Acta Physica Academiae Scientiarum Hungaricae, Tomus 48 (2- -3) , pp. 169--185 (1980)

EXACT SOLUTION OF A QUASI-ONE- DIMENSIONAL MODEL WITH LONG RANGE

INTERACTION (COUPLED TOMONAGA CHAINS)

By

NGUYEN MINH KHUE and J. S‰

crNT~U.r RSSr~~CH ~~STIT~r roR rHYSICS. SVVArEST, HU~r

(Received 14. XII . 1979)

The novel method proposed by one of the authors to calculate exactly the response functions of the one-dimensional Tomonaga model is described in more detail. The method is generalized for the case of a system of conpled chains where both the intrachain and inter- chain interactions have forward scatte¡ components only. The model does not show real phase transition at any finite temperature indicating that the interchain backward scattering or hopping is needed to have ah ordering of the chains at finite temperatnre.

1. lntroduction

Reeently quasi-one-dimensional (quasi-l-d) conductors a t t raeted a great deal of interest because of their unusual properties. F o r a review of works done in the past few years see [1]--[3]. These systems are built usually of large, flat molecules which ate relatively closely stacked in one direction to forro ehains, the adj aeent chains being a t a relatively larger distante. Thus the motion of electrons is confined predominantly to motion along the chains with rare hoppings between them.

Various theoretical models have been worked out to s tudy the properties of these systems. One of these models is the Fermi gas model which has been investigated in the strictly 1-d case in great detail. A review of the properties of this model can be found in [4]. Since a 1-d system may ¡ specific pro- perties of behaviour (such as absence of phase transition at finite temperature) which ate characteristic for 1-d systems but ate not neeessarily true in higher dimensions, any realistic model of quasi-l-d conductors should contain some kind of interaction between the chains. Generalizations of the 1-d Fermi gas model in this direction have indeed been a t tempted [5]--[9]. The Fermi gas model is a model with infrared singularities difficult to t reat in a satisfactory manner at low temperatures and low energies even in its strietly 1-d version already. The t rea tment becomes increasingly more complicated when interchain interactions ate taken into account. Therefore any simplified model which can be solved exactly is of great value in finding out what the effect of the interaction terms is.

Acta Physica .,4cademiae Scientiarum Hungarieoz 48, 1980

170 NGUYEI~ MINH KHUE and J. S‰

The 1-d TOMOr~ACA model [10] is a well known exactly soluble model of a 1-d interaeting eleetron system. The two important features whieh make it soluble are the linear dispersion relation and the negleet of backward seatter- ing. In the present paper we will eonsider a generalization of the model to a set of coupled ehains but keeping these two features intact, i.e. a one-dimensio- nal linear dispersion relation is taken for every chain which means that no hopping between the chains is allowed and only forward scattering processes are taken into aceount for both the intraehain and interehain scatterings.

There ate various ways to solve the 1-d TOMONAGA model. The excita- tions of the model have been shown to he the boson-like charge- and spin- density excitations [11]. Therefore the ealculation of the one-particle Green's function and the response funetions of two-particle correlation func- tions in which the fermion operators cannot be easily expressed in terms of the eharge- of spin-density operators was a formidable task [12]. I t was shown, however, by LUTHER and PESCHEL [13] and by MATTIS [14] tha t an exaet operator identity can be found which allows to represent the fermion opera- tors in terms of the boson-]Jke density operators. Since the TOMO~AGA Hamil- tonian can be transformed to a diagonal form of the boson operators, the Greens' function and response functions can be calculated by calculating har- monic averages. The operator identity involves, however, a delicate limiting procedure, whieh is not always easy to perform and therefore other methods are also of interest. FOG~DBY [15] used funetional integra]s to solve the 1-d TOMOr~AGA model. Another very elegant method was used by DZYALOSHII~SKY and LAnKIr~ [16] to ealculate the Green's function. They could derive a Ward ident i ty between the Green's function and the vertex appearing in the Dyson equation. Making use of this relation, the Dyson equation cou]d be solved exaet]y. This method has been generalized by one of the authors [17] to cal- culate the response functions. In this paper we will make a further generaliza- t ion to study the behaviour of the quasi-l-d system of TOMOr~ACA chains.

First in Seetion 2 the method of DZYALOSEXr~SKY and LARKI~ [16] is briefly presented sinee it is the basis of all further eonsiderations. Then the generalization of the method to ealculate response functions is described in Section 3. Section 4 eontains the results obtained for the quasi-l-d system of TOMOr~Ar ehains. Finally the results ate diseussed in Seetion 5.

2. The Dzyaloshinsky--Larkin method of sl]mmiug diagrams in the Tomonaga model

The TOr~Orr model was defined originally a s a 1-d model of interaet- ing electrons

k 2 1 H = ~y--c~+=ck= + ~ ~(p) + ck +p= c+'-pp ck'~ ck~, (1)

2m k, k,k',p

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EXACT SOLUTION OF A QUASI:ONE-DIMEN$IONAL MODEL 171

whereL is the length of the system and the interaction potential is supposed to be of long range and therefore the Fourier components X(p) are non-zero for small momenta only. I f the cutoff on the momentum transfer Ais much smal- ler than the Fermi momentum k m one can distinguish between particles near the Fermi point ~-k F and particles near the other Fermi point --kF, since in the absence of large momentum transfer interaction (backward scattering) the electrons s tay always in the neighbourhood of the same Fermi point after any interaction process. In order to make this distinction more explicit, we will denote the creation (annihilation) operators of eleetrons near the right and left

+ + Fermi points by a k (ak) and b k (bk), respectively.

The kinetic energy term can be approximated by a linear relationship near the Fermi points and the free part of the Hamiltonian can be written as

H o = , ~ v p ( k - - kp) a+~ ak~ -f- . ~ v F ( - - k - - kF) b+~ bk, ~. (2) k,= k,=

The interaction part of the Hamiltonian can be generalized by allowing for different coupling strengths for the tases when the scattered electrons ate from the same of different branches of the spectrum and for parallel of anti- parallel orientation of the scattered eleetrons. In the most general case the Hamiltonian reads:

1 Hin t = ,.,~,. (klll ~=~ + ~.~ ~=,-~) (a++ w a+'_p~ ak'p ak~ ~-

2L , ,

+ bk~) 1

ak+p= + ~- 2 (~~,, ~~~ + ~= ~~,-~) + b~,_~p b~,~ ~~~. (3) k,k',p

To make contact with other works on the Fermi gas model [4] where back- ward scattering is also included, the present choice of couplings corresponds in the language of. 'g'-ology to kx = g~ and k2 = gz" The scattering processes can be represented diagrammatically as

o f

where the solid and dashed lines correspond to electrons on the right or left branch of the spectrum.

DZYALOSHI~SKY and LAnKIN [16] recognized tha t the two particular features of the TOMOI~AGA model, namely the linearized dispersion relation

Acta P¡ Avademiae Scientiarum Htmgaricae 48, 1980

1 7 2 NGUYEN MINH KHUE ~md J. $‰

and the negleet of backward seattering terms lead to ah enormous simplifi- cation in the contribution of diagrams. They have found that all diagrams which contain electron bubbles with more than two interaction legs ate mutu- ally caneelled and only simple bubbles with two interaction legs and series of simple bubbles remain.

Effective interactions can be introduced by summing these series of bubbles, in the same way as the sereened interaction is introduced in a RPA ealculation, but here working with these interactions will lead to an exact procedure. The diagrammatie equations for the effective interaetions denoted b y DII I , Dli, D211 and D2• ate as follows:

D 1 o r D 1 1 = = , § 4, ~,

D21orD21 = = + + ~ 7

The diagrammatic equations ate the same for parallel and antiparallel spin orientations. Ir is worth mentioning that though the bare ~1 coupling is the same for eleetrons on the right of left branehes, the effeetive couplings will be different.

The analytie forro of these equations is

Dn, = Ax,, + ~i,, II+Dm + ;(ii 1-[+DLr + Ÿ H-D2,, + 22•

Dl• -= tr~ + 2111/~+Dll "3 L ~I.LH+Dll, -4- ~211 li-D21 -~ ~2• ,

D2, I = ~.~,, + ~211/'/'+Dlu -~- q • -4- 9 .n l / - /_D2. + ~1• D 2 i ,

D2-L = '~21 -q '/~211T/-t-Dll + ~'21//+Dl l l + ~- I I I / / -D21 + '/~11-~-D211 ,

(4)

where / /+ a n d / / _ are the polarization bubbles for electrons on the right and left branches, for one spin orientation. These equations can be solved using the expressions

k k //+(k, co) = , r L ( k , ~) = -- (5)

2~(~ - ~~k) 2~(~ + v~k)

.4~ Physica Aa~emiao S~n t iarum Hungarica~ 48, 1980

EXACT SOLUTION OF A QUASLONE-DIMENSIONAL MODEL 173

giving e.g. for Dll I which couples two electrons on the r ight b ranch

D m ( k , to) ----- (to - - vFk)[ A + B -4- A - - B to - - uok + i~ sign k to -4- u.k - - i~ sign k

C + D C - - D + -t-

o~ - - uek + i~ sign k ~o "4- uok - - i 6 sign k

where

=.~ = ~ + ~ ( ~ ~ , , - ~~~) -

2 1 1 (~2,1 + A21) u . = v,~ +~( ;~~, , + ;L~~) - ~

_ { [ + ! ( ~ . , , _ ~ ~ ) ] } A = 1 (~m--21.L)' B = ~ u~--vF VF 4 2Uo 2~t

1 l [ ]} C = ~ - ( R x , , + 2~• D = ~ u~ -- vp VF + 1----(~~,, + 21i) 2ue 2~

+

], (6)

and for D211

(7)

1 ~2 _ v ~ k ~ D2n(k, co) --= - - (3.2, - - 3.2~ +

2 co 2 - u ~ k 2 + i ~

+ __I ~~ - - '~~' ~ (~~,, + ;~,.,). (8) 2 o~ ~ - u ~ k 2 + i a

Once the effeetive interact ions are used, the Dyson equation can be wri t ten d iagrammat ica l ly as

O+(p,E) = > : > + -" - ~ J ' r -"

in which the three-leg ver tex has two solid line legs interactions in all possible ways.

P+ ( p,�91 k,~)

coupled by effective

p-k p . ~

4- 4- § §

.4aa Phydca A ~ m i a o Sdr Hungarir 48, 1980

174 NGUYEN MINH KHUE and J. S‰

A further consequence of the particular features of the TOMO~ACA model is that , as shown by DZYALOSHINSKY and LAax*r~ [16], a Ward ident i ty can be found for general energy and momentum variables by use of which the vertex / '+ can be expressed in terms of the Green's function

1 r + ( p . ~. k. 09)

09 - - ~ F k

and ah analogous relation

1

09 + v pk

- - [ G u e) - - G u - - k , ~ - - 09)1 ( 9 )

[G::(p, ~) -- G:X(p - - k, e - - 09)1 (10)

holds for the vertex in which the electrons on the external legs belong to the left braneh of the speetrum.

By making use of the Ward identi ty the Dyson equation for the Green's function can be written in a elosed integral equation forro

[E - - ' V F ( . p - - kF) ] G+(p, 8 ) = 1 "gr" --~~,~dk do) Dxll(k' 09) ~ - v~-k c + ( . - k . . - 09).

(11)

where the term leading to Fermi energy renormalization has been negleeted. In real space and time represcntation wc get

I--~t -4- V~~x I G+(x, t ) = K: , , (x , t)G+(x, t) (12)

with

Kl, t (x , t) = ~ ~dk d09 D:, ,(k, 09) e_i(~t_kx). (13) 4z~2J 09 - - v pk

This equation can be solved in the forro

{~f; } c§ t) = c~>(r) exp K~,,(r, s') ds' f§ (14)

where the new variables r = x - - vpt and s = x + vFt were introduced.

1 1 G<~)(x, t) = (15)

2z~ x - - vpt -4- i~(t)

with ~(t) = ~ sign t and f+(r) has to be chosen in a way to ensure the correet analytic properties. Performing the integrations in Eqs. (13) and (14) with

Acta Physir Ar Scier~iarum Hungaricae 48, 1980

EXACT SOLUTION OF A QUASI-ONE-DIMENSIONAL MODEL 175

the cut-off factor exp (--[kl/A ) for the momentum transfer, the final result is

1 G+(x, t) : - -

2~r

with

1 x - - v~t A- i /A(t) • x - - vpt -k i~(t) [x - - Uot + i/A(t)] 112 [x - - uqt A- i/A(t)] 11~

X [A2(x - - u , t q- i /A(t)) (x -k u j - - i/A(t))]-~~, (16)

X [A~(x - - uqt q- i /A(t)) (x -4- uqt - - i /A(t))]-~~

% - - v~ - - (17) 4u~ 2~r

and A(t) = A sign t. The velocities u~ and u~ ate given in Eq. (7). The Green's function of the electrons on the left branch can be calculated

similarly leading to the result:

G_(x, t) -~ G+(x, --t). (18)

3. Genera l i zed W a r d ident i t ies and r e s p o n s e f u n c t i o n s o f the 1 -d m o d e l

One of the best ways to get information about the possibility of phase transition to ah ordered state is to calculate the response of the system to various external perturbations. A singularity in the response function is ah indication that a spontaneous ordering can take place. We will consider the density and pairing responses since they eontain logarithmically singular terms in every order ofperturbation theory and therefore there is a high chance for them to be singular when ah exact summation of aH contributions is performed.

These funetions can be defined as

w h e r e

(0

R(k , o~) = - - i S dt ei~ <T{ O(k, t) O+(k, 0)}>,

O(k, t) = ~ ~ b ~ ( t ) ap+kt(t )

(19)

(20)

for the charge-density response function N(k , o~) with large momentum (k ~ 2kF),

1 (ii) O(k, t) - - ~ b~(t) ap+ k ~ (t) (21)

Liq

Acta Physica Academiae Scientiarum Hungarieae 48, 1980

176 NGU'YEN MINH KHUE and J . S‰

for the spin-densi ty response funet ion • (k, to) with large m o m e n t u m (k N 2kF).

1 (iii) O(k, t) = L 112 . ~ bp t (t) a_p+ k ~ (t) (22)

P

for the s ingle t -supereonduetor response ZŸ to) with small m o m e n t u m k, and

1 (iv) O(k, t) -- L 1/2 . ~ bp f (t) a-p+k t (t) (23)

P

for the t r ip le t -superconductor response zJt(k, to). The diagrams represent ing these response funet ions have one solid and

one dashed line running f rom one ex te rna l ve r t ex to the o ther and these lines are dressed and eonneeted b y effeetive interaet ions in aH possible ways. The eharge-densi ty response funet ion can be represented as

N(k,~) =

p-k, t~-~,

where the two lines are renormal ized lines and the ver tex

k , t ~

p~ ~~ p=k

is analogous to the vert iees P+ and -P_, bu t here the two legs belong to difs ent branehes.

Unfor tuna te ly , there is no Ward iden t i ty whieh eould relate this ve r tex to the Green's funet ions. There is, however , ah a l ternat ive way to write the response funetions. All the diagrams can be elassified into th ree elasses. E i the r there is no in te rae t ion line eoupling to the solid line, or ir there is one, the first of t hem can eouple baek to the solid line in whieh case this is an effeetive Dll I in te rae t ion or it eouples to the dashed line in whieh case it is an effeetive D211 in teraet ion. Aeeordingly

Acta Phyd~'a Aca~mias Scisntiarum Hungari~as 48, 1980

EXACT SOLUTION OF A QUASI-ONE-DIMEN$IONAL MODEL 177

where two new ver t ices have been in t roduced:

§

r F/l�91 I Ÿ " +-~.=,-,.,~,~.=,=/ \ - t ',,- , , . . .

in which the smal l m o m e n t u m t rans fe r in te rac t ion eouples to the solid line, and

p-q p-k �91191 o..y~

in which the smal l m o m e n t u m t r ans fe r in te rac t ion couples to the dashed line. The two legs and the in ter inedia te line a te connec ted b y effect ive couplings in aH possible ways and self energy correct ions on the i n t e r m e d i a t e line should also be eonsidered. The d i a g r a m m a t i c r ep resen ta t ion of the sp in -dens i ty res- ponse is s imilar , b u t ins tead of D211 the effeet ive in t e rac t ion D2• should appear . I n the p a i ¡ responses the ver t ices have two ineoming lines, o ther- wise the represen ta t ions ate similar.

As was po in ted out b y S£ [17], the pa r t i cu la r fea tures of the To~so- NACA model allow to derive general ized W a r d ident i t ies re la t ing the four- leg vert iees/ ' (~) and q 4) to three- leg ver t iees r + _ wi th large m o m e n t u m t ransfer .

These re la t ions are the consequence of the eonserva t ion law for par t ic les in eaeh b r anch and for eaeh spin o r ien ta t ion and a te s t r a igh t fo rward general iza- t ions of the W a r d ident i t ies given in Eqs. (9) and (10). T h e y can be wr i t t en ana ly t iea l ly as

l X . ~ ) ( p _ _ q , t o 1 _ _ ~ , g , ~ , k , r - - - - 1

e - - v F q [F+_(p - -q , o~ 1 - - e, k - - q, o~ - - e)--

- c z ~ ( p - q, ~1 - ~) c + ( # , ~1) F + _ ( p , ~1, k, ~)1 (24)

and

/~4)(p __ q, to 1 __ e, k, (o, q, e) - - - - --I

e + v ~ / [ /"+-(P - - q, ~1 - - e, k - - ~/, co - e) - -

_ c - l ( p _ k, ~1 - ~ ) C _ ( p - h - l , ~~ - ~ - ,0)I '+_(1, - i , ~ 1 - ' ~ , k, ~ ) ] (25)

Acta Physi~ Ar162 S�9 Hungarir162 48, 1980

178 NGD'YEN MINH KHUE mad J. S‰ M

or diagrammatica]ly as

+ q,e k.~ k~q k,o.,

i \

k,ar -q,~ k,a~ ~Ek-

&-_~ p-_kr p-c~ p-k-q ~-�91 ~-Cle. p-k o.rl- ~'

where the external renormalized Green's function legs should also be taken into account.

The analytie expressions for the charge-density response fmaetion N ( k , (o), when the two representations ate used, are

D A &ol N(k, ~) = - - i i -~ C+(,, ~~) r+_( , , ~1, k, ~) G_(, -- k, ~1 -- ~) J 2 z t 2zt

and

N(k. ~) = - i ( " d . d~~ G(~)(,. o.1) G_( . -- k. ~ol -- ~.) + )2zt 2~

(26)

+ f ~ do,____~~ V(~@, ~oOf d~ de Dl, ,____(q_, ~ ) • 2z~ 2~ 2~ e -- vp~

• v+ (p - ~. ~ol - ~) r ( ~ @ - ~. ~ol - ~. q. ~. k. ~o) •

x G _ ( p - - k , a~ 1 - - w ) f l - f - ~ ~ d(~ G(~)(P' wl)X

f d~ D~,.(~_ ~_) G+(p - - ~, ~ol -- ~) • ?_ X . . ~ 2zt e - - v F ~l

• _ r ~ @ _ ~, ~o 1 _ ~, k , ~o, ~, ~) G _ ( p - - k , ~o 1 - - o,). (27)

Introducing the quant i ty

N(p, ~ol. k, ~o) = V+(p, ~o) r+_(p, ~o. k, ~o) C_(~ -- k, ~ol -- ~o) (28)

~4eta Physiea Jlr Sr Hungaric~~ 48, 1980

EXACT SOLUTION OF A QUASI-ONE-DIMENSIONAL MODEL 179

from which N(k, w) is obtained by simple integration, making use of the gene- ralized Ward identities in Eq. (27) and comparing Eqs. (26) and (27) gives

N(p, o,1, k, o,) c~>- l (p , o,1) = G_(p - k, ~ol - ~o) +

.2f_ i f d~ dE D111.___(9...~~ f} [ N ( p - - 9, (DI - - �91 k - - 9, w -- e) - - ) 2 : t 2 z t e - - V F 9

- N ( z , , o~~, k , o~)J + q a~ D~,,(q,~) [ :V(p - - ~ , o~~ - - ~, t~, o~) - - J2z~ 2 ~ e - - v p 9

- - N ( p - - 9 , o~x - - e , k - - 9 , o~ - - e)]. (29)

with This equation can be solved by writing it in Fourier transformed forro

H(p, to 1, k, w) = S dxl dh dx dt 1V(x 1, tl, x, t) ei(~ e i(~ (30)

since then

Ir ,q~ i [~tl @ vF~x l ] N(xl , tz, x , t ) = ~(x-~- Xl) ~(t + tl) G_(--x , - - t ) -4-

+ i Kl , , (x + x 1, t -4- ti) N(xl , tv x, t) - -

- - i[K2,,(x + x 1, t + t 0 - - K2t,(x 1, tO] N(x 1, ti, x, t),

(31)

where the term leading to a Fermi energy renormalization has again been neglected and

K2,,(x, t) = f dq de D2,,( 9, e)e_i(st_qx) " (32) 2~ 2~ e - - vp q

The solution can be looked for in the forro

Nl(x 1, t I, x, t) = Nl(x -4- xi, t + ti) N2(x 1, q)N3(x, t). (33)

Eq. (31) can be separated into two equations for N 1 and N2, and furthermore N3(x, t) = G_(- -x , - - t ) N ; l ( - - x , --t) . These equations are solved in the same way as Eq. (12). Put t ing everything together, taking into aceount that N(k, a~) is obtained from N(p , w 1, k, ~o) by a simple integration, whieh means that the Fourier transform N(k, t) of N(k, ~o) is obtained by put t ing x 1 ---- t 1 = 0 in N(x 1, t 1, x, t), we finally get

Ir(x, t) -= - - i G+(x, t) G_(--x , -- t) [A2(x - - u J + i/A(t)) (x -4- ttot - - i/A(t))]P~•

X [A2(x - - uet + i/A(t)) (x -4- uet -- i/A(t))]ae (34)

Acta Physiea Academiae Scientiarum Hungaricae 48, 1980

180 NCUYEN MINH KHUE aur J. S~LYOM

with 1

# . = - - (a=,, - ; tu) . 4~tu a

1 ~t = (a~,, + a , , ) . (35)

4~tu t

Since the spin-density response function contains D~• instead of D21 I, a similar expression is obtained for X(x, t) with --~~ instead of ~. in the expo- nent.

The calculation of the pairing responses can be done in a similar way. The generalized Ward identities are also similar with one overali sign difference in the case when the small momentum transfer interaction leg couples to the dashed line due to the opposite orientation of this line. The restdt for the singlet-superconductor response is

As(x, t) = iG+(x, t) G_(x, t) [A'(x - - u~t + i/A(t)) (x + u.t - - i/A(t))]a. •

• [A~(x -- uet + i/A(t)) (x + uet -- i/A(t))]-&. (36)

The triplet-superconductor response has --/~~ in the exponent instead of ~r Writing in the expressions for the Green's functions from Eqs. (16) and

(18) we finally get

where

(i)

with

R(x, t) = T i x -- vF* + i/A(t) . x + vp t - - i /A ( t ) A ' • x - ,.,pt + l a ( t ) . + Vpt - i /~ ( t )

x [(~ - ~ot + i / a ( O ) (x + ,.,vt - i / A ( O ) ] ~ •

X [(x -- u,t + i/A(t)) (x q- u,t -- i/A(t))]'e,

1 1 ~,. = T T . , ~,, = T ~ , , ,

v p + ~-g= (~~,, - - E~-= (,h,, -

1 1 - - / ' v. + g(a~, , - a~~) + g ( a . , , - a..) j

1 1/2

vF + ~I--L {J'n'zzt + J, r O - ~-(i2,,1 + J.u)]

1 (a~,, + a**) + 2 - 7 ( a , , , + a , , )

(37)

(38)

(39)

A ~ Ph3~~ A~~emt~ Sc~rdmrum H u n ~ r ~ 48, 1980

EXACT SOLUTION OF A QUASI.ONE-DIMENSlONAL MODEL 181

for the charge-density response function,

1 1 (ii) P*-- 2~o ' P, = - ~ V , (40)

for the spin-density resl)onse function,

1 1 (¡ p. ------~-~,o, p i = 27e' (41)

for the singlet-superconductor response, and

1 1 (iv) ~'~ - 2 v . ' ~~ = (42)

2~,~

for the triplet-superconduetor response. In the case of spin-independent interaetion these results agree will those

obtained by FOGEDBY [15].

4. System of conpled Tomonaga chains

I t was shown in the previous Sections how the strictly 1-d TOMONAO~ model can be so]ved exactly by summing all diagrams by the use of Ward identities. Now this method will be extended to s tudy the properties of a quasi-l-d system, in which 3-d couplings are taken into account.

The model is defined as follows: a set of 1-d TOMONAOA chains with linear dispersion relation and intrachain forward scattering is coupled together by intcrchain coupling which has forward scattering eomponents only. Assigning a chain index i to the e]ectrons on the chain at position R;, the Hamiltonian of the system can be written as

with

and

H = H o + Hin t

l,k,~t i,k#

I-~int = 1 .~ O~,,u ~}.o + xu~j *.,-p) (a~+.. a~k._., alk. ~ a~k. +

2L i,j k,k',p

b fk~ b jk ?J blk,) 1 + --2" (R~,,u a.p + ;~~u a.,_o) a~+p. b~._po bj~.pa~~..

(43)

(44)

(4s)

Acre PAysi~ A�91 Sr H u n g a r ~ 48, 1980

182 NGUYEN MINH KHUE and J. S‰

where Xlq, and X2/j are the coupling constants between eleetrons on ith and jth chains. For simplicity equivalent chains are considered, i.e. v~ and k~ ate taken to be the same for all chains.

The method of solution is a straightforward generalization of the method used in the strictly 1-d case. Since the quasi-l-d model has the two main characteristic features of the 1-d ToMo~~oA model, namely the linearity of the dispersion relation and the absence of large momentum transfer (backward scattering) terms, diagrams with simple bubbles and series of bubbles will only contribute, the effeet of more eomplicated diagrams being caneelled by each other. A s a consequence, effective interactions can again be introduced, satis- fying the same diagrammatic equations as in the 1-d case, but the electron- hule pair in the intermediate state can be on any chain. The analytic equations for the effective interactions ate

Dll q = ~11 li j + , ~ [ ~1! Iq161 +Dll ilj-~ ).LLllFf +Dlj.l j-}-~t21 l i l17-D 211U -4- ,Tt2illH_D 2ilj], l

Du.tj : ~lj.ij .q ~ [~-11 iiq -4- Ÿ q "~- ~'21 q -b 2~HII_D~I ti i i , 1

D211ij = ~211/j -~- . ~ [Ÿ237 1

D2.L/] = Ÿ -~- 2 [~21 ffl[-[+Dl• ~2.Lil[I+Dll IU-k ~J.I lilH-D2.uj + Ÿ ' (46)

These effective interar are of course functions of the momentum compo- nent kll which is parallel to the chain direction, and the energy variable co thxough the dependence of H on these variables. Performing a Fourier trans- formation in the perpendicular direction with

1 .~Ds(k, , , ki, co) e'k• q ----- XII, lA_, 211, 2 1 (47) D~tj(k.,co)= �9 k.

we have

D~,,(k.) = ~~,,(k~) + ~~,,(k.)~+D~,,(k~.) + ~~.(k3 ~r+D,,(k.) +

+ ~2,,(k,) 17_D~,,(k3 + ~~.(k.) ~_D~~(k3 (48)

and three similar equations with

2~(k,) = . ~ a~q e-ikl(R,-R, ), = 111, 1• 211, 2_L. (49)

The ki-dependent effective interactions obey exactly the same equations as the effective interactions in the 1-d case, except that everywhere Ÿ should

Acta Physi~ A~&~mi~ SdenJian~ Hungari~~ 48, 1980

EXACT SOLUTION OF A QUASI-ONE-DIMENSIONAL MODEL 1 8 3

be replaced by ~(k~). Keeping this in mind, the solution of these equations is obtained from Eqs. (6)--(8) if the dependence on k• is taken into account. The velocities u~ and u~ will also depend on k• e.g.

uo(ki) = {[VF -F ~ 1 (;~l,,(k~) -- 21•177 (;~2,,(ki)- 22•177 (50)

A very important consequence of the neglect of hopping and backward scattering is that perturbations created on any chain can propagate along that chain only. The neighbouring chains can influence this propagation, but there is no response on other chains. The Green's function

G+q(p, e) -~ - - i S dt e"t (T{aip(t) a~p(0)}> (51)

will be diagonal in the chain index, G+ij = G+li �9 ~q, and similarly all the response functions

Rq(k, co) = -- i ~ dt ei•t (T{ Oi(k, t) Oy+(k, 0))> (52) with e.g.

1 Oi(k, t) = L1/~ ~ b~ t (t) a,p+k t (t) (53)

P

for the charge-density response function, will be diagonal in the chain index. This is easily seen since in the density responses a large ( ~ 2 kF) momen-

tum is given to the electrons on one chain and this momentum cannot be transferred to other chains if only small momentum transfer processes ate allowed. In the superconducting responses the Cooper pair added to a chain cannot propagate to other chains if hopping is not allowed.

Due to this property, the vertices appearing in the Dyson equation for the Green's function and in the two alternative representations of the response functions have identical chain index on the two electron legs. The Ward iden- tities used in the 1-d case will be valid here as well, because the problem is reduced to a one chain problem with effective interactions

1 D~,,(k,,, ~) = -~~~D~(k, , , k., co), # = 111, I J-, 21t, 2_L. (54)

.Lu kj.

Using these expressions for the effective intrachain interactions, the Dyson equation (11), Eq. (29) for the charge-density response and the analo- gous equations for the other response functions can be solved by taking over the results of the 1-d case to get

c+~j(~, t) - - ~ , j / ] [c+(~, t, k,)]~/~ (55) k•

5* ~lcta Physica ~4r Sr Hungaricae 48, 1980

184 NGUYEN MINH KHUE ~md J. S‰

and Rq(x, t) = t~q I I [R(x, t, kx)] *IN , (56)

k i

where N is the number of ehains, G+(x, t, kl) and R(x, t, k• have the same forro as the eorresponding expressions for a single chain with ls(ki) instead of ~~. E.g.

R(x, t, kx) = T i x -- vFt + i /A ( t ) x -4- vFt -- i /A(t) A, • x - - vFt + i~(t) x + v F t - - i~(t)

• [ ( x - uo(k~)t + i /A(t)) (x + uo(k~) t - - i /A( t )]"~ k,~ ( 5 7 )

• [ ( x - u~(k,) t + i /A(t)) (x + u~(k,) t -- i/A(t))]"e (k~>,

1 1 where for the charge-density response pr162177 and /~e(k• ~?e(ki)

with

~ ~ ( k i ) =

1 1

1 1 VF + Z ~ [21"(k')- 2u)k)] + 2~ [~2"(k~) -- ~2"(k')]

2~ " 1 11/2 ~~ + [al,,(k.) + a~.(k,)] -- ~ [x2,,(k~) + a . . (kd

v~ [Xl,,(k.) + ~ . ( k 0 ] + - - [~2,,(ki) + ~2~(kl)] 2= 2:r

I. (58)

and the exponents of the other response functions are obtained by the rules given in Eqs. (40)--(42).

In the limit of weak nearest neighbour interchain interaction this restdt agrees with that obtained by KLE~M and GUTFREUND [7].

5. Discussion

We have given an exact solution of a quasi-l-d model of interar electrons. Although the model contains explicitly a 3-d interchain coupling, due to the particular choice of this coupling (only forward scattering terms were considered), this model turns out to be equivalent to a set of effectively decoupled chains. The perturbations can propagate along the chains only, no response is obtained on other chains. Earlier approximate treatments of the backward scattering model [6]--[8] have indicated already that backward scattering terms or interchain hopping ate important to have a phase transi- tion in quasi-l-d system. Our exact calculation proves that without these terms no ordering is possible at any finite temperature, since the ordered phaae would have a non-decaying response function.

Acta Physitm Atmi~miaa Seitntiarwm Htmgarieat 48, 1980

EXACT SOLUTION OF A QUASLONE-DIMENSIONAL MODEL |85

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Acre PhyJ/as Aa~bm/ae ~/mmkwum Hun/jar/aw 48, 1980