Spin Freezing Transition and Non-Fermi-Liquid Self-Energy in a Three-Orbital Model

A single-site dynamical mean-field study of a three band model with the rotationally invariant interactions appropriate to the $t_{2g}$ levels of a transition metal oxide reveals a quantum phase transition between a paramagnetic metallic phase and an incoherent metallic phase with frozen moments. The Mott transitions occurring at electron densities $n=2$, 3 per site take place inside the frozen moment phase. The critical line separating the two phases is characterized by a self-energy with the frequency dependence $\Sigma (\omega )\sim \sqrt{\omega }$ and a broad quantum critical regime. The findings are discussed in the context of the power law observed in the optical conductivity of ${\mathrm{SrRuO}}_3$.

Figure 1

Phase diagram for $J/U=1/6$ and $\beta t=50$, 100 in the space of density $n$ and interaction strength $U$. The light line with circles or diamonds indicates a phase transition between a Fermi liquid metal and a “frozen-moment” metal. The black lines mark the regions of Mott insulating behavior.

Figure 2

Upper panel: imaginary time dependence of the spin-spin correlation function $⟨S_z(0)S_z(\tau )⟩$ (positive correlation function, full symbols) and orbital correlation function $⟨n_1(0)n_2(\tau )⟩$ (negative correlation function, open symbols) for $U=8t$ and carrier concentrations $n$ indicated. Lower panel: variation with doping of the temperature dependence of the spin-spin correlation at $\tau =\beta /2$. The error bars are large at smaller $n$ because the midpoint spin correlator is very small. The black line indicates the $n$-value of the phase transition deduced from an analysis of the self-energy.

Figure 3

Doping dependence of the imaginary part of the Matsubara axis self-energy for $U/t=8$, $\beta t=50$ (full symbols) and 100 (open symbols) at dopings indicated. The dashed lines are proportional to $({\omega }_n/t{)}^{\alpha }$, $\alpha =0.5$, 0.54, 0.62 (from top to bottom).

Figure 4

Results of a fit of the computed Matsubara-axis self-energy to the scaling form $-\mathrm{Im}\Sigma /t=C+A({\omega }_n/t{)}^{\alpha }$ at temperatures $T=t/50$ (diamonds; blue online) and $T=t/100$ (circles; red online) at density $n=2$ and interaction strengths indicated. The plot illustrates the wide quantum critical regime of the spin-freezing transition.

Figure 5

Self-energy for the majority spin as a function of magnetic field at $U/t=7$, $\beta t=50$, and $n=2$, slightly above the critical point for the glass transition.