particle localization length

λ_2

on disorder and interaction strength is investigated. Our results demonstrate that the recently proposed enhancement of

λ_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 onsite interaction and also a long-range interaction.

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 $\ensuremath{\Delta}=4.0\ifmmode\pm\else\textpm\fi{}0.5.$

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 ${\ensuremath{\xi}}_{2}/{\ensuremath{\xi}}_{1}=F(u{\ensuremath{\xi}}_{1})$.

We investigate Anderson transitions for a system of two particles moving in a three-dimensional disordered lattice and subject to on-site (Hubbard) interactions of strength

U

. The two-body problem is exactly mapped into an effective single-particle equation for the center-of-mass motion, whose localization properties are studied numerically. We show that, for zero total energy of the pair, the transition occurs in a regime where all single-particle states are localized. In particular, the critical disorder strength exhibits a nonmonotonic behavior as a function of

|U|

, increasing sharply for weak interactions and converging to a finite value in the strong-coupling limit. Within our numerical accuracy, short-range interactions do not affect the universality class of the transition.