We study a class of one-dimensional chains whose topology leads to flatbands in the electronic spectrum. Using the Hubbard model, we find that these materials should exhibit ferrimagnetic ordering for a half-filled band, in agreement with a theorem by Lieb. Away from half filling the system displays a very rich magnetic phase diagram. Possible experimental realizations are suggested.74101851185

Coutinho-Filho M.D.

Dos Santos M.C.

Macedo A.M.S.

Macedo C.A.

Repositorio da Producao Cientifica e Intelectual da Unicamp

VOLUME 74, NUMBER 10 PHYSICAL REVIEW LETTERS 6 MARcH 1995
Magnetism and Phase Separation in Polymeric Hubbard Chains
A. M. S. Macedo, ' M. C. dos Santos, M. D. Coutinho-Filho, ' and C. A. Macedo
Departamento de Fisica, Universidade Federal de Pernambuco, 50760-90l Recife PE,-Brazil
Instituto de Fisica "Gleb Wataghin, " Universidade Estadual de Campinas, 13081-970 Campinas-SP, Brazil
Departamento de Fisica, Universidade Federal de Sergipe, 49001 Aracaju-SE, Brazil
(Received 11 January 1994)
We study a class of one-dimensional chains whose topology leads to flatbands in the electronic
spectrum. Using the Hubbard model, we find that these materials should exhibit femmagnetic ordering
for a half-filled band, in agreement with a theorem by Lieb. Away from half filling the system displays
a very rich magnetic phase diagram. Possible experimental realizations are suggested.
PACS numbers: 75.10.Lp, 71.30.+h, 75,30.KX
The possibility that the single-orbital version of the
Hubbard model might exhibit, under any circumstance, a
ferromagnetic ground state has been a longstanding highly
controversial issue [1]. Recently, Lieb presented [2] the
first proven example of unsaturated itinerant-electron fer-
romagnetism based on the repulsive Hubbard model, thus
taking a significant step forward. He showed that a bi-
partite lattice at half filled band has total ground-state spin
2Sr = [(B( —(A~(, where (B( ([A[) is the number of sites
in the B (A) sublattice. The theorem holds in all dimen-
sions without even the need of a periodic structure. Ex-
tensions of these ideas have been put forward by Mielke
and Tasaki [3], who showed that such lattices, whether
bipartite or not, might, in fact, exhibit a multiband struc-
ture containing a flatband, i.e., a highly (bulk) degenerate
single-electron state. The mechanism for ferromagnetism
works most effectively when this flatband is nearly "satu-
rated" but becomes ineffective when the electron filling
factor is too small. Specific examples of two- and three-
dimensional lattices have been given on which ferromag-
netism is sustained in a finite range of electron filling
factor. Notice that on "symmetric" bipartite lattices hav-
ing ~A~ = ~B~, ferromagnetism might only be found away
from half filling. In this case the so-called Nagoaka theo-
rem [4] asserts that the ground state on square and cubic
finite lattices with one hole in a half-filled band is a fully
aligned ferromagnetic state (Nagoaka state) in the infinite
on-site Hubbard-Coulomb repulsion (U) limit. Several
studies and attempts to verify and to extend Nagoaka's
theorem to finite values of hole density, Coulomb repul-
sion, and temperature have recently been reported [5].
In this work we study the very interesting possibility of
the existence of unsaturated ferromagnetic ground states
in complex polymeric organic and inorganic chains. Two
of the possible structures leading to highly degenerate
one-particle states are depicted in Figs. 1(a) and 1(b):
The model polymer of Fig. 1(a) is a well-known structure
in inorganic chemistry [6], where M and L represent,
respectively, a metal atom and a suitable ligand, and
Fig. 1(b) shows one example of an organic conjugated
polymer designed to possess aligned spins due to the
presence of molecular radicals R*.
~M~ ~M~ ~M~ ~M~
L L L L
(b)
(c)
l
B t B t B
"-A~ ~A~ ~A~ ~A~ ~A".
B B B B
FIG. 1. (a) Bipartite inorganic metal (M) —ligand (L) chain;
(b) substituted polyacetylene with lateral radical R"; (c)
bipartite AB2 chain and spin configuration for a half-filled band.
Organic conjugated polymers have been extensively
studied since the discovery that some of these materi-
als undergo a dramatic change in conductivity on doping
[7]. However, very few examples of organic ferromag-
nets have been reported in the literature [8,9]. The syn-
thesis and characterization of metal-free poly(m-aniline),
a conjugated polymer that reacts with acids (proton dop-
ing), have recently been reported [8]. ESR and magnetic
susceptibility measurements on this material are consis-
tent with ferrimagnetic interaction between unpaired ~
electrons.
We focus our attention on the bipartite lozenge chain
shown in Fig. 1(c) on which the unit cell contains two
sublattices A and B, the sites of A being connected by
hopping t to four sites of B, whereas the sites of B have
only two nearest-neighbor sites of A. On-site Hubbard-
Coulomb repulsion U is assumed on every site.
The tight-binding (TB) energy spectrum is com-
posed of two distinct level structures: The first one
has nondegenerate levels at e = ~2J2t and twofold
degenerate levels at e = ~2~2t cos(3 ver/N), v =
1, 2, 3. . . , (N/3 —1)/2, for N odd, and an extra twofold
degenerate level at e = 0 (v = N/6), for N even. The
second level structure is an (N/3)-fold degenerate
0031-9007/95/74(10)/1851(4)$06. 00 1995 The American Physical Society 1851
VOLUME 74, NUMBER 10 PHYSICAL REVIEW LETTERS 6 MARcH 1995
level at e = 0. In the thermodynamic limit, the re-
sulting density of states (DOS) for one spin species
is given by p(e) = ~6(e) + ~NQ„B(e—ek), where
ei, = 2+—2tcoska, ka = (0, 7r), and a is the lattice
1
parameter. We thus easily obtain p(e) = 36(e) +
(2/3~) [I/(8t —e )'~ ]. It is clear that the presence
of the "off" sites, added to a linear chain in order to
build the bipartite lattice structure of Fig. 1(c), leads to
the formation of a highly degenerate energy level at the
center of the band [see Fig. 2(a)], containing 1/3 of the
levels.
In the Hartree-Fock (HF) approximation, the Hubbard
Hamiltonian is written as
(k, a-) = (U/2t) (nA + nii )
+ 8 cos2(ka/2), (2)
ei„,(k, a-) = (U/t) ns
= (U/2t) (ng —nii ) .
In the above expressions, the subscripts A and B refer to
the corresponding atoms of Fig. 1(c). The dispersionless
band e&„(k,a.) arises from the topology of this polymer
and corresponds to states localized on B atoms only.
Equations for the densities n& and n& are found by
application of the standard variational procedure and are
solved self-consistently. For electronic densities such that
the flatband is occupied, the equation for n~ is increased
by a factor of f/2, where f is the fraction of occupied
states of the appropriate spin. The total energy per unit
t
(cilTci+,1cr + ,ci+1cr cia, ) +, U ~ ni—roc, i, l7ci, cr
I )t7 1,0 (1)
where c; (c; ) are electron operators and n; is thet
mean electron density of spin o at site i. Periodic
boundary conditions are applied, resulting in the following
band structure:
cell can be evaluated by
kF
e(k, rr)dk —U P n; n; . (5)
i E eel 1
At half filling, the HF DOS is shown in Fig. 2(b) for U =
2r. The system is insulating, the gap given by k(U) =
~+(7r/a, f) —~ (7r/a, J,) ()0 for any U ) 0); the fulfilled
localized level lies on the top of the e (k, t') band, i.e.,
«o. (k t') = & —(7r/& t) We then find 2S~ = n~t —n~t =
—0.36 and 25~ = n~~ —n~~ = 0.68, and thus an average
total spin per unit cell 2S = 4Sii —~2S~~ = 1, in fact,
valid for any U & 0, as required by Lieb's theorem. A
salient feature here is that the spin magnitudes on sites A
and B are unequal, indicating a quit general ferrimagnetic
ordering.
We account for electron correlations by means of
the exact diagonalization (ED) of finite clusters and by
Monte Carlo (MC) simulation using periodic boundary
conditions (PBC). The spatial ordering is studied by
calculating the longitudinal spin correlation functions
I-q = (1/N) g;(S S,'+q), where S,' + z(n;t —n;t) and
defines the order of neighborhood of the thermodynamic
correlation function. Our MC data are shown in Fig. 3
for a cluster of three unit cells using PBC, U = 2t,
and half-filled band. At the lowest temperature one can
reach, the local moment Lo is already about 70% of the
value for a completely localized spin, L() = 0.25. This
result is consistent with the estimated value of Lo =
0.2 at T = 0 found by ED of a cluster of two unit
cells. Note also that the system sustains some degree of
short-range magnetic correlations up to temperatures of
about kT = 5.0t. As L~ ( 0, L2 ) 0, and L3 ( 0 for all
temperatures investigated, we conclude that in the ground
state each sublattice is, on the average, ferromagnetically
il
0.3 -'.
0.0—
0.17
O.
0.16 - '' ~
0.16-
~ I I I 1 I
-1.0 -0.5 0.0 0.5 1.0 1.5
~ ~ ~
a ~
I
4
I
5
FIG. 2. Densities of states (DOS) for the AB& chain: (a) TB
(dotted lines) and (b) HF (full lines) for U = 2t at half filling.
Vertical arrows indicate localized bands (energies in units of
2v 2r).
FIG. 3. Monte Carlo data for the temperature dependence of
the first (1.&,~ ), second (I, ~), and third (L, ,a) neighbor
correlation functions. The inset shows the average site local
moment.
VOLUME 74, NUMBER 10 PHYSICAL REVIEW LETTERS 6 MARcH 1995
ordered but in opposite directions, thus confirming the
ferrimagnetic ordering found in the HF approximation.
We now turn to investigate the rich and complex be-
havior of the system away from half filling and for the
three limiting cases of interest, namely, U « t, U —t,
and U » t. The HF predictions are as follows. For
U/t ~ 0 (TB limit) the HF dispersive bands are degen-
erate, and we find n~t = n~t —= k F/2~, night = kF/4~, and
nial = kF/4~ + f/2, resulting in 2S = f, where f is the
fraction of occupied orbital states in the e~„(k,t) level.
The total spin per unit cell is nonzero for 2/3 ( n ~ 1, as
shown by the full straight line in Fig. 4(a). Figure 4(b)
shows the TB energy per unit cell (U ~ 0) and evidences
the/at portion for 2/3 ( n ( 1, corresponding to states
of zero energy on the e~„(k,t) level. It suggests that
for 2/3 ( n ( 1 the system phase separates into a rich
ferrimagnetic region, characterized by a completely filled
el„(k,t) level, and a hole-rich paramagnetic region with
an empty el„(k,l') level.
On the other hand, for U = 2t, say, holes first appear
in the dispersive band e (k, l,), causing a major decrease
of the local spin on site A and thus an increase of the
total spin per unit cell, as shown in Fig. 4(a). This spin
increase persists until holes appear in the e~„(k,]') level.
For values of n leading to a partially filled et„(k,t) level
the charge compressibility, sc ' = n 8p, /Bn, is negative,
where p, is the chemical potential whose n dependence is
displayed in Fig. 4(b). It also predicts phase separation,
as confirmed by the Maxwell construction shown in
0.8
0.6-
L 0 4
CL
M~ 0.2-
0 8»'
2
0.2 0.4
0&
Q)
CL
~~-2I «3
UJ
4.0 0.2 0.4
0.6 0.8 t.0
2
- '1
-0
llI
CD
3
o
~g g—Ok HF
~O~MO TB
0.6 0.8 1.0
electrons per site
FIG. 4. (a) Average spin per unit cell [TB results (—), HF
(~ ), and exact diagonalization (ED) (~)], and (b) total energies
per unit cell (left axis) and Fermi energy (right axis), of the
AB2 chain, for U = 2t, as a function of site electronic density.
Local spin per cell from ED indicated by (0). Energies in units
of t.
1 I I -2.4
- -2.5
~ 0.4-
Q
E
—0.2-
LL
0.0-
- -2.6 g
2 7
2.8 oz
- -2.9
0.6 0.7 0.8 0.9
electrons per site
1.0 0.6 0.7 0.8 0.9 1.0
electrons per site
FIG. 5. Maxwell construction of HF results for U = 2t: (a)
Fermi energy, and (b) total energy. Densities n„and n, 2define.
the range in which phase separation is predicted.
Fig. 5, devised to fix the wrong upward convexity of
the energy exhibited in Fig. 5(b). It should be noticed,
however, that now there is a finite range of doping, near
the half-filled band, for which itinerant ferrimagnetism
persists in the system before phase separation takes place
for higher values of doping.
Finally, we study the limit U » t for which dou-
ble occupancy of sites is avoided. In this case care
must be taken for n —= 1, since now there are two com-
peting mechanisms: the one leading to antiferromag-
netic ordering between sublattices A and B (and also
to Lieb's theorem at n = 1) and that of Nagaoka forc-
ing the system to fully polarize. If the former mecha-
nism dominates, we find from Eqs. (2) and (4) the simple
band structure s (k, t) = e (k, l) —= 8(t /U—)cos ka/2,
e+(k, l) = a~(k, l) —= U + 8(t /U) cos ka/2, and e~„(k,
l') = e&„(k,l) —U = 0. Similarly to the case U/t ~ 0,
the dispersive bands are spin degenerate, but, in contrast, a
strong narrowing effect occurs together with the expected
Hubbard splitting. We also have n~t = n~~ = 0, ng~ =
kF/7r, nest =—kF/27r + f/2, and 25 = f, thus resulting
in phase separation for 2/3 ( n ( 1 because the e~„(k,'t)
level is partially filled. If, on the other hand, Nagaoka's
mechanism dominates, the band structure is TB-like with
strong spin splitting U. For n(1 —n) ) t/U, where n is
of the order of unity, we expect full polarization, though
the argument of Nagaoka, and extensions of it, cannot en-
sure a unique ground state (see Tasaki, Ref. [5]).
In Fig. 6 are plotted HF data of a site-to-site simulation
for a chain of 150 sites using PBC and U = 103t. Indeed,
we find that the average total spin per cell is saturated
down to n = 0. The n dependence of the energy exhibits
a quite symmetric curve, except very near n = 1 where
Lieb's theorem is valid. Both features characterize the
behavior of a collection of noninteracting TB spinless
fermions. Notice that for 1/3 ( n ( 2/3 the energy
curve is Oat, suggesting phase separation for doping in
this interval. Moreover, for n ( 1/3 and 2/3 ( n ( 1 we
find very small (of the order of 0.3%—0.7%) modulation
of the unit-cell spin distribution on top of the average fully
polarized ferromagnetic background.
1853
VOLUME 74, NUMBER 10 PHYSICAL REVIEW LETTERS 6 MARcH 1995
1.5
g &.0
C
g) 0.5 4 .~ ~ ~ ..P
0.5
- 0.0
ITl
- -0.5 g
--1.0 ~
5 o
--20=
I I 1 I
'S.O 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0 2.5
electrons per site electrons per site
FIG. 6. (a) Average spin per unit cell versus electronic
density: HF results for U = 10't (~ ) and ED for U = 10~t
(j). Local spin per cell from ED indicated by (0). (b) Total
energy per unit cell from HF calculation (~ ) and ED (j).
In Fig. 4 we compare the HF results with those from
ED of a cluster containing two unit cells and U = 2t.
The n dependence of the total spin and energy per unit
cell follows the same trends found in HF approximation.
In particular, for n ( 2/3 the total spin has the minimum
possible value. We also calculate the local value of
the spin per unit cell (notice that quantum fiuctuation
effects are present) and find that its n dependence suggests
that for n ( n, 2 the unit cells are antiferromagnetically
coupled, leading to a singlet (S = 0) ground state. In the
HF approximation this configuration has energy only 1%
higher than that with unit cells ferromagnetically coupled.
For such a small system we find no indication of phase
separation instabilities for U —t.
For U = 100t, we see from Fig. 6 that Nagaoka's
condition is satisfied for one hole and, furthermore, that
each unit cell is fully polarized down to n = 0. For
n ( n,.q the spin arrangement is again such as to attain the
minimum total spin value. In the HF approximation this
configuration has energy about 20% higher than that with
full polarization. The energy curve has similar trends,
except in the region where HF predicts phase separation.
For higher values of U we find numerical instabilities for
I/3 ( n ( 2/3, which might indicate a region of phase
separation. However, we cannot exclude the possibility
that in the bulk system the formation of a more complex
spiral spin density wave is realized and that, in the high
doping regime, the destruction of long-range order occurs.
In summary, we have studied the magnetic properties
of a class of one-dimensional Hubbard chains whose
topology produces Aatbands in the electronic spectrum.
It has been found that this class of materials should
exhibit ferrimagnetic order at half filling, for any U )
0, in agreement with Lieb's theorem [2]. The above
reported results also point to the possibility of producing
metal-free, fully organic polymers having a rich magnetic
behavior. This could be experimentally monitored by
electrochemical techniques, or by fixing p H conditions,
in the same way that conductivity can be controlled in
conventional conjugated polymers.
This work was supported in part by CNPq, CAPES, and
FINEP (Brazilian Government Agencies).
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[7] For recent developments see Synth. Met. 55—57 (1993).
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1854


Magnetism And Phase Separation In Polymeric Hubbard Chains

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Magnetism And Phase Separation In Polymeric Hubbard Chains

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