Classical dipoles on the kagome lattice

Motivated by recent developments in magnetic materials, frustrated nanoarrays, and cold atomic systems, we investigate the behavior of dipolar spins on the frustrated two-dimensional kagome lattice. By combining the Luttinger-Tisza approach, numerical energy minimization, spin-wave analysis, and parallel tempering Monte Carlo, we study long-range ordering and finite-temperature phase transitions for a Hamiltonian containing both dipolar and nearest-neighbor interactions. For antiferromagnetic exchange and both weak and moderate dipolar interactions, the system enters a three-sublattice long-range ordered state with each triangle having vanishing dipole and quadrupole moments; whereas for dominating dipolar interactions we uncover ferrimagnetic three-sublattice order. These are also the ground states for $XY$ spins. We discuss excitations of, as well as phase transitions into, these states. We find behavior consistent with Ising criticality for the ${120}^{\circ }$ state, whereas the ferrimagnetic state appears to be associated with drifting exponents. The celebrated flat band of zero-energy excitations of the kagome nearest-neighbor Heisenberg model is lifted to finite energies but acquires only minimal dispersion as dipolar interactions are added.

Figure 1

The kagome lattice (left) and the ground-state phase diagram of the model consisting of nearest-neighbor exchange $J=sin\theta$ and dipolar interactions of strength $D=cos\theta$ (right). $A_1$ and $A_2$ are the basis vectors of the triangular Bravais lattice, and a dark green triangle denotes the unit cell of three sites labeled by ${\mathbf{r}}_0=(0,0),{\mathbf{r}}_1=(1/2,0)$, and ${\mathbf{r}}_2=(0,1/2)$ in this basis. The system exhibits long-range ${120}^{\circ }$ order for $\pi /2\ge \theta >{\theta }_1$ with ${\theta }_1=10.{01}^{\circ }$ and ferrimagnetic order for $-\pi /2<\theta <{\theta }_2$ where ${\theta }_2=1.{03}^{\circ }$. The latter has two spins inclined with respect to one of the unit-cell edges by angle $\pm \varphi \left(\theta \right)$. The area between two phases possibly contains an incommensurate intermediate regime.

Figure 2

Eigenvalues of the interaction matrix $J_{ij}^{\alpha \beta }\left(\mathbf{k}\right)$ along lines in the Brillouin zone. (a) Spectra for dominant exchange case $\theta =\pi /4$. (b) Spectra for $D=1,\theta =0$. The eigenvalue lowest in energy is generally at the $\mathrm{\Gamma }$ point ${\mathbf{k}}_0$.

Figure 3

Inclination angle $\pm \varphi$ of two of the spins in the unit cell (Fig. 1) as a function of $\theta$ [Eq. (4)] in the ferrimagnetic phase.

Figure 4

Spin-wave spectra for (a) the ${120}^{\circ }$ state at $\theta =\pi /4$ and for (b) the ferrimagnetic ground state obtained by energy minimization at $\theta =0$. In (a), the lowest branch remains almost perfectly flat while acquiring a gap $\propto \sqrt{D}$ (the inset).

Figure 5

(a) Specific heat scaled by the temperature and (b) temperature dependence of the reduction of sublattice magnetization due to quantum fluctuations calculated using linear spin-wave theory, which is controlled in the limit of small $T$.

Figure 6

Monte Carlo results for specific heat $C/N$, Binder cumulant $U_L$, magnetization $M$, and susceptibility $\chi$ for $\theta =0$ ($D=1$ and $J=0$).

Figure 7

Monte Carlo results for specific heat $C/N$, Binder cumulant $U_L$, chiral order parameter $M_{\chi }$, and corresponding susceptibility $\chi$ for $\theta =35.6^{\circ }$.

Figure 8

Scaling collapse for the magnetic (chiral) order parameter and its susceptibility for the transition into the ${120}^{\circ }$ state (left) and the ferrimagnetic state (right).