Quantum paramagnetism in the decorated square-kagome antiferromagnet ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left({\mathrm{PO}}_4\right)}_4{\mathrm{Cl}}_3$

The square-kagome lattice Heisenberg antiferromagnet is a highly frustrated Hamiltonian whose material realizations have been scarce. We theoretically investigate the recently synthesized ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left(\mathrm{PO}\right)}_4{)}_4{\mathrm{Cl}}_3$ where a ${\mathrm{Cu}}^{2+}$ spin-$1/2$ square-kagome lattice (with a six site unit cell) is decorated by a seventh magnetic site alternatingly above and below the layers. The material does not show any sign of long-range magnetic order down to 50 mK despite a Curie-Weiss temperature of $-212\mathrm{K}$ indicating a quantum paramagnetic phase. Our DFT energy mapping elicits a purely antiferromagnetic Hamiltonian that features longer range exchange interactions beyond the pure square-kagome model and, importantly, we find the seventh site to be strongly coupled to the plane. We combine two variational Monte Carlo approaches, pseudofermion/Majorana functional renormalization group and Schwinger-Boson mean field calculations to show that the complex Hamiltonian of ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left(\mathrm{PO}\right)}_4{)}_4{\mathrm{Cl}}_3$ still features a nonmagnetic ground state. We explain how the seventh ${\mathrm{Cu}}^{2+}$ site actually aids the stabilization of the disordered state. We predict static and dynamic spin structure factors to guide future neutron scattering experiments.

Figure 1

(a) Heisenberg Hamiltonian parameters of ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left(\mathrm{PO}\right)}_4{)}_4{\mathrm{Cl}}_3$ determined by DFT energy mapping as function of onsite interaction strength $U$ (negligible couplings are not shown). The vertical line indicates the $U$ value where the exchange couplings match the experimental [18] Curie-Weiss temperature. The resulting nonnegligible exchange couplings are $J_1=109.1\left(8\right)\mathrm{K}$, $J_2=186.2\left(7\right)\mathrm{K}$, $J_3=155.3\left(1.4\right)\mathrm{K}$, $J_4=46.9\left(4\right)\mathrm{K}$, and $J_{10}=64.6\left(2\right)\mathrm{K}$. (b) Relevant exchange paths of ${\mathrm{Na}}_6{\mathrm{Cu}}_7{\mathrm{BiO}}_4{\left(\mathrm{PO}\right)}_4{)}_4{\mathrm{Cl}}_3$. (c) Finite-size scaling of the maxima of the equal-time structure factor $S\left(\mathbf{Q}\right)/N_s$ from many-variable VMC (mVMC), VMC, pseudofermion FRG (PFFRG), pseudo-Majorana FRG (PMFRG) (at $T=0.2J_2$), and the Schwinger-Boson mean-field theory method. Note that, compared to the other methods, PFFRG and PMFRG use a different definition of the system size $N_s$, which counts the number of correlated sites around a reference site, likely explaining the quantitative differences of our results. Furthermore, PMFRG results are obtained at a finite temperature $T=0.2J_2$.

Figure 2

(a)–(c) The pattern of real space (equal-time) spin-spin correlations $\langle {\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_j\rangle$ from mVMC measured with respect to the three symmetry inequivalent sites. The radius of the circle is proportional to the magnitude of the correlator, and blue (red) denotes antiferromagnetic (ferromagnetic) correlations. The largest red circle corresponds to $i=j$. (d) The pattern of $\langle {\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_j\rangle$ within a $2\times 2$ unit cell showing the pattern of strong/weak bonds in the VBC ground state. The thickness is proportional to $|\langle {\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_j\rangle |$ and blue (red) denotes antiferromagnetic (ferromagnetic) bonds, while the dashed lines denote the $J_{10}$ bonds. Note that the idealized 2D unit cell shown here is rotated by ${45}^{\circ }$ with respect to Fig. 1.

Figure 3

(a) Static (equal-time) structure factor from mVMC [Eq. (1)] obtained w.r.t. true crystal lattice site positions [25] obtained on a $8\times 8\times 7$ site cluster [Note: the ($S\mathbf{q}$) is not periodic, and the Brillouin zones and high-symmetry points of the ideal geometrical lattice are only drawn for illustrative purposes]. (b) The corresponding powder average after accounting for the form factor.

Figure 4

Dynamical structure factors as function of $S$ and $J_{10}$. The other parameters are as given in the caption of Fig. 1. As $J_{10}$ increases and/or $S$ decreases, a gap opens and a quantum paramagnet is stabilized. The Bose condensations appear at incommensurate $q$ vectors. In the condensed state, the Cu(3) spins are ordered but the other spins in the square-kagome lattice remain very weakly ordered. This is reflected by the gap between the lower branch excitations and the continuum in the lower panels.