Effective field theory approach for the $S=\frac{3}{2}$ bilayer honeycomb antiferromagnet

The spin-$\frac{3}{2}$ Heisenberg antiferromagnet on the bilayer honeycomb lattice is a minimal model to describe the magnetic behavior of ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$. We study this model with frustrating interlayer second-neighbor couplings taking into account quantum and thermal fluctuations. We use a path integral formulation in terms of coherent states to describe the low-energy physics of the model. We show that for a particular point in the parameter space, close to the experimental estimated couplings, a continuum classical degeneracy is lifted by both quantum and thermal fluctuations, and a collinear state is then selected by an order by disorder mechanism. Our results provide a global perspective in the understanding of the experimental observations.

Figure 1

The bilayer geometry considered here. Pannel (a) shows the experimental structure corresponding to ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}$(${\mathrm{NO}}_3$). (b) Bilayer honeycomb lattice corresponding to model (1). (c) Four $S=\frac{3}{2}$ spins unit cell. The interactions in the Hamiltonian correspond to the in-plane nearest-neighbor coupling $J_1$, the interlayer nearest-neighbor coupling $J_0$, and the interlayer next-nearest-neighbor coupling $J_x$.

Figure 2

Schematic representation of the zero modes corresponding to (a) $J_x=J_0/3$ and (b) $J_1=J_x$.

Figure 3

Free energy corresponding to Eq. (18), with $J_1=1, J_x=1/10, J_0=3J_x, {\theta }^0=\pi /4$, and $\beta =5$. The free energy presents minimums at $\alpha =0,2\pi$ corresponding to antiparallel configuration in the vertical bonds. The maximum at $\alpha =\pi$ corresponds to a parrallel configuration in the vertical bonds. For the presentation, we have normalized the free energy such that $F(\alpha =0)=0$ and the maximum value of $F$ is 1