Heisenberg antiferromagnet on the kagome lattice with arbitrary spin: A higher-order coupled cluster treatment

Starting with the $\sqrt{3}\times \sqrt{3}$ and the $q=0$ states as reference states, we use the coupled cluster method to high orders of approximation to investigate the ground state of the Heisenberg antiferromagnet on the kagome lattice for spin quantum numbers $s=1/2,$ 1, 3/2, 2, 5/2, and 3. Our data for the ground-state energy for $s=1/2$ are in good agreement with recent large-scale density-matrix renormalization group and exact diagonalization data. We find that the ground-state selection depends on the spin quantum number $s$. While for the extreme quantum case, $s=1/2$, the $q=0$ state is energetically favored by quantum fluctuations, for any $s>1/2$ the $\sqrt{3}\times \sqrt{3}$ state is selected. For both the $\sqrt{3}\times \sqrt{3}$ and the $q=0$ states the magnetic order is strongly suppressed by quantum fluctuations. Within our coupled cluster method we get vanishing values for the order parameter (sublattice magnetization) $M$ for $s=1/2$ and $s=1$, but (small) nonzero values for $M$ for $s>1$. Using the data for the ground-state energy and the order parameter for $s=3/2$, 2, 5/2, and 3 we also estimate the leading quantum corrections to the classical values.

Figure 1

Illustration of the $\sqrt{3}\times \sqrt{3}$ (left) and the $q=0$ (right) classical GS of the kagome HAFM.

Figure 2

Difference of GS energies per site, $\Delta e\equiv e_0^{\sqrt{3}\times \sqrt{3}}-e_0^{q=0}$, between the $\sqrt{3}\times \sqrt{3}$ and $q=0$ states of the spin-$1/2$ kagome HAFM, for various CCM LSUB$n$ approximations and spin quantum number $s=1/2$.

Figure 3

CCM-LSUB$n$ data for the magnetic order parameter $M$ versus $\lambda$ for the spin-$1/2$ kagome HAFM, for (a) the $\sqrt{3}\times \sqrt{3}$ reference state and (b) the $q=0$ reference state. For the extrapolations to $n\to \infty$ according to $M\left(n\right)=b_0+b_1{(1/n)}^{1/2}+b_2{(1/n)}^{3/2}$ we have used LSUB$n$ data for $n=4,5,\cdots ,10$ as well as for $n=6,7,\cdots ,10$.

Figure 4

Illustration of the extrapolation of the CCM-LSUB$n$ data (symbols) for (a) the ground-state energy $e_0$ and (b) the magnetic order parameter $M$ for the $q=0$ state of the spin-$1/2$ kagome HAFM. For the extrapolations to $n\to \infty$ according to $e_0\left(n\right)=a_0+a_1{(1/n)}^2+a_2{(1/n)}^4$ and $M\left(n\right)=b_0+b_1{(1/n)}^{1/2}+b_2{(1/n)}^{3/2}$ we have used LSUB$n$ data for $n=4,5,\cdots ,10$ (solid lines) as well as for $n=6,7,\cdots ,10$ (dashed lines).

Figure 5

Difference of the GS energies per site, $\Delta e\equiv e_0^{\sqrt{3}\times \sqrt{3}}-e_0^{q=0}$, between the $\sqrt{3}\times \sqrt{3}$ and $q=0$ states of the kagome HAFM, calculated for the CCM SUB8-8 approximation and for spin quantum numbers $s=1/2$, 1, 3/2, 2, 5/2, and 3.

Figure 6

Extrapolated magnetic order parameter $M/s$ versus $\lambda$ for (a) the $\sqrt{3}\times \sqrt{3}$ reference state and (b) the $q=0$ reference state of the kagome HAFM, for various values of the spin quantum number $s$. For the extrapolations to $n\to \infty$ according to $M\left(n\right)=b_0+b_1{(1/n)}^{1/2}+b_2{(1/n)}^{3/2}$ we have used SUB$n$-$n$ data for $n=4,5,\cdots ,8$. [Note that even for the $s=1/2$ case we have excluded the (available) LSUB9 and LSUB10 data here to be consistent with the $s>1/2$ cases.]

Figure 7

Dependence on the spin quantum number $s$ of (a) the extrapolated scaled GS energy per spin, $e_0/s^2$, and (b) the extrapolated scaled magnetic order parameter, $M/s$, of the kagome HAFM. For $M/s$ the exponent $\alpha =1/2$ for the $\sqrt{3}\times \sqrt{3}$ state and $\alpha =2/3$ for the $q=0$ state. (Symbols are CCM data points; lines are fits to the data points).