Dzyaloshinskii-Moriya-driven instabilities in square-kagome quantum antiferromagnets

Abstract

Decorated square-kagome quantum antiferromagnets provide a natural setting in which strong frustration, lattice decoration, and spin-orbit-induced anisotropy compete on comparable energy scales. Here we show that in Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ the coupling ($J_{10}$) which links the decorating Cu(3) sites to the square-kagome backbone, stabilizes the gapped quantum-paramagnetic regime, while symmetry-allowed Dzyaloshinskii-Moriya (DM) interactions systematically suppress the minimum spinon gap $Δ_{\mathrm{spinon}}$ and drive the system toward magnetic condensation. To establish this, we combine ab initio calculation of the DM vectors with a generalized Schwinger-boson self-consistent mean-field theory that treats singlet and triplet hopping/pairing channels on equal footing. As a benchmark, the isotropic square-kagome Heisenberg model exhibits four competing low-energy saddle points distinguished by their Wilson-loop fluxes and by characteristic static and dynamical structure-factor fingerprints. A minimal DM perturbation does not qualitatively reshape this competing landscape, but already enhances the tendency towards order. For the realistic decorated Hamiltonian, finite-size scaling of $Δ_{\mathrm{spinon}}$ together with momentum-resolved structure factors identifies $J_{10}$ (exchange with decorating Cu) as the control parameter of the gapped regime and shows that the full symmetry-allowed DM pattern shifts the system further toward condensation. Our results place Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ in close proximity to a magnetic instability and provide experimentally testable predictions for anisotropy-enhanced soft modes in decorated square-kagome materials.

Figures (12)

• Figure 1: Crystal structure of Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$. The Cu$^{2+}$ ions occupy three crystallographically inequivalent positions, Cu(1), Cu(2), and Cu(3). Cu(1) and Cu(2) form the square-kagome backbone, while Cu(3) decorates it in a checkerboard manner above and below the square-kagome plane.
• Figure 2: Band structure of Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ from nonmagnetic GGA calculations. Dashed red lines show the DFT bands, while solid black lines show the Wannier-interpolated bands obtained from the Cu $d_{x^2-y^2}$ orbitals used to estimate the exchange parameters via hopping integrals. The isolated 14-band manifold near the Fermi level supports the one-band low-energy description.
• Figure 3: Relevant exchange paths in Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ with Cu sites labeled for clarity. Cu(1) and Cu(2) form the square-kagome backbone, while Cu(3) denotes the decorating sites. Colored bonds distinguish the different isotropic exchange couplings entering the microscopic spin Hamiltonian.
• Figure 4: First Brillouin zones corresponding to the different unit cells considered in this work: the square--kagome lattice with $n_u=6$ sites per unit cell (blue), the enlarged unit cell with $n_u=14$ sites (red), and the underlying square lattice (black dashed). High-symmetry points are indicated, and the gray arrows denote the momentum path used to compute the dynamical spin structure factors.
• Figure 5: Equal-time (left) and dynamical (right) spin structure factors obtained within SBMFT for the four lowest-energy Ansätze of the isotropic square-kagome model at $S=0.3$ without Dzyaloshinskii--Moriya interactions. Panels (a)--(d) correspond to the Wilson-loop flux patterns $\Phi=(\pi,\pi,\pi,\pi)$, $(0,\pi,0,\pi)$, $(0,0,0,0)$, and $(\pi,0,0,\pi)$, identified respectively as a gapped quantum spin liquid, the $\sqrt{3}\!\times\!\sqrt{3}$ order, the $Q=0$ order, and a second gapped quantum spin liquid. The percentage of the total spectral weight displayed is indicated in the plots. The first (solid line) and extended (dashed line) Brillouin zones of the square-kagome lattice showing the location of the Bragg peaks with the fraction of the total spectral weight of the classical (a) $\mathbf{Q}=0$ 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., $\lambda/\Lambda=0.36$, of the spectral weight of the dominant ones note_peaks. The $\sqrt{3}\times\sqrt{3}$ order breaks the four-fold rotational symmetry.