Thermodynamics of the kagome-lattice Heisenberg antiferromagnet with arbitrary spin $S$

We use a second-order rotational invariant Green's-function method (RGM) and the high-temperature expansion (HTE) to calculate the thermodynamic properties of the kagome-lattice spin-$S$ Heisenberg antiferromagnet with nearest-neighbor exchange $J$. While the HTE yields accurate results down to temperatures of about $T/S(S+1)\sim J$, the RGM provides data for arbitrary $T\ge 0$. For the ground state we use the RGM data to analyze the $S$ dependence of the excitation spectrum, the excitation velocity, the uniform susceptibility, the spin-spin correlation functions, the correlation length, and the structure factor. We found that the so-called $\sqrt{3}\times \sqrt{3}$ ordering is more pronounced than the $q=0$ ordering for all values of $S$. In the extreme quantum case $S=1/2$, the zero-temperature correlation length is only of the order of the nearest-neighbor separation. Then we study the temperature dependence of several physical quantities for spin quantum numbers $S=1/2,1,\cdots ,7/2$. As $S$ increases, the typical maximum in the specific heat and that in the uniform susceptibility are shifted toward lower values of $T/S(S+1)$, and the height of the maximum is growing. The structure factor $\mathcal{S}\left(\mathbf{q}\right)$ exhibits two maxima at magnetic wave vectors $\mathbf{q}={\mathbf{Q}}_i,i=0,1,$ corresponding to the $q=0$ and $\sqrt{3}\times \sqrt{3}$ state. We find that the $\sqrt{3}\times \sqrt{3}$ short-range order is more pronounced than the $q=0$ short-range order for all temperatures $T\ge 0$. For the spin-spin correlation functions, the correlation lengths, and the structure factors, we find a finite low-temperature region $0\le T<T^{*}\approx a/S(S+1), a\approx 0.2$, where these quantities are almost independent of $T$.

Figure 1

Illustration of the two most relevant classical states. Left: $q=0$ state with magnetic wave vector ${\mathbf{Q}}_0=(2\pi /\sqrt{3},0)$. Right: $\sqrt{3}\times \sqrt{3}$ state with magnetic wave vector ${\mathbf{Q}}_1=(0,4\pi /3)$. “$+$” and “$-$” symbols denote plaquettes of different vector spin chirality. The arrows indicate the basis vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$.

Figure 2

RGM GS energy per site $E_0/S^2$ in dependence on the inverse spin quantum number $1/S$ (minimal vs extended version). Note that $E_0$ for the extended version naturally coincides with the CCM data of [26, 39].

Figure 3

RGM GS static uniform susceptibility ${\chi }_0$ in dependence on the inverse spin quantum number $1/S$ (minimal vs extended version).

Figure 4

RGM GS results for the dispersion of the magnetic excitations ${\omega }_{\gamma \mathbf{q}}/S$ ($\gamma =1,2,3$) for $S=1/2$ (red lines) and $S=3$ (blue lines) compared with data of the LSWT (black lines) along a typical path in the first Brillouin zone (see the inset). LSWT formulas for ${\omega }_{\gamma \mathbf{q}}/S$ can be found, e.g., in [68].

Figure 5

Main: normalized RGM GS excitation velocity $v\left(0\right)/S$ in dependence on the inverse spin quantum number $1/S$. Inset: position $E_{\mathrm{flat}}/S$ of the flat band in dependence on the inverse spin quantum number $1/S$.

Figure 6

(a) Magnitude of the GS correlation functions $|\langle {\widehat{\mathbf{S}}}_0{\widehat{\mathbf{S}}}_{\mathbf{R}}\rangle |/S(S+1)$ within a range of separation $\left|\mathbf{R}\right|\le 6$ for spin quantum numbers $S=1/2$, 1, and $7/2$. (b) Comparison of $\langle {\widehat{\mathbf{S}}}_0{\widehat{\mathbf{S}}}_{\mathbf{R}}\rangle /S(S+1)$ of the minimal and extended version of the RGM for $S=1/2$ (nearest-neighbor correlation not included).

Figure 7

RGM GS correlation lengths corresponding to $q=0$ (${\xi }_{{\mathbf{Q}}_0}$) and $\sqrt{3}\times \sqrt{3}$ (${\xi }_{{\mathbf{Q}}_1}$) ordering.

Figure 8

RGM GS structure factor $\mathcal{S}\left(\mathbf{q}\right)/S(S+1)$. (a) Brillouin zones: the solid and dashed lines show the first and extended Brillouin zones; the red (black) circles indicate the expected maxima for a classical $\sqrt{3}\times \sqrt{3}$ ($q=0$) state. (b) $S=1/2$, (c) $S=7/2$, and (d) $S=20$.

Figure 9

Normalized RGM GS structure factor $\mathcal{S}\left(\mathbf{q}\right)/S(S+1)$ along the path $\mathrm{\Gamma }\to {\mathbf{Q}}_1\to {\mathbf{Q}}_0\to \mathrm{\Gamma }$ (see the inset) for $S=1/2$ and $7/2$.

Figure 10

Main panel: magnitude of the normalized spin-spin correlation functions $|\langle {\widehat{\mathbf{S}}}_0{\widehat{\mathbf{S}}}_{\mathbf{R}}\rangle |/S(S+1)$ as a function of the normalized temperature $T/S(S+1)$ (logarithmic scale) for spin quantum numbers $S=1/2$, 1, and $7/2$ (NN, nearest neighbors; NNN, next-nearest neighbors; NNNN, next-next-nearest neighbors straight along two $J_1$ bonds). Inset: magnitude of the spin-spin correlation functions $|\langle {\widehat{\mathbf{S}}}_0{\widehat{\mathbf{S}}}_{\mathbf{R}}\rangle |$ for $S=1/2$ as a function of the temperature $T$ (linear scale). Comparison of the extended (solid) and minimal (dashed) versions of the RGM as well as the 11th-order HTE with subsequent Padé (dashed-dotted).

Figure 11

Main: specific heat $C$ for various values of the spin quantum number $S$ as a function of the normalized temperature $T/S(S+1)$. The Monte Carlo data for the classical limit are taken from [5, 15] (MC I) and from [22] (MC II). Inset: low-temperature behavior of $C$ using a logarithmic temperature scale.

Figure 12

Uniform static susceptibility ${\chi }_0$ for various values of the spin $S$ as a function of the normalized temperature $T/S(S+1)$. The Monte Carlo data for the classical limit are taken from [15].

Figure 13

Intensity plot of the normalized structure factor $\mathcal{S}\left(\mathbf{q}\right)/S(S+1)$ within the first and extended Brillouin zones [cf. Fig. 8] for $S=1/2$ and 3 at $T/S(S+1)=1.3$ (left, ninth-order HTE; right, RGM). The red (black) circles indicate the expected maxima for a classical $\sqrt{3}\times \sqrt{3}$ ($q=0$) state.

Figure 14

RGM and HTE data for the normalized structure factor $\mathcal{S}\left(\mathbf{q}\right)/S(S+1)$ along the path $\mathrm{\Gamma }\to {\mathbf{Q}}_1\to {\mathbf{Q}}_0\to \mathrm{\Gamma }$ for $S=1/2$ and $7/2$ at $T/S(S+1)=1.5$. For comparison we also present RGM data for $T=0$ (dashed-dotted lines).

Figure 15

Normalized powder-averaged structure factor ${\mathcal{S}}^{\mathrm{av}}\left(\right|\mathbf{q}\left|\right)/S(S+1)$ for spin $S=1/2$ (solid lines) and $S=7/2$ (dashed lines) using HTE of ninth order.

Figure 16

RGM data for the normalized structure factor $\mathcal{S}\left({\mathbf{Q}}_i\right)/S(S+1)$ at the magnetic wave vector ${\mathbf{Q}}_i$ (dashed, ${\mathbf{Q}}_0$; solid, ${\mathbf{Q}}_1$) for various values of the spin $S$ as a function of the normalized temperature $T/S(S+1)$.

Figure 17

Main: RGM data for the correlation length ${\xi }_{{\mathbf{Q}}_i}$ (dashed, ${\mathbf{Q}}_0$; solid, ${\mathbf{Q}}_1$) for various values of the spin $S$ as a function of the normalized temperature $T/S(S+1)$. Inset: correlation lengths ${\xi }_{{\mathbf{Q}}_i}$ (dashed, ${\mathbf{Q}}_0$; solid, ${\mathbf{Q}}_1$) for $S=1/2$ and 1 using an enlarged $y$ axis.

Figure 18

RGM data for the temperature region $T^{*}/S(S+1)$ of almost constant ($p=99\%$; see the text) correlation lengths and structure factors as a function of $1/S$. The green line corresponds to the fit function $f\left(S\right)=0.2/S(S+1)$.