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.

Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

“…The first rigorous results in this direction appeared recently in [35]; until then the results consisted of numerical (and quite accurate, in fact, as our rigorous results confirmed) computations and some soft (but nontrivial) analysis. In [35], the previously postulated multifractal structure of the zeros in the thermodynamic limit of the Ising partition function with Fibonacci-modulated couplings was proved; however, an open problem remained: how does the above result depend on the choice of a sequence from the subshift generated by the Fibonacci substitution sequence? One way to attack this problem would be to relate it to a spectral problem of some operator (such as a Jacobi operator or a CMV matrix).…”

“…Then for a given ω ∈ Ω, we take J i = p(ω i ). The case where ω = u has been considered in a number of papers (see, for example, [3,4] and references therein), and the more general case (with variable ω) was recently considered in [35].…”

“…(2) In the case ω = u, the problem was studied numerically in [3], heuristically in [4], and rigorously in [35]; however, the dependence (or lack thereof ) on ω remained a mystery. (3) A connection between the Lee-Yang zeros and spectra of linear operators has been postulated by a few authors.…”

Mathematics of Aperiodic Order

“…Aside from the context described here, this map appears in a natural way in problems related to dynamics of mapping classes [85], Fuchsian groups [20], number theory [19], Painlevé sixth equations [31,103], the Ising model for quasicrystals [15,91,175,176], the Fibonacci quantum walk [154,155], among others [7,65,167,177]. See [30] or [17] for an algebraic explanation of this universality.…”

On the Spectrum of 1D Quantum Ising Quasicrystal

“…Not to get sidetracked too far, let us conclude this introduction by noting that we study and apply the trace map as a real analytic map; study of the complexified version (though within a different context) was carried out by S. Cantat in [11]. We have applied the trace map as a holomorphic map on C 3 in [80] as a renormalization map for the classical one-dimensional Ising model with quasi-periodic nearest neighbor interaction and magnetic field, and were able to relate the analyticity of the free energy function to the analyticity of the escape rate of orbits under the action of the trace map, proving absence of phase transitions. Also in [80] we applied the techniques from the present paper to obtain precise description of the Lee-Yang zeros of the classical model in the thermodynamic limit.…”