Gapless spin-liquid phase in the kagome spin-$\frac{1}{2}$ Heisenberg antiferromagnet

We study the energy and the static spin structure factor of the ground state of the spin-$1/2$ quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green’s function Monte Carlo technique, we find that a gapless (algebraic) $U\left(1\right)$ Dirac spin liquid is competitive with previously proposed gapped (topological) ${\mathbb{Z}}_2$ spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site $E/J=-0.4365\left(2\right)$, which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.

Figure 1

Variational energies of the 48-site cluster as a function of the variance of the energy, for zero, one, and two Lanczos steps. The ground-state energy is estimated by extrapolating the three variational results to the zero-variance limit by a quadratic fit. Three different starting wave functions are used. The $U\left(1\right)$ Dirac spin liquid has also been studied using FN approximation.

Figure 2

The same as in Fig. 1 for the 192-site cluster. Here, only ${\mathbb{Z}}_2[0,\pi ]\beta$ and $U\left(1\right)$ Dirac states have been considered.

Figure 3

The thermodynamic estimate of the ground-state energy obtained by a finite-size extrapolation of the estimated ground-state energies (see Table ). The linear (solid line) and quadratic (dashed line) fits give essentially the same estimate. The energy on the 36-site cluster is from exact diagonalization. Comparison is also made with recent DMRG estimates.[13, 14]

Figure 4

Intensity plot of the static spin structure factor $S$(q) on the 192-site cluster.