Orbitals, reduced dimensionality and spin gaps in correlated oxides

Due to directional character of orbitals very often orbital ordering leads to an effective dimensionality reduction. Especially strong are these effects in systems with triply-degenerate t$_{2g}$ electrons. One of the striking consequences thereof is the possibility to form spin-gap states instead of long-range magnetic ~ ordering. Such spin gap states may be formed on dimers, on trimers, on bigger clusters and even on chains. In the talk I will review this question, discuss different situations and different mechanisms of spin gap formation, and illustrate these general ideas on several examples (perovskite KCuF3; spinels MgTi2O4 and CuIr2S4 [1]; pyroxene NaTiSi2O6 [2]; layered systems La4Ru2O10 with square [3], LiVO2 with triangular [4], Li2RuO3 with honeycomb lattices [5], pyrochlore Tl2Ru2O7 [6]) \\[4pt] [1] D.I.Khomskii and T.Mizokawa, Phys.Rev.Lett. \textbf{94}, 156402 (2005); \\[0pt] [2] S.V.Streltsov, O.A.Popova and D.I.Khomskii, Phys. Rev. Lett. 96, 249701 (2006) \\[0pt] [3] Hua Wu et al., Phys.Rev.Lett. 96, 256402 (2006) \\[0pt] [4] H. Pen et al., Phys. Rev. Lett. 78, 1323 (1997) \\[0pt] [5] G.Jackeli and D.I.Khomskii, Phys. Rev. Lett. 100, 147203 (2008) \\[0pt] [6] Seongsu Lee et al., Nature Mater. 5, 471 (2006)