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Contributed Papers
Metal Insulator Transition in Strongly Correlated Electron
Systems
H. Monien and J. K. Freericka
Theoretische
Physik, ETH H6nggerberg,
CH-8093 Zurich, Switzerland
*Physics Department
Davis, University of California, Davis, California
95616
Strongly interacting
quantum
systems exhibit same of the most interesting
phenomena
in condensed
matter physi=. Until very recently the study of these
systems has heen confined to either numerical simulations of rather small systems
(which still have an enormous number of q“ant”m
stat=)
or approximate
~na.
lytical methods which are uncontrolled.
Mat of the effort concentrated
on the
fermionic systems. In the last few yearn the hmonic systems have attracted
a lot of
interest.l ,2,3,4,5 Example
of the bmonic strongly correlated systems inc]ude ~h~rt
coherence length superconductors,
Jc6ephson junction arrays, the dynamics of flux
lines in high temperature
superconductors
and the quantum hall effect.6 The ~ntial physics of all of these problems is the competition
between the hopping matr~
elements and the short range strong repulsion. The minimum mdel which contains
the bmicphysics
of theproblem
istheso
called Bme Hubbard model:
Addison-Wesley Publishing Company, 1994
320
H. Monien and J. K. Freericks
H=–t~b~bJ–p~fii
i.i
where b< is the bmn
annihilation
operator at site i, t is the hopping matrix element between the site i and j, U is the strength of the on-site repulsion, and p
isthecbemical
potential. Tbe approximate
form of the zero temperature
(’T=O)
ph=
d]agram can be understood
hy starting from the strong- mupling or “atomic”
limit, ‘,7,8 In this limit, the kinetic energy vanishes (t = O) and every site is occupied
byafixed
number of bo30ns, no. The ground-state
bwonoccupancy
(no) is then
chmn in such a way a.3 to minimize the on-site energy. For a finite deviation from
integer filiing, afinite amount ofenergy (gap) isrequired tomovea
particle through
the lattice. The bmns
are localized, producing aMott insulator. This energy gap
decreas= with increming strength of the hopping matrix elements until it vankhe
and the bmons condenw into the superfluid ph=e. A3 the strength of the hop
ping matrix elements increms,
therange
of the chemical potential pabout
wbih
the system is incompr~sible
decrea3=. The Mott-insulator
phase will completely
disappear at a critical value of thehopping
matrix elements.
The Hamiltonian,
Eq. (l), h= been studied numerically with the Quantum
Monte Carlo Method, e.g.,2 ,4,5 and analytically
by mean field approximatiom,g
renormalisation
gr0up3,7 and other technique.s.5 Here we prewnt an analyti~l
approachl”
which fallows w to calculate the Mott-insulator
. superfluid
ph33e
boundary
at zero temperature
with an accuracy comparable to the best available
numerical results. We proceed % follows. We determine theenergy
the of the Mott
insulator in many bcdy perturbation
theory in power3 of the kinetic energy. We
compare the energy of the Mott - insulator with the energy of a m calld
“defect
state” in wbi& one additional
boson (or one additional
hole) is in an extendd
state. The energy of the defect state is alwdetermind
with many body perturbation theory. The Mott - insulator superfluid transition occurs when the energy of
the’’defect
state” and the Mott - insulator state are equal. Tozerothorderin
t/U
the Mott insulating
state is given by. Tozeroth
order in t/f/ the Mott imulating
state is given by
lwMott(no))(”)
= fi ~
,=,m
where no is the
and 10) is the
particle (hole)
“defwt state”
theory
@!)no
10)
(2)
number of b~mon
each site, Nisthenumber
ofsites in the lattice
vacuum state. The defect pha3e is characterized
by one additional
which moves coherently throughout
the lattice. Theansatz
for tbe
to zeroth order in t/fJis determined
by degenerate perturbation
l~De,(nO)):}t,c,e
‘0)
‘vDef(nO))hO’e
=
&~
fib~l~~ott(no))(o)
‘A~fib,l~Mott(no))(0)
- A
,
(3)
322
3.0 ~
H. Monien and J. K. Freerbks
1
3.0
b)
2.0
F
dtflt
FIGURE
1. Ph&
diagram of the Bc6e Hubbard model in one (left) and two
(right)dimension.
The WUarES (left) are the r-ults
of the Quantum
Monte Carlo
calculation
by Scalettar et al. for the one dimensional BHubbard model. The
square (right) is the r=ult of the Quantum Monte Carlo calculation
of Wlvedi et
al. The hopping matrix element is scaled by the dimension d.
where the fi is the eigenvector of the hopping matr~ tij with the 10w~t eigenvalue.
The energiof three state can be calculated with Rayleigh-SchrHlnger
perturbation theory. The result of the a third order perturbation
calculation
spialized
to
the hypercubic Iattim in one and two dimensions is shown in the Figure 1.
We have repeatedly
compared a strong-coupling
expansion to the numerid
QMC simulations
for tbe incomprmsible-compr-ible
phase boundary of the hose
Hubbard model. A mem-field treatment of the hose Hubbard model (e.g. 1,9) cannot
capture the physim of the one dimensional system which is completely dominated by
fluctuations.
The dimensionality
only enters m a triviaf prefactor in integrals over
the phase space. For this reason, mean-field theories will always give a concave shape
H. klonieIl atld J. K. Reericks
323
to the Mott-insulator
lobes independent of the dimension. A strong-coupling
expmsion, on the other hand, e~ily distinguishthe shape difference from one dimension
to higher dimensions md shows that a proper treatment of density fluctuations
is
critical in determining
the Mott-insulator
to superfluid transition.
In conclusion
we have described an analytical method to accurately calculate the ph~e diagram
of the bme Hubbard model in my dimension. Extensions of these techniqum
to
include disorder will be prmented separately.
ACKNOWLEDGMENTS
We would like to thank R. Scalettar, G. Batrouni and K. Singh, for providing
us with the Quantum
Monte Culo data for the one dimensional bme Hubbard
mndel and for many &ful discussions. This r=arch
was supported in part by the
NSF under Grant No. PHY89-04035 and DMR90-02492.
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11, The hopping matrix is -umed
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