Orbital-dependent singlet dimers and orbital-selective Peierls transitions in transition-metal compounds

We show that in transition-metal compounds containing structural metal dimers there may exist in the presence of different orbitals a special state with partial formation of singlets by electrons on one orbital, while others are effectively decoupled and may give, e.g., long-range magnetic order or stay paramagnetic. A similar situation can be realized in dimers spontaneously formed at structural phase transitions, which can be called the orbital-selective Peierls transition. This can occur in the case of strongly nonuniform hopping integrals for different orbitals and small intra-atomic Hund's rule coupling $J_H$. Yet another consequence of this picture is that for an odd number of electrons per dimer there exists competition between the double-exchange mechanism of ferromagnetism and the formation of a singlet dimer by the electron on one orbital, with the remaining electrons giving a net spin of a dimer. The first case is realized for strong Hund's rule coupling, typical for $3d$ compounds, whereas the second is more plausible for 4$d$-5$d$ compounds. We discuss some implications of these phenomena, and consider examples of real systems, in which the orbital-selective phase seems to be realized.

Figure 1

(a) The sketch illustrating how different hopping integrals may appear in the system. The $t_{dd\sigma }$ hopping between white orbitals is larger than between gray $t_{dd\delta }$. (b) Corresponding level splitting.

Figure 2

The total and partial magnetization per dimer, calculated in C-DMFT. $t^{\prime }=0.1$ eV, $t_d=0.2$ eV, $t_c=6t_d$, $J_H=t_d/2$, $U=5t_d$, $T=0.1$ eV. Inset shows dependence of total magnetization on Hund's rule exchange.

Figure 3

Uniform magnetic susceptibility, calculated in C-DMFT as $\chi =M/B_{\mathrm{ext}}$, where $M$ is magnetization per dimer, and $B_{\mathrm{ext}}$ external magnetic field. $t^{\prime }=0.1$ eV, $t_d=0.4$ eV, $B_{\mathrm{ext}}=0.1$ eV, $U=5.25t^{\prime }$, $t_c=3t_d$, $J_H=1.25t_d$.

Figure 4

Possible orbital states in the case of the dimer with 2 orbitals per site and 3 electrons per dimer.