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Localization of quantum particles with long-range hopping induced by reducing lattice size

Joshua T. Cantin, Tianrui Xu, Roman V. Krems

Published Apr 6, 2016• 0 citations
OpenAlex

Abstract

Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. Here, we show that such particles undergo localization in lattices of finite size, even macroscopically finite. This leads to a rather unusual phenomenon: quantum particles can transverse a lattice of size $10A$, but not a lattice of size $A$.

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Localization of quantum particles with long-range hopping in 3D lattices of finite size

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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 $10A$, but not through a lattice of size $A$.

•Apr 6, 2016•0 citations

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Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. Here, we show that such particles undergo localization in lattices of finite size, even macroscopically finite. This leads to a rather unusual phenomenon: quantum particles can transverse a lattice of size 10A10A10A, but not a lattice of size AAA.
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.
It has become widely accepted that particles with long-range hopping do not undergo Anderson localization. However, several recent studies demonstrated localization of particles with long-range hopping. In particular, it was recently shown that the effect of long-range hopping in one-dimensional (1D) lattices can be mitigated by cooperative shielding, which makes the system behave effectively as one with short-range hopping. Here, we show that cooperative shielding, demonstrated previously for 1D lattices, extends to 3D lattices with isotropic long-range ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ hopping, but not to 3D cubic lattices with anisotropic long-range hopping. The specific anisotropy we consider corresponds to the interaction between dipoles aligned along one of the principal axes of the lattice. We demonstrate the presence of localization in 3D lattices with uniform ($\ensuremath{\alpha}=0$) isotropic long-range hopping and the absence of localization with uniform anisotropic long-range hopping by using the scaling behavior of eigenstate participation ratios. We use the scaling behavior of participation ratios and energy-level statistics to show that the existence of delocalized, extended nonergodic, or localized states in the presence of disorder depends on both the exponents $\ensuremath{\alpha}$ and the anisotropy of the long-range hopping amplitudes.
We present a theory of Anderson localization on a one-dimensional lattice with translation-invariant hopping. We find by analytical calculation the localization length for arbitrary finite-range hopping in the single propagating channel regime. Then by examining the convergence of the localization length, in the limit of infinite hopping range, we revisit the problem of localization criteria in this model and investigate the conditions under which it can be violated. Our results reveal possibilities of having delocalized states by tuning the long-range hopping.
We study the effects of long-range hopping and long-range interparticle interactions on the quantum walk of hard-core bosons in ideal and disordered one-dimensional lattices. We find that the range of hopping has a much more significant effect on the particle correlation dynamics than the range of interactions. We illustrate that long-range hopping makes the correlation diagrams asymmetric with respect to the sign of the interaction. We examine the relative role of repulsive and attractive interactions on the dynamics of scattering by isolated impurities and Anderson localization in disordered lattices. We show that weakly repulsive interactions increase the probability of tunneling through isolated impurities and decrease the localization.
We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t)\ensuremath{\sim}lnt$ and the saturated entanglement entropy scales with system size LLL as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.
Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/{R}^{\ensuremath{\alpha}}$ is investigated combining analytical theory considering resonant interactions and a finite-size scaling of exact numerical solutions with number of spins NNN. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for a ddd-dimensional system including the absence of localization in the infinite system at $\ensuremath{\alpha}<2d$ and a universal scaling of a critical energy disordering ${W}_{c}\ensuremath{\propto}{N}^{\frac{2d\ensuremath{-}\ensuremath{\alpha}}{d}}$.