Finite-temperature Gutzwiller approximation and the phase diagram of a toy model for V${}_2$O${}_3$

We exploit exact inequalities that refer to the entropy of a distribution to derive a simple variational principle at finite temperature for trial density matrices of Gutzwiller and Jastrow type. We use the result to extend at finite temperature the Gutzwiller approximation, which we apply to study a two-orbital model that we believe captures some essential features of V${}_2$O${}_3$. We indeed find that the phase diagram of the model bears many similarities to that of real vanadium sesquioxide. In addition, we show that in a Bethe lattice, where the finite-temperature Gutzwiller approximation provides a rigorous upper bound of the actual free energy, the results compare well with the exact phase diagram obtained by dynamical mean-field theory.

Figure 1

$T=0$ phase diagram for model in Eq. (31). The solid line indicates the MIT transition; the dashed/dotted line separates the two-band paramagnetic metal (2-band PM) from the one-band metal (1-band PM). The vertical dashed red lines indicate the values of $\Delta$ which are used in Fig. 2.

Figure 2

$n_{>}-n_{<}$ (blue diamonds), renormalization factors $R_1$ (black squares) and $R_2$ (red circles) as a function of $U$ for fixed $\Delta =0.025W$ (left panel) and $\Delta =0.3W$ (right panel). The vertical dashed lines indicate the two-band $\to$ one-band metal transition and the MIT.

Figure 3

Zero-temperature phase diagram allowing for magnetism. The black line separates the two-band paramagnetic metal (2-band PM) from the one-band antiferromagnetic insulator (1-band AFI). In the inset we plot the orbital polarization $n_1-n_2$ and the staggered magnetization for a fixed value of $\Delta =0.025W$ (red dashed line).

Figure 4

Upper panel: Phase diagram in the paramagnetic domain. The black line separates the PM phase from the PI one. The red vertical lines indicate the values of $U$ plotted in the lower panel. Lower panel: Temperature dependence of the quasiparticle renormalization factor for the majority orbital (left) and of the orbital polarization (right) for different values of $U$.

Figure 5

Finite-temperature phase diagram as a function of $U$ and $T$, for a fixed value of $\Delta =0.025W$. The paramagnetic solution is continued also within the AFM domain (dotted line). The vertical red dashed line indicates the temperature cut represented in Fig. 6.

Figure 6

The blue (to the left in each panel) and red areas (to the right) indicate respectively the AFI phase and the paramagnetic phases as a function of temperature at fixed values of $U=1.22W$ (left panel) and $U=1.8W$ (right panel). At low temperatures the orbital polarization (blue squares) and the staggered magnetization (black squares) are practically equal to the zero-temperature values and display a discontinuous jump at the AFI-paramagnetic transition. In the paramagnetic phase we show also the behavior of the renormalization factors whose jump indicates the PM-PI transition.

Figure 7

Top panel: Finite $T$ phase diagram within the Gutzwiller approximation at $\Delta =0.025W$, $t^{\prime }=0$, and a semicircular DOS. Bottom panel: Same as before but within DMFT, which is exact.