Pinch-points to half-moons and up in the stars: The kagome skymap

Pinch point singularities, associated with flat band magnetic excitations, are tell-tale signatures of Coulomb spin liquids. While their properties in the presence of quantum fluctuations have been widely studied, the fate of the complementary nonanalytic features—shaped as half moons and stars—arising from adjacent shallow dispersive bands has remained unexplored. Here, we address this question for the spin $S=1/2$ Heisenberg antiferromagnet on the kagome lattice with second and third neighbor couplings, which allows one to tune the classical ground state characterized by flat bands to one that is governed by shallow dispersive bands for intermediate coupling strengths. Employing the complementary strengths of variational Monte Carlo, pseudofermion functional renormalization group, and density-matrix renormalization group, we establish the quantum phase diagram of the model. The $U\left(1\right)$ Dirac spin liquid ground state of the nearest-neighbor antiferromagnet remains remarkably robust till intermediate coupling strengths when it transitions into a pinwheel valence bond crystal displaying signatures of half moons in its structure factor. Our Letter thus identifies a microscopic setting that realizes one of the proximate orders of the Dirac spin liquid identified in a recent work [Song, Wang, Vishwanath, and He, Nat. Commun. 10, 4254 (2019)]. For larger couplings, we obtain a collinear magnetically ordered ground state characterized by starlike patterns.

Figure 1

The kagome skymap. (a) Illustration of first ($J_1$), second ($J_2$), and third neighbor interactions along edges ($J_{3a}$) of the kagome lattice for the model (1). (b) The $S=1/2$ quantum phase diagram with (top panel) representative real space spin-spin correlation profiles, with red (blue) bonds denoting antiferromagnetic (ferromagnetic) correlations, and (lower panel) spin structure factors of the different phases evaluated at $J/J_1=0.1$ (DSL), $J/J_1=0.4$ (pinwheel VBC), and $J/J_1=0.9$ (collinear order) from pf-FRG. (c) Estimates of the phase boundaries ($g_1$ and $g_2$) obtained from the various approaches employed in this Letter. While we see good agreement within error bars for $g_1$ for all approaches, the pf-FRG result for $g_2$ (marked by an asterisk) shows a notable deviation whose origin is discussed in the Supplemental Material [59].

Figure 2

Transition into half-moon phase. (a) From VMC, the evolution with $J/J_1$ of the energy per site of the DSL and VBC states ($3\times 12\times 12$ lattice). DMRG energies are also shown for comparison. (b) From pf-FRG, the variation of the spectral measure $\mathrm{\Delta }{\chi }_{\mathit{k}}$ (see text below) with $J/J_1$ evaluated at the lowest simulated RG cutoff $\mathrm{\Lambda }/Z=0.01$ where $Z=\sqrt{J_1^2+2J^2}$.

Figure 3

Half-moon radii. From pf-FRG, we show for different values of spin $S$ [43] the evolution with $J/J_1$ of the radius of the half moons characterizing the pinwheel VBC. The large-$S$ (classical) result is from Ref. [6].

Figure 4

Transition into the star phase. The behavior of the square of the sublattice magnetization $m^2$ with $J/J_1$ near the transition from the pinwheel VBC into collinear magnetic order (illustrated in the inset, with blue and red spins pointing in opposite directions). The results from VMC and mVMC are for a $3\times 8\times 8$ site cluster (see Fig. S11 in Supplemental Material [59] for finite-size scaling results of $m^2$ from VMC to the thermodynamic limit), while those from DMRG are obtained on a YC8-8 cylinder (see Supplemental Material [59] for further details). In all the three cases, $m^2$ is estimated by computing the isotropic equal-time spin-spin correlation between lattice sites separated by eight nearest-neighbor distances.