Features of the spectral density of a spin system
type: Item
instance of: scholarly article
title: Features of the spectral density of a spin system
zbMATH Open document ID: 1434.82020
DOI: 10.1134/S1064562418020126
published in: Doklady Mathematics
publication date: 2018-07-20T00:00:00Z
review text: The article deals with thermodynamic properties of systems with classical spins. In particular, in the limit of large number \(N\) of spins the author finds a connection between the free energy \(f(\beta )\) (at temperature \(T=1/k \beta\)) and the `spectral density' \(D(E)\), i.e., the degeneracy of a generic energy level \(E\). The connection is summarised in the two relations (6) of the paper: \(\Psi (E)=bE-f(b)\), \(E=\frac{df}{db}\). After establishing this connection in Section 1, in Section 2 the author tests such relations for some spin models: one-dimensional Ising model, Ising model on a square lattice, Ising model on a Bethe lattice with an arbitrary number of nearest neighbours and a mean-field model. Plots of the function \(\Psi (E)=\frac{\ln D(E)}{N}\) and of its first and second derivatives versus the energy \(E\) are provided in Fig. 1 of the paper. With the aid of these plots, in Section 3 the author discusses the results, with a particular attention paid to possible phase transitions of the models. Indeed, using the connection spectral density/free energy he found and similar formulæ for the derivatives \(\frac{d\Psi}{dE}=b\), \(\frac{d^2\Psi}{dE^2}=\left ( \frac{d^2f}{db^2} \right )^{-1}\) (Formula (7) of the paper) one can state that properties of the phase transitions depend not only on the spectral function, but also (and especially) on its derivatives. Therefore, in approximate modeling of systems, the first and second derivatives of the spectral function have to be `reproduced with high accuracy'. For instance, from Formula (7) of the paper, it follows that if the second derivative of the spectral energy vanishes, then a second-order phase transition occurs. In the case of an inflection, at the phase transition the heat capacity diverges, in the case of a corner, the heat capacity is discontinuous. Discussion of these results for the 2d Ising model shows that for this model there is a second-order phase transition with logarithmic divergence …
reviewed by: Marco Rossi
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B23
Mathematics Subject Classification ID: 82B26
zbMATH DE Number: 6908965
zbMATH Keywords: thermodynamics of spin systems
zbMATH Keywords: phase transitions
zbMATH Keywords: modeling of spin systems
author: B. V. Kryzhanovskii
MaRDI profile type: Publication
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full work available at URL: s1064562418020126
OpenAlex ID: W2804464034
Wikidata QID: Q129765941
description: scientific article; zbMATH DE number 6908965