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Spectral density of a one-dimensional quantum Ising model: Gaussian and multi-Gaussian approximations. (English) Zbl 1304.60045
The purpose of the paper is to discuss the spectral densities of one-dimensional spin chains. As a typical example, the authors consider the quantum spin-\({1\over 2}\) Ising model in transverse and longitudinal fields. Let \(N\) be the number of spins. It is confirmed that, in the thermodynamic limit \(N\to \infty\), the spectral densities of the Ising model, when all coupling constants are of the order of unity, attain a Gaussian form, even at a relatively small \(N\). Moreover, different types of corrections at large but finite \(N\) are discussed. The simplest are corrections to the equation of a Gaussian form in inverse powers of \(N\), in close analogy with the usual corrections to the central limit theorem. The second kind of correction is more drastic, because it manifests as pronounced peaks in the spectral density. These peaks are related to strong degeneracies of the eigenvalues in certain limits of the coupling constants. A simple method is developed which permits to obtain approximate formulae describing well multi-peaks spectral densities that are in good agreement with numerical calculations for different values of the coupling constants. Limiting densities are represented as a sum of different Gaussian functions (multi-Gaussians), whose parameters are calculated analytically from the Hamiltonian without the full solution of the problem.

60G15 Gaussian processes
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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