Possibility of a Metallic Phase in theHalf-Filled Hubbard-Holstein Model
For the last few decades, the competition between
the phonon-mediated electron-electron attraction andthe local Coulomb repulsion has been discussed invarious contexts of condensed matter physics. Initiallythe problem was raised in relation to the quasi-one
dimensional organic charge-transfer salts. In recentyears, it has also become an important issue instudying the perovskite oxides such as cuprates andmanganites.
Theoretically the Hubbard-Holstein (HH) modelserves as a useful framework to investigate thecompetition. In addition to the kinetic-energy term
represented by t the nearest-neighbor electron hopping
integral, it consists of two interaction parts. One is theterm describing the electron correlation effects
through U the on-site Coulomb repulsion as in theHubbard model. The other is the term to couple anelectron with local phonons as in the Holstein model,
leading to
0 the polaron stabilization energy with
the non-dimensional electron-phonon coupling
constant and 0 the optic phonon energy.
For this model, there have already been many
previous works using various methods. These studies
have revealed that the HH model can afford
interesting phase diagrams due to the competition
among charge-density wave (CDW), spin-density
wave (SDW), and superconductivity as we change theparameters involved in the system, i.e., U, , t, 0,and n the electron filling. Roughly speaking, the CDW(SDW) nature manifests itself in the ground state at
half filling (n=1), if the effective electron-electroninteraction Ueff (~ U-2 0) is negative (positive).
However, the detail of the CDW-SDW transition is
not known well. In fact, the appearance of asuperconducting phase is suggested [1] in theexact-diagonalization study on the two-site HH model.
Thus we carefully consider the transition region athalf-filling by approaching the HH model from theantiadiabatic region (t<< 0) where the variabledisplacement Lang Firsov canonical transformation isuseful. In order to avoid any error that might enter
through the approximations involved in dealing with
the Coulomb correlation term, we have investigatedthe one-dimensional system so that the exact solutionof Lieb and Wu can be taken advantage of. By
determining the displacement variationally, we haveobtained fairly accurate results for the properties of
the ground state. [2]In Fig. 1, we show the obtained phase diagram inthe ( ,U) plane at t=0.2 0. The metallic phasecharacterized by the local spin moment L0 ~ 3/8 and asmall effective mass of polarons appears along the lineof U = 2 0, namely, in the CDW-SDW crossover
region. This phase seems to continuously convert into
the superconducting phase in the attractive Hubbard
model (or the system with =0 and U<0).
We do not believe that this behavior is specific to

one dimension (1D); a similar metallic behavior will
be seen in higher dimensions as well, as long as |Ueff|
is less than the polaron bandwidth in the crossover
region. We may even conjecture that such a metallic
phase will be realized more easily in higher
dimensions in which we can avoid unusual
phenomena specific to 1D like the Mott transition at
U=0 irrespective of t in the 1D Hubbard model.

Fig. 1. Phase diagram in the ( ,U) plane for thehalf-filled Hubbard-Holstein model.

It is not known for sure whether this kind of ametallic phase has already been observed in actual
materials or not, but it has been pointed out [3,4] thatphysical properties including superconductivity in thealkali-doped fullerenes A3C60 can be reproduced in a
unified way in terms of this picture. For example, thesuperconducting transition temperature Tc in the HH
model with suitably chosen parameters is obtained asa function of the lattice constant a0 as shown in Fig. 2.

Fig. 2. Tc as a function of the lattice constant a0 in A3C60
and its dependence on the crystal structure in comparison
with experimental results.

References[1]Y. Takada, J. Phys. Soc. Jpn. 65, 1544 (1996).
[2]Y. Takada and A. Chatterjee, Phys. Rev. B 67,081102(R) (2003).
[3]Y. Takada, J. Phys. Soc. Jpn. 65, 3134 (1996).
[4]Y. Takada and T. Hotta, Int. J. of Mod. Phys. B 12,3042 (1998).
Authors
Y. Takada and A. Chatterjeea
aUniversity of Hyderabad.