Order by disorder in a four-flavor Mott insulator on the fcc lattice

The classical ground states of the SU(4) Heisenberg model on the face-centered-cubic lattice constitute a highly degenerate manifold. We explicitly construct all the classical ground states of the model. To describe quantum fluctuations above these classical states, we apply linear flavor-wave theory. At zero temperature, the bosonic flavor waves select the simplest of these SU(4) symmetry-breaking states, the four-sublattice-ordered state defined by the cubic unit cell of the fcc lattice. Due to geometrical constraints, flavor waves interact along specific planes only, thus rendering the system effectively two dimensional and forbidding ordering at finite temperatures. We argue that longer-range interactions generated by quantum fluctuations can shift the transition to finite temperatures.

Figure 1

The cubic unit cell of the four-sublattice-ordered state. The coloring of the sites on the figure corresponds to the ${\vartheta }_z=0$ case in Eq. (12).

Figure 2

A selection tree illustrating the degeneracy of the classical configurations on an octahedron. The figure shows how the classical fields on the sites of an octahedron formed by the face centers of the unit cell can be chosen one by one, in a way to minimize $E^{\left(0\right)}$ (see the text for details). The gray planes denote the planes colored by two flavors only.

Figure 3

Ordering vectors of the helical state in the Brillouin zone for three different values of $\vartheta$. (a) The Brillouin zone and high-symmetry points of the fcc lattice. (b) The $\vartheta =0$ (the four-sublattice-ordered) case: the ordering vectors ${\mathbf{Q}}_j$ ($j=1,2,3$) are located at the three distinct $X$ high-symmetry points. (c) The $\vartheta =\pi /4$ case: two of the ordering vectors ${\mathbf{Q}}_i$ split into ${\mathbf{Q}}_i^{+}$ and ${\mathbf{Q}}_i^{-}$ ($i=1,2$). These roots move towards the closest $W$ points as $\vartheta$ increases. (d) At $\vartheta =\pi /2$, the ordering wave vectors merge again at the $W_1$ and $W_2$ points, but now ${\mathbf{Q}}_1:={\mathbf{Q}}_1^{+}={\mathbf{Q}}_2^{-}$ and ${\mathbf{Q}}_2:={\mathbf{Q}}_1^{-}={\mathbf{Q}}_2^{+}$.

Figure 4

(a) Quantum correction to the energy of the helical state as a function of $\vartheta$. The energy takes its minimum value for $\vartheta =0$ and $\pi$, which correspond to the four-sublattice state shown in Fig. 1. The red squares and the diamond correspond to the values given in Eqs. (41) and (A4), respectively. (b) The coefficients $e_d$ of the Fourier series of the energy, defined as $E^{\left(2\right)}\left(\vartheta \right)/N_s=e_0+{\sum }_{d=1}^{\infty }{e}_{d}cos2d\vartheta$, plotted on a log-log scale. The first two coefficients are $e_0=-1.843MJ$ and $e_1=-0.043MJ$. Further coefficients scale with the index $d$ as $\propto d^{-3.5}$ (straight line) showing a nonanalytical behavior at $\vartheta =0$.

Figure 5

Dispersion relation of the flavor waves for the Heisenberg model with nearest-neighbor interactions only (a), and with a finite second-neighbor ferromagnetic exchange $J_2/J=-0.086$ (b), along the path in the Brillouin zone shown in the inset. The degeneracy ($\times 1, \times 2$, and $\times 3$) of each mode is proportional to the width of the lines and the color becomes lighter with increasing degeneracy.

Figure 6

The spin reduction vs temperature for different values of the ferromagnetic, next-to-nearest-neighbor coupling $J_2$. For $J_2=0$, the spin reduction is finite only at zero temperature, otherwise it diverges. For nonzero $J_2$, the spin reduction is finite also at nonzero temperatures. The horizontal gray line at ${\mu }_{\mathbf{r},A}=1$ indicates an upper limit of the model's applicability.

Figure 7

The twisted four-sublattice state. The coloring of the sites on the figure corresponds to $\vartheta =\pi /2$ in Eq. (15).