Effective band-structure in the insulating phase versus strong dynamical
  correlations in metallic VO2
s_Im_U3.eps
Effective band-structure in the insulating phase versus
strong dynamical correlations in metallic VO2
Jan M. Tomczak,1 Ferdi Aryasetiawan,2 and Silke Biermann1
Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France
Research Institute for Computational Sciences, AIST,
Umezono 1-1-1, Tsukuba Central 2, Tsukuba Ibaraki 305-8568, Japan
Using a general analytical continuation scheme for cluster dynamical mean field calculations,
we analyze real-frequency self-energies, momentum-resolved spectral functions, and one-particle
excitations of the metallic and insulating phases of VO2. While for the former dynamical correlations
and lifetime effects prevent a description in terms of quasi-particles, the excitations of the latter
allow for an effective band-structure. We construct an orbital-dependent, but static one-particle
potential that reproduces the full many-body spectrum. Yet, the ground state is well beyond a
static one-particle description. The emerging picture gives a non-trivial answer to the decade-old
question of the nature of the insulator, which we characterize as a “many-body Peierls” state.
PACS numbers: 71.27.+a, 71.30.+h, 71.15.Ap
Describing electronic correlations is a challenge for
modern condensed matter physics. While weak corre-
lations slightly modify quasi-particle states, by broad-
ening them with lifetime effects and shifting their ener-
gies, strong enough correlations can entirely invalidate
the band picture by inducing a Mott insulating state.
In a half-filled one-band model, an insulator is re-
alized above a critical ratio of interaction to band-
width. Though more complex scenarios exist in realis-
tic multi-band cases, a common feature of compounds
that undergo a metal-insulator transition (MIT) upon
the change of an external parameter, such as temper-
ature or pressure, is that the respective insulator feels
stronger correlations than the metal, since it is precisely
their enhancement that drives the system insulating.
In this paper we discuss a material where this rule of
thumb is inverted : We argue that in VO2 it is the insu-
lator that is less correlated, in the sense that band-like
excitations are better defined and have longer lifetimes
than in the metal. Albeit, neither phase is well described
by standard band-structure techniques. Using an an-
alytical continuation scheme for quantum Monte Carlo
solutions to Dynamical Mean Field Theory (DMFT) [1],
we discuss quasi-particle lifetimes, k-resolved spectra (for
comparison with future angle resolved photoemission ex-
periments) and effective band-structures. While dynam-
ical effects are crucial in the metal, the excitations of
the insulator are well described within a static picture :
For the insulator we devise an effective one-particle po-
tential that captures the interacting excitation spectrum.
Still, the corresponding ground state is far from a Slater
determinant, leading us to introduce the concept of a
“many-body Peierls” insulator.
The MIT of VO2 has intrigued solid state physicists
for decades [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. A
high temperature metallic rutile (R) phase transforms at
Tc=340 K into an insulating monoclinic structure (M1),
in which vanadium atoms pair up to form tilted dimers
along the c-axis. The resistivity jumps up by two orders
of magnitude, yet no local moments form. Despite exten-
sive efforts, the mechanism of the transition is still under
debate [6, 7, 8, 9, 10, 11, 12]. Two scenarios compete : In
the Peierls picture the structural aspect (unit-cell dou-
bling) causes the MIT, while in the Mott picture local
correlations predominate.
VO2 has a d
1 configuration and the crystal field splits
the 3d-manifold into ⁀2g and empty eσg components. The
former further split into eπg and a1g orbitals, which
overlap in R-VO2, accounting for the metallic charac-
ter. Still, the quasi-particle peak seen in photoemission
(PES) [9, 10, 11] is much narrower than the Kohn-Sham
spectrum of density functional theory (DFT) in the local
density approximation (LDA) [7], and eminent satellite
features evidenced in PES are absent. In M1-VO2, the
a1g form bonding/antibonding orbitals, due to the dimer-
ization. As discussed by Goodenough [3], this also pushes
up the eπg relative to the a1g. Yet, the LDA [7] yields a
metal. Non-local correlations beyond LDA were shown
to be essential [15, 16, 17]. Indeed, recent Cluster DMFT
(CDMFT) calculations [15], in which a two-site vanadium
dimer constituted the DMFT impurity, opened a gap,
agreeing well with PES and x-ray experiments [11, 12].
Starting from these LDA + CDMFT results [15] for
the Matsubara ⁀2g Green’s function G(ıωn) we deduce the
real frequency Green’s function G(ω) by the maximum
entropy method [18] and a Kramers-Kronig transform.
The self-energy matrix Σ(ω) we obtain by numerical in-
version of G(ω) =
[ω + µ−Hk − Σ(ω)]−1 [1], with
the LDA Hamiltonian H , and the chemical potential µ.
Fig. 1 shows (a) the diagonal elements of the R-
VO2 self-energy, and (b) the resulting k-resolved spec-
trum. Notwithstanding minor details, the a1g and e
self-energies exhibit a similar dynamical behavior. The
real-parts at zero energy, ℜΣ(0), entailing relative shifts
of quasi-particle bands, are almost equal, congruent with
the low changes in their occupations vis-à-vis LDA [15],
http://arxiv.org/abs/0704.0902v2
-2 -1  0  1  2  3
ω [eV]
  a1g 
-2 -1  0  1  2  3
ω [eV]
  a1g 
-2 -1  0  1  2  3
ω [eV]
  a1g 
-2 -1  0  1  2  3
ω [eV]
  a1g 
-2 -1  0  1  2  3
ω [eV]
-2 -1  0  1  2  3
ω [eV]
M1 a1g
M1 a1g-a1g 
ΓZCYΓ
FIG. 1: (color online) Rutile VO2 : (a) self-energy (Σ − µ).
Real (imaginary) parts are solid (dashed). As comparison M1
ℑΣa1g , ℑΣa1g−a1g are shown. (b) spectral function A(k,ω)
and solutions of the QPE (blue). The LHB is the (yellow)
region at -1.7 eV, the broad UHB appears (yellow) at ∼2.5 eV.
and with the isotropy evidenced in experiment [19].
Neglecting lifetime effects (i.e. ℑΣ≈0), one-particle ex-
citations are given by the poles of G(ω) : det[ωk + µ −
Hk − ℜΣ(ωk)] = 0. We shall refer to this as the quasi-
particle equation (QPE) [23]. For static or absentℜΣ this
reduces to a simple eigenvalue problem. In regions of low
ℑΣ, the QPE solutions will give an accurate description
of the position of spectral weight and constitute an effec-
tive band-structure of the interacting system. Yet, due
to the frequency dependence, the number of solutions is
no longer bounded to the number of orbitals.
Below (above) -0.5 (0.2) eV, the imaginary parts of
the self-energy – the inverse lifetime – of R-VO2 is con-
siderable. Due to our limited precision for ℑΣ(0), we
have not attempted a temperature dependent study to
assess the experimental bad metal behavior, but the re-
sistivity exceeding the Ioffe-Regel-Mott limit [20] indi-
cates that even close to the Fermi level, coherence is not
fully reached. At low energy, the QPE solutions (dots
in Fig. 1b) closely follow the spectral weight. Above
0.2 eV, regions of high intensity appear, howbeit, the
larger ℑΣ broadens the excitations, and no coherent fea-
tures emerge, though the positions of some eπg derived
excitations are discernible. At high energies, positive and
negative, distinctive features appear in ℑΣ(ω) that are
responsible for lower (upper) Hubbard bands (L/UHB),
seen in the spectrum at around -1.7 (2.5) eV. The UHB
exhibits a pole-structure that reminds of the low-energy
quasi-particle band-structure. Hence, an effective band
picture is limited to the close vicinity of the Fermi level,
and R-VO2 has to be considered as a strongly correlated
metal (the weight of the quasi-particle peak is of the or-
der of 0.6). This is experimentally corroborated by the
fact that an increase in the lattice spacing by Nb-doping
results in a Mott insulator of rutile structure [4].
The imaginary parts of the M1 a1g on-site, and a1g–
a1g intra-dimer self-energies, Fig. 1a, are larger than
in R-VO2, usually a hallmark of increased correla-
tions. However, we shall argue that correlations are in
fact weaker than in the metal. Indeed, the dimeriza-
tion in M1 leads to strong inter-site fluctuations, evi-
denced by the significant intra-dimer a1g–a1g self-energy.
Fig. 2 displays the M1-VO2 self-energy in the a1g bond-
ing/antibonding (bab) basis, Σb/ab = Σa1g ± Σa1g−a1g .
The a1g (anti)bonding imaginary part is low and varies
-3 -2 -1  0  1  2  3  4
ω [eV]
a1g b
a1g ab
-3 -2 -1  0  1  2  3  4
ω [eV]
FIG. 2: (color online) Self-energy (Σ−µ) of M1-VO2 in the a1g
bab–basis : (a) real parts. The black stripes delimit the a1g
LDA bandwidths, dashed horizontal lines indicate the values
of the static potential ∆. (b) imaginary parts. Self-energy
elements are dotted in regions irrelevant for the spectrum.
little with frequency in the (un)occupied part of the
spectrum, thus allowing for coherent weight. In the
opposite regions, the imaginary parts reach huge val-
ues. The eπg elements are flat, and their imaginary parts
tiny. This is a direct consequence of the drastically re-
duced eπg occupancy which drops to merely 0.14. These
almost empty orbitals feel only weak correlations, and
sharp bands are expected at all energies. A first idea
for the a1g excitations is obtained from the intersections
ω+µ−ǫb/ab(k)=ℜΣb/ab(ω) as depicted in Fig. 2a, where
the black stripes delimit the LDA a1g bandwidths. The
(anti)bonding band appears as the crossing of the (blue)
red solid line with the stripe at (positive) negative en-
ergy. Hence, the (anti)bonding band emerges at (2.5)
−0.75 eV. Still, the antibonding band is much broadened
since ℑΣab reaches -1 eV. To confirm this, we solved the
QPE and calculated the k-resolved spectrum (Fig. 3a).
As expected, reasonably coherent weight appears over
nearly the entire spectrum from -1 to +2 eV, whose po-
sition coincides with the QPE poles : The filled bands
correspond to the a1g bonding orbitals, while above the
gap, the eπg bands give rise to sharp features. The anti-
bonding a1g is not clearly distinguished since e
g weight
prevails in this range. The L/UHB have faded : a mere
shoulder at -1.5 eV reminds of the LHB. Finally, con-
trary to R-VO2, the number of poles equals the orbital
dimension. Since, moreover, the real-parts of the M1-
VO2 self-energy are almost constant for relevant ener-
gies [24], we construct a static potential, ∆, by evaluat-
ing the dynamical self-energy at the LDA band centers
(pole energies) for the eπg (a1g), see Fig. 2a [25]. Fig. 3b
shows the band-structure ofHk+∆ : The agreement with
the DMFT poles is excellent. Our one-particle potential,
albeit static, depends on the orbital, and is thus non-
local. We emphasize the conceptual difference to the
Kohn-Sham (KS) potential of DFT : The latter gener-
ates an effective one-particle problem with the ground
state density of the true system. The KS energies and
states are auxiliary quantities. Our one-particle poten-
tial, ∆, on the contrary, was designed to reproduce the
interacting excitations. The eigenvalues of Hk+∆ are
thus not artificial. Still, like in DFT, the eigenstates are
SDs by construction, although the true states are not (see
below). The crucial point for M1-VO2 is that spectral
properties are capturable with this effective one-particle
description. It is in this sense that M1-VO2 exhibits only
weak correlation effects. The weight of the bonding ex-
citation is Z=(1 − ∂ωℜΣb(ω))−1ω=−0.7eV≈0.75, and thus
larger than the rutile quasi-particle weight (see above).
What is at the origin of this overall surprising coher-
ence? For the eπg orbitals, this simply owes to their deple-
tion. For the nearly half-filled a1g orbitals the situation is
more intricate. It is a joint effect of charge transfer into
the a1g bands, and the bonding/antibonding–splitting.
Indeed, the filled bonding band experiences only weak
fluctuations, due to its separation of several eV from the
antibonding one. To substantiate these qualitative argu-
ments, we resort to the following model, which treats the
solid as a collection of Hubbard dimers :
H = −t
l1σcl2σ+h.c.
i=1,2
〈l,l′〉
liσcl′iσ+U
nli↑nli↓
Here, c
liσ (cliσ) creates (destroys) an electron with
spin σ on site i of the lth dimer. t is the intra-dimer,
t⊥ the inter-dimer hopping, U the on-site Coulomb
repulsion, and we assume half-filling. First, we discuss
Γ Y C Z Γ
Γ Y C Z Γ
scissors
FIG. 3: (color online) M1-VO2 : (a) spectral function A(k,ω).
(blue) dots ((a) & (b)) are solutions of the QPE. (b) The (red)
dots are the eigenvalues of Hk+∆. See text for discussion.
the t⊥→ 0 limit, which is an isolated dimer : the
Hubbard molecule. We choose t=0.7 eV, the LDA
intra-dimer a1g–a1g hopping, and U=4.0 eV [15] for
all evaluations. The bonding/antibonding–splitting,
∆bab=−2t +
16t2 + U2=3.48 eV, gets enhanced with
respect to the U=0 case. In M1-VO2, the embedding
into the solid, and the hybridization with the eπg reduce
the splitting to ∼3 eV, as can be inferred from the one-
particle poles (Fig. 3), consistent with experiment [11].
The ground state of the dimer is given by |ψ0〉 =
{4t/ (c− U) (| ↓ ↑〉 − | ↑ ↓〉) + (| ↑↓ 0〉+ |0 ↑↓〉)} /a [26]
which is intermediate to the Slater determinant (SD)
(the four states having equal weight), and the Heitler-
London (HL) limit (double occupancies projected out).
With the VO2 parameters, the model dimer is close
to the HL limit [5]. The inset of Fig. 4b shows the
projections of the ground state onto the SD and the
HL state. The former, |〈SD|ψ0〉|2, equals the weight
of the band-derived features in the spectrum (for U>0
satellites appear), while the other measures the double
occupancy
i〈ni↑ni↓〉 = 1 − |〈HL|ψ0〉|
. For U=4.0 eV
the latter is largely suppressed, as a consequence of the
interaction : The N-particle state is clearly not a SD.
Still, the overlap with the SD, and thus the coherent
weight, remains significant, i.e. one-particle excitations
survive and lifetimes are large. To do justice to the
seemingly opposing tendencies of correlation driven
non-SD-behavior, coexisting with a band-like spectrum,
we introduce the notion of a “many-body Peierls” state.
The charge transfer from the eπg into the then almost
half-filled a1g orbitals, finds its origin in the effective re-
duction of the local interaction in the bab–configuration :
While for U=4 eV, 〈SD|H |SD〉 = 2.0 eV in the SD limit,
it reduces to merely 〈ψ0|H |ψ0〉 = 0.91 eV in the ground
state. In fact, inter-site fluctuations are an efficient way
to avoid the on-site Coulomb repulsion. In M1-VO2, this
effect manifests itself in a close cancellation of the local
and inter-site self-energies in the (un-)occupied parts of
the spectrum for the (anti)bonding a1g orbitals.
The gap-opening in VO2 thus owes to two effects : The
self-energy enhancement of the a1g bab–splitting, and
a charge transfer from the eπg orbitals. The difference
in ℜΣ corresponds to this depopulation, seen in exper-
iments [19] and theoretical studies [8, 15], and leads to
the separation of the a1g and e
g at the Fermi level. The
local interactions thus amplify Goodenough’s scenario.
To show that the embedding of the dimer into the
solid does not qualitatively alter our picture of the M1
phase, we solve the model, Eq. (1), using CDMFT. This
moreover allows to study the essentials of the rutile to
M1 MIT by scanning through the degree of dimeriza-
tion t at constant interaction strength U and embedding,
or inter-dimer hopping, t⊥. For the latter we assume
a semi-circular density of states D⊥(ω) of bandwidth
W=4t⊥. In M1-VO2, the t⊥ for direct a1g-a1g hopping
is rather small, yet eπg -hybridizations lead to an effective
D⊥-bandwidth of about 1 eV. We choose U=4t⊥, and an
inverse temperature β=10/t⊥. Fig. 4a displays the or-
bital traced local spectral function A(ω)=Ab(ω)+Aab(ω)
(b,ab denoting again the bonding/antibonding combi-
nations) and the bonding self-energy Σb(ω) for differ-
ent intra-dimer hoppings t : In the absence of t, the
result equals by construction the single site DMFT so-
lution (Σb=Σab), which, for our parameters, is a corre-
lated metal, analog to R-VO2. The spectral weight at the
Fermi level is given by Ab/ab(0) = D⊥(±t − ℜΣb/ab(0)),
with ℜΣb/ab(0)=∓ℜΣab(0). Thus a MIT occurs at
t + ℜΣab(0) = 2t⊥, when all spectral weight has been
shifted out of the bandwidth : Above t/t⊥=0.5 we find a
many-body Peierls phase corresponding to M1-VO2. In
Fig. 4a we have indicated again the graphical QPE ap-
proach : The system evolves from three solutions per or-
bital (Kondo resonance, L/UHB) at t=0 to a single one at
t/t⊥=0.6. Hence the peaks in the insulator are not Hub-
bard satellites, but just shifted bands. The embedding,
t⊥, broadens the excitations and washes out the satellites
of the isolated dimer, like for M1-VO2. Still, as a function
of t, the coherence of the spectrum increases, since the
imaginary part of the (anti-)bonding self-energy subsides
at the renormalized (anti-)bonding excitation energies.
Our model thus captures the essence of the rutile to M1
transition, reproducing both, the dimerization induced
increase in coherence, and the shifting of excitations.
-4 -2  0  2  4
ω / t⊥
t / t⊥
-4 -2  0  2  4
ω / t⊥
t / t⊥
-4 -2  0  2  4
ω / t⊥
t / t⊥
-4 -2  0  2  4
ω / t⊥
t / t⊥
     
     
     
     
     
     
     
-4 -2  0  2  4
     
-4 -2  0  2  4
     
-4 -2  0  2  4
     
-4 -2  0  2  4
 0  1  2  3  4  5
iω / t⊥
 0  1  2  3  4  5
iω / t⊥
 0  1  2  3  4  5
iω / t⊥
 0  1  2  3  4  5
iω / t⊥
 0  1  2  3  4  5
iω / t⊥
 0  1  2  3  4  5
iω / t⊥
 0  1  2  3  4  5
iω / t⊥
t / t⊥=0.0
t / t⊥=0.2
t / t⊥=0.4
t / t⊥=0.6
t / t⊥=0.8
 0  2  4
U [eV]
t=0.7eV
 0  2  4
|<SD|Ψ0>|
|<HL|Ψ0>|
U [eV]
t=0.7eV
FIG. 4: (color online) (a) spectral function (top), real
(middle), imaginary (bottom) bonding self-energy Σb of the
CDMFT solution to Eq. (1) for U=4.0t⊥, β=10/t⊥, and
varying intra-dimer hopping t/t⊥. ℜΣb(ω)=−ℜΣab(−ω),
ℑΣb(ω)=ℑΣab(−ω) by symmetry. (b) Imaginary Matsuba-
ra self-energy, ℑΣb(ıω)=ℑΣab(ıω), for U=6t⊥, β=10/t⊥ and
varying t. Inset: Projection of the SD and HL limit on the
Hubbard molecule ground state (t=0.7 eV, t⊥=0) versus U.
Under uni-axial pressure or Cr-doping, VO2 develops
the insulating M2 phase [4] in which every second vana-
dium chain along the c-axis consists of untilted dimers,
whereas in the others only the tilting occurs. We may
now speculate that the dimerized pairs in M2 form a1g
Peierls singlets as in M1, while the tilted pairs are in a
Mott state. Hence, we interpret the seminal work of [4]
as the observation of a Mott to many-body Peierls tran-
sition taking place on the tilted chains when going from
M2 to M1. To illustrate this, we solve again Eq. (1) for
appropriate parameters. The tilted M2 chains are akin
to the rutile phase, yet with a reduced a1g bandwidth [7].
Thus we now choose U=6t⊥, β=10/t⊥, and vary t. All
solutions shown in Fig. 4b are insulating, however, the
diverging self-energy at vanishing intra-dimer coupling
(t=0, tilted “M2” chains) becomes regularized with the
bond enhancement (t>0, “M1”). The imaginary part of
the self-energy gets flatter and the system thus more co-
herent. The above is consistent with the finding of (S=0)
S=1/2 for the (dimerized) tilted pairs in M2-VO2 [4].
While our results do not exclude surprises in the direct
vicinity of Tc [22], the nature of insulating VO2 is shown
to be rather “band-like” in the above sense. Our analyti-
cal continuation scheme allowed us to explicitly calculate
this band-structure. The latter can also be derived from
a static one-particle potential. Yet, this does not im-
ply a one-particle picture for quantities other than the
spectrum. Above all, the ground state is not a Slater de-
terminant. Hence, we qualify M1-VO2 as a “many-body
Peierls” phase. We argue that the weakness of lifetime
effects results from strong inter-site fluctuations that cir-
cumvent local interactions in an otherwise strongly cor-
related solid. This is in striking contrast to the strong
dynamical correlations in the metal, which is dominated
by important lifetime effects and incoherent features.
We thank H. T. Kim, J. P. Pouget, M. M. Qazil-
bash, and A. Tanaka for valuable discussions and A. I.
Poteryaev, A. Georges and A. I. Lichtenstein for discus-
sions and the collaboration [15] that was our starting
point. We thank AIST, Tsukuba, for hospitality. JMT
was supported by a JSPS fellowship. Computer time was
provided by IDRIS, Orsay (project No. 071393).
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[23] We solve the equation numerically by iterating until self-
consistency within an accuracy of 0.05 eV.
[24] Explaining why LDA+U opens a gap [14, 16], yet while
missing the correct bonding/antibonding splitting.
[25] ∆eπ
=0.48eV, ∆eπ
=0.54eV, ∆b=−0.32eV, ∆ab=1.2eV
[26] a =
2 (16t2/(c− U)2 + 1), c =
16t2 + U2
