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Annalen der Physik

Quantum diffusion and scaling of localized interacting electrons

Ole Halfpap

Published Jul 6, 2001• 1 citation
Publisher

Abstract

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.

DOI
10.1002/andp.20015130801

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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.
Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. It is thus natural to assume that such particles can traverse any finite-size lattice. Here, we demonstrate that this is not generally true. The delocalization mechanism is induced by resonances between distant lattice sites. The number of such resonances diverges with the system size. For a finite-size lattice the number of resonances is finite and may not be sufficient to result in delocalization. We consider particles with long-range (dipolar) hopping in three-dimensional lattices with diagonal disorder and random dilution. We compute the wavepacket dynamics of particles placed in an individual lattice site, the inverse participation ratios and the fluctuations of the wavefunctions with disorder. We show that, for certain finite-size disordered lattices, particles remain localized within a finite volume much smaller than the lattice size, and that the wavefunctions exhibit the log-normal fluctuations characteristic of Anderson localization. We characterize the localization properties as functions of dilution and diagonal disorder. We combine our results with scaling theory to obtain the size dependence of the localization--diffusion crossover. Our results indicate that particles with long-range hopping undergo exponential localization in lattices of finite size, even macroscopically finite. Our phase diagrams illustrate a rather unusual phenomenon: quantum particles can diffuse through a lattice of size 10A10A10A, but not through a lattice of size AAA.
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 .
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 numerically investigate the transport properties of interacting spinless electrons in disordered systems. We use an efficient method which is based on the diagonalization of the Hamiltonian in the subspace of the many-particle Hilbert space which is spanned by the low-energy Slater states. Low-energy properties can be calculated with an accuracy comparable to that of exact diagonalization but for larger system sizes. The method works well in the entire parameter space, and it can handle long-range as well as short-range interactions. Using this method we calculate the combined effect of disorder and interactions on the Kubo-Greenwood conductance and on the sensitivity of the ground state energy to a twist in the boundary conditions. We find that the influence of the interactions on the transport properties is opposite for large and small disorder. In the strongly localized regime (small kinetic energy, large disorder) interactions increase the transport whereas for weak disorder (large kinetic energy) interactions decrease the transport.
The properties of the entanglement entropy (EE) in one-dimensional disordered interacting systems are studied. Anderson localization leaves a clear signature on the average EE, as it saturates on the length scale exceeding the localization length. This is verified by numerically calculating the EE for an ensemble of disordered realizations using the density matrix renormalization group method. A heuristic expression describing the dependence of the EE on the localization length, which takes into account finite-size effects, is proposed. This is used to extract the localization length as a function of the interaction strength. The localization length dependence on the interaction fits nicely with the expectations.
We develop a numerical technique to study Anderson localization in interacting electronic systems. The ground state of the disordered system is calculated with quantum Monte Carlo simulations while the localization properties are extracted from the "Thouless conductance" g, i.e., the curvature of the energy with respect to an Aharonov-Bohm flux. We apply our method to polarized electrons in a two-dimensional system of size L. We recover the well-known universal beta(g)=dlogg/dlogL one parameter scaling function without interaction. Upon switching on the interaction, we find that beta(g) is unchanged while the system flows toward the insulating limit. We conclude that polarized electrons in two dimensions stay in an insulating state in the presence of weak to moderate electron-electron correlations.