Frustrated Antiferromagnets with Entanglement Renormalization: Ground State of the Spin-$\frac{1}{2}$ Heisenberg Model on a Kagome Lattice

Entanglement renormalization techniques are applied to numerically investigate the ground state of the spin-$\frac{1}{2}$ Heisenberg model on a kagome lattice. Lattices of $N=\{36,144,\infty \}$ sites with periodic boundary conditions are considered. For the infinite lattice, the best approximation to the ground state is found to be a valence bond crystal with a 36-site unit cell, compatible with a previous proposal. Its energy per site, $E=-0.43221$, is an exact upper bound and is lower than the energy of any previous (gapped or algebraic) spin liquid candidate for the ground state.

Figure 1

Coarse-graining transformation that maps (i) a kagome lattice ${\mathcal{L}}_0$ of $N$ sites into (vi) a coarser lattice ${\mathcal{L}}_1$ of $N/36$ sites. (ii) The lattice is first partitioned into blocks of 36 sites. (iii) Disentanglers $u$ are applied across the corners of three blocks, followed by (iv) disentanglers $v$ applied across the sides of two neighboring blocks. (v) Isometries $w$ map blocks to an effective site of the coarse-grained lattice. (vii) Tensors $u$, $v$ and $w$, have a varying number of incoming and outgoing indices according to Eq. (1).

Figure 2

The 36-site unit cell for the honeycomb VBC, strong bonds are drawn with thick lines. Three different types of strong bonds can be identified; the six bonds belonging to the pinwheels (red), six bonds belonging to each “perfect hexagon” (blue), and the parallel bonds between perfect hexagons (green). Dotted arrows indicate the axis where spin-spin correlators have been computed. Bond-bond correlators have been computed between the reference bond (1) and the other numbered bonds.

Figure 3

(top) Bond energies for the 36-site unit cell of infinite MERA wave functions, for two different values of $\tilde{\chi }$, as compared to those of an exact honeycomb VBC, $|h\mathrm{\text{−}}\mathrm{VBC}⟩$, and those of a spin liquid, which by definition has all equal strength bonds. The MERA wave functions clearly match the proposed honeycomb VBC; we identify (i) the six strong “pinwheel” bonds (red bonds), the six “parallel” bonds (green bonds), and (iii) the 12 “perfect hexagon” bonds (blue bonds). The (iv) remaining 48 bonds are the weak bonds of the unit cell. (bottom) Bond energies for the 36-site lattice. Here a randomly initialized MERA converges to a dimerized state that does not match the honeycomb VBC pattern, but gives lower overall energy than a honeycomb VBC initialized MERA of the same $\tilde{\chi }$.

Figure 4

Spin-spin correlators along arrows “A” and “B” of Fig. 2 for infinite lattice MERA of $\tilde{\chi }=4$ and $\tilde{\chi }=16$. Although along both lattice directions considered the correlators decay exponentially, the decay along arrow “A” (a line joining two perfect hexagons) is seen to be slower than along arrow “B” (a line joining perfect hexagon to pinwheel). The plateaus marked (i), (ii), and (iii) show the correlation is the same with both spins of a strong bond.