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Rom. Journ. Phys., Vol. 53, Nos. 910, P. 10451052, Bucharest, 2008
SITE DENSITY WAVES VS. BOND DENSITY WAVES
IN THE ONE-DIMENSIONAL IONIC HUBBARD MODEL
IN THE HIGH IONICITY LIMIT
FLORIN D. BUZATU1,2, DANIELA BUZATU
1 Institute of Atomic Physics, Mgurele-Bucharest, 077125, Romania2 Dept. of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear
Engineering, Mgurele-Bucharest, 077125, Romania3 Dept. of Physics, Politehnica University, Bucharest, 060040, Romania
Received August 30, 2008
We consider a Hubbard chain with an energy difference between the odd and
the even sites (ionicity). For a sufficiently large ionicity and electron concentrations
less than half-filling, the system can be described by the one-band Hubbard model
with a bond-site interaction. We investigate the competition between the density
waves localized on sites and on bonds at different band fillings by solving an
appropriate Bethe-Salpeter equation. Our results indicate the occurrence of the bond
density waves at quarter-filling.
1. INTRODUCTION
The organic charge-transfer (CT) solids are of two types: (i) separatestacks, with donor (D) or acceptor (A) molecules; (ii) mixed stacks with
alternating D and A molecules. The CT insulators, which are mixed stacks, are
either neutral or ionic, depending on the size of the orbital overlap: neutral at
small overlap and ionic at large overlap. Torrance et al. [1] discovered that by
increasing the pressure on an organic CT compound, such as TTF-CA
(tetrathiafulvalene-chloranil), it could pass from a neutral state to an ionic one,
and the transition is reversible. The same effect was observed by decreasing the
temperature, with the formation of a coexistence neutral-ionic region [2]. In
order to explain the neutral-to-ionic transition, Hubbard and Torrance proposed
a microscopic model for TTF-CA with alternating on-site energies [3]. Based
on that model, Nagaosa and Takimoto introduced the ionic Hubbard model
(IHM) [4]:
( ) 1 ( 1)j IHM j jj j j j j j
H t c c H c n U n n+ , ,, , ,, ,
= + . . + + . (1)
The first term in the Eq. (1) describes free one-dimensional (1D) electrons in the
tight binding approximation, the second one introduces the alternating on-site
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energies of the electrons, and the last one represents the on-site (Hubbard)
interaction between the electrons. A slightly modified Hamiltonian was used to
describe the displacive-type ferroelectric transition in ferroelectric transition
metal oxides such as BaTiO3 [57].
At half-filling and U= 0 the system is obviously a band insulator (BI),
characterized by the existence of a gap in both charge and spin excitations; for
= 0, the model reduces to the 1D Hubbard model (HM) and at half-filling it isa Mott insulator(MI), with no gap in the spin excitations [8]. In the atomic limit
(t 0), the transition is expected to occur at 2 ,U a critical line separatingtwo configurations: double-occupied (D) and empty (A) sites for U< 2 and
single-occupied sites for U> 2. The interest for the 1D IHM has been renewedonce with the work of Fabrizio, Gogolin, and Nersesyan [9], who proved that atwo-step transition should take place at half-filling between the BI and MI phases,
with the occurrence of an intermediate spontaneously dimerized insulator(SDI)
phase. The SDI phase is like the BI phase (with gaps in both charge and spin
excitations) in which a bond-charge-density-wave (BCDW)state occurs, with a
non-zero average value of the operator ( ) 1( 1) .ii ii D c c H c+ ,,= + . . Byincreasing Ufrom zero at half-filling, the transition between the BI and the SDI
is given by the place where the charge gap vanishes (it opens again in the SDI
phase) and iD takes some finite value; by increasing U further, the system
undergoes the transition to the MI phase where both the spin gap and iD
become (and remain) zero [9]. This scenario was confirmed by numerical studies
[10] and exact results for an effective model [11]. It is worth mentioning that the
occurrence of the BCDW phase is also predicted for the 1D extended Hubbard
model (on-site and inter-site interactions) at half-filling [12, 13].
In the present work we investigate the competition between the site density
waves (charge CDW, spin SDW) and the bond density waves (charge
BCDW, spin BSDW) in the ground-state of the 1D IHM at high ionicities2 2( 2 1)U t| |< , / and for electron densities less than half-filling.
2. EFFECTIVE MODEL AT HIGH IONICITIES
It was shown in the Ref. [14] that the Hamiltonian composed of the first
two terms in the Eq. (1)
( ) ( ) 20 2 cos 2iak
k k kk k k
k k
akH t e a b H c a a b b / , , ,, , ,, ,
= + . . + , (2)
where the Fermi operators in the Bloch representation a(b) correspond to the
even (odd) lattice sites, can be easily diagonalized by the following canonical
transformation:
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3 One-dimensional ionic Hubbard model 1047
(1) (2)2
(1) (2)2
( ) ( )
( ) ( )
iakk k k
iakk k k
a A k e c B k c
b B k c A k e c
/, ,
/
, ,
= +
= (3)
with
12
12
1( ) 1( )2
1( ) 1( )2
A kk
B kk
= +
=
(4)
and
2 2 2( ) 4 cos ( 2)k t ak = + / . (5)
The result is a two-band model and the last term in the Eq. (1) introduces both
intra-band and inter-band electron-electron interactions.
In the high ionicity limit
2 22 1U t| |< , / (6)
and for electron concentrations less than half-filling, it was also shown in the
Ref. [14] that we can restrict the considerations to a one-band model with the
same dispersion law as for free electrons in an usual non-alternating chain
2( ) 2 cos( )
2tk t ak t ,
(7)
and the following two-particle interaction
[ ]1 1 4 1 3( ) 4 cos( ) cos( )V k k U X ak ak ,.., + + (8)
with
2 2
2 21
8
t tU U X U , .
(9)
In the high ionicity regime and below half-filling, the 1D IHM can be thusapproximated by a one-band model with a narrower bandwidth, a reduced on-
site interaction, and a small bond-site coupling term proportional to the on-site
interaction and with opposite sign. The model is know as the 1D (t, U, X)-model
[15] and it is relevant to quasi-1D materials with large bandwidth like
conducting polymers. The ground-state instabilities (charge and spin density
waves, singlet superconductivity) of the 1D (t, U, X)-model have been analyzed
in a mean-field-like approach in the Ref. [16].
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3. SITE DENSITY WAVES VS. BOND DENSITY WAVES
In investigating the competition between the site and the bond density
waves in the 1D IHM, we use the same method applied to determine the ground-
state phase diagrams of the 1D (t, U, X)-model [16] and 1D Penson-Kolb-
Hubbard model [17]. Without going into details, the recipe is the following:
(i) start with the Bethe-Salpeter equation in the particle-hole channel in the
simplest approximation; (ii) find the solution (K, ); (iii) fix the totalmomentum 2 ;FK k= (iv) take of the form excE T+ with the excitation
energy Eexc = 0; (v) find Tc corresponding to the poles of 1( cT giving the
relaxation time of the unstable ground-state); (vi) determine the regions wherethe instabilities can occur from the well-behavior condition Tc 0 atno-interaction; (vii) find the dominant instability in the new ground-state, i.e.,
that one with the shortest relaxation time. The method amounts to a mean-field-
type approach to the ground state instabilities for 1D Hubbard-like models.
In the case of an alternating chain, the relevant operators in investigating
the occurrence of the density wave instabilities are (we use the notation DW for
the site density waves and BDW for the bond density waves):
DW ( ) ( ) ( ) ( ) ( )j jj jj t a t a t b t b t , ,, ,
, (10)
{ } BDW 1( ) ( ) ( ) ( ) ( )j jj jj t a t b t H c b t a t H c, + ,, , , + . . + . . (11)
where
1 charge sector
1 spin sectorji i j i j
c c c c c c,, , , , ,
, = +
(12)
We look now for the poles of the response function ( ) ( 0) ;K t K, , in
the high ionicity limit,
2
2 22 2
1( ) ( 0) ( )
( ) ( ) (0) (0)
k k
k K k K k K k K
K t K V k k K N
c t c t c c
,
+ / , + / , / , / , ,
, , , ;
,
(13)
where V2 is the potential generated by the site/bond response function and has
the form
2
2( ) 1 sin( 2) 2cos( 2) cos cos
2DW
tV k k K aK aK ak ak , ; = / / + +(14)
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for the site density waves, or
22
2
4( ) sin ( 2)cos cosBDWtV k k K aK ak ak , ; = / .
(15)
for the bond density waves. When 0,t/ 1DWV and 0,BDWV as for aregular (non-alternating) chain.
The approximate Bethe-Salpeter equation for the investigated instabilities
has the following form:
( ) ( ) ( ) ( ) ( )2
k
ik k K V k k K V k k K G k K k k K
, ; , = , ; + , ; ; , , ; , (16)
with
1 2( ) ( ) ( )V k k K V k k K V k k K , ; = , ; , ; . (17)
V1 in the Eq. (17) is given by the Eq. (8) but in the new variables 1 2k k K= + /and 3 2 :k k K = + /
( )1( ) 4 cos( 2) cos cosV k k K U X aK ak ak , ; = + / + , (18)
V2 is either VDW or VBDW from above, and ( )k K; , has the form
( ) ( )0 0( ) 2 2 2 2 2i K Kk K d G k G k
+
; , = + , + , . (19)
In the second order of ,t/ the potential Vgiven by the Eq. (17) becomes
( )2
22
( ) 4 ( ) cos cos , DW
( )4 sin ( 2)cos cos , BDW
U K X K ak ak
V k k K t U aK ak ak
+ +, ;
/
(20)
where
2
2( ) 1 (2 sin )
2
tU K U aK +
(21)
[ ]2
2( ) sin( 2) cos( 2)
8
U t X K aK aK / + /
(22)
The approximate Bethe-Salpeter equation (16) ca be solved exactly. The
poles ofare given by the zeros of
2
20 1 1 0 2 2
22
1 8 16( ) ,( )
8 sin ( 2),
U X Xc c c c c DW t t tD K
Uc aK BDW
+ + +
, = /
(23)
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where
( ) cos ( ) 0 1 2jnk
tc K ak k K nN
, ; , , = , , (24)
can be calculated by elementary integrations. For :T
0D lnFg T
= + (25)
where2
21 ln cos 1 , DW2 tan
4 sin , BDW
F
F
F
Uakak
U ak
+ | | + =
(26)
( )
2
2
2
2 2
2
11 sin2 sin cos2
ln cos1 1sin 1 sin2 DW4 2 sin
4 sin , BDW
F F F
FF F
F
F
U ak ak ak t
ak Uak ak ak
U ak
| |
= + + + ,
(27)
20 8 sin cosF Ft ak ak = / (28)
and
( )1
2 sinF Fg ak= , (29)
Fg t/ being the density of states at the Fermi level.The transition to an ordered state occurs for 0/< at
0 expcF
Tg
= | | (30)
which has the same form as the BCS critical temperature.The competition between the site and bond density waves for the 1D IHM
in the high ionicity regime can be visualized in the phase diagram from Fig. 1. In
the validity range of the used approximations, at half-filling (akF= ) only sitedensity waves occur. Going away from half-filling, the bond density waves also
occur but the site density waves are still dominant. By decreasing the density
further, the bond density waves become dominant in an increasing area and,
approaching the quarter-filling, a separation U| | / -line appears between the
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7 One-dimensional ionic Hubbard model 1051
site and the bond spin (charge) density waves for U> 0 (U< 0). At exactly
quarter-filling, only bond density waves occur in the ground-state of the system.
It can be also remarked in Fig. 1 the spin-charge symmetry of the phase diagram
when Uchanges its sign.
Fig. 1 Site density waves vs. bond density waves for the 1D IHM at high ionicities.
In conclusion, our results show that, for the 1D IHM in the high ionicity
limit, the bond density waves are likely to occur for a quarter-filled band.
Although the 1D IHM was much less investigated away from half-filling, some
interesting properties have been reported in this case [18, 19].
Acknowledgments. This work was supported by the Romanian research project CERES
C4-163.
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REFERENCES
1. J. B. Torrance, J. E. Vazquez, J. J. Mayerle, V. Y. Lee, Phys. Rev. Lett. 46, 253 (1981).
2. J. B. Torrance, A. Girlando, J. J. Mayerle, J. I. Crowley, V. Y. Lee, P. Batail, Phys. Rev. Lett.
47, 1747 (1981).
3. J. Hubbard, J. B Torrance, Phys. Rev. Lett. 47, 1750 (1981).
4. N. Nagaosa, J. Takimoto, J. Phys. Soc. Jpn. 55, 2735 (1986).
5. T. Egami, S. Ishihara, M. Tachiki, Science 261, 1307 (1993); S. Ishihara, M. Tachiki, and
T. Egami, Phys. Rev. B 49, 8944 (1994).
6. G. Ortiz, R. Martin, Phys. Rev. B 49, 14202 (1994); G. Ortiz, P. Ordejn, R. Martin, and
G. Chiappe, ibid., 54, 13515 (1996).
7. R. Resta, S. Sorella, Phys. Rev. Lett. 74, 4738 (1995); ibidem, 82, 370 (1999).
8. E. H. Lieb, F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).
9. M. Fabrizio, A. O. Gogolin, A. A. Nersesyan, Phys. Rev. Lett. 83, 2014 (1999).
10. H. Otsuka, M. Nakamura, Phys. Rev. B 71, 155105 (2005).
11. A. A. Aligia, C. D. Batista, Phys. Rev. B 71, 125110 (2005).
12. M. Nakamura, J. Phys. Soc. Jpn. 68, 3123 (1999); Phys. Rev. B 61, 16377 (2000), Erratum
ibid. 65, 209902 (2002).
13. M. Tsuchiizu, A. Furusaki, Phys. Rev. B 69, 035103 (2004).
14. F. D. Buzatu, D. Buzatu, Rom. Rep. Phys. 59, 351 (2007).
15. A. Painelli, A. Girlando, Phys. Rev. B 39, 2830 (1989).
16. F. D. Buzatu, Phys. Rev. B 49, 10176 (1994).
17. A. Belkasri, F. D. Buzatu, Phys. Rev. B 53, 7171 (1996).
18. K. Penc, F. Mila, Phys. Rev. B 50, 11429 (1994).
19. M. E. Torio, A. A. Aligia, G. I. Japaridze, B. Normand, Phys. Rev. B 73, 115109 (2006).