High-spin to low-spin and orbital polarization transitions in multiorbital Mott systems

TL;DR

This paper investigates how crystal field splitting and Hund coupling influence phase transitions in a two-orbital Mott system, revealing novel insulating states and orbital selectivity through advanced dynamical mean field theory.

Contribution

It introduces a detailed study of phase boundaries and orbital polarization transitions in multiorbital systems using a new impurity solver at strong coupling and low temperatures.

Findings

Hund coupling induces a novel Mott insulating phase with zero orbital susceptibility.

Crystal field splitting can lead to orbital selective Mott states away from half-filling.

Phase diagrams as functions of Coulomb repulsion, exchange coupling, and crystal field splitting are mapped.

Abstract

We study the interplay of crystal field splitting and Hund coupling in a two-orbital model which captures the essential physics of systems with two electrons or holes in the e_g shell. We use single site dynamical mean field theory with a recently developed impurity solver which is able to access strong couplings and low temperatures. The fillings of the orbitals and the location of phase boundaries are computed as a function of Coulomb repulsion, exchange coupling and crystal field splitting. We find that the Hund coupling can drive the system into a novel Mott insulating phase with vanishing orbital susceptibility. Away from half-filling, the crystal field splitting can induce an orbital selective Mott state.

Figures

Equations

$$H_{\text{loc}}=-\sum _{\alpha =1,2}\sum _{\sigma }\mu n_{\alpha ,\sigma }+\sum _{\sigma }\Delta (n_{1,\sigma }-n_{2,\sigma })$$

$$+\sum _{\alpha =1,2}U{n}_{\alpha ,\uparrow }{n}_{\alpha ,\downarrow }+\sum _{\sigma }{U}^{\prime }{n}_{1,\sigma }{n}_{2,-\sigma }$$

$$+\sum _{\sigma }(U^{\prime }-J)n_{1,\sigma }{n}_{2,\sigma }$$

$$-J({\psi }_{1,\downarrow }^{†}{\psi }_{2,\uparrow }^{†}{\psi }_{2,\downarrow }{\psi }_{1,\uparrow }+{\psi }_{2,\uparrow }^{†}{\psi }_{2,\downarrow }^{†}{\psi }_{1,\uparrow }{\psi }_{1,\downarrow }+h.c.),$$

Full text

The Mott metal-insulator transition plays a fundamental role in electronic condensed matter physics Imada98 . Much attention has focused on the one-orbital case, in part because of its presumed relevance to high temperature superconductivity Anderson86 and in part because appropriate theoretical tools for the multiorbital case have until recently not been available. In most Mott systems, however, more than one orbital is relevant Tokura00 and the redistribution of electrons among different orbitals leads to new phenomena such as orbital ordering or “orbital selective” Mott transitions. Recent studies of nickelates Kunes , titanates Ulrich06 , cobaltates Ishida05 , manganates Yamasaki06 , vanadates Biermann05 ; Lechermann05 ; Yoshida05 and ruthenates Tokura00 ; Koga04 ; Okamoto04 ; Liebsch07 have focused interest on the interplay between the Mott metal-insulator transition and orbital degeneracy. A fundamental question in this field, relevant in particular to the issue of lattice distortions in strongly correlated materials, is the response of multi-orbital systems to a perturbation which breaks the orbital degeneracy. In this paper, we show that a two orbital model with Hund coupling and crystal field splitting exhibits two fundamentally different Mott phases, one characterized by a vanishing orbital susceptibility, and one adiabatically connected to the band insulating state. We characterize these phases in terms of the atomic ground states.

Multiorbital models are more difficult to study both because of the larger number of degrees of freedom, and because the physically important exchange and pair-hopping terms are not easy to treat by standard Hubbard-Stratonovich methods Sakai06 . Weak coupling approaches Okamoto04 have been used to show that exchange and pair hopping interactions act to suppress the response to a crystal field splitting, and some authors have studied the model without exchange and pair hopping terms Lechermann05 , but a reliable extension of these results to physically relevant Slater-Kanamori interactions and the strong coupling regime has been lacking.

Dynamical mean field theory (DMFT) provides a non-perturbative and computationally tractable framework to study correlation effects and has allowed insights into the Mott metal-insulator transition Georges96 . In its single site version, DMFT ignores the momentum dependence of the self-energy and reduces the original lattice problem to the self-consistent solution of a quantum impurity model given by the Hamiltonian $H_{\text{QI}}=H_{\text{loc}}+H_{\text{hyb}}+H_{\text{bath}}$. For multi-orbital models $H_{\text{loc}}=\sum_{m}\epsilon_{m}c^{\dagger}_{m}c_{m}+\sum_{j,k,l,m}U^{jklm}c^{\dagger}_{j}c_{k}c^{\dagger}_{l}c_{m}$, where $m=(i,\sigma)$ denotes both orbital and spin indices, and $U^{jklm}$ some general four-fermion interaction. $H_{\text{hyb}}$ and $H_{\text{bath}}$ are the impurity-bath mixing and bath Hamiltonians, respectively. While the DMFT approximation simplifies the problem enormously (replacing a $3+1$ dimensional field theory by a quantum impurity model plus a self consistency condition), the extra complications associated with exchange couplings in multiorbital systems have until recently prohibited extensive numerical work. Interesting progress has been made using a finite temperature exact diagonalization technique Ishida05 ; Liebsch07 , but this approach requires a truncation of $H_{\text{bath}}$ to a small number of levels. In Refs. Werner05 ; Werner06 we have introduced a continuous-time impurity solver which can handle the general interactions in $H_{\text{loc}}$. The method, which is free from systematic errors, is based on a diagrammatic expansion of the partition function in the impurity-bath hybridization $H_{\text{hyb}}$.

Here, we employ this solver to study the physically relevant case in which the number of electrons matches the number of orbitals. The local Hamiltonian is

[TABLE]

with $\mu$ the chemical potential, $\Delta$ the crystal field splitting, $U$ the intra-orbital and $U^{\prime}$ the inter-orbital Coulomb interaction, and $J$ the coefficient of the Hund coupling. We adopt the conventional choice of parameters, $U^{\prime}=U-2J$, which follows from symmetry considerations for $d$-orbitals in free space and is also assumed to hold in solids. With this choice the Hamiltonian (1) is rotationally invariant in orbital space and the condition for half-filling becomes $\mu=\mu_{1/2}\equiv\frac{3}{2}U-\frac{5}{2}J$. In the DMFT self-consistency loop we use a semi-circular density of states of bandwith $4t$ (Bethe lattice). The temperature, unless otherwise noted, is $T/t=0.02$ and we suppress magnetic order by averaging over spin up and down in each orbital. No sign problem is encountered in the simulations.

The main result is shown in Fig. 1, which for several values of $\Delta$ and $J/U$ plots the filling per spin, $n_{1}$, of orbital 1 as a function of interaction strength. The half filling, non-magnetic condition implies $n_{2}=1-n_{1}$. In the $T\rightarrow 0$ limit, three phases are found: a metallic phase (which may have any value of $n_{1}$ between [math] and $0.5$), an orbitally polarized insulator favored by large $\Delta$ and small $J$, and a Mott insulator (with $n_{1}=0.5=n_{2}$) favored by large $U$, small $\Delta$ and large $J$. If $U$ is increased from zero to a small value, the orbital splitting either increases (small $J/U$) or decreases (large $J/U$), consistent with the findings of Ref. Okamoto04 . As interaction strength is further increased, one of several things may happen: at very small $J/U$, $n_{1}$ continues to decrease, and the system eventually undergoes a transition to an orbitally polarized insulator (for large $\Delta$ essentially a band insulator). For somewhat larger $J/U$, the occupancy $n_{1}$, after an initial decrease, goes through a minimum and begins to increase. At even stronger interactions, one then observes a transition either to an orbitally polarized insulator (where $n_{1}$ may take a range of values) or into a special type of insulator with $n_{1}=0.5$.

Figure 2 shows the metal-insulator phase diagram in the space of crystal field splitting and Coulomb repulsion for several values of $J/U$. In the absence of a crystal field splitting ($\Delta=0$), we observe a metal-insulator transition at a strongly $J$-dependent critical $U$. This finding is consistent with data presented in Ref. Koga02 . As $\Delta$ is increased, the critical $U$ changes. For $J=0$ and fixed $U$, $n_{1}$ decreases until the band is emptied and a metal-insulator transition occurs. The monotonic decrease of the critical $U$ with $\Delta$ at $J=0$ is a special case. For $J>0$, the first effect of a small $\Delta$ is to stabilize the metallic phase. Then, at larger $\Delta$, a reentrant insulating phase occurs. We shall show below that this behavior arises from the unusual nature of the insulating state at $J>0$ and small $\Delta$, which is characterized at $T=0$ by a strictly vanishing orbital susceptibility. If $\Delta$ is increased at large $U$, this state makes a transition to an orbitally polarized insulator at $\Delta\approx\sqrt{2}J$. We therefore plot in Fig. 2 the curves $\Delta=\sqrt{2}J$ as dotted lines, and suggest that they correspond to the $T=0$ phase boundary between two distinct insulating states.

Figure 3 plots the filling of orbital 1 as a function of crystal field splitting for fixed $U/t$ and several values of $J/U$. The leftmost curve in the upper panel shows the density variation for $J/U=0.02$. At $\Delta=0$, the model is metallic. The rapid variation of $n_{1}$ with $\Delta$ reflects the large, but finite orbital susceptibility of the metal, which for this small value of $J$ is strongly enhanced by $U$. At $\Delta/t\approx 0.325>\sqrt{2}J$, an apparently first order transition occurs to the orbitally polarized insulating state, which then evolves smoothly (as $\Delta$ is increased) to the band insulator ($n_{1}\rightarrow 0$). The two larger $J$ values reveal a different behavior. For $\Delta<\Delta_{c}\approx\sqrt{2}J$ the insulating state is characterized by an orbital occupancy which is independent of crystal field splitting. Then, an apparently discontinuous transition occurs to a metallic state with a large orbital susceptibility, which at even larger $\Delta$ exhibits a first order transition to the orbitally polarized insulating state. The lower panel of Fig. 3 shows the behavior for larger $U$, where the model is always insulating. Our data for $J=0$ exhibit a rapid variation of $n_{1}$ with $\Delta$. The slope is set by the inverse of the Kugel-Khomskii superexchange $\sim t^{2}/U$; thermal effects are unimportant at $\beta t=50$. For $J>0$ and small $\Delta$, the model is insulating, with a vanishing orbital susceptibility, then (near $\Delta=\sqrt{2}J$) makes a transition to the orbitally polarized insulating phase with a differential susceptibility $\partial n_{1}/\partial\Delta$ determined by Kugel-Khomskii physics. Note that the transition between the two insulators is sharp only at $T=0$; for $T>0$ a rapid (but smooth) crossover occurs.

To gain insight into these phenomena, we look at the contribution to the partition function from the different eigenstates of the local Hamiltonian. $H_{\text{loc}}$ has 16 eigenstates, which we number essentially as in Table II of Ref. Werner06 . For the following discussion it is important to note that $|6\rangle$, $|7\rangle$ and $|8\rangle$ are the three spin triplet states (with energy $U-3J-2\mu$), while $|10\rangle$ and $|11\rangle$ are linear combinations of the states $|\!\!\uparrow\downarrow,0\rangle$ and $|0,\!\uparrow\downarrow\rangle$ with two electrons in one orbital and none in the other. The latter two states are coupled by the pair hopping and affected by the crystal field splitting. Here, we choose them to be eigenstates of $H_{\text{loc}}$ corresponding to the eigenenergies $U\pm\sqrt{J^{2}+4\Delta^{2}}-2\mu$: $(1+\alpha_{\pm}^{2})^{-1/2}(|\!\uparrow\downarrow,0\rangle+\alpha_{\pm}|0,\!\uparrow\downarrow\rangle)$, $\alpha_{\pm}=\pm(\sqrt{J^{2}+4\Delta^{2}}\mp 2\Delta)/J$. In particular, we choose $|10\rangle$ to be the eigenstate with lower energy.

Figure 4 shows the weights of these states for the three phases found at $U/t=6$, $J/U=0.05$ (see Fig. 3). In the small-$\Delta$ phase, the triplet states are occupied, with small excursions into states with occupancy 1 or 3. We therefore call this phase the triplet Mott insulator. The triplet states of course have one electron in each orbital and gain no energy from orbital polarization (the remarkable fact is that this feature is preserved after coupling to the lattice). In the metallic phase, a large number of states is visited, while in the orbitally polarized insulator, the dominant local state (whose weight increases continuously with $\Delta$) is a singlet ($|10\rangle$). The triplet states are almost completely suppressed in the orbitally polarized phase. The large-$U$ insulator-insulator transition exhibits the same features, but without the intermediate metallic phase, and is therefore also a transition between high and low spin states. Comparison of the eigenenergies of the spin triplet states and $|10\rangle$ show that these levels cross at $\Delta=\sqrt{2}J$. Thus, the transition from triplet Mott insulator to orbitally polarized insulator occurs at $\Delta_{c}=\sqrt{2}J$, consistent with our numerical data. We also note that the wave-function of state $|10\rangle$ depends on the ratio $J/\Delta$, leading in the large-$\Delta$ limit to $n_{1}(\Delta)\approx(J/4\Delta)^{2}$. In the low spin phase, the orbital susceptibility has therefore two contributions: one originating from Kugel-Khomskii physics and one of order $J^{2}/\Delta^{3}$ from $H_{\text{loc}}$. The latter explains the roundings seen in the right most curve of Fig. 3.

We briefly address the issue of the orbital selective Mott transition, which provides a mechanism for local moment formation in correlated materials, and has been the subject of much recent debate Koga04 . Previous studies focused on two-orbital models with different band-widths and integer number of electrons. We find that in the presence of a crystal field splitting, shifting the chemical potential can drive the system into an orbital selective Mott state, even if the band-widths are equal. Figure 5 shows the filling per spin in both orbitals as a function of $\mu$, for $U/t=4$, $J/U=0.25$ and $\Delta/t=0.4$. Doping occurs first in one of the bands, leaving the other in a Mott state with a magnetic moment. Further change of the chemical potential drives the second band metallic.

In conclusion, we have shown that multiorbital impurity models with realistic couplings can be efficiently simulated with the method of Ref. Werner06 . We have presented numerical evidence, based on single site DMFT calculations, for the existence of two distinct Mott insulating phases in a half-filled two-orbital model with Hund coupling and crystal field splitting. At strong interactions and $J<\sqrt{2}\Delta$, the system, in the $T\rightarrow 0$ limit, is in a phase characterized by a vanishing orbital susceptibility, and a spin $1$ moment on each site. For $J>\sqrt{2}\Delta$ an orbitally polarized insulator is found. The exchange terms promote insulating behavior at $\Delta=0$ but can stabilize a metallic phase at values of $\Delta$ for which the non-interacting model is a band-insulator.

It is interesting to compare our results to recent work on the bilayer Hubbard model Fuhrmann06 ; Kancharla07 . The model which these authors study is equivalent to our model with $U=U^{\prime}=J$, and $\Delta$ replaced by the interlayer hopping. In the low energy sector of this model, only four states (essentially our three triplets and the pair hopping state $|10\rangle$) are relevant, and what these authors describe as the Mott insulator to band insulator crossover corresponds to our transition (apparently sharp at $T=0$) between triplet Mott insulator and orbitally polarized insulator.

The existence of two distinct insulating phases raises many interesting questions including the theory of an insulator with strictly vanishing orbital susceptibility (which should exhibit an orbital gauge symmetry) and the nature and properties of the different metal-insulator transitions. The physics near the “triple point” remains to be studied. Our results away from half-filling suggest that lightly doped La2NiO4 is in an orbitally selective Mott phase.