High-spin to low-spin and orbital polarization transitions in
  multiorbital Mott systems
High-spin to low-spin and orbital polarization transitions in multiorbital Mott systems
Philipp Werner and Andrew J. Millis
Columbia University, 538 West, 120th Street, New York, NY 10027, USA
(Dated: June 30, 2007)
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 eg 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.
PACS numbers: 71.10.Fd, 71.10.Fd, 71.28.+d, 71.30.+h
The Mott metal-insulator transition plays a fundamen-
tal role in electronic condensed matter physics [1]. Much
attention has focused on the one-orbital case, in part be-
cause of its presumed relevance to high temperature su-
perconductivity [2] and in part because appropriate the-
oretical tools for the multiorbital case have until recently
not been available. In most Mott systems, however, more
than one orbital is relevant [3] and the redistribution of
electrons among different orbitals leads to new phenom-
ena such as orbital ordering or “orbital selective” Mott
transitions. Recent studies of nickelates [4], titanates [5],
cobaltates [6], manganates [7], vanadates [8, 9, 10] and
ruthenates [3, 11, 12, 13] have focused interest on the in-
terplay 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 ex-
hibits two fundamentally different Mott phases, one char-
acterized 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 [14]. Weak coupling ap-
proaches [12] 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 [9],
but a reliable extension of these results to physically rel-
evant Slater-Kanamori interactions and the strong cou-
pling 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 [15]. In its sin-
gle site version, DMFT ignores the momentum depen-
dence of the self-energy and reduces the original lat-
tice problem to the self-consistent solution of a quan-
tum impurity model given by the Hamiltonian HQI =
Hloc + Hhyb + Hbath. For multi-orbital models Hloc =∑
m ǫmc
j,k,l,m U
jklmc
l cm, where m = (i, σ)
denotes both orbital and spin indices, and U jklm some
general four-fermion interaction. Hhyb and Hbath are
the impurity-bath mixing and bath Hamiltonians, respec-
tively. While the DMFT approximation simplifies the
problem enormously (replacing a 3 + 1 dimensional field
theory by a quantum impurity model plus a self consis-
tency condition), the extra complications associated with
exchange couplings in multiorbital systems have until re-
cently prohibited extensive numerical work. Interesting
progress has been made using a finite temperature exact
diagonalization technique [6, 13], but this approach re-
quires a truncation of Hbath to a small number of levels.
In Refs. [16, 17] we have introduced a continuous-time
impurity solver which can handle the general interactions
in Hloc. The method, which is free from systematic er-
rors, is based on a diagrammatic expansion of the parti-
tion function in the impurity-bath hybridization Hhyb.
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
Hloc = −
α=1,2
µnα,σ +
∆(n1,σ − n2,σ)
α=1,2
Unα,↑nα,↓ +
U ′n1,σn2,−σ
(U ′ − J)n1,σn2,σ
− J(ψ†
2,↑ψ2,↓ψ1,↑ + ψ
2,↓ψ1,↑ψ1,↓ + h.c.), (1)
with µ the chemical potential, ∆ the crystal field split-
ting, U the intra-orbital and U ′ the inter-orbital Coulomb
interaction, and J the coefficient of the Hund coupling.
We adopt the conventional choice of parameters, U ′ =
http://arxiv.org/abs/0704.0057v2
 0  1  2  3  4  5  6  7  8  9
∆/t=0.2
∆/t=0.6
∆/t=1
FIG. 1: Filling of orbital 1 as a function of U for ∆/t =
0.2, 0.6, 1 and several values of J/U . The different curves
for given ∆ correspond (from bottom to top) to J/U = 0,
(0.01), (0.02), 0.05, 0.1, 0.15, 0.25, respectively. Open (full)
symbols correspond to metallic (insulating) solutions. The
metal-insulator transition is characterized by a jump in filling
and a coexistence region where both insulating and metallic
solutions exist. Our data show the region of stability of the
metallic phase.
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 rotation-
ally invariant in orbital space and the condition for half-
filling becomes µ = µ1/2 ≡ 32U −
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 ∆ and J/U plots the filling per spin, n1, of or-
bital 1 as a function of interaction strength. The half
filling, non-magnetic condition implies n2 = 1 − n1. In
the T → 0 limit, three phases are found: a metallic phase
(which may have any value of n1 between 0 and 0.5), an
orbitally polarized insulator favored by large ∆ and small
J , and a Mott insulator (with n1 = 0.5 = n2) favored
by large U , small ∆ 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. [12]. As interaction strength is further in-
creased, one of several things may happen: at very small
J/U , n1 continues to decrease, and the system eventually
undergoes a transition to an orbitally polarized insulator
(for large ∆ essentially a band insulator). For somewhat
larger J/U , the occupancy n1, after an initial decrease,
goes through a minimum and begins to increase. At even
stronger interactions, one then observes a transition ei-
ther to an orbitally polarized insulator (where n1 may
take a range of values) or into a special type of insulator
 0  0.2  0.4  0.6  0.8  1
J/U=0
0.05 0.1
metal
insulator
 0  0.5  1  1.5  2  2.5  3  3.5
J/U=0 J/U=0.25
Mott insulator
(spin triplet for J/U=0.25)
metal
orbitally polarized
insulator
FIG. 2: Phase diagram in the plane of crystal field splitting ∆
and intraorbital Coulomb repulsion U for indicated values of
J/U . For J = 0 the phase boundary is a monotonic function
of ∆, whereas for J/U > 0 it peaks near ∆ =
2J (indicated
by the dotted lines). The insulating state in the region ∆ .√
2J is characterized by a vanishing orbital susceptibility.
with n1 = 0.5.
Figure 2 shows the metal-insulator phase diagram in
the space of crystal field splitting and Coulomb repul-
sion for several values of J/U . In the absence of a crystal
field splitting (∆ = 0), we observe a metal-insulator tran-
sition at a strongly J-dependent critical U . This finding
is consistent with data presented in Ref. [18]. As ∆ is
increased, the critical U changes. For J = 0 and fixed
U , n1 decreases until the band is emptied and a metal-
insulator transition occurs. The monotonic decrease of
the critical U with ∆ at J = 0 is a special case. For
J > 0, the first effect of a small ∆ is to stabilize the
metallic phase. Then, at larger ∆, 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 ∆, which is characterized at T = 0 by a
strictly vanishing orbital susceptibility. If ∆ is increased
at large U , this state makes a transition to an orbitally
polarized insulator at ∆ ≈
2J . We therefore plot in
Fig. 2 the curves ∆ =
2J as dotted lines, and sug-
gest that they correspond to the T = 0 phase boundary
 0.05
 0.15
 0.25
 0.35
 0.45
 0  0.2  0.4  0.6  0.8  1  1.2
J/U=0.1
 0  0.05  0.1  0.15  0.2
J/U=0
J/U=0.002
J/U=0.005
J/U=0.010
J/U=0.020
FIG. 3: Filling of orbital 1 as a function of ∆ for fixed U
and indicated values of J/U . Top panel: U/t = 6. Open
(full) symbols correspond to metallic (insulating) solutions.
Bottom panel: U/t = 9. Here, all solutions are insulating.
For crystal field splittings smaller than ∆c =
2J (indicated
by a vertical line) the orbital susceptibility in the T → 0 limit
is completely suppressed. Solid lines are for βt = 50, dotted
lines show results for βt = 12.5, 25 and 100, respectively.
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 ∆ = 0, the model
is metallic. The rapid variation of n1 with ∆ reflects the
large, but finite orbital susceptibility of the metal, which
for this small value of J is strongly enhanced by U . At
∆/t ≈ 0.325 >
2J , an apparently first order transition
occurs to the orbitally polarized insulating state, which
then evolves smoothly (as ∆ is increased) to the band
insulator (n1 → 0). The two larger J values reveal a
different behavior. For ∆ < ∆c ≈
2J the insulating
state is characterized by an orbital occupancy which is
independent of crystal field splitting. Then, an appar-
ently discontinuous transition occurs to a metallic state
with a large orbital susceptibility, which at even larger ∆
exhibits a first order transition to the orbitally polarized
insulating state. The lower panel of Fig. 3 shows the be-
 2  4  6  8  10  12  14  16
basis state
∆/t=0.3
∆/t=0.5
∆/t=0.7
FIG. 4: Weight of the different eigenstates of Hloc for U/t =
6, J/U = 0.05 and ∆/t = 0.3, 0.5 and 0.7. The smallest
crystal field splitting corresponds to an insulating state with
suppressed orbital susceptibility, the intermediate value to a
metallic state and the largest splitting to a “band insulator”
(see Fig. 3).
havior for larger U , where the model is always insulating.
Our data for J = 0 exhibit a rapid variation of n1 with
∆. The slope is set by the inverse of the Kugel-Khomskii
superexchange ∼ t2/U ; thermal effects are unimportant
at βt = 50. For J > 0 and small ∆, the model is insu-
lating, with a vanishing orbital susceptibility, then (near
2J) makes a transition to the orbitally polarized
insulating phase with a differential susceptibility ∂n1/∂∆
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 differ-
ent eigenstates of the local Hamiltonian. Hloc has 16
eigenstates, which we number essentially as in Table II
of Ref. [17]. For the following discussion it is important to
note that |6〉, |7〉 and |8〉 are the three spin triplet states
(with energy U − 3J − 2µ), while |10〉 and |11〉 are lin-
ear combinations of the states |↑↓, 0〉 and |0,↑↓〉 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 Hloc corresponding to the eigenenergies
J2 + 4∆2 − 2µ: (1 + α2±)−1/2(| ↑↓, 0〉+ α±|0,↑↓〉),
α± = ±(
J2 + 4∆2 ∓ 2∆)/J . In particular, we choose
|10〉 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-∆ phase, the triplet states are occupied,
with small excursions into states with occupancy 1 or
3. We therefore call this phase the triplet Mott insu-
lator. 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 af-
 0.55
 0.65
 0.75
 3.5  4  4.5  5  5.5  6
orbital 1
orbital 2
FIG. 5: Filling n1(µ), n2(µ) for U/t = 4, J/U = 0.25 and
∆/t = 0.4. Full (open) symbols correspond to insulating
(metallic) solutions. At half-filling (µ/t = 3.5), the system
is in a triplet Mott insulating state, for 3.9 . µ/t . 4.6 in
an orbital selective Mott state, and for µ/t & 4.6 metallic in
both bands.
ter coupling to the lattice). In the metallic phase, a large
number of states is visited, while in the orbitally polar-
ized insulator, the dominant local state (whose weight
increases continuously with ∆) is a singlet (|10〉). The
triplet states are almost completely suppressed in the or-
bitally polarized phase. The large-U insulator-insulator
transition exhibits the same features, but without the
intermediate metallic phase, and is therefore also a tran-
sition between high and low spin states. Comparison of
the eigenenergies of the spin triplet states and |10〉 show
that these levels cross at ∆ =
2J . Thus, the transition
from triplet Mott insulator to orbitally polarized insula-
tor occurs at ∆c =
2J , consistent with our numerical
data. We also note that the wave-function of state |10〉
depends on the ratio J/∆, leading in the large-∆ limit
to n1(∆) ≈ (J/4∆)2. In the low spin phase, the or-
bital susceptibility has therefore two contributions: one
originating from Kugel-Khomskii physics and one of or-
der J2/∆3 from Hloc. 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 [11]. 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 µ, for U/t = 4, J/U = 0.25 and ∆/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 impu-
rity models with realistic couplings can be efficiently sim-
ulated with the method of Ref. [17]. We have presented
numerical evidence, based on single site DMFT calcula-
tions, for the existence of two distinct Mott insulating
phases in a half-filled two-orbital model with Hund cou-
pling and crystal field splitting. At strong interactions
and J <
2∆, the system, in the T → 0 limit, is in
a phase characterized by a vanishing orbital susceptibil-
ity, and a spin 1 moment on each site. For J >
an orbitally polarized insulator is found. The exchange
terms promote insulating behavior at ∆ = 0 but can
stabilize a metallic phase at values of ∆ 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 [19, 20]. The model
which these authors study is equivalent to our model with
U = U ′ = J , and ∆ 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〉) 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.
The calculations have been performed on the Hreidar
beowulf cluster at ETH Zürich, using the ALPS-library
[21]. We thank M. Troyer for the generous allocation of
computer time, A. Georges and A. Poteryaev for stimu-
lating discussions and NSF-DMR-040135 for support.
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