Finding spectral gaps in quasicrystals

Abstract

We show how the spectrum of normal discrete short-range infinite-volume operators can be approximated with two-sided error control using only data from finite-sized local patches. As a corollary, we prove the computability of the spectrum of such infinite-volume operators with the additional property of finite local complexity and provide an explicit algorithm. Such operators appear in many applications, e.g. as discretizations of differential operators on unbounded domains or as so-called tight-binding Hamiltonians in solid state physics. For a large class of such operators, our result allows for the first time to establish computationally also the absence of spectrum, i.e. the existence and the size of spectral gaps. We extend our results to the ε-pseudospectrum of non-normal operators, proving that also the pseudospectrum of such operators is computable.

The Mass Gap of the Space‐time and its Shape

“…Recently, it was shown that spectral gaps exist in Hamiltonian with quasicrystalline order. [63] Quasicrystal considerations in Holography, the basic structure of nature, and cosmology are discussed in ref. [64][65][66][67][68][69][70][71].…”

Super band gaps and periodic approximants of generalised Fibonacci tilings

“…Characterizing the spectra of quasi-periodic differential operators is a longstanding and fascinating problem [2,3]. In particular, one-dimensional Schrödinger operators with quasi-periodic potentials have been widely studied.…”

Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

“…Gapless edge modes are also expected to be relevant there. So far, they have been studied only in the single-particle setting with the goal to numerically distinguish the behavior of edge modes from bulk spectrum [39] or to prove the existence of a bulk gap in the presence of gapless edge modes [28]. These approaches have yet to be generalized to the many-body setting.…”