Spin Singlet State in Heptamers Emerging in Spinel Oxide AlV$_2$O$_4$

We present our theoretical results on the electronic state in vanadium spinel oxide AlV$_2$O$_4$. The material is a mixed-valent system with the average valence V$^{2.5+}$, and V cations constitute a frustrated pyrochlore structure. It shows a structural transition at $\sim 700$ K, leading to the formation of seven V-sites clusters -- heptamers. We study the electronic state of the heptamer by explicitly taking account of orbital degree of freedom as well as electron correlations. We show that the ground state of the heptamer for realistic parameters becomes spin-singlet because of strong $\sigma$-type bonding states of $t_{2g}$ orbitals. The temperature dependence of the magnetic susceptibility in experiments is naturally understood by this singlet formation in heptamers. Our results indicate that in AlV$_2$O$_4$ orbital physics is relevant to stabilize a cluster-type singlet state instead of a previously proposed charge-ordered state with valence skipping.