On the spectrum of 1D quantum Ising quasicrystal

TL;DR

This paper rigorously analyzes the energy spectrum of a 1D quantum Ising quasicrystal, revealing it as a zero-measure Cantor set with a continuous, nonconstant local Hausdorff dimension, confirming numerical predictions.

Contribution

It provides a rigorous mathematical proof of the spectral properties of 1D quantum Ising quasicrystals, complementing previous numerical studies.

Findings

Energy spectrum is a Cantor set of zero Lebesgue measure.

Local Hausdorff dimension varies continuously over the spectrum.

Spectrum properties are rigorously established, confirming numerical results.

Abstract

We consider one dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan-Wigner transformation of the spin operators to spinless fermions, the energy spectrum can be computed exactly on a finite lattice. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and nonconstant over the spectrum. This forms a rigorous counterpart of numerous numerical studies.