Gapless Spin-Liquid Ground State in the $S=1/2$ Kagome Antiferromagnet

The defining problem in frustrated quantum magnetism, the ground state of the nearest-neighbor $S=1/2$ antiferromagnetic Heisenberg model on the kagome lattice, has defied all theoretical and numerical methods employed to date. We apply the formalism of tensor-network states, specifically the method of projected entangled simplex states, which combines infinite system size with a correct accounting for multipartite entanglement. By studying the ground-state energy, the finite magnetic order appearing at finite tensor bond dimensions, and the effects of a next-nearest-neighbor coupling, we demonstrate that the ground state is a gapless spin liquid. We discuss the comparison with other numerical studies and the physical interpretation of this result.

Figure 1

Ground-state energy of the KHAF. (a) $E_0$ as a function of $D$, shown for the 3-PESS and simple-update method up to $D=25$, 3-PESS by full update to $D=13$, and 9-PESS with simple update to $D=15$. Shown for comparison are results from other numerical studies. (b) $E_0(D)$ for the 3-PESS ansatz, shown as a function of $1/D$ and compared with results obtained for the Husimi lattice [57]. (c) Convergence of $E_0(D)$ as a function of $D_{\mathrm{mps}}$, shown for several values of $D$.

Figure 2

Staggered magnetization of the KHAF at finite $D$. (a) $M$ as a function of $D$, shown for the 3-PESS and simple-update method up to $D=20$, 3-PESS by full update to $D=13$, and 9-PESS with simple update to $D=15$. Shown for comparison are results obtained for the Husimi lattice [57]. (b) $M$ as a function of $1/D^{0.588}$, the power-law form obtained for the Husimi lattice. (c) Convergence of $M(D)$ as a function of $D_{\mathrm{mps}}$, shown for $D=15$ and $D=20$.

Figure 3

Energy and staggered magnetization of the KHAF with next-nearest-neighbor coupling. (a) $E_0(J_2)$ calculated with a 9-PESS using $D=15$. Shown for comparison are results for the Husimi lattice with $D=15$ and $D=\infty$, as well as the VMC results of Ref. [14]. (b) Detail of $E_0(J_2)$ near $J_2=0$; with increasing $D$, a cusp-type discontinuity emerges near $J_2=-0.03$ (red solid line). The shaded region denotes the location of the continuous transition at small positive $J_2$, deduced from the magnetization of panel (d). (c) $M(J_2)$ calculated using $D=9$, 12, and 15, compared with results for the Husimi lattice at $D=\infty$. Insets represent the $\sqrt{3}\times \sqrt{3}$ (left) and $q=0$ ordered phases (right). (d) $M$ as a function of $1/D^{0.588}$, with the Husimi result indicating $J_2$ values for which $M$ may extrapolate to zero within the error bars.