Hyperbolicity of the trace map for a strongly coupled quasiperiodic Schrödinger operator

Abstract. We study the spectral properties of the Sturm Hamiltolian of eventually constant type, which includes the Fibonacci Hamiltonian. Let s be the Hausdorff dimension of the spectrum. For V > 20, we show that the restriction of the s-dimensional Hausdorff measure to the spectrum is a Gibbs type measure; the density of states measure is a Markov measure. Based on the fine structures of these measures, we show that both measures are exact dimensional; we obtain exact asymptotic behaviors for the optimal Hölder exponent and the Hausdorff dimension of the density of states measure and for the Hausdorff dimension of the spectrum. As a consequence, if the frequency is not silver number type, then for V big enough, we establish strict inequalities between these three spectral characteristics. We achieve them by introducing an auxiliary symbolic dynamical system and applying the thermodynamical and multifractal formalisms of almost additive potentials.

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 97%

The Spectral Properties of the Strongly Coupled Sturm Hamiltonian of Eventually Constant Type

Qu¹

2016

“…We can now apply Theorem 2.1 with G : Ω → GL(2, R) given by G(ω) = T ω,1 , in which case T (N ) ω = G(T N −1 ω)G(T N −2 ω) · · · G(ω) (recall that T : Ω → Ω is the left shift). That G is continuous is obvious, since G depends only on the first element of the sequence ω; also every entry of G(ω) is strictly positive, as is evident from (22). Now applying Theorem 2.1 on both sides of the inequality in (24), we see that…”

Section: 3mentioning

confidence: 82%

“…That G is continuous is obvious, since G depends only on the first element of the sequence ω; also every entry of G(ω) is strictly positive, as is evident from (22). Now applying Theorem 2.1 on both sides of the inequality in (24), we see that…”

Section: 3mentioning

confidence: 82%

Properties of 1D Classical and Quantum Ising Models: Rigorous Results

Yessen

2013

In this paper we consider one-dimensional classical and quantum spin-1/2 quasi-periodic Ising chains, with two-valued nearest neighbor interaction modulated by a Fibonacci substitution sequence on two letters. In the quantum case, we investigate the energy spectrum of the Ising Hamiltonian, in presence of constant transverse magnetic field. In the classical case, we investigate and prove analyticity of the free energy function when the magnetic field, together with interaction strength couplings, is modulated by the same Fibonacci substitution (thus proving absence of phase transitions of any order at finite temperature). We also investigate the distribution of Lee-Yang zeros of the partition function in the complex magnetic field regime, and prove its Cantor set structure (together with some additional qualitative properties), thus providing a rigorous justification for the observations in some previous works. In both, quantum and classical models, we concentrate on the ferromagnetic class.

“…Sinai on thermodynamic formalism]). The method of trace maps has led, for example, to fundamental results in spectral theory of discrete Schrödinger operators and Ising models on one-dimensional quasi-periodic lattices (for Schrödinger operators: [5,12,14,15,17,18,20,23,48,60,71], for Ising models: [6,7,13,24,33,74,76,84], and references therein).…”

Section: Introductionmentioning

confidence: 99%

On the Spectrum of 1D Quantum Ising Quasicrystal

Yessen

2013

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 non-constant over the spectrum. This forms a rigorous counterpart of numerous numerical studies.