Orbital-selective bad metals due to Hund's rule and orbital anisotropy: A finite-temperature slave-spin treatment of the two-band Hubbard model

We study the finite-temperature properties of the half-filled two-band Hubbard model in the presence of Hund's rule coupling and orbital anisotropy. We use the mean-field treatment of the $Z_2$ slave-spin theory with a finite-temperature extension of the zero-temperature gauge variable previously developed by Hassan and de' Medici [Phys. Rev. B 81, 035106 (2010)]. We consider the instability of the Fermi-liquid phases and how it is enhanced by the Hund's rule. We identify paramagnetic solutions that have zero quasiparticle weight with a bad metal, and the first-order transition temperature between the bad metal and the Fermi-liquid phase as a coherence temperature that signals the crossover to the bad metallic state. When orbital anisotropy is present, we found an intermediate transition to an orbital-selective bad metal (OSBM), where the narrow band becomes a bad metal while the wide band remains a renormalized Fermi liquid. The temperatures $T_{\text{coh}}$ and $T_{\text{OSBM}}$ at which the system transitions to the bad metal phases can be orders of magnitude less than the Fermi temperature associated with the noninteracting band. The parameter dependence of the temperature at which the OSBM is destroyed can be understood in terms of a ferromagnetic Kondo-Hubbard lattice model. In general, Hund's rule coupling enhances the bad metallic phases, reduces interorbital charge fluctuations, and increases spin fluctuations. The qualitative difference found in the ground state whether the Hund's rule is present or not, related to the degeneracy of the low-energy manifold, is also maintained for finite temperatures.

Figure 1

Phase diagram for $T$ vs $J/U$. Stabilization of the orbital-selective bad metal by Hund's rule interaction. The system is at half-filling, the interaction strength is $U/U_{\text{c1}}=0.5$, and orbital anisotropy $W_2/W_1=0.4$.

Figure 2

Destruction of the Fermi liquid by Hund's coupling. Top: quasiparticle weight for half-filling as a function of temperature, with $U/U_{\text{c1}}=0.5$ (red) for different $J/U$. See Appendix pp2 for details of the construction of the physical solution. An increase in temperature from $T=0$ slightly reduces the quasiparticle weight $Z_m$ on each band, and at $T=T_{\text{coh}}$ a first-order transition to the trivial state with $Z=0$ occurs. We identify the trivial solution with a bad metal state, and $T_{\text{coh}}$ with the coherence temperature associated with the crossover to the bad metal regime. Also in thin solid blue line, we show the case $U/U_{\text{c1}}=0.62$ and $J=0$ that is comparable to the $U/U_{\text{c1}}=0.5$ and $J/U=0.2$ one. Bottom: phase diagram for temperature versus interaction. We plot the coherence temperature $T_{\text{coh}}$ as function of the interaction $U$, for different values of the Hund's coupling $J/U=0.0$, 0.1, 0.25, and 0.5. An increase in Hund's coupling $J$ increases the correlations, driving the system closer to a MIT, significantly decreasing $T_{\text{coh}}$, and enhancing the stability of the bad metal state. Inset: same plot with the interaction $U$ normalized with the value at the metal-insulator transition $U_{\text{MIT}}$. The universal behavior of $T_{\text{coh}}$ when varying $J$ reflects that changes in the Hund's rule interaction affect the system only through $U_{\text{MIT}}$.

Figure 3

Top: interorbital charge correlations for half-filling with $U/U_{\text{c1}}=0.5$, varying $J/U$. For $T<T_{\text{coh}}$, increasing $J$ increases localization, the system prefers to have one electron per orbital, and the charge fluctuation approaches to zero. At the transition $T=T_{\text{coh}}$ the charge fluctuations jump to zero for $J>0$. For $J=0$ it jumps to $-\frac{1}{3}$, which is the same limit value when $U\to U_{\text{MIT}}$ (see main text). High-temperature limit for all the cases is zero (not shown). Bottom: interorbital physical spin fluctuations $\langle 4{\widehat{\vec{\mathbb{S}}}}_1·{\widehat{\vec{\mathbb{S}}}}_2\rangle$, same parameters as above. For $T<T_{\text{coh}}$, increasing $J$ increases electron localization and polarizes their spins (one electron per orbital and fully parallel spins). The limit value $\langle 4{\widehat{\vec{\mathbb{S}}}}_1·{\widehat{\vec{\mathbb{S}}}}_2\rangle =1$ is achieved at the transition temperature $T=T_{\text{coh}}$. For $J=0$ there is no interorbital effective interaction and the correlation between orbital spins is zero in the trivial state, at $T>T_{\text{coh}}$. High-temperature limit for all the cases is zero (not shown).

Figure 4

Hund's rule coupling produces an orbital-selective bad metal. Quasiparticle weights $Z_1$ and $Z_2$ for half-filling and $W_2/W_1=0.8$, with $U/U_{\text{c1}}=0.5$. From top to bottom, $J/U=0,0.2$ and 0.4. For $J/U=0.4$ an intermediate transition to an orbital-selective bad metal (OSBM) occurs at $T_{\text{OSBM}}$, where $Z_2$ vanishes while $Z_1$ is further renormalized. After this transition, the interorbital charge fluctuations vanish (cf. Fig. 5, top) while spin fluctuations increase (cf. Fig. 5, bottom), in a similar manner to the first-order transition in the isotropic case.

Figure 5

Charge and spin fluctuations for $n=1$, with $W_2/W_1=0.8$ and $U/U_{\text{c1}}=0.5$. Black, red, and blue are for $J/U=0$, 0.2, and 0.4, respectively. Top: interorbital charge correlations. Same as in the isotropic case, increasing $J$ with $T<(T_{\text{coh}},T_{\text{OSBM}})$ increases electronic localization, and the charge fluctuation approaches to zero. In the OSBM phase there are no charge fluctuations between orbitals ($J/U=0.4$ for $T>T_{\text{OSBM}}$). Middle: intraorbital charge fluctuations. Solid (dashed) line refers to the wide (narrow) orbital. In the OSBM phase, the wide orbital has local charge fluctuations, while no fluctuations of the charge in the other orbital occur. This means that in the OSBM phase, the wide orbital remains a metal with renormalized quasiparticles, while the narrow orbital is fully localized. Bottom: interorbital spin fluctuations. An increase in spin fluctuations occurs when transitioning to the OSBM phase.

Figure 6

Phase diagrams for $T$ vs $U/U_{\text{MIT}}$, for different values of Hund's coupling, with $W_2/W_1=0.8$. The transition temperatures $T_{\text{OSBM}}$ and $T_{\text{coh}}$ are in blue and black solid lines, respectively. For simplicity, OSBM phase regions are shaded gray. Red arrows mark $U=0.5U_{c1}$ on each case. In addition to the strong reduction of the critical interaction $U_{\text{MIT}}$ for finite $J$, Hund's coupling enhances the strong interaction OSBM region and diminishes the weak interaction OSBM region.

Figure 7

Stabilization of the OSBM with increasing $J$ and $W_2/W_1$. Phase diagrams $T$ vs $J$ for $U/U_{\text{c1}}=0.5$ and different anisotropies. For $W_2/W_1<1$ exists a critical $J_c$ where for $J>J_c$ and increasing temperature, the system goes a first-order transition at $T_{\text{OSBM}}$ (blue) toward an OSBM phase that is stable up to $T_{\text{coh}}$. At $T_{\text{coh}}$ (black) the quasiparticle weight of the wide band vanishes, and we have a bad metal in both bands. In the OSBM phase, the narrow orbital is localized, in a high-spin configuration, and no charge fluctuations between orbitals are present. The effective model of the system is a metallic band coupled to a localized band through only a spin-spin interaction, i.e., a ferromagnetic Kondo-Hubbard problem. This manifests in the fact that the $T_{\text{coh}}$ curves that are above in temperature to an OSBM phase become independent of the anisotropy, and has a decay that depends exponentially with the coupling $J$. We plot this temperature $T_K$ with a dashed line for $W_2/W_1=1$, 0.8, and 0.6, while for $W_2/W_1=0.4$ it coincides with $T_{\text{coh}}$ (black solid).

Figure 8

Entropy for the isotropic case with $J=0$. In red, blue, and green we plot the total, fermionic, and slave-spin entropy, respectively. Left: for $T<T_{\text{coh}}$ the slave-spin contribution is very small, and the fermionic degrees of freedom contribute almost all of the total entropy. Right: for high $T$, each of the 16-dimensional subspaces (fermions/slave spin) reach the corresponding value $ln\left(16\right)$. The total entropy reflects the expansion of the Hilbert space to ${16}^2$ degrees of freedom per site and the approximation in the implementation of the constraint. Dashed green line is the slave-spin contribution to the entropy at the trivial solution for $J/U=0.4$.

Figure 9

Entropy for the anisotropic case where the OSBM phase occurs. In red, blue, and green we plot the total, fermionic, and slave-spin entropy, respectively. Left: for $T<T_{\text{coh}}$ the slave-spin contribution is very small, even when the OSBM phase occurs and $Z_2=0$. In this case, the fermionic degrees of freedom contribute almost all of the total entropy in the FL and OSBM phases. Right: for high $T$, each of the 16-dimensional subspaces (fermions/slave spin) reach the corresponding value $ln\left(16\right)$. The total entropy reflects the expansion of the original Hilbert space to ${16}^2$ degrees of freedom per site and the approximated treatment of the constraint.

Figure 10

Construction of the physical solution in the isotropic case. Top: several solutions to the self-consistent equations (18) and (19) for the isotropic case $W_2=W_1$ and $J/U=0$ and 0.2. Bottom: the corresponding free-energy value of each solution (same colors used). The temperature where each finite-$Z$ solution free energy crosses the $Z=0$ one is the corresponding $T_{\text{coh}}$ (dotted lines). The solutions shown in the main text are constructed by concatenating the solutions with the lower free energy on each temperature range.

Figure 11

Construction of the physical solution in the anisotropic case. Each color is a different solution. Top: several solutions to the self-consistent equations (18) and (19) for $W_2=0.8W_1, U=0.5U_{\text{c1}}$, and $J=0.4U$. Solid and dashed lines are $Z_1$ and $Z_2$, respectively. Bottom: the corresponding free-energy value of each solution (same colors used). Two transition temperatures exist in this case, namely, $T_{\text{OSBM}}$ and $T_{\text{coh}}$. The former signals the first-order transition to the OSBM red solution, while the latter the transition to the $Z_1=Z_2=0$ trivial state (orange). The solutions shown in the main text are constructed similarly to the isotropic case, concatenating those with the lower free energy.

Figure 12

Different contributions to the total free energy. For $W_2=0.8W_1, U=0.5U_{\text{c1}}$, and $J=0.4U$, we plot the fermionic (top), slave-spin (middle), and total (bottom) free energies (see main text). We use the same colors as in Fig. 11 for the three solutions with the lowest free energy: renormalized Fermi liquid with $Z_1$ and $Z_2$ finite (black), OSBM with $Z_1$ finite and $Z_2=0$ (red), and bad metal with $Z_1=Z_2=0$ (orange).

Figure 13

Different contributions to the total free energy in the noninteracting, isotropic case. (Top) fermionic contribution, (middle) slave-spin contribution, and (bottom) total free energy. The slave-spin contribution to the free energy for the $Z=1$ solution is lower than the corresponding to $Z=0$ for all temperatures. The addition of the fermionic contribution $f_f$ to obtain the total free energy results in a crossing of the free energy of both solutions.

Figure 14

Dependence of the energies of the atomic configurations in function of $J/U$. The numbers in parentheses denote the degeneracy of the state. For $J=0$ we can see that the $S=0$ and 1 sectors are degenerated. This degeneracy is lifted as soon as $J$ becomes finite, and the $S=1$ sector becomes the lowest in energy. The next in energy states are those corresponding to the $S=0$ sector for $0<J<U/3$, and those corresponding to the total spin $S=\frac{1}{2}$ for $U/3<J<U$.