Features of the spectral density of a spin system
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 of the heat-capacity at some critical value \(E_c\) of the energy, because of the presence of `narrow peaks' of \(d^2\Psi (E)/dE^2\) near \(E_c\). On the one hand, this agrees with Onsager old results; on the other hand, the analysis of the paper shows that this result depends strongly on the properties of the spectral density of the model: a very small change in the modeling of the interaction of the spins can lead to a loss of previously mentioned `narrow peaks', and, consequently, of the logarithmic divergence of the heat capacity. As a final remark, the connection between free energy and spectral density outlined in this paper is reminiscent of the Thermodynamic Bethe Ansatz (TBA) method firstly developed by \textit{C. N. Yang} and \textit{C. P. Yang} [J. Math. Phys. 10, 1115--1122 (1969; Zbl 0987.82503)]. It would be of interest to interpret the conclusions of the paper in the framework of the well-known TBA technique.
scientific article; zbMATH DE number 6908965