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arXiv:1811.08703 · Optical Memory and Neural Networks

n-vicinity Method and 1D Ising Model

Boris Kryzhanovsky, Leonid B. Litinskii

Published Oct 1, 2018• 3 citations
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Abstract

Abstract For a 1D Ising model, we obtained an exact expression for the spectral density in an n-vicinity of the ground state and explained why our n-vicinity method with the Gaussian approximation of the spectral density did not applicable in this case. We also found an analytical expression for the distribution of magnetization at an arbitrary temperature. When the temperature tends to zero the distribution of magnetization gradually flattens.

DOI
10.3103/s1060992x18040033

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Abstract For a 1D Ising model, we obtained an exact expression for the spectral density in an n-vicinity of the ground state and explained why our n-vicinity method with the Gaussian approximation of the spectral density did not applicable in this case. We also found an analytical expression for the distribution of magnetization at an arbitrary temperature. When the temperature tends to zero the distribution of magnetization gradually flattens.
The spectral density of a quantum Ising model in transverse and longitudinal fields for a large but finite number of spins N is discussed in detail. When both fields are of the same order, the spectral densities in the bulk are well approximated by a Gaussian function, which is typical behaviour for many-body models with short-range interactions. The main part of the paper is devoted to the investigation of a different characteristic case where spectral densities have peaks related with strong degeneracies of unperturbed states in certain limits of field amplitudes. In the strict limit peaks overlap and disappear but for values of N accessible in numerical calculations they often strongly influence spectral densities as well as other quantities. A simple method is developed that permits us to find general approximation formulae for the multi-peak structure of spectral density, which is in good agreement with numerics.
We formulate the spectral-density method in classical statistical mechanics in strict analogy with the known quantum version, and we apply it to the classical Heisenberg ferromagnetic model in an external field. A new formula for the magnetization in the classical formulation of spin-vector Green functions is derived for arbitrary spatial dimensionality. Furthermore, the static properties of the one- and three-dimensional cases are considered in detail. We obtain accurate results over a wide range of temperatures with the external field for the one-dimensional model and without field for the three-dimensional model. In the first case very good agreement with the exact numerical transfer-matrix data is also found in the region of higher temperature where the interacting-boson approach fails. In particular the zero-field susceptibility results are found to be consistent with the exact results obtained by Fisher.
Spectral density of quantum Ising model in two fields for large but finite number of spins NNN, is discussed in detail. When all coupling constants are of the same order, spectral densities in the bulk are well approximated by a Gaussian function which is typical behaviour for many-body models with short-range interactions. The main part of the paper is devoted to the investigation of a different characteristic case when spectral densities have peaks related with strong degeneracies of unperturbed states in certain limits of coupling constants. In the strict limit N→∞N\to\inftyN→∞, peaks overlap and disappear but for values of NNN accessible in numerical calculations they often strongly influence spectral densities and other quantities as well. A simple method is developed which permits to find general approximation formulae for multi-peak structure of spectral density in good agreement with numerics.