Pinwheel valence bond crystal ground state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the shuriken lattice

We investigate the nature of the ground state of the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the shuriken lattice by complementary state-of-the-art numerical techniques, such as variational Monte Carlo (VMC) with versatile Gutzwiller-projected Jastrow wave functions, unconstrained multivariable variational Monte Carlo (mVMC), and pseudofermion/pseudo-Majorana functional renormalization group (PFFRG/PMFRG) methods. We establish the presence of a quantum paramagnetic ground state and investigate its nature, by classifying symmetric and chiral quantum spin liquids, and inspecting their instabilities towards competing valence bond crystal (VBC) orders. Our VMC analysis reveals that a VBC with a pinwheel structure emerges as the lowest-energy variational ground state, and it is obtained as an instability of the U(1) Dirac spin liquid. Analogous conclusions are drawn from mVMC calculations employing accurate BCS pairing states supplemented by symmetry projectors, which confirm the presence of pinwheel VBC order by a thorough analysis of dimer-dimer correlation functions. Our work highlights the nontrivial role of quantum fluctuations via the Gutzwiller projector in resolving the subtle interplay between competing orders.

Figure 1

(a), (b) Top row: Illustration of two types of magnetic orders within the ground-state manifold of the classical Heisenberg antiferromagnet on the shuriken lattice [4, 5]. The two symmetry inequivalent nearest-neighbor bonds are labeled as $J_{△}$ and $J_{\square }$. The $\mathbf{q}=\mathbf{0}$ ($\sqrt{3}\times \sqrt{3}$) order has an angle of ${120}^{\circ }$ between neighboring spins, and a magnetic unit cell which is identical (three times enlarged) compared to the six-site geometrical unit cell is marked by a dashed line. The green ellipses depict further degrees of freedom present in the classically degenerate ground-state manifold. Bottom row: The first (solid line) and extended (dashed line) Brillouin zones of the shuriken lattice showing the location of the Bragg peaks with the fraction of the total spectral weight of the classical (a) $\mathbf{q}=\mathbf{0}$ order, at $(4\pi ,0)$ and $(2\pi ,2\pi )$ (and symmetry related points) with equal spectral weight, and (b) $\sqrt{3}\times \sqrt{3}$ order, at $(2\pi \pm q,2\pi \mp q)$ with $q=4\pi /3$, and leading subdominant peaks at $(q,-q)$ with 36%, i.e., $\lambda /\mathrm{\Lambda }=0.36$, of the spectral weight of the dominant ones [6]. The $\sqrt{3}\times \sqrt{3}$ order breaks the fourfold rotational symmetry. From (c) VMC, the size scaling of the $h_{\mathrm{AF}}$ parameter (fictitious Zeeman field [7]) for $\mathbf{q}=\mathbf{0}$ and $\sqrt{3}\times \sqrt{3}$ orders on finite clusters of $N_{\mathrm{sites}}=6L^2$ sites, with $L=6$, 9, 12, 15, and 18 (quadratic fit), (d) mVMC, the size scaling of the sublattice magnetization for the $\mathbf{q}=\mathbf{0}$ order finite clusters of $N_{\mathrm{sites}}=6L^2$ sites, with $L=2$, 4, 6, and 8 (quadratic fit), and (e) PFFRG, the RG flow of the susceptibility tracked at the dominant ordering vectors of the two classical orders.

Figure 2

Two competing dimer orders, (a) PW-VBC and (b) L6-VBC on the shuriken lattice. Both dimer states are eightfold degenerate. (c) The evolution of the energy per site during a typical VMC optimization for the VBCs, here shown for Hamiltonian (1) on the $L=20$ cluster. The inset shows the finite-size scaling of the energy gain of the PW-VBC state with respect to the U(1) DSL.

Figure 3

(a) Dimer-dimer correlations between the base bond (between sublattice $B$ and $C$ sites [7]), marked by a black dotted line, and other bonds measured within mVMC on the $L=8$ cluster. Solid red (dashed blue) lines represent positive (negative) correlation values. An analogous figure with the base bond on the side of a triangle can be found in SM [7]. (b) Long-range correlations between the base bond and other bonds labeled by $\alpha$. Only correlations between the (0,0) and $(2n,2m)$ unit cells are shown due to the alternating $2\times 2$ pattern in bond correlations. (c) Infinite-volume extrapolation of the largest eigenvalues of ${\chi }^D$. (d) The dominant eigenvectors of ${\chi }^D$: Red bonds are positive, while blue are negative and other bonds are zero. In the middle of each drawing, the high-symmetry momentum of the representation is shown, as well as the corresponding irreducible representation label of the $D_4$ point symmetry group.

Figure 4

The ground-state energies on the $L=4$, 6, and 8 clusters obtained from VMC and mVMC together with the inset showing the finite-size scaling. The blue star denotes the exact diagonalization (ED) energy on the $D_4$ symmetric $6\times 2^2$ cluster, and the iPEPS energy (obtained by a quadratic fit for the three largest bond dimensions) is marked by a horizontal dashedline [7].

Figure 5

The static (equal-time) spin structure factor $S\left(\mathbf{q}\right)$ obtained within (a) VMC and (b) mVMC on the $L=8$ cluster. The solid (dashed) white lines mark the first (extended) Brillouin zones.