Simplex Crystal Ground State and Magnetization Plateaus in the Spin-$1/2$ Heisenberg Model on the Ruby Lattice

TL;DR

This work investigates the spin-$\tfrac{1}{2}$ Heisenberg antiferromagnet on the ruby lattice, employing infinite projected entangled pair states (iPEPS) to access the thermodynamic limit. At zero field it finds a gapped, threefold-degenerate simplex-crystal ground state with strong simplex singlets, and under a Zeeman field it discovers magnetization plateaus at $m/m_s = 0,\ 1/3,\ 1/2,\ 2/3$ accompanied by intermediate supersolid phases, all breaking the lattice's sixfold rotational symmetry. The results are interpreted via an effective spin-chirality Hamiltonian on a honeycomb network of trimer supersites in the $J_t \gg J_h = J_d$ limit, linking the observed order to dimer-covering physics and distinguishing these plateaus from localized-magnon scenarios. The study advances understanding of frustrated quantum magnets on the ruby lattice and offers insight into complex, symmetry-broken quantum phases with potential realizations in Rydberg-atom platforms and related systems.

Abstract

We investigate the spin-$1/2$ Heisenberg antiferromagnet on the ruby lattice with uniform first- and second-neighbor interactions, which forms a two-dimensional network of corner-sharing tetrahedra. Using infinite projected entangled pair states (iPEPS), we study the ground state of the system to find that it assumes a gapped threefold-degenerate simplex crystal ground state, with strong singlets formed on pairs of neighboring triangles. We argue that the formation of the simplex singlet ground state at the isotropic point relates to the weak inter-triangle coupling limit where an effective spin-chirality Hamiltonian on the honeycomb lattice exhibits an extensively degenerate ground state manifold of singlet coverings at the mean-field level. Under an applied Zeeman field, the iPEPS simulations uncover magnetization plateaus at $m/m_s = 0, 1/3, 1/2,$ and $2/3$, separated by intermediate supersolid phases, all breaking the sixfold rotational symmetry of the lattice. Unlike the checkerboard lattice, these plateaus cannot be described by strongly localized magnons.

Figures (7)

• Figure 1: (a) Checkerboard lattice with first-neighbor ($J$) and second-neighbor ($J_d$) interactions. Empty plaquettes are labeled with A, B, and C, regions where localized magnon states can harbor. (b) Ruby lattice with nearest-neighbor ($J_t$ and $J_h$) and next-nearest-neighbor ($J_d$) interactions. In the isotropic interaction limit, both lattices can be viewed as networks of corner-sharing tetrahedra, but with different connectivity.
• Figure 2: (a) Coarse-grained three-site tensors. Each $J_t$-trimer of the ruby lattice is represented by a tensor with physical dimension $d=8$. Thin dotted lines show the bonds of the underlying lattice, while thick black lines indicate virtual bonds of dimension $D$. (b) 2D iPEPS wavefunction on the honeycomb network with a six-site unit cell. Black lines represent virtual legs and gray lines the physical legs. In the simple update scheme, a two-site Suzuki-Trotter gate $e^{-\tau H_e}$ is applied to each nearest-neighbor pair of tensors, followed by a singular value decomposition (SVD) across the dashed orange line separating the two physical indices. The resulting $dD^3 \times dD^2$ tensor is truncated to retain the $D$ largest singular values. (c) The wavefunction overlap is represented as the contraction of the infinite tensor network. The overlapped tensor $\alpha^{[\mathbf{x}]\dagger}\alpha^{[\mathbf{x}]}$ is mapped to a local tensor $A^{[\mathbf{x}]}$ with dimensions $D^2 \times D^2 \times D^2$. (d) This contraction is approximated using an environment composed of row and corner tensors; thicker black lines indicate environment bonds of dimension $\chi$. (e) Schematic of the environment tensor update scheme. (f) Two-trimer unit-cell setup, ordered as $a$-$b$-$R(a)$-$R(b)$-$R^2(a)$-$R^2(b)$ in the counterclockwise direction, where $R$ denotes a rotation by $2\pi/3$ about the center of the tensors.
• Figure 3: (a) Product state of singlets on the hexagons, a possible candidate ground state of the system. (b) Scaling of the ground state energies per site, $e_g$, obtained in iPEPS as a function of the truncation error from SVD at $J_h = J_t = J_d= 1$. The solid line shows the extrapolation for $D\to\infty$ obtained via an exponential fitting of the data obtained for $D=8$-$12$. The horizontal dashed line corresponds to the energy of the state shown in (a). The horizontal dotted line shows the per-site energy of an isolated simplex. (c) The spin-spin correlations on the first- and second-neighbor bonds of the ground state of \ref{['eq:hmail']}. This state is a simplex crystal, where strong singlets form on simplexes composed of two neighboring $J_t$ triangles. The state is threefold degenerate. We mark the three inequivalent hexagonal plaquettes with $H_1$, $H_2$, and $H_3$. The legend at the extreme right indicates the spin-spin correlation values, shown as a false-color scale, and applies to both panel (a) and (c).
• Figure 4: (a) Localized one magnon state on the ruby lattice. The numbers $\pm 1$ at the corners of the central hexagon denote the coefficients $c_i$ in \ref{['eq:magnon']}. (b) Magnetization curve of \ref{['eq:hmail']} with $J_t = J_h = J_d = 1$ under a Zeeman term $-h\sum_i S_i^z$. Plateaus appear at $m/m_s = 0, 1/3, 1/2,$ and $2/3$, with the stability of the $m/m_s = 0$ and $1/3$ plateaus indicating substantial spin gaps. (c) Energies obtained from iPEPS calculations for various bond dimensions $D$. (d) and (e) Zoomed-in views of the magnetization curve near the $m/m_s = 1/2$ and $2/3$ plateaus, respectively. (i) Zoomed-in view of the energies near the saturation field. The red dashed line shows the energy of the fully polarized state, while the green dotted lines correspond to localized-magnon eigenstates with $m/m_s = 2/3$ (see text).
• Figure 5: (a) Relevant symmetries of the ruby lattice: $C_6$ rotational symmetry through the center of the hexagons and reflection symmetry with respect to the $\sigma_d$ planes passing through opposite edges of a hexagon. (b)-(d) Magnetization and spin-spin correlations on the first- and second-neighbor bonds of the ruby lattice for $m/m_s = 1/3$, $1/2$, and $2/3$ plateau states, all of which break the $C_6$ symmetry down to $C_3$ and the reflection symmetry about the $\sigma_d$ planes, making them sixfold degenerate. The plot legends indicate the values of the spin-spin correlations and local magnetizations, represented as false-color scales. The radius of the circles is also indicative of local magnetizations. The dark-blue hexagons in panels (c) and (d) indicate the simplex patterns.