Competing Paramagnetic Phases in the Maple-Leaf Heisenberg Antiferromagnet

Abstract

We establish a remarkably rich ground state phase diagram in the maple-leaf lattice spin-$1/2$ Heisenberg antiferromagnet as a function of the three symmetry-inequivalent nearest-neighbor bonds using exact diagonalization and tower-of-states analysis on clusters up to $N=36$ sites. Besides a hexagonal plaquette state, a star-shaped valence bond solid state is discovered in close vicinity to the (canted) $120^\circ$ magnetic phase, strongly reminiscent of a de-confined critical point or Dirac spin liquid scenario on the triangular lattice antiferromagnets. Moreover, an exact dimer product-state is observed next to a collinear Néel-state, similar to the Shastry-Sutherland model. All identified phases compete in a parameter regime close to the isotropic point, providing a promising region for spin liquids to emerge. By analyzing Gutzwiller-projected wave-functions we identify a sliver of parameter regime where a gapped $\mathbb{Z}_{2}$ spin liquid Ansatz is in astonishing agreement with the exact $N=36$ ground state. This rich competition of paramagnetic phases demonstrates that the maple-leaf antiferromagnet is a promising platform for exotic states of matter and quantum critical phenomena.

Figures (8)

• Figure 1: Maple-leaf lattice with three NN bonds: $J_t$ (red triangles), $J_h$ (blue hexagons), $J_d$ (green "dimers"); possible six-atom units cells are highlighted.
• Figure 2: Phase diagram of the HAF \ref{['eq:H']} on the MLL based on $N=36$ ED data (Voronoi tesselation), where $0 \leq J_t, J_h, J_d \leq \infty$ (AFM octant on the $J_t^2 + J_h^2 + J_d^2=\mathrm{const.}$ sphere). The three external boundaries correspond to the Heisenberg models on the ruby (R), star (S) and honeycomb (H) lattices, respectively obtained by setting $J_d$, $J_h$ or $J_t$ to zero. Lines where two couplings coincide are drawn in white, marking the isotropic point $J_t=J_h=J_d$ at the center. Prototypical states for the color-coded phases are illustrated, including a novel "star VBS" state on $J_t, J_h$ bonds where the dimer motives on $J_h$-hexagons can resonate in two variations (light blue and pink, a 36-site pattern being shown here). Two proximate points, called P1 and P2 below, where Gutzwiller-projected gapped $\mathbb{Z}_2$ QSL ansätze are in strong agreement with the ED ground state, are drawn in black.
• Figure 3: ED ground state diagnostics on the $N=36$ cluster. (a): Average spin-spin correlator $C_{ss}(J_d)$ on $J_d$-bonds; (b): staggered magnetization per site; (c): (equal time) spin-spin structure factor $\mathcal{S}$ at the K point, $\mathrm{K}_\mathrm{TRI}$, of the underlying triangular lattice; (d): bond-averaged "twist correlator" $\langle D_{01} D_{ij} \rangle$ (see Eq. \ref{['eq:Dij']}); (e): relative deviation of NN spin-spin correlator on $J_h$-bonds from their plaquette value (see Eq. \ref{['eq:DeltaPlaquette']}); (f): connected bond-bond correlator $C_{bb}$ (see Eq. \ref{['eq:Cbb']}) for the "star VBS"; (g): NN spin-spin correlator at the point marked in (e); (h): connected bond-bond correlator at the point marked in (f) with reference bond $\langle 0,1\rangle$ shown in yellow.
• Figure 4: (a): Lowest eigenstates near the transition into the $J_d$-dimer phase along the $J=J_t=J_h$ line (vertical dashed line in the phase diagram from Fig. \ref{['fig:phaseDiagram']}). The inset shows the first singlet excitations as the gap closes. (b) and (c): Tower of states of the $N=36$ and $N=18$ clusters at the isotropic point $J_t = J_h = J_d = J$. Dashed lines show the least-squares fit to the lowest eigenstate for each $S$. Other levels are merely drawn as lines for clarity. The energies of the lowest eigenstate for each $S$ are found to scale as $S(S+1)/N$ in agreement with a magnetically-ordered system.
• Figure 5: The two finite clusters used in all ED computations as rhombuses on the MLL; (a) for $N=18$ and (b) for $N=36$ spins. The same color-coding as in Fig. \ref{['fig:lattice']} is assumed.