Specific heat of the $S=\frac{1}{2}$ Heisenberg model on the kagome lattice: High-temperature series expansion analysis

We compute specific heat of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model on the kagome lattice. We use a recently introduced technique to analyze high-temperature series expansion based on the knowledge of high-temperature series expansions, the total entropy of the system and the low-temperature expected behavior of the specific heat, as well as the ground-state energy. In the case of the kagome-lattice antiferromagnet, this method predicts a low-temperature peak at $T∕J\lesssim 0.1$.

Figure 1

(Color online) Specific heat obtained from Padé approximants to the high-temperature series of $c_v\left(T\right)$. Only approximations of degrees $[u,u+1]$, ($[u,u+2]$ and $[u,u+3]$) are considered. They vanish at zero temperature as $T$ ($T^2$ and $T^3$). The only six such approximants from order 9 to 17 which are positive on the positive real axis are shown. The ground-state energy per site $e_0$ and entropy $s_0$ obtained by integrating these Padé approximations is indicated.

Figure 2

(Color online) Left: Series expansion for $s\left(e\right)$ (entropy per site) as a function of the energy per site $e$. The result of the bare series are displayed for orders from ${\beta }^{11}$ to ${\beta }^{17}$. The dashed vertical lines indicate upper $(e=-0.84267)$ and lower $(e=-0.909952)$ rigorous bounds on the ground-state energy in the thermodynamic limit (Ref. 19 Right: Same data as on the left panel but plotted as a function of temperature $T=1∕s^{\prime }\left(e\right)$. For each curve the lowest temperature corresponds to $e=-0.909952$ (lower bound). The black segment corresponds to $e=-0.84267$ (upper bound).

Figure 3

(Color online) Zero temperature limit of $c_0$ [see Eq. (4)] for the different Padé approximants as a function of the ground-state energy. Top: a $c_v\sim (T∕c_0)$ behavior is assumed at low temperature. Bottom: $c_v\sim {(T∕c_0)}^2$. The degree $u$ of the numerator of each approximant is given. The degree of the denominator is $d=n-u$ where $n$ is the order of the series. Left: order $n=16$. Right: order $n=17$. The number of physical approximant $\left(N_{\text{Pade}}\right)$ is plotted as a function of $e_0$ in each lower panel.

Figure 4

(Color online) Specific heat computed at order ${\beta }^{17}$ with the ground-state energies $e_0=-0.89$ (top), $e_0=-0.88$ (center), and $e_0=-0.865$ (bottom). Left: $c_v\sim T$. Right: $c_v\sim T^2$. The different curves correspond to all the physical Padés approximants. The degree $u$ of the numerator is indicated (the denominator has degree $d=17-u$).

Figure 5

(Color online) Specific heat curves from order ${\beta }^{13}$ to ${\beta }^{17}$ with $e_0=-0.89$ (top), $e_0=-0.88$ (center), and $e_0=-0.865$ (bottom). The degree of the numerator of each approximant is indicated and each column corresponds to a given order $\left({\beta }^n\right)$ of the series ($n=13$, 14, 15, 16, and 17 from left to right). Left panels: $c_v\sim T$. Right panels: $c_v\sim T^2$. In all cases the specific heat shows a maximum around $T\simeq 1.3$ [corresponding to $e\left(T\right)\simeq -0.7$, see Fig. 2] and a low-temperature peak (or shoulder).