Quantum Liquid with Strong Orbital Fluctuations: The Case of a Pyroxene Family

We discuss quasi-one-dimensional magnetic Mott insulators from the pyroxene family where spin and orbital degrees of freedom remain tightly bound. We analyze their excitation spectrum and outline the conditions under which the orbital degrees of freedom become liberated so that the corresponding excitations become dispersive and the spectral weight shifts to energies much smaller than the exchange integral.

Figure 1

Crystal structure of Na $M{\mathrm{Si}}_2{\mathrm{O}}_6$.

Figure 2

Spin spectral function in the folded Brillouin Zone for various values of $t_2/t_1$ and the crystal field. With increase of $\mathrm{\Delta }/J$ or the anisotropy the spectral weight shifts toward the dimerized configuration where the singlet-triplet gap is equal to $2J=4t_2^2/(U-\mathrm{\Delta })$ corresponding to the breaking of a dimer.

Figure 3

(a) A graphic description of a dimerized state for an isolated $\mathrm{Na}M{\mathrm{Si}}_2{\mathrm{O}}_6$ chain. Only $t_{2g}$ orbital of $M$ ions are depicted. (b) The original tunneling scheme. (c) The tunneling scheme with relabeled orbitals. We relabel the orbitals on different sites to make the tunneling diagonal. The solid lines correspond to matrix element $t_1$, the dashed lines corresponds to matrix element $t_2$. (d) The dimerization pattern in the presence of crystal field. The orbitals on which spin singlets form are shown by thick lines.

Figure 4

Numerical results for the lowest spectral gaps for various values of the crystal field $\mathrm{\Delta }$ and $t_2=t_1$.

Figure 5

Dimerization for various values of the crystal field $\mathrm{\Delta }$ and $t_2=t_1$. (a) Nearest neighbor spin-spin correlation; (b) density-density correlation in the orbital channel; (c) schematic illustration of the spin-orbital order in the limit of large $\mathrm{\Delta }$: dashed lines represent orbitals that are projected out. Charge fluctuations are suppressed and charge is frozen in the depicted pattern. Spin is only allowed to interact in pairs forming independent singlets.