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

Abstract

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. © 2013 Springer Basel.