arXiv subjects

Samuel Savitz

Publications and source records attributed to Samuel Savitz.

4 recordsLinked to original sources

arXiv2019

Anderson Localization on the Bethe Lattice using Cages and the Wegner Flow

Anderson localization on tree-like graphs such as the Bethe lattice, Cayley tree, or random regular graphs has attracted attention due to its apparent mathematical tractability, hypothesized connections to many-body localization, and the possibility of non-ergodic extended regimes. This behavior has been conjectured to also appear in many-body localization as a "bad metal" phase, and constitutes an intermediate possibility between the extremes of ergodic quantum chaos and integrable localization. Despite decades of research, a complete consensus understanding of this model remains elusive. Here, we use cages, maximally tree-like structures from extremal graph theory; and numerical continuous unitary Wegner flows of the Anderson Hamiltonian to develop an intuitive picture which, after extrapolating to the infinite Bethe lattice, appears to capture ergodic, non-ergodic extended, and fully localized behavior.

cond-mat.dis-nn↗

arXiv2017

Exceptional Lattice Green's Functions

The three exceptional lattices, $E_6$, $E_7$, and $E_8$, have attracted much attention due to their anomalously dense and symmetric structures which are of critical importance in modern theoretical physics. Here, we study the electronic band structure of a single spinless quantum particle hopping between their nearest-neighbor lattice points in the tight-binding limit. Using Markov chain Monte Carlo methods, we numerically sample their lattice Green's functions, densities of states, and random walk return probabilities. We find and tabulate a plethora of Van Hove singularities in the densities of states, including degenerate ones in $E_6$ and $E_7$. Finally, we use brute force enumeration to count the number of distinct closed walks of length up to eight, which gives the first eight moments of the densities of states.

math-ph↗

arXiv2017

Multiple-scale structures: from Faraday waves to soft-matter quasicrystals

For many years, quasicrystals were observed only as solid-state metallic alloys, yet current research is now actively exploring their formation in a variety of soft materials, including systems of macromolecules, nanoparticles and colloids. Much effort is being invested in understanding the thermodynamic properties of these soft-matter quasicrystals in order to predict and possibly control the structures that form, and hopefully to shed light on the broader yet unresolved general questions of quasicrystal formation and stability. Moreover, the ability to control the self-assembly of soft quasicrystals may contribute to the development of novel photonic or other applications based on self-assembled metamaterials. Here a path is followed, leading to quantitative stability predictions, that starts with a model developed two decades ago to treat the formation of multiple-scale quasiperiodic Faraday waves (standing wave patterns in vibrated fluid surfaces) and which was later mapped onto systems of soft particles, interacting via multiple-scale pair potentials. The article reviews, and substantially expands, the quantitative predictions of these models, while correcting a few discrepancies in earlier calculations, and presents new analytical methods for treating the models. In so doing, a number of new stable quasicrystalline structures with octagonal, octadecagonal and higher-order symmetries, some of which may, it is hoped, be observed in future experiments.

cond-mat.soft↗

arXiv2017

Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations

The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present a number of approaches to resolving these issues including a choice of generator which induces what we call the "uniform tangent decay flow" and three numerical integrators specifically designed to perform continuous unitary transformations efficiently while preserving the unitarity of flow. We conclude by applying one of the flow algorithms to a simple calculation that visually demonstrates the many-body localization transition.

cond-mat.dis-nn↗