A Unified Approach to Filling- and Bandwidth-Control Mott Transitions
The nature of the
system in which quantum fluctuations and electron correlations play essential
roles is one of the main subjects in condensed matter physics. When the kinetic
energy and the Coulomb repulsion compete severely, the ground state of
many-body electron systems can be highly nontrivial. Metal-insulator
transitions driven by the electron correlation postulated by Mott [1] provide a
typical example of such a nontrivial behavior.
In the transition to
the Mott insulator, it is known that there exist two different basic routes to
control the competition of the interaction and kinetic energies. One is the
control by bandwidth (relative to Coulomb repulsion) and the other is filling
(averaged number of carrier electrons in a unit cell). Controls by these two
parameters can be found in a lot of examples in real materials including
transition metal compound, organic materials and 3He systems. In spite of
plenty of the experimental results, phase diagrams of the Mott insulator and
metals have not been fully elucidated in microscopic theoretical descriptions.
The Hubbard model with
nearest- and next-nearest-neighbor transfers is a minimal model to describe the
essence of the Mott insulator and metals with their transitions on the
bandwidth- and filling-control routes.
The filling-control Mott transition (FCMT) was studied at zero
temperature by the quantum
Mott has originally
proposed a first-order BCMT because of the role of the long-ranged part of the
Coulomb interaction [1]. However, the numerical result shows that the
first-order transition takes place even with the onsite interaction only. In
this circumstance, it has been greatly desired to clarify basic properties of
the BCMT and the FCMT in a unified way to further elucidate the contrast.
In this study, we have
developed the grand-canonical path-integral renormalization group method [4]
which is useful to calculate chemical potential dependence of physical
quantities. The path-integral renormalization group method has been originally
developed in the canonical ensemble. This algorithm does not suffer from
negative-sign problem and can be applied to any lattice structure with any boundary
conditions. Extension of this
algorithm to the grand-canonical framework makes it possible to study the
correlated electron systems whose control parameters are filling, bandwidth and
lattice structure in a unified way. In the grand-canonical path-integral
renormalization group calculations, metal-insulator transitions are carefully
examined with finite-size scalings and extrapolations
to the thermodynamic limit.
By using the newly developed
method, the ground-state phase diagram of the Hubbard model with
next-nearest-neighbor transfer scaled by nearest-neighbor transfer t’/t
=-0.2 on the square lattice has been determined in the plane of chemical
potential m and Coulomb interaction U [4](Figure 1). The Mott-insulator phase is drawn as the
pink area and the metallic phase is drawn as the yellow area. The remarkable
result is that the V-shaped Mott insulator phase appears. At the corner of the
V-shaped metal-insulator boundary, the BCMT occurs and at the edges except the
corner, the FCMT occurs. We have confirmed that the continuous character of the
FCMT with diverging charge compressibility and the first-order BCMT with a jump
of the double occupancy.
To analyze the relation
between the shape of the phase boundary and the order of the metal-insulator
transition, we have analytically derived a general relation of the slope of the
metal-insulator transition line in the m-U phase diagram
and physical quantities [4]: In the case of the first-order metal-insulator
transition, dU/dm is expressed by the ratio of the jumps in the
filling and the double occupancy Dn/DD. In the case of the continuous transition, dU/dm is expressed by the
ratio of the compressibility and dn/dU
in the metallic phase. These relations can be regarded as generalizations of Clausius-Clapeyron and Ehrenfest equations
in the first- and second-order transitions at finite temperature, respectively,
to the quantum phase transitions.
These relations support
that the V-shaped phase boundary is resulted from the first-order BCMT
coexisting with the continuous FCMT with diverging compressibility. Namely, it
is shown that the V-shaped metal-insulator transition line together with the
first-order BCMT is not compatible with the presence of the first-order FCMT
near the BCMT.
When V-shaped Mott insulator phase appears, the charge
gap Dc opens at U=Uc and shows marked linear dependence on U
for U>Uc,
namely, Dc~ U-Uc. It is
noted that the linear opening of the Mott gap has been actually observed in the
perovskite compounds, R1-xCaxTiO3 [5], which is consistent with our results.
We have also shown the
possible and impossible shapes of the metal-insulator boundary with first-order
and continuous transitions by using the thermodynamic relations derived above:
The U-shaped structure of the insulator phase with the first-order BCMT
coexisting with the first-order FCMT can be realized in contrast with the
V-shaped case. On the other hand, in the case of the U-shaped insulator phase
(for example, the case with an essential singular form of the charge gap as Dc ~ exp[at/(U-Uc)]), if the
first-order BCMT exists at the corner, the first-order transition cannot be
retained at the FCMT, which is classified to the same class as the V-shaped
case.
References
[1] N. F. Mott and R. Peierls:
Proc. Phys. Soc. London. A49 72 (1937).
[2] N. Furukawa and M. Imada:
J. Phys. Soc. Jpn. 61 3331 (1992).
[3] T. Kashima and M. Imada: J.
Phys. Soc. Jpn. 70 3052 (2001).
[4] S. Watanabe and M.
Imada: J. Phys. Soc. Jpn. 73
1251 (2004).
[5] T. Katsufuji,
Y. Okimoto and Y. Tokura:
Phys. Rev. Lett. 75 3497 (1995).
Authors
S. Watanabe and M. Imada
Fig. 1:
Ground-state phase diagram in the plane
of chemical potential m
and Coulomb interaction U for nearest-neighbor transfer t=1.0 and next-nearest-neighbor
transfer t'=-0.2 on the
square lattice in the thermodynamic limit. Open circles represent the
metal-insulator boundary extrapolated to the thermodynamic limit by using the
data of N=4×4, 6×6,
8×8, 10×10
lattice systems. The solid black lines represent the least-square fit of
metal-insulator boundary for U=4.0, 5.0, 6.0, 7.0 and 8.0. The V-shaped
Mott insulator phase is drawn as pink area and the metallic phase is drawn as
the yellow area. The corner of the V-shaped metal-insulator boundary is the bandwidth-control
Mott transition point and the edges are the filling-control Mott transition
points. The red diamonds represent chemical potentials for half filling for U=0.0
and U=2.0 in the thermodynamic limit. By connecting those points to the
bandwidth-control Mott transition point, the blue-dashed line represents the
half-filled density, which is the bandwidth-control route in the metallic
phase.