High-temperature series expansion of the Helmholtz free energy of the quantum spin-$S$ $XYZ$ chain

We consider the $XYZ$ chain model of arbitrary spin $S$ 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 $S$, which may range from $1∕2$ to the classical limit of infinite spin $(S\to \infty )$. Our expansion is carried out up to order ${\left(J\beta \right)}^5$. Our results agree with numerical results of the specific heat per site for $S=1∕2$, obtained by the Bethe ansatz, with $h=0$ and $D=0$. 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.

Figure 1

Specif heat of spin-$1∕2$ $XYZ$ chain with $h=0$, $D=0$, $J_x=-1.0$, $J_y=1.2$, and $J_z=2.0$. The solid line corresponds to the numerical solution of Takahashi and the dashed line represents the $\beta$ expansion of this function derived from (3).

Figure 2

In (a), we compare the classical $(S\to \infty )$ and the quantum ($S=1∕2$, 1, $3∕2$ and 4) magnetic susceptibility per site as a function of $\left(J\beta \right)$. In (b), we present the relative percent difference between the quantum and classical results. We let $\Delta =1$, $\delta =1$, $h∕J=0.3$, and $D∕J=-0.5$.

Figure 3

The magnetization versus $h∕J$ at $J\beta =0.4$ is presented in (a) for $S=1∕2$, 1, $3∕2$, 4 and $S\to \infty$. (b) shows the percent difference of those quantum magnetization curves with respect to the classical one. We let $\Delta =1$, $\delta =1$, and $D∕J=-0.5$.