Tenth-order high-temperature expansion for the susceptibility and the specific heat of spin-$s$ Heisenberg models with arbitrary exchange patterns: Application to pyrochlore and kagome magnets

We present the high-temperature expansion (HTE) up to tenth order of the specific heat $C$ and the uniform susceptibility $\chi$ for Heisenberg models with arbitrary exchange patterns and arbitrary spin quantum number $s$. We encode the algorithm in a c++ program provided in the Supplemental Material [http://link.aps.org/supplemental/10.1103/PhysRevB.89.014415] which allows to explicitly get the HTE series for concrete Heisenberg models. We apply our algorithm to pyrochlore ferromagnets and kagome antiferromagnets using several Padé approximants for the HTE series. For the pyrochlore ferromagnet, we use the HTE data for $\chi$ to estimate the Curie temperature $T_c$ as a function of the spin quantum number $s$. We find that $T_c$ is smaller than that for the simple-cubic lattice, although both lattices have the same coordination number. For the kagome antiferromagnet, the influence of the spin quantum number $s$ on the susceptibility as a function of renormalized temperature $T/s(s+1)$ is rather weak for temperatures down to $T/s(s+1)\sim 0.3$. On the other hand, the specific heat as a function of $T/s(s+1)$ noticeably depends on $s$. The characteristic maximum in $C\left(T\right)$ is monotonously shifted to lower values of $T/s(s+1)$ when increasing $s$.

Figure 1

Padé approximant [4,6] of the inverse susceptibility $1/\chi$ of the pyrochlore Heisenberg ferromagnet for various spin quantum numbers $s$.

Figure 2

Curie temperature $T_c$ in dependence on the inverse spin quantum number $1/s$ of the pyrochlore Heisenberg ferromagnet for $s=1/2,1,3/2,2,5/2,15/2$, and $s\to \infty$. For comparison, we also show the $T_c$ values for the simple-cubic Heisenberg ferromagnet. The Monte Carlo data (MC) for $s=1/2$ and $s\to \infty$ are taken from Refs. [48] and [49], respectively.

Figure 3

(a) Specific heat $C$ and (b) susceptibility $\chi$ of the $s=1/2$ kagome Heisenberg antiferromagnet. For comparison, we show the raw data of the 15th/16th-order HTEs and the corresponding Padé [7,8] approximant taken from Ref. [32].

Figure 4

(a) Specific heat $C$ and (b) susceptibility $\chi$ of the classical kagome Heisenberg antiferromagnet. For comparison, we show the Monte Carlo data (MC) taken from Ref. [20].

Figure 5

HTE data for the susceptibility $\chi$ of the spin-$s$ kagome Heisenberg antiferromagnet, (a) Padé [4,6] and (b) Padé [5,5]. For comparison, we show the Monte Carlo data (MC) taken from Ref. [20].

Figure 6

Specific heat $C$ of the spin-$s$ kagome Heisenberg antiferromagnet, (a) Padé [4,6] and (b) Padé [5,5]. For comparison, we show the Monte Carlo data (MC) taken from Ref. [20].

Figure 7

(a) Height $C_{max}$ and (b) position $T_{max}$ of the maximum in the specific heat $C$ in dependence on the inverse spin quantum number $1/s$ of the kagome Heisenberg antiferromagnet. The Monte Carlo results for $s\to \infty$ are taken from Ref. [20] (MC I) and Ref. [21] (MC II). Note, however, that the maximum in $C\left(T\right)$ for the classical model is not well pronounced, rather there is a fairly broad region of high values of $C$.[21]