Tensor network study of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the shuriken lattice

We investigate the ground state of the spin $S=\frac{1}{2}$ Heisenberg antiferromagnet on the shuriken lattice, also in the presence of an external magnetic field. To this end, we employ two-dimensional tensor network techniques based on infinite projected entangled pair and simplex states considering states with different sizes of the unit cells. We show that a valence bond crystal with resonances over length six loops emerges as the ground state (at any given finite bond dimension) yielding the lowest reported estimate of the ground state energy $E_0/J=-0.4410\pm 0.0001$ for this model, estimated in the thermodynamic limit. We also study the model in the presence of an external magnetic field and find the emergence of 0, $\frac{1}{3}$, and $\frac{2}{3}$ magnetization plateaus. The $\frac{1}{3}$ and $\frac{2}{3}$ plateau states respect translation and point group symmetries and feature loop-four plaquette resonances.

Figure 1

Nearest-neighbor spin-spin correlations for the ground state configuration of the pinwheel VBC (a) and the loop-six VBC (b) states at ${\chi }_B=12$. The expectation values of the spin-$z$ component are typically $<{10}^{-3}$ and only shown for completeness. (c) Shuriken lattice with spin-$\frac{1}{2}$ degrees of freedom on the sites. The elementary unit cell (gray rectangles) consists of six sites, which are coarse-grained to map the shuriken lattice to a regular square lattice. Straight lines denote virtual bond indices; curly lines denote physical indices in the TN structure.

Figure 2

Ground state energy without magnetic field [$h=0$ in Eq. (1)] versus the inverse of the bond dimension ${\chi }_B$. iPESS simulations are denoted by $\left(S\right)$, iPEPS simulations by $\left(P\right)$.

Figure 3

Comparison of unconstrained and constrained ground state simulations up to ${\chi }_B=12$ for the different configurations imprinted. A first-order polynomial fit is used to extract the infinite bond dimension limit for the three largest bond dimensions.

Figure 4

Magnetization curve of the Heisenberg model for ${\chi }_B=10$. Upon tuning the magnetic field, two magnetization plateaus at $\frac{1}{3}$ and $\frac{2}{3}$ of the saturated magnetization $m_S=\frac{1}{2}$ appear. Additionally, we find the presence of a small plateau at $m_z=0$ indicative of the gapped nature of the ground state.

Figure 5

Ground state configurations of magnetic plateaus for $h=1.4$ (top) and $h=2.8$ (bottom) at ${\chi }_B=10$. Both the $\frac{1}{3}$ plateau state and the $\frac{2}{3}$ plateau state have strong spin-spin correlations on the squares, whose spins appear to be in an entangled superposition. In contrast, spins on the shuriken sites are isolated and (almost) fully polarized.

Figure 6

On-site magnetization of the triangle and square sites versus the magnetic field $h$. Dotted vertical lines represent the four magnetization plateaus.

Figure 7

Shuriken lattice shown in black and its dual lattice [the so-called $(4,8^2)$ Archimedean lattice] shown in green. Since the spin-$\frac{1}{2}$'s live on the links of the dual lattice, additional three-index simplex tensors are introduced on the vertices for the iPESS Ansatz.

Figure 8

Shuriken lattice shown on the left and iPEPS Ansatz shown on the right. The four sites on each square are coarse-grained into an effective site, highlighted by the green dotted area. The resulting structure is a square lattice with missing links.

Figure 9

Simple update in the iPESS simulations of the shuriken lattice. The Trotterized three-body Hamiltonian gate is absorbed into a triangle configuration. A truncated higher-order SVD is used to decompose the resulting six-index tensor back into the separate iPESS tensors. The same procedure is applied to the other simplex tensors along with the different lattice tensors.

Figure 10

Simple update in the iPEPS simulations of the shuriken lattice. The Trotterized six-body Hamiltonian gate is absorbed into a bottom-left corner of the deformed square lattice. The individual tensors are restored using two successive SVDs with truncation to a fixed bond dimension. The update of a top-right corner is performed similarly.

Figure 11

A CTMRG routine is used to approximate the contraction of the infinite square lattice by a set of fixed-point environment tensors.