Density Matrix Renormalization Group Numerical Study of the Kagome Antiferromagnet

We numerically study the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the kagome lattice using the density-matrix renormalization group method. We find that the ground state is a magnetically disordered spin liquid, characterized by an exponential decay of spin-spin correlation function in real space and a magnetic structure factor showing system-size independent peaks at commensurate magnetic wave vectors. We obtain a spin triplet excitation gap $\Delta E(S=1)=0.055\pm 0.005$ by extrapolation based on the large size results, and confirm the presence of gapless singlet excitations. The physical nature of such an exotic spin liquid is also discussed.

Figure 1

Sketch of a three-leg kagome lattice with total number of sites $N=3\times N_1\times N_2$ and number of unit cells $N_1\times N_2=4\times 3$. Here ${\mathbf{a}}_1=(2,0)$ and ${\mathbf{a}}_2=(1,\sqrt{3})$ are two primitive vectors of the unit cell including three inequivalent sites (e.g., 1, 2, 3).

Figure 2

(a) The ground state energy per site $E_0/N$ (solid circles) and the excitation energy $E_1/N$ (solid stars) in the total spin $S=1$ sector are shown as a function of $m$ (the number of states kept in each block) for the system with $N=3\times 4\times 4$; (b) for a system $N=3\times 12\times 3$. The energy gap for spin-1 excitation $\Delta E=E_1-E_0$ (open circles) as a function of $m$ for these two systems are shown in the insets.

Figure 3

The spin gap $\Delta E(S=1)$ for squarelike systems (see text) at different system sizes obtained from the ED [7] (open circles) and DMRG (solid squares) with $N=36–108$. The singlet excitation gap $\Delta E(S=0)=E_1(S=0)-E_0$ is also given (solid stars). Inset shows the spin gap for both 3-leg and 4-leg systems.

Figure 4

The spin-spin correlations $|⟨S_0^z{S}_r^z⟩|$ along the ${\mathbf{a}}_1$ direction with $N=3\times 10\times 3$ and $N=3\times 10\times 4$. The error bar represents the mean square deviation of all the equivalent sites. The straight line is a fitting to an exponential function $|⟨S_0^z{S}_r^z⟩|=A\exp (-r/\tau )$. The system size dependence of the correlation length $\tau$ is shown in the lower left inset. The spin-spin correlations $⟨S_0^z{S}_r^z⟩$ are also given in the insets.

Figure 5

(a) The static structure factors $S^z(q)$ for the kagome system with $N=3\times 6\times 6$ as a function of ($q_1$, $q_2$) in units of two primitive basis vectors (${\mathbf{b}}_{\mathbf{1}}$, ${\mathbf{b}}_{\mathbf{2}}$) in the reciprocal lattice. (b) The locations of three peaks in $S^z(q)$ marked by solid circles in the extended kagome BZ. (c) The peak values $S^z(1,1)$ vs $1/N$.

Figure 6

The static structure factor $S^D$ for the dimer-dimer correlation for a system with $N=3\times 4\times 4$ in (a). The peak value of $S^D$ vs $1/N$ is plotted in (b).