Chiral spin states in the pyrochlore Heisenberg magnet: Fermionic mean-field theory and variational Monte Carlo calculations

Fermionic mean-field theory and variational Monte Carlo calculations are employed to shed light on the possible uniform ground states of the Heisenberg model on the pyrochlore lattice. Among the various flux configurations, we find the chiral spin states carrying $\pm \pi /2$ flux through each triangular face to be the most stable both within the mean-field theory and the projected wave-function studies. Properties of the spin-spin correlation function and the chirality order parameter are calculated for the projected wave functions. Mean-field band structures are examined.

Figure 1

(Color online) ${\chi }_{ij}$ bond configurations producing (a) $\left[\frac{\pi }{2},\frac{\pi }{2},0\right]$, (b) $\left[\frac{\pi }{2},-\frac{\pi }{2},0\right]$, and (c) $\left[0,0,\pi \right]$ flux states. Setting $\left|{\chi }_{ij}\right|=1$, the direction of the arrows indicates ${\chi }_{ij}=+i$ going from $i$ to $j$. The shaded bonds are lying on the back sides. Independent sites are labeled from 1 through 4 (one up tetrahedron in a unit cell) in (a) and from 1 through 16 (four up tetrahedra in a unit cell) in (b) and (c). In (c), the thick (dotted) lines represent bonds with ${\chi }_{ij}=1\left(-1\right)$.

Figure 2

(Color online) The two states, $|s_{+}⟩$ and $|s_{-}⟩$, produced by the chirality operation on a state $|s⟩$.

Figure 3

Mean-field energy bands for the $\left[\frac{\pi }{2},-\frac{\pi }{2},0\right]$ state along $k_{\alpha }=\mathbf{k}\cdot e_{\alpha }$ directions, $0\le k_{\alpha }<2\pi$. Each band is twofold degenerate. The occupied bands shown here are symmetric with the upper unoccupied bands due to particle-hole symmetry. The horizontal line $E_F=0$ is the Fermi level. Dispersion along other directions (not shown) also shows a full gap.