VOLUME 49, NUMBER 6
PHYSICAL REVIEW B
Magnetism
1
and pairing in a C6o molecule: A variational
FEBRUARY 1994-II
Monte Carlo study
D. N. Sheng, Z. Y. Weng, and C. S. Ting
Texas Center for Superconductivity,
University
of Houston, , Houston, Tesas 7720$-5508
J. M. Dong
Department of Physics, Nanjing University, Nanking, China
(Received 4 December 1992; revised manuscript received 31 August 1993)
The ground-state properties of electrons in the C60 geometry is studied based on the t-J-like
Hamiltonian. By using a variational Monte Carlo method, it is found that the Gutzwiller projected
Fermi-sea state and the pure RVB state have the lowest variational energies in the undoped case as
compared to other spin singlet states. For a doped C60 molecule, the electron pairing and itinerant
ferromagnetism are discussed. A comparison with related works is also presented.
I. INTRODUCTION
The discovery of superconductivity in the alkali-doped
C60 system naturally leads to the question. of mechanism: whether the superconductivity
is due to the conventional electron-phonon interaction or to the electronelectron correlations. It is still a controversial
subject,
and one of the important issues is how the special band
structure of C60 plays a role in the renormalization of the
Coulomb pseudopotential. ' '
In the C60 crystal, the band structure calculations
show that those energy levels which have considerable
dispersion below and above the Ferxni level mainly come
&om the x-bond states of a Cso molecule.
These ~electron states can be described by the tight-binding
model
and form many narrow subbands with an overall bandwidth
10 eV. With the presence of a moderate
bare on-site Coulomb interaction4 s ( U 6—12 eV), the
electron-electron correlations would become crucial if the
system was a square lattice, like in the copper oxide materials. In the latter case, the low-energy physics in the
weak- and intermediate-coupling
regixnes can be continuously scaled down to the strong-coupling regixne, without
encountering a Mott-insulator phase-transition point U,
(at the half-filling). In the Cso system, however, a finite
U, could exist and if U & U„one expects that the usual
screening-retardation
effect in determining the Coulomb
pseudopotential is valid. ' But when U & U„ the onsite electron-electron correlations would become so predominant that their effect should be included before any
screening effects are considered.
In this paper, instead of determining whether the realistic regime lies at U
U„we will con6ne
U, or U
ourselves to a less ambitious goal by assuming U
U,
and then asking what the possible ground state is in this
strong-coupling cas~. Since the hopping inside the C60
molecule is xnuch stronger than between the molecules,
we will consider the Hubbard model on a single C60
molecule.
This model has been previously discussed
by Chakravarty, Gelfand, and Kivelson in the weak-
)
0163-1829/94/49(6)/4279(6)/$06. 00
(
)
49
coupling regime. We will be interested in its large-U
version, which is similar to the well-known t- model on
the square lattice,
J
II= Jg)
S, Ss+ J2)
(ij)
S; Ss
(ij)
—tg) PgC,+ Cs~Pg —t2)
(ij)
PdC,+ Cs~Pg
+ H. c. ,
(ij)
where the primed sum runs over all of the 30 interpentagon bonds and the double-primed sum runs over all
of the 60 intrapentagon bonds on the surface of the C60
molecule. In the undoped case only the 6rst two terms,
namely, the superexchange coupling, in (1.1) exist. S;
is the spin operator of the x electron. Spin exchange Jq
will usually be slightly larger than J2 because the pentagon bonds are slightly longer. The remaining terms in
(1.1) describe the hopping processes, with Pd being the
projection operators which impose the constraint of no
vacant sites under the electron doping.
For the undoped case, (1.1) has been studied recently
by Coffey and Trugman
by using the classical spin
(large-S) method with the zero-point spin fiuctuations
included in the second-order perturbation.
In that calculation, they found the long-range spin order to survive
quantum Quctuations on the C60 sphere. However, the
antiferromagnetic &ustration within an elementary pentagon xnay make the C60 molecule a proper candidate to
have a spin singlet (S = 0 ) ground state. In the present
paper, various possible spin singlet states on the C60
geometry will be systematically studied by the numerical variational Monte Carlo (VMC) method. ' Furthermore, electron-doping sects, including pairing and
itinerant ferromagnetism, will be discussed for a range
of parameters. In Sec. II, we consider the undoped C6p
molecule described by (1.1). Various resonating-valencebond (RVB) states are reexamined for the present spherical lattice system. In this systexn they exhibit the differ4279
1994
The American Physical Society
D. N. SHENG, Z. Y. WENG, C. S. TING, AND J. M. DONG
4280
ent trends as compared to the square lattice case. In Sec.
III, the electron-doped case is investigated and a rich variety of possible phases is found in the metallic regime.
Finally, Sec. IV is devoted to the summary and a brief
dlsc uss1011.
II. UNDOPED
C60 MOLECULE
For a square lattice, various RVB states with shortrange spin-spin correlations have been proposed as the
possible ground states for the t-J model. In the undoped case, however, the long-range antiferromagnetic
state turns out to have the lowest energy. Nevertheless,
the RVB states are believed to become energetically competitive once some proper &ustration is introduced. In
the present C6g geometry, the frustration of antiferromagnetic coupling is obvious: Unlike in the square lattice
case, within a pentagon (cf. Fig. 1) there exists no spin
configuration that would allow every bond to reach the
maximum classical energy of —
Ji/4. In the following, we
will re-examine various RVB states in the C60 molecule
by using the variational Monte Carlo method.
For the undoped case, there is one electron per site, and
only the superexchange coupling terms in (1.1) remain.
The Hamiltonian (1.1) then reduces to the form
Hg
= Ji)
—C, C;
Pg
(ij ) o
+~gt C. gt2
)
(
—C,t C,
Pg
)
0.'C,
,
Hilbert space is restricted by the condition that there is
one and only one electron per site, i.e. ,
)
Cj
)
O'Ct, Cj
gled
2
2
(2 1)
2
after the spin operator S; is expressed by the electron
Ct (0) C; . Here, 0 is
operators C; as S; = 2
the Pauli matrix and the spin index 0 (cr') = +1. The
P,
HMF
=
Jlgopi
)
Ci~oCjo
J2gop2
)
(ij)o
g
C;oCjo
C,.oC,g+—o ——J242)
CT
8
/1
C,oC.
+ H. c.
o+ H. c. —p
(ij)o
where N, = N at half-filling, with N being the total number of electrons and N = 60 being the total
number of sites per molecule.
The order parameters
pi(Ai) and p2 (E2 ) are defined by pi — (C; Cjo)
(b, i —Q' (C, C; )) and p2 = g"(C, Cjo) (A2 ——
(Cj C; )), corresponding to nonpentagon and pentagon bonds, respectively.
By using the renormalized
mean-field approximation,
the projection operator Pp
g
g"
(2.2)
(ij)cr
(v)
——Jinni
)
CtC, =1,
through the Gutzwiller projection operator P& = II;(1—
C,&C,tCt&C;g). The Hainiltonian (2.1) with the projection operator P~ enforcing the constraint (2.2) is quite
difFicult to deal with. Like in the square lattice case, one
could obtain various interesting spin singlet states under
the mean-Geld approximation by relaxing the constraint
(2.2). These mean-field states can be used to construct
variational wave functions for the Hamiltonian (2.1) and
the single-occupancy constraint (2.2) will then be exactly
implemented in the VMC calculation.
The mean-field Hamiltonian is obtained by decoupling the four-operator coupling in (2. 1). We introduce
(Ct Cj ) and the RVB
the nearest-neighbor hopping
pairing (C;tCjt) order parameters to get
2
2
+J2
)
FIG. 1. Geometry of the molecule C60. The pentagon
bonds (solid lines) are slightly longer than the hexagon bonds
(dashed lines).
) C,~C;o —
W,
k
io
)
(2.3)
has been replaced in (2.3) by the so-called renormalization factor g, , in the following way:
(P~S' SjP~)
=g
(S*
(2 4)
where g, = g, for nonpentagon bond and g, = g, for
pentagon bond. (- . )p is the expectation value under the
ground state of the mean-field Hamiltonian (2.3) without
MAGNETISM AND PAIRING IN A
49
imposing the no-double-occupancy constraint.
In the following, we will take g, pq
pq and g, p2 -+ p2
as two independent variational parameters and construct
variational wave functions based on (2.3). We will minimize the energy expectation value of the original Hamiltonian (2.1) under these trial wave functions by performing VMC calculation to exactly implement the constraint
~
(2.2).
A. Gutzvriller-projected
Fermi-sea state
The simplest spin singlet state is the so-called projected Fermi-sea state with only the nearest-neighbor
hopping order parameters p~ and p2 (putting b, ~ = b, 2 ——
0). Diagonalizing (2.3) to get the ground-state wave func-
tion l%'o), we obtain
I@o) =
~.'t~.'gl0)
(2.5)
(s)
where pt is the creating operator of an eigenstate of (2.3),
. a;,
pt
The product in (2.5) runs over 30
'&
V~ a
eigenstates with the lowest energy. Thus the variational
wave function, after projecting out double occupancy, is
the product of two determinants:
=,
l4')
g, Ct.
= Pgl@o) =
) det(a;„) det(az„)Ct&Ct&
(i)o)
XCt~Ct, ~
Ct
&
C,t ~l0). (2.6)
Both sq and s2 run over 30 occupied eigenstates. The
sum includes only the configurations in which all the
sites occupied by the spin-up electrons (i) (30 sites)
and the spin-down electrons
(30 sites) have no overlap. Given the variational wave function (2.6), the energy expectation value of original Hamiltonian (2.1), i.e. ,
E(p~/p2) = (4'lHl4'), can be evaluated by the Monte
Carlo (MC) method
In Fig. 2 we plot the variational energy E(p~/p2)
as a function of p~/y2. The two curves correspond to
1.0 and 1.1 respectively. The energy minimum
J~/J2 —
0.9, which is a very broad miniappears around p'/p2
mum, as shown in Fig. 2. At the minimal energy point,
(j)
C6p
MOLECULE:
4281
0.3425660.0001
for J~ = J2 — case, (S; S~) = —
30.83J. This is much
and the total energy is about —
lower than the fully dimerized state (y~ = 1 and p2 —0),
and it compares very well with the energy of the CofFey~ugman state with a long-range spin order (our energy
is only 0.7% higher). ~ We have performed about 80000
MC steps per site in order to gain high precision.
The spin-spin correlation (@lSo.S;l@) as a function of
the position of site i relative to site 0 is also shown in Fig.
3. The sites 0 and i for i = 1, . . . , 11 have been defined
in Fig. 1. The spin correlations are not sensitive to the
exchange parameters Jq and J2. Figure 3 shows that the
local spin structure is quite similar to the classical spin
array obtained in Ref. 12. Considering this similarity
and the closeness of the energies, one may find it amazing
that these are, superficially, two quite different states:
One is a spin singlet state while the other has a longrange spin order which survives the quantum Quctuation.
Given such an extremely small energy difFerence (0.
between the two states, it is hard to tell which of them is
more stable. A possibility of a deep connection between
these states needs further exploration.
J
7')
B. Flux state
and other RVB states
We have shown that the projected "Fermi-sea" state
has a rather favorable energy. However, in the square
lattice case, it is well known that there are two other
kinds of spin liquid states which have even lower energy:
i.e. , the vr-Qux state and the d-wave RVB state. These
two states are degenerate and actually related to each
other by a SU(2) transformation at half-filling. We shall
consider the generalization of these states in the C60 case
below.
In the Qux state, it is an important fact that the particles' hopping amplitudes are always accompanied by
phase factors. If one particle goes through a closed loop
and returns to its original position, it can gain an extra phase which is proportional to the average number
of particles enclosed in the loop. We will consider the
vr-Qux state in which each electron contributes electively
a x Qux. For the geometry of the C60 molecule, a di0.4
0.2—
-0.20
J1= J2= J
-0.25
A. . .
&
0.0
M
C)
M
-0.30
-0.2
V
-0.35
-0.4
-0.40
-0.6
0.0
0.5
1.0
2
4
6
8
10
12
p'/p2
FIG. 2. The energy of the Gutzwiller-projected Fermisea state versus the variational parameter p'/ps for different
J'/Js cases.
FIG. 3. The spin-spin correlation function at the
Fermi-sea state. 0 and i refer to the
Gutzwiller-projected
spin sites marked on the sphere of Cep in Fig. 1.
D. N. SHENG, Z. Y. WENG, C. S. TING, AND J. M. DONG
4282
rect counting gives a 2m fIux penetrating through each
nonpentagon and a 5z/3 Hux going through each penThe effective mean-field Hamiltotagon, respectively.
nian (2.3) is then changed to the following form (with
Ai ——A2 ——0):
HMF
— 3 Jipi
)
(ij)cr
(2.7)
the
redefined
as pq
parameters
e'
=
and
"C, C, ~)2.
"C; C~~)i
p2
g, (P
g,
4;~ is a phase factor connected to t;he Gctitious magnetic
. A dl with the aforementioned
Hux
Held A by 4';z —J"
with
'(P
order
e*
2
given by
2x for a hexagon,
for a pentagon.
49
plitude pq —p2 —0. This is the pure "RVB" state which
resembles the 8-wave RVB in the square lattice case. We
note that the s-wave RVB state is equivalent to the projected Fermi-sea state through a SU(2) transformation
on a square lattice. In the present pure "RVB" state,
(S, . S~) = 0.342 56+0.000 16 which is almost identical to
that of the projected "Fermi-sea" state, even though we
have no analytical proof for their equivalency here. We
have performed around 6 x 10 —10 x 10 MC steps per
site. The error bar is about +0.00016. No other minimum is found in the region where pq, p2 as well as Aq
and A2 are all finite.
In summary, we have carried out the VMC calculations
for three kinds of states: the projected "Fermi-sea" state,
the Bux state, and the RVB states with the order parameter (C~tC, g). The results are quite different from those
of a square lattice case. The "Fermi-sea" state and the
pure "RVB" (no hopping order parameters) are found to
have the lowest energy among these states.
(2.8)
5'/3
Treating pi/p2 as the variational parameter, we put a
phase 4;~. satisfying the constraint (2.8) on each bond
and then diagonalize Hamiltonian (2.7). The variational
energy of (2. 1) under such a trial wave function is calculated as a function of pi/p2 and is shown in Fig. 4. At
the energy minimum, (S; S~) = —0.339 for Ji ——J2 case,
which is about 1% higher than the projected "Fermi-sea"
state. For J~ & J2, the energy minimum is always a little bit higher than the projected "Fermi-sea" state. We
have also checked other flux states (each particle carrying a Hux other than x), and found them to always give
higher energies.
Next we consider the RVB state with nonzero order
oc (C~gC;t). Now we need to vary five variparameter
ational parameters pq, p2, Aq, A2, and the chemical potential p in (2.3) and try to find the energy minimum of
Hamiltonian (2.1). We may always rescale one of these
parameters to unity (for example, set p2 —
1) so that we
have only four parameters left.
Basically two regions containing local energy minima
0 but
have been found. One corresponds to pi
p2
with Az —A2 —0 which is just the Gutzwiller projected Fermi-sea state discussed in Sec. II A. There is an
another minimum at Ai/A2 —0.90 but the hopping am-
6
j
III. ELECTRON-DOPED
Cgo
MOLECULE
In the previous section, we have discussed the possible
ground states for the undoped C60 molecule in the largeU limit. Of them, the most energetically favorable are
found to be the projected "Fermi-sea" state and the pure
"RVB" state. In the present section, we shall study the
doping effect after some additional electrons are added
into the C60 molecule.
In the doped case, the electron hopping terms will
emerge, as in the Hamiltonian (1.1). Each site is either
singly occupied or doubly occupied due to the doped electrons. The corresponding mean-field Hamiltonian is the
same as (2.3) with N, = N + Kg and N~ being the total number of the doped electrons on the molecule. The
mean Gelds pq and p2 now include both the electron hopping and the spin superexchange effects. The procedure
for solving this variational problem is the same as in the
undoped case.
For the case of two added electrons, we plot the energy difFerence Egg —Ett (i.e. , the difFerence between the
total energies of one molecule with two doped electrons
carrying opposite spins and parallel spins, respectively)
in Fig. 5 as a function of t/J For simp. licity, we have
-0.20
-0.25
Q
-0.30—
-0.35—
-0.40
0.0
1.0
0.5
1.5
pi/p2
FIG. 4. The energy per bond for the generalized
state versus the variational parameter pi/p2.
vr-Bux
FIG. 5. The energy di8'erence between two-electrondoped states with antiparallel spins and parallel spins.
MAGNETISM A ND PAIRING IN A C MQLECULE: A
49
= J2 —J, neglecting
0.3
the small
Ellll —E
0.2
there is a transi i
E
(
the spin super exchange
the doped electrons pre er
the other an,
g
/
'll b do
t d
opp g
lectrons preferr too have para e
two doped elec
the nonzero order p
1' htl
h
d
1
th
tion is
'
o
4283
&&
ET111 —ET~71
Oq
0. 1
0.0
-0.1
0.0
6.0
4.0
2.0
8.0
10.0
state with both hopping annd
FIG. 7. Th e energy differences
or thee four-electron- -do p ed
nces for
cases.
6
A
t
h
Int
two Hipped spins. Th e'
g
is
stal n-electron dopeed molecule
electrons m y
11
e
eno
so(
, th
are three o
jection.
hhih
ne
other sta es
'
'
er
g
h
h
enstate.
n. InFig. 7wepo
functions of h e
b
locally sta
e,
J. The
60
h lo west unoccup
mini an
it is about severaal times smaller and
'
' '
e
ra
2. For sma er
ndt/J
a spin sing let
e is preferre w e or
/
favored.
The above resul ts sugges t that for smalll11lert J, t e
g
p referred, w hilee for larger t/1 &2
e
m
'
to the
the doped C
h
q
t d singlet stat
u ests a low transition em
'
agreement wit h thee experiment.
0, 1 to
1) and Cs—
o rt
Hkl
al.. Inth
n
t' crysta
lt
d
to
J reglOIl
t h emse ives so that
f 1
n the
o redistribute
second-lowest unoccup' ied
'
n
ma-
nitu e ar
eg p
itineran
gy
do ed electron
1
b
lues of t/
g
molecule C60 2 may not
the absolute value offfA
6
J
J as shown
e sta
'
ve
gisve
very
AE„&0
in Fig. 8.
y
e
close to the error bar
0.013t) there.
0.8
;
o
a=1
0.4
0.2
0. 1
0.0
0.0
ye
=
-0.4
-0. 1
-0.8
-0.2
0.0
2.0
4.0
6.0
8.0
0.0
10.0
FIG. 6. The energy difFerence
x er
pro three-electronbetween taro
i
doped states vnith different
spins.
2.0
4.0
The binding energy Q
AB
versus t/
1 for n = 1, 2, an
6.0
8.0
10.0
= 2E„—(E„+g+
~)
D. N. SHENG, Z. Y. %'ENG, C. S. TING, AND J. M. DONG
IV. SUMMARY AND DISCUSSION
We have studied the Hubbard type model for the C60
molecule proposed in Ref. 4 in the limit of large on-site
Coulomb interaction. On the C60 geometry, the &ustration of the antiferromagnetic coupling naturally occurs
so that the short-range singlet spin states become energetically favorable. We have investigated the variational
energies of these states on the C60 geometry by using the
VMC numerical method.
In the undoped C6p molecule, the Gutzwiller-projected
"Fermi-sea" state and the pure "RVB" state have been
shown to have the lowest variational energies among the
various spin singlet states. In the square lattice case, the
projected "Fermi-sea" state and the pure "RVB" state
are actually equivalent due to the SU(2) invariance. For
C60 geometry, they are also found to be degenerate within
the error bar of our numerical computation, even though
an analytic proof is absent. In contrast to the square
lattice, however, the counterparts of the Aux state and
the d-wave RVB state in C60 have relatively higher energies than the projected "Fermi-sea" state. This is due
to the antiferromagnetic &ustration on the C60 sphere.
The local spin-spin correlations have shown a very similar
structure to that given by the approach base on the clasThis fact may make it possible to construct
sic spins.
a path-integral theory by de6ning a local spin reference
axis, which would be useful for studying the breakdown
point U, of the present large-U approach.
When two electrons are added to the C60 molecule, it
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2 and the doped electrons are in singlet pairt/1
ing. If t/I
2, the triplet state is more stable where
Aq —42 —0 in most regime. For one-electron, threeelectron, or four-electron doped cases, A~ and A2 seem
too small to be determined in our work.
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larger spin state will be favored once t/ is large enough.
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have erst discussed such an
Gelfand, and Kivelson
electron-electron interaction mechanism for itinerant ferromagnetism by using a second-order perturbation theory for U. Nevertheless, our phase diagram does not seem
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the weak-coupling regime. Whether the present strongcoupling approach can be continuously extended down to
the realistic region for C6o system, without encountering
a phase boundary, still needs further investigation.
(
)
J
V. ACKNOWLEDGMENTS
The authors would like to thank G. Baskaran, B. Friedman, and J. Yang for the helpful discussions. The critical
reading of the manuscript by D. Frenkel is gratefully acknowledged. The present work is supported by a grant
from Robert A. Welch Foundation and the Texas Center
for Superconductivity at the University of Houston.
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