We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes $t(r)\propto r^{-a}$. In contrast to the standard one-dimensional Anderson model ($a\to\infty$), in which all states are localized and the localization length is minimal at the band edge, the long-range model with $1<a<3/2$ exhibits a disorder-driven transition at the band edge, while high-energy states remain localized at arbitrary disorder strength. We investigate this transition for the ground state in momentum space. In the weak-disorder regime, we derive perturbative expressions for the characteristic functions and moments of the momentum-space wave function, as well as for its fractal dimensions. Our results demonstrate that the ground state exhibits $\it semilocalization$ rather than conventional localization, thereby extending the class of models displaying the unusual $\it semifractality$ of wave functions.
In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane''spin rotation symmetry. True long-range order of $(S^x,S^y)$ arises at the phase transition between tw...
Nearest-neighbor (NN) tight-binding models (TBMs) on plaquette-centered $\{ 10,3\}$ (Schl\"afli symbol), $\{ 8,3\}$, and $\{ 8,4\}$ hyperbolic lattices on a Poincar\'e disk with open boundary conditions display a vanishing, a finite, and a diverging density of states (DOS) near zero energy (band center), respectively,...
Sheersh Sen, Christopher A. Leong, Bitan Roy· 0 citations
We propose a $q$-deformation in the single-particle kinetic energy and investigate how it modifies the localization in the one-dimensional Aubry-Andr\'e (AA) model. We construct a Hermitian $q$-deformed kinetic operator as a nonlinear function of the lattice translation operator, preserving the uniform lattice and reco...
We study local eigenvalue statistics and dynamical (de)-localization properties for long-range Anderson models, including the fractional Anderson model. These systems are random Anderson-type perturbations of operators with long-range (non random) hopping terms of the form $|T(n,m)| \sim \|n-m\|^{-(d+2\beta)}$ for $\be...
P. Hislop, Rodrigo Matos, C. Rojas-Molina· 0 citations
We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as $1/r^{2+\sigma}$. At the marginal case $\sigma=2$, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a...
We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances $h(x)$ either summable or with a critical decay, satisfying $h(x) \sim (\log |x|)^a/|x|^2$ as $|x|\to\infty$ for some $a>-1$. We determine the precise high-temperature as...
Quentin Berger, Francesca Cottini· 0 citations
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