florin d. buzatu and daniela buzatu- site density waves vs bond density waves in the one-dimensional...

8/3/2019 Florin D. Buzatu and Daniela Buzatu- Site Density Waves vs Bond Density Waves in the One-Dimensional Ionic Hubb

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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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1046 Florin D. Buzatu, Daniela Buzatu 2

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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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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