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The European Physical Journal Plus

The XYZ model by the series expansion of exponentials for spin matrices in the mean-field approximation

Erhan Albayrak

Published Mar 10, 2023• 2 citations
Publisher
DOI
10.1140/epjp/s13360-023-03802-y

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We consider the XYZXYZXYZ chain model of arbitrary spin SSS in the high-temperature region, with external magnetic field and single-ion anisotropy term. Our high-temperature expansion of the Helmholtz free energy is analytic in the parameters of the model for SSS, which may range from 1∕21∕21∕2 to the classical limit of infinite spin $(S\ensuremath{\rightarrow}\ensuremath{\infty})$. Our expansion is carried out up to order ${(J\ensuremath{\beta})}^{5}$. Our results agree with numerical results of the specific heat per site for S=1∕2S=1∕2S=1∕2, obtained by the Bethe ansatz, with h=0h=0h=0 and D=0D=0D=0. Finally, we show that the magnetic susceptibility and magnetization of the quantum model can be well approximated by their classical analog in this region of temperature.
Using renormalization group methods, we study the Heisenberg–Ising XYZ chain in an external magnetic field directed as the z axis, in the case of small coupling J3 in the z direction. In particular, we focus our attention on the asymptotic behaviour of the spin correlation function in the direction of the magnetic field and the singularities of its Fourier transform. An expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. Moreover, by using hidden symmetries of the model, we show that this condition is indeed verified, if J3 is small enough, and we derive an expansion for the spin correlation function. We also prove, by means of an approximate Ward identity, that a critical index, related with the asymptotic behaviour of the correlation function, is exactly vanishing, together with other properties, so obtaining a rather detailed description of the XYZ correlation function.
Uses high-field series expansions for the square lattice Ising model to investigate the physical singularity in the magnetisation as a function of the field. High-field series were obtained to order 35 at temperatures T approximately=0.5 T c and T approximately=0.766T c using series expansion techniques based on corner transfer matrices. At neither temperature is there any evidence of a spinodal line; the behaviour is consistent with the predictions of the droplet model, suggesting that the first-order transition line is a line of infinitely differentiable singularities.