influence of coulomb interactions on the phase diagram of quasi-one dimensional metals

Z. Phys. B - Condensed M a t t e r 69, 263-278 (1987) Condensed Matter ffir Physik B

�9 Springer-Verlag 1987

Influence of Coulomb Interactions on the Phase Diagram of Quasi-One Dimensional Metals

R. Brendle and R. Zeyher

Max-Planck-Institut ffir Festk6rperforschung, Stuttgart, Federal Republic of Germany

Received July 6, 1987

The influence of only partially screened Coulomb interactions on the phase diagram and the order parameter of quasi-one dimensional metals is investigated. Using a s tandard microscopic model, the free energy functional is derived by means of the heat kernel method. It is assumed that the Peierls gap and the mismatch are small compared to the band width and the reciprocal lattice vectors, respectively. Furthermore we neglect interchain to intrachain hopping elements. The resulting mean field phase diagram and the properties of the order parameter are discussed. We show in particular that the Coulomb forces are responsible for: a) a first-order transition between the incommensurate and the commensurate phase; b) only small deviations of the order parameter from a single plane wave over the entire incommensurate phase; and c) the approximate temperature independence of the wavelength of the modulation throughout the incommensurate phase. The possible relevance of these results for quasi-one dimensional systems exhibiting nonlinear conduction is pointed out.

1. Introduction

Many metals with a quasi-one or -two dimensional Fermi surface show modulated phases at lower temperatures which may be either incommensurate or commensurate with the underlying lattice [1-3]. There are, however, several properties which depend sensitively on the dimensionality. The ohmic conduc- tivity of modulated phases of quasi-two dimensional metals (examples are TaS%, NbSez) is quite large and screening effects are important. The order parameter consists of many higher harmonics in the incommensurate phase far away from the normal phase. Near the transition to the commensurate phase the phase of the order parameter describes a soliton lattice which vanishes continuously at the commensurate phase boundary because its lattice constant tends to infinity. All these details agree well with the predic- tions of the phenomenological Landau theory for modulated phases [zk6].

In contrast to two-dimensional metals quasi-one dimensional metals (examples are NbSe3, TaS 3, NbTe4, the blue bronzes) are often bad ohmic con- ductors at low temperatures and exhibit rather imper-

fect static screening properties. The Debye screening length, for instance, may become comparable to the length scale for typical variations of the order parameter [7]. Higher harmonics in the order parameter are rarely, if at all, observed. The periodicity length of the modulation does not show any substantial temperature dependence and there is no evidence for an incommensurate-commensurate transition via a soliton lattice. Most of these properties cannot be ac- counted for by a conventional Landau theory. A possible hint for these difficulties has recently been put forward in [7]. The authors of this reference show by a simple energy comparison that a charged soliton lattice is unstable against a first-order transition to the commensurate phase if the lattice constant exceeds a certain value.

In view of the failure of the conventional Landau theory in the case of quasi-one dimensional metals, we have carried out a microscopic derivation for the free energy taking into account explicitly the Cou- lomb interaction between the electrons as well as Umklapp processes. Usually it is argued that the long-range Coulomb interactions need not be considered explicitly because they simply change the values

264 R. Brendle and R. Zeyher: Influence of Coulomb Interactions

of the Landau parameters which are unknown any- way. Such a statement may be true if the Debye screening length is much shorter than the typical length scale of the modulation such as the lattice constant of the soliton lattice. This condition is, according to [7], not fulfilled in many quasi-one dimensional metals and one expects severe modifications. The study of these modifications due to imperfectly screened Coulomb forces is the subject of this paper. Questions to be answered are: What does the free energy functional look like if the Coulomb interaction between electrons is taken into account? What is the effect of the Coulomb forces on the phase diagram? Can solitons appear in the order parameter near the incommensurate-commensurate phase boundary or are higher harmonics always strongly suppressed in the incommensurate phase?

A serious problem in deriving free energy functionals for low-dimensional systems concerns the expansion of the free energy in powers of the order parameter. It is known that such an expansion does not exist in general in quasi-one dimensional problems [8]. We will circumvent this problem by using a gradient expansion instead of a power expansion of the free energy. Our expressions show that the power expansion cannot be considered even as an asymptotic series: For instances, the second-order coefficient determining the elastic energy is given by VF/2 T in the power expansion, where Vv is the Fermi velocity and T the temperature. The corresponding coefficient in the gradient expansion (B/fl, see (65)) has actually a maximum near T ~ A (A is the Peierls gap) and vanishes exponentially for T < A instead of diverging! These considerations are especially rele- vant in our case because Coulomb effects are most important near the incommensurate-commensurate phase boundary, i.e., at low temperatures.

In our approach, the microscopic model as well as the approximations used in evaluating the free energy have to be specified. We assume a solid consist- ing of chains arranged on a simple quadratic lattice perpendicular to the chains direction. The electrons propagate only along the chains within one band. However, the electrons on the same chain, as well as on different chains, interact via the Coulomb force. The basic assumption in evaluating the free energy is that the length scale over which the order parameter (which is proportional to the electronic density) varies is large compared to the lattice constant so that a gradient expansion makes sense. We also take into account only the leading contributions in the ratio of the Peierls gap to the width of the electronic band. This means that all terms are neglected which merely represent corrections to already existing terms. Finally, the phase diagram is calculated from the free energy functional within a mean field approximation.

Technically we use the path integral approach with anticommuting fields [9, 10]. This technique is rather convenient for our purpose because it allows an easy elimination of variables such as macroscopic density fluctuations or phonons which do not appear in the final free energy functional. Furthermore, the necessary early decision for one of the two ways to decouple the Coulomb interaction by means of a Hubbard-Stratanovich transformation [11] is not a liability in our case: The electronic density is the natu- ral order parameter (and not the exchange density) which selects one choice for the decoupling.

As mentioned above we do not want to represent the free energy as a power expansion in the order parameter. The alternative is a gradient expansion with prefactors which are in general nonanalytic in the order parameter and terms containing higher and higher derivatives of the order parameter. Clearly, such an expansion is more general than a power expansion and assumes only that the length scale for order parameter variations is much larger than the lattice constant. A very convenient way to carry out such an expansion is the heat kernel or zeta function technique well known in field theory [12]. The appli- cation of this method to our rather complicated solid state problem is quite novel and justifies a self-contained presentation.

2. Theory

a) Effective Action

The partition function Z for the interacting electron- phonon system can be written as a path integral [9, 10]

Z = ~ D O Dc* Dc e st~*'~'ol (:)

with the action

S = S El + S c + S Ph + S', (2)

s ~ = Y~ ( ico . - ~ + ~) c* (o~.) c~(a).), (3)

S c = - �89 ~ v (q) nq (co.) n_ q (-- oo.), (4) n,q

sPh : -- 1 2 (0")2 -~ c~ (k)) qS* (co,) qSq(co.), (5) n,k

S '= -- ~, g(q) nq(co.) gb _q(--(.On) , (6)

nq (('On) ~- Z eke1 ((J).l) Ckl +q(('Onl + (On)" (7) tll,kl

c*, c are anticommuting Grassmann numbers which replace in the path integral the electron opera- tors. q5 is a real field due to the phonons. S El is the

R. Brendle and R. Zeyher: Influence of Cou lomb Interact ions 265

free electron part describing one band of Bloch electrons with quasimomentum k, energy ek, chemical potential p, and inverse temperature fl = 1/T. We choose a system of units such that h = kB = vv = A = 1. VF is the Fermi velocity and A a symmetric momentum cutoff around each Fermi point. The energy unit is then vvA which is of the order of the width of the electronic band. S Ph describes one branch of optical phonons with eigenfrequencies e)(k). The Matsubara frequencies o9, are equal to (2n+ 1)~zT for electrons and 2nnT for phonons, respectively, where n is an integer. S c stands for the Coulomb interaction between the electrons with v given by

4he 2 v (q) = e v~ q2, (8)

where e is a background dielectric constant and v~ the volume of the primitive cell. Finally, S' represents the electron-phonon interaction with the coupling function g. The measure of the functional integral is

D 4 Dc* D c = ( ~ d qSk(co.))(y[ dc* (co.)d%(co.)). (9) n , k n , k

It is convenient to perform the integral over the phonon field first. Z becomes then

Z=~ Dc* De e sE1 +~:c (10)

where ~c is given by S c with v replaced by ~:

ff = v (q) g (q) g* (q) 0)2 + 0) 2 (q) = e - 1 (q, co.) ~ v(q) (11)

with the dielectric function

e(q, co . )=(~ v~q2 g(q)g*(q)~ t 4ne~ ~ o~(q)] (12)

The exact elimination of the phonons thus leads to a screening of the bare Coulomb potential v(q) and a reduction of the repulsion between the electrons. At long wavelengths, the repulsion always dominates. However, at short wavelengths and low frequencies, the second term in (12) may be more important, leading to an effective attraction between the electrons.

Next we apply a Hubbard-Stratanovich transformation to the effective two-particle interaction ft. This transformation makes use of the identity

I ~ %(0)~ ~(q. ~,,) n q(-Co.) e . . . . . f D 0

1 _ _ q l 2 2 Z r 5(q, C%) S (CO.) I~'_q(--m,,)+ Z nq(COn) s(q, O%) 0 q(--COn)

�9 e n , q n , q (13)

which can be verified by carrying out the integrations over the field ~ on the right-hand side of (13). s is defined by

i if e(q, co,)> 0 s(q, co,)= (14)

1 if e (q, o9,) < 0'

The introduction of s guarantees that all the Gaussian integrals on the right-hand side of (13) are well-defined.

Inserting (13) into (10) the actions become bilinear in the Grassmann numbers c*, c so that the integrations over c*, c can be carried out. Using also the identity d e t A = e x p ( T r logA) for an operator A (Tr denotes the trace) the result is

Z = f D O e x p ( � 8 9 log G -1) (15)

where the operator G- 1 is defined by

G- 1 (k 1 n l ; k z n2 ) = ~k, k2 6 . . . . (ico.i -- ekl + #)

-k S (k 1 - - k 2 , cons - - COn2) ~/kl k2 (COn z - - con2)" (16)

In the first term in the exponent of (15) the vectors s, ~ should be interpreted as the diagonal elements of matrices whose nondiagonal elements are zero.

Equations (15) and (16) have a simple physical meaning�9 It is evident from (16) that 0 describes a real, space- and time-dependent one-particle potential. It is caused by density fluctuations and mediated by the effective interaction ~ which contains the screening by phonons. Z is then obtained by integra- ting over all potential values weighted with the exponential of (15). The expression (15) is exact and also suitable for an approximate evaluation as will be shown in the following.

b) Continuum Approximation and Umklapp terms

As discussed in the introduction the electrons are assumed to propagate only along, but not perpendicular to, the chains in our model. This means that the one- particle energies ek depend only on the component of k parallel to the chains, which we denote simply by k in the following. The Fermi surface is a square prism, which is bounded by the two planes k = _ k F

corresponding to the two Fermi points for each chain. In the following we also assume that we deal with

a nearly commensurate situation. This means that the Fermi wave number kv is near G/(2n) where G is the primitive reciprocal lattice vector along the chain direction and n an integer determining the order of commensurability. It is well known that the important k-values in the electronic spectrum are located near the values k,~--_+(2m+ 1)G/2n. The allowed values

for m are m=0, 1,... ~ - 1 for even n. For odd n we

n - 1 have m = 0, 1 .. . . - - and either k~+_l or k~-_ 1 must

2 2 2

266 R. Brendle and R. Zeyher : Influence of C o u l o m b In te rac t ions

be omitted. Larger values of m need not be considered because of the periodicity of the electronic band and the corresponding wave functions. Quantities with wave numbers near k~ can be obtained by means of Taylor expansions around k,~ which amounts to performing a continuum approximation for each fixed value of (+ , m). Due to the Peierls instability, the leading wave numbers of A are located near the values +_ Gin. The presence of nonlinearities allows, however, also a coupling to k ~ 0 components of A, i.e., to macroscopic density fluctuations. These are the degrees of freedom affected most by the long-range nature of the Coulomb force. Therefore, we will keep also the k ~ 0 components of A but neglect all the higher wave numbers k~ +_mG/n with m > 1.

The matrix G -1 of (16) can now be arranged in blocks with block labels (+ , m) corresponding to the k-values k,-+,. The diagonal elements of G- 1 are given by A + +iAo with

A + = G (~ '(k + + q, co.) = i co . - ~k~ + q +/~, (17)

A o = A0(q, o9.) = ~q (co.). (18)

q is restricted to small values, i.e., I q[ < A where A ~ G. The only non-diagonal elements of G-1 are A and A* which lie just above and below the diagonal, respectively. A is given by

A = A (q, 09.)= ~_+ q(~,,). (19)

The two elements which would be outside of the range of the matrix according to the above prescription have to be shifted up or down by n rows similar as in problems with cyclic boundary conditions. In (17-

also used the facts that s(~+q, o3./=1 and 19) w e

s (q, co,) = i. \ n /

In calculating the term Tr log G- 1 in (15) use can be made of the fact that the mean value of A (the Peierls gap) is much smaller than the width of the electronic band. This means that the submatrix con- sisting of the blocks with m = 0 is only weakly coupled to the other blocks. In order to exploit this fact we divide the total matrix G-~ into a submatrix consist- ing of the two m = 0 blocks (this submatrix is labeled by the index 1 in the following) and the remaining m 4=0 blocks labeled by the index 2. We then solve the trivial relations

(G-1)11 Gll +(G-1)12 G21 = 1 (20)

(G-1)21 Gll +(G-1)22 G21 = 0 (21)

(1 is here the 2 x 2 unit matrix and 0 a ( n - 2 ) x 2 rectangular matrix containing only zeros) for GII:

(GlO-l=(G-~)11--(G-1)lz(G)22(G-1)zl. (22)

Defining a self-energy $22 by

( O - 1)22 = (O (~ 1)22 - X22 (23)

(the elements of ~22 are either A, A*, or 0) we obtain the expression

G 2 2 = ~ ( o ) l ~ ( o ) ~ 2 2 / 7 ( 0 ) v 2 2 q- u 2 2 "- '22

+ G(2~ ) X22 ,-(0) Z22 ~0) • (24) I-r22 ,022 T . . . .

Inserting (24) into (22) one obtains an expansion for (G10 -1. Using the fact that ]ql~G/n and that only small frequencies 09, will contribute, one can easily verify that the above expansion is indeed an expansion characterized by the small ratio of the Peierls gap to the band width. Furthermore, it is evident that the Tr log operation in (15) may be confined to the part (Gll)-1. Our method of calculating the Tr log term means that we take the trace over the exact eigenstates of the two m = 0 blocks, i.e., near the Fermi level.

The various contributions to (Gtl) 1 can be clas- sified by considering the momenta k~ appearing in these expressions. If the initial and final momenta are the same, momentum is conserved as in a system with infinitesimal translational invariance. These contributions are ~ (A*)l (A)~ where I is a positive integer. These terms represent small corrections to terms contained already in (G-1)11 and thus will be neglected. If the initial and final momenta, taken in the extended zone scheme, differ by a reciprocal lattice vector, by rG for example, the contributions are ~(A*)t(A) ~+~". Clearly, the l o e0 terms represent again corrections to the / = 0 contributions. Similarly the [ r [> l terms are higher-order commensurability terms which are roughly smaller by ( r - 1 ) n powers in the expansion parameter compared to the Irl = 1 terms. Though the Irl = 1 terms are also smaller by n powers in the expansion parameter compared to the direct terms they nevertheless are important near the incommensurate- commensurate phase boundary causing, for instance, the transition from a sinusoidal to a soliton lattice behavior of the order parameter.

Using (22) and (24) a short calculation shows that the [ r ]= l contributions to (Gtt) -1 have, for even n, the form

( o - A * , ~ ; Z*,~? A* ... ;o~_1A*X+_I A* ) -- A ,~ ~ A ,~ ; A ... "~ +~ -1 A ,~ ~_ _ 1A 0

(25)

R. Brendle and R. Zeyher: Influence of Coulomb Interactions 267

with

L+= ico,--~k~ + iz + iA o "

(26)

Since e k ~ - # is of the order of the electronic band width the co, - and the q-dependence as well as A 0 may be neglected in (26).

We represent the one-particle energies near the Fermi level by a linear law,

ek3 + q -- tt = -t- (q + C~) = __ C] (27)

choosing units such that the Fermi velocity is equal G

to one. 5 is the mismatch parameter }~n-/~. (G~a) -1 becomes then

l={ico,+Fl+iAo A--h(A*)"-i] (Gll)- \A ._h(A) , -1 ico_(l+iAo]

with

1 h= I~

( - eka +/~)' m = l . . . ~ - i

_+

(28)

(29)

(A) "- t denotes the product of the n - 1 matrices A with respect to co, and q. h is a real constant which depends on the detail of the electronic bands far away from the Fermi level. We consider h therefore as a parameter. Transforming from (co,, q) space to (xo, x~) space by means of a Fourier transformation we obtain with x = (x o, X 1 )

1 = / 8 0 -- i~1 + iA o (x) (G10- f*(x)

with

f ( x ) = A (x ) - h (A* (x))"- 1,

f(x) ] (30) 8o+i~1 + iAo(x)]

(31)

where (A*)"-a is now just the (n-1)- th power of A*. ~o, 81 are abbreviations for 8/8Xo and O/Sxi, respectively. ~1 denotes the operator 8~- i5 in agreement with the definition of q in (27).

c) Gradient Expansion for the Free Energy

The Tr log operation cannot applied to (G10 - i in a simple way if f depends on x. However, even in the mean field approximation f depends on xx in the incommensurate phase. One standard way is to expand log(G11)-~ in powers o f f yielding a free energy functional as a power series in f similar to phenomenological Landau theory. Later, we will see that such an expansion is not well behaved�9 A better way to proceed is to perform a gradient expansion for Tr log(G10 - i assuming that f varies sufficiently

slowly in x. Such an assumption has actually been made already earlier with respect to x l since the Xl-dependence in (30) is due to wave numbers Iql <A ~ G/n. A convenient and elegant tool to perform the gradient expansion is the (-function technique. In the following, we will adapt the standard procedure described for instance in [12] to our case.

The (-function of the operator G-~ is defined by

((x, x'; s)= <xl (G-1)-s Ix'> c~

_ 1 j 'dzz "- lW(x ,x ' ;z ) (32) r ( s ) o

with

W(x, x'; ~)= <xle-*G-' Ix' >. (33)

( actually is a 2 • 2 matrix and it is tacitly assumed in the following that G -~ means ( G~ ) - i as defined in (30). s is real, (G-i) ~ is defined by exp(s log G-l), and F(s) is the gamma function. Using (30), one can easily verify that Wsatisfies the matrix equation

+8o- - i8 i +iA o

\ f*(x)

�9 W(x,x';z)=0

d f (x) o l

~ + O o + i J l +iA

(34)

with the initial condition W(x x'; z = 0 ) = b ( x - x ' ) . 1 is the unit 2 x 2 matrix. Differentiating (32) with respect to s and putting s equal to zero we obtain

d (xl log G- 'Ix'> = - d s ((x, x'; s)J~= o. (35)

In order to obtain the term Tr log G -~ of (15), one integrates (35) with respect to x ' = x over the in- tervals 0<Xo<f l ; O<xi<L, i= 1, 2, 3. The Xo integration, however, diverges, as can be seen most easily in frequency space. On the other hand, one is free to subtract a constant in the exponent of the right- hand side of (15) because this changes only the nor- malization of Z. Doing this, we replace the term Tr logG - : in (15) by L 2 F~nt,

Tr log G- 1 ~ - - L 2/iint, (36)

with

Fi.t =Yr S d2x lira lim d (((x, x'; s)-(o(X, x'; s)). (37) x '~x s ~ 0 d s

The factor L z in (34) is due to the x2, x3 integrations perpendicular to the chains. We also used a cube with length L and applied periodic boundary conditions. Tr denotes now a trace over a 2 • 2 matrix, (o is the ~-function with G-1 replaced by 1 8o.


It is not possible to solve (34) in a straightforward way if the fields A o and f depend on x. For slowly varying fields, we make the Ansatz

W(x, x'; z)= U(x, x', Ao(x),f(x); z)

�9 c~(x, x', Ao(x),f(x))C . (38) v

U is the function which satisfies (34) for constant fields A o and A with the same initial conditions as W. Insert- ing (38) into (34), the resulting equation must hold for each power in z separately. The resulting equations can be solved for the unknown functions c,. The details of this calculation are given in the Appen- dix. The three lowest coefficients for equal arguments x = x' are, omitting A o and f as arguments for simplic- ity,

co (x, x) = 1, q ( x , x ) = O ,

, 1 [iD_ A o c2(x' x ) = 2 ~ D+f

(39)

(40)

D_f*], iD+ Ao] (41)

_l . { iD z _ Ao--f*(D + f) + f (D_ f*)-- 2if*(fflf ) C3 (X~ X) =

D2 f--4f(~l Ao)

with the abbreviations

with

"C)--Z i dkl ~(k-l,f; z) e ik'~', (49) ?(xl ,f ; - 2 n --1

~(<,f ; z) = cosh (* ~ ) - i

s i n h ( ' c ~ ) ( 9 f ~ ) . (50,

The momentum cutoff A has been put equal to one in agreement with the conventions made at the begin- ning of Sect�9 2. Equation (48) shows that the dependence of U on the variables Xo and x 1 factorizes, which will simplify greatly the calculation of F~m.

d) Calculation of the Local Part of F~.,

The local part of ~int, /7(Or ), is obtained by inserting the expression for U instead of Winto (32) and by evaluating (37) with the resultant C (~ The difference of C-functions, appearing in (37), becomes:

D2- f* +4f*(ffl Ao) ) iD 2 Ao + f* (D + f)-- f(D_ f*) + 2if(fflf*)

(42)

D + = ~o - iffl. (43)

Furthermore, we need the expression for U. For constant fields A o and f G ~ is diagonal in frequency- momentum space (co,, k) = k = (ko, kl):

<k'G-l 'k '>=~Sk'k '( ik~ iko+e+0 ) (44)

with the energies

8=~ = i A o ~ ~]//~lz+lfl 2. (45)

It follows that

(k le - ~ - ' t k'> = 6k. k, ~ e-~(~ + ~k~ P_ v , (46) -v

where P~ are the projectors on the eigenstate of G-1, i.e.,

1 1 P~_ = ~ - 1 ~ ~ ( ~ f ; 1 ) " (47,

Transforming from k- to x-space we obtain

U(x-- x', Ao,f; z)= e-i~a~ 1 -x'l,f; z)

L (-1)"6(Xo-X'o-nfl - z ) (48)

1 1 ((~ x'; s)-Co(X, x'; s)= 2rc/'(s~ S 1

�9 ; d z C - ~ (e-i~o(~)~7(k-l,f (x); z ) - 1) 0

�9 . ~ o ~ ( - 1 )"6 (Xo-X'o -n~-~ ) .

dk I el< (x~-x;)

(51)

Carrying out the z-integration in (51) and using the identity

a ~ - 1 - 1 ~ ; d y y - ~ e - % (52) F(1--s) o

which is valid for s < 1, a>0 , we obtain for (51) after summing over n

((~ 0; s)-Co(X, 0; s) = -~ sin(rcs) fls- 1 i dkl elk,xl - - rc - 1

�9 ~ dyy ~ e-X~ o l+e~-(Y+~+) P+

e -xo~- 1 "1 -t- 1 + e -v (y +~-) P- - 1 +e~y~. (53)

The upper and lower signs hold if 0 < z < fl and 0 < - z < fl, respectively. The derivative with respect to


S and, subsequently, the limit s ~ 0 can be applied immediately to (53). The limit x~ ~ 0 is also straightforward. A short calculation shows that the limits xoT0 and Xo+0 agree with each other so that the limit x o ---, 0 is also well-defined. Taking also the spin of the electrons into account one obtains

2 2 1 dk 1 r(o)_ ~ i d x I 2~ l i n t - - - -

- - 1

, , + ) + l o g t 2 ~--)} (54)

where e+ depend on kx as well as on Ao(x ) and f (x) according to the definition Eq. (45). The expression (54) consists of all contributions to F~nt which do not contain derivatives of the order parameters with respect to time or space.

If the induced order parameter A o is small, we can expand F}~ in powers of A o:

F(O)_ i dkl (fl -2 ) i n t - - - ~ I d 2x 2 ~ l o g c o s h ~ / k l +l/(x)l 2

- 1

- -2 i~d2xA0(x )

2 ~ d2x A~(x) i d2~cosh-2 - 1

The second term in (55) can be omitted because the total charge is conserved and equal to zero, so that the ka --0 component of the corresponding potential Ao(x) must also be zero. The first term in (55) contains all local contributions caused by density fluctuations with components near G/n. The Peierls instability is obtained by expanding this term in powers of f(x). The prefactor of the quadratic term exceeds a certain value below some temperature such that the total prefactor of the quadratic term in the exponent of the partition function becomes positive, indicating an instability. The last term in (55) describes the screening of the potential Ao by the electrons within the RPA as a functional of the slowly varying order parameter f.

e) Calculation of the Nonlocal Part of liint

It is convenient to introduce the function

oo

1 ~ dz.cS_le_~U ~, (Xo, S) = F~(s) o

(56)

Inserting (38) into (32) one then obtains for the m-th order contribution to (:

lim ((m)(x, x'; s) = (s + rn-- 1)... (s + 1) x ' --* x

i dk �9 s ~ - - ( l i r a ~iAo+E(X o ' s+m))P+c,.(x,x) + -1 2~ x'o xo - -Xo, _ (57)

with the abbreviation E=]~2i2+l f l 2. From (56) one t may verify that the limit Xo Xo in the function

is well-defined for s + m = 1, 2, 3... as the second argu- ment. Therefore, we have

d~(m)(x, x; s ) = o

dkl- ~ 1.2.(m-- 1)Z 2~ ~o• cm(x,x). (58) J

_+ - 1

Taking derivatives with respect to # in (56) one obtains expressions for ('~(0, m) for m = 1, 2, 3 etc. For instance,

and

fi2 tanh ( ~ ) 60)

Carrying out the trace over the 2 x 2 matrices (58) reads for m = 2:

Trds~(2)(x,x;s)s=o=~ i d k l - 0 -a 4 u ~i~o_+E(,2)

( ._ U*D+f T D _ f * ) .~.16oAo+ 2 ~ + ~ - ~. (61)

Using the expansion

~Ao+E(0, 2)= ~• 2)--2i('+~(0, 3) 'Ao+ ... (62)

as well as the expansions (59) and (60), we obtain

d x;s) s = 0 - i Tr ~ s ~ (2 ) (x , - - - ~ A [ f ] c ~ o A o

+ 2 B[ fJ Ao(f*(D + f )+ f ( d - f*)) (63)

with

i A [ f ] = f l ~ c o s h 2 - 1

(64)


tanh( ) dk~ 1

B [ f ] = ~ '

(65)

The last term in (63) describes a coupling between the k ~ 0 and k ~ +2kv density fluctuations, which is linear in Ao. Keeping higher orders in the expansion Eq. (62), one obtains higher-order couplings in A o, which we will however neglect in the following.

Carrying out the trace in the m = 3 term in (58) and putting A o = 0 to be consistent with the treatment of the m = 2 term, we obtain

Tr dd s ~(3)(x, x; s)s=~ = 61- B[f] { f * (D2+ f ) + f ( D 2_ f*)}. (66)

Going back to the unbarred quantities (i.e., to the situation where the origin of momenta is -4-_ Gin and not _+ 2kF) we obtain for the total nonlocal part of

/iint

F(2) " r (3 ) - f az ," { - 2 A [ f ] ~o Ao + 2 B [ f ] A o int ~ ~int - - J ~

�9 I f * (0o + i 0 ~ + 6 ) f + f ( (?o - - i ~ , - - b ) f * ]

+ 1 B [ f ] [f*(~o + i O~ + b)z f+f (~o- - i01 -- 8)2f*]}.

(67)

The functionals A and B a re given by (64) and (65) where k-~ in the square roots may be replaced by kl because of [ ~ I ~ A.

f ) Elimination of Macroscopic Density Fluctuations

Keeping only quantities which depend on A0, the functional integral over A o becomes

S D A o e - ~ Tr(Z~ l + A)~t~ + zia~ = e r ' l f l , (68)

where a is given by

a = �89 B I f ] ( f* (~0 + i01 + 6 ) f

+f(O0 + i ~ 1 - 6)f*) + 0o A I f ] . (69)

Tr in (68) denotes now the trace over 3 spatial and one temporal coordinates. Fc, defined by (68), is given by

F c I f ] = �89 Tr (a (~- ~ + A) -a a). (70)

According to (69), a is of first-order in the derivatives with respect to x. Since we keep only terms up to the second order in the gradient expansion, it is sufficient to evaluate the operator (~- 1 _ A)- 1 in the local approximation. We obtain then, using three spatial coordinates

e 2 F c I f ] = - - ~ d 4 x d 4 x' a (x) 5 (Xo - x;)

2 e V c

_ /X+X'\

�9 e 4 ~ a ( x ' ) ( 71 ) I x - x ' l 2 / x + x ' \ 3

with the inverse Debye-Hueckel screening length

/ 4 ~ e 2 qDH(X) = V ~ A I f ] . (72)

The second term in the parantheses on the right-hand side of (71) is due to charge neutrality. The vectors x, x' in (71) consist of one time component x0 and three spatial components xl, x2, x3 where x2, x3 are the coordinates perpendicular to the direction of the chains.

Equation (71) represents a general expression for the Coulomb contribution to the free energy functional. a(x) describes induced macroscopic charge density fluctuations due to a general distortion of the complex order parameter A (x). a is therefore quadratic in f and not linear as was assumed in the phenomenological approach of Ref. [71. Imposing a general order parameter field f (x), the expectation value of the macroscopic charge density is zero and only fluctuations in the macroscopic charge density can contribute to the free energy�9 The charge densities interact via the screened, instantaneous Coulomb interaction among themselves and via the neutralizing background.

The above expressions contain two special cases: Just below the normal-incommensurate phase boundary Ifl is small and the Debye-Hueckel screening

1 length - - is also small compared to the length of

qDH typical spatial variations of the order parameter. The expression inside the parantheses on the right-hand side of (71) can then approximated by

+3) 73, In this limit the first term in { } becomes local but the second one keeps its nonlocal character. This means that the usual free energy functional cannot account for F c even if the screening is good. However,


the prefactor of F c is smaller by the factor q2H'62 compared to other terms in the free energy functional so that F c becomes less important. The second special case occurs for temperatures much lower than the Peierls gap. Then qDn ~ 1/6, { } can be approximated

e 2 by the bare Coulomb interaction ~ Z ~ 7 T and charge

neutrality is satisfied if a(x) is replaced by a(x) - - L - 3 f d x 1 d x 2 d x 3 a ( x ) . Strictly speaking, this case does not apply for T ~ 0: We have then qDH ~ 0, however also B ~ 0 so that Fc vanishes altogether exponentially.

We expand the function 2g- ~ in the Gauss• weight (first term in the exponent in (15)) as

2 2 m

+2-i=~2'3 Qq2 g( G + q ' 0 q=O

M + l y i ( q 2 + q ~ ) + . . . . TC

(74)

Terms proportional to powers of con or qa are omitted in (74) because they are in general smaller compared to the analogous terms in r ~z) and r(3) Moreover, ~int l i n t �9 we have considered in (74) the most simple case where the effective interaction ~7 does not produce a modulation perpendicular to the chains. Then y• > 0 and tin- ear terms in qz,q3 are absent. Otherwise, one has to expand around a finite transverse modulation vector. The partition function can be written in the form

Z =~ DA exp(--flF(A)), (75)

F(A) = FL(A,f) + FNL(f) + Fc (f), (76)

4 { M 12 2 i dk~ FL(A,f)= T ~ d x 2~n ]A - -~ 27z

- 1

T FNL(f) =~- ~ d 4x B [ f ]

�9 (f*(ao+i~?a+6)2f+f(Oo--iO1--6)2f *) T 4 * 2 - ~ - 7 • x f (02 -[- ~2)f~ (78)

Fc(f) = TFc(f), (79)

where Fc is given by (71) and f is a functional of A according to (31).

3. Mean Field Phase Diagram and Behavior of the Order Parameter

a) General Remarks

Equations (75-79) represent the leading contribution to the free energy functional with-respect to three small parameters: the ratio of the Peierls gap and the width of the electronic band, the ratio of the mismatch 3 to 2kv, and the ratio of interband to intra- band hopping terms. Expressions (75-79) differ from standard phenomenological free energy functionals for Peierls systems mainly with respect to two things: 1) They are not power series expansions in the order parameter A. If one expands them in a power series in A, one encounters infrared divergences in the expansion coefficients of higher orders. For instance, the integrals for the expansion coefficients of A and B diverge near k~ ,-~ 0 from the third and second orders in ]A J: on, respectively�9 This holds for all temperatures. As a result, the free energy is nonanalytic in A and the usual Landau expansion does not exist�9 One consequence of this is the appearance of two different commensurate phases as will be discussed in subsection b); 2) The Coulomb interaction between the electrons contributes the term Fc to the free energy. Equations (71) and (69) show that Fc is zero if the order parameter is either a constant or a single plane wave assuming a time-independent order parameter. Thus F c does not affect the normal, the commensurate and the sinusoidal incommensurate phases. Fc should, however, be important for the incommensurate phase near the boundary to the commensurate phase where the order parameter is no lon- ger sinusoidal.

In the following, we treat our free energy functional F classically and by mean field theory. This means that the disregard any time dependence of the order parameter and determine it by minimizing the functional under the constraint that the extremum represents an absolute minimum of the free energy. We will also use the simplified form (73) for the Coulomb interaction. We found that this approximation is not serious down to rather low temperatures.

The functional F contains altogether 6 dimension- less parameters: M, 71, T, h, e2/(~vc), and 6. The parameter 7• does not enter in mean field theory, eZ/(evc) drops out of Fc within the approximation of (73). In is convenient to replace M by T c defined by

0FL 4= = 0 (80) 0A* 0

assuming a homogeneous order parameter A and h = 0. Using the approximation

272 R. Brendle and R. Zeyher : Inf luence of C o u l o m b In te rac t ions

dkl 0 ]/k2+lAI 2

~-21og( f l+/~2+f l2 lAl2)- log( l+f i2 lAI 2) (81)

(80) yields

M ~ - = log(2/Tc). (82)

A convenient choice for independent variables is then T/Tc, T c, h, and ~. In the following we will consider Tc and 6 mainly as fixed parameters for which sensible values lie in the range 0.005-0.05 and 0.001-0.01, respectively.

b) Commensurate Phases

It is convenient to use

A=fl(A - -hA*" - ' )= f l f (83)

as an independent variable instead of A. The extremal condition for F as well as the condition of positive definiteness of the second derivative of F is not affected by this change of variables. Because h r 1 (its value is proportional to some powers of the small ratio Peierls gap over band width), (83) may be solved for A by perturbation theory in h yielding

A = T ~ + h T n - ~ * "-'

+(n-- 1) hZ(TA*)"-2(TA) "-1 + . . . . (84)

Using the abbreviations

6=~/~, (85)

= B/f l 2, ( 8 6 )

as well as the abbreviations

r = T/(T c tcc), (88)

the total free energy F becomes, for a constant A:

~ L T 2 [ Ihl 2 F(A)=m~-[ \ S d p l o g [ ( l + p ) r 2]

0

M ~ n + ~ - f l [ - - h ( A T ) +c.c.

h 2 (2 n -- 1) I TA 12 (, - 1)]/. (89) + /

In the commensurate phase, z~ is a constant

z l ----= C e i~ (90)

characterized by an amplitude C and a phase ~. For positive h, the phase q~ = rc/n makes F minimal. The extremal condition for C reads

log [(1 + C 2) r z]

M n 2 + ~ - h(rC) - [ ( 2 n - 1)(n- 1) h(rC) n - z - n] =0. (91)

In the following we consider the case n=4. Equa- tion (91) has then in general two stable solutions separated by a first-order line which terminates at a critical point. Elementary considerations show that this critical point exists if Tc>2 exp(--21/4) and its coordinates (denoted by CP as an subindex) are

1 /1 2M'~ h c e - ( T c t c c ) 2 2 ] / ~ M e x p ~ - 21 ), (92)

{Tce~ 2 2 t~c ( 1 _ ] / ~ ) ( 2 M _ I] \ Tc ] = exp \ 21 2]' (93)

where M is to be replaced by Tc via (82). The coex- istence region of the two commensurate phases is rather small. The phase boundary between the two commensurate phases is then apprximately deter- mined by the condition that the arithmetic mean of the two points of inflection forms an extremum. This yields

[ T'~2 2 [1 e x p k ~ - k ~ - ~ - h T4)). (94)

The curve T(h) defined by (94) extends from the critical point towards higher temperatures and separates a small amplitude commensurate phase for small values of h from a large amplitude commensurate phase for large values of h. The appearance of two stable commensurate phases is rather unusual and an outcome of the nonpolynomial form of F. Illustra- tions of (92-94) can be found in Sect. d).

c) Inadequacy of the Phase-Only Approximation Near the Incommensurate-Commensurate Phase Boundary

The normal-incommensurate phase boundary is given by T-- T c. The incommensurate-commensurate phase boundary can usually be obtained in the phase-only approximation: The transition occurs if, on the incommensurate side, the lattice constant of the soliton lattice tends to infinity, or, on the commensurate side, one soliton appears spontaneously. In this section,


we discuss the phase-only approximat ion for our functional F and solve the corresponding extremal condition. The result is that the incommensurate- commensurate boundary always becomes a first- order line. This conclusion is the same as in [7], al- though we treat a different case: [7] uses an un- screened Coulomb interaction which is realistic at very low temperatures. In contrast to that, we assume that screening always dominates and that the spatial variations due to the 1/r Coulomb law which appear only in higher than second-order terms in the gradient expansion of F can be neglected.

Using (90) with a spatially varying phase ~ (x) we obtain for the 4Ldependent part of the free energy functional

L

F' fib) = F o ~ d x {(8x ~b-- 6) 2 + g) 2 Wcosn 0

+ Q(ax ~)2} _ OLFo 82 p2 (95)


/~C z F o - 3 ' (96)

3 /~C 2 Q - 2 A ' (97)

3 M T " h C " - 2 W 2re B 6 2 ' (98)

( L ) - �9 (0 ) P - 6 L (99)

A and B are given by (65) and L is the length of the solid. Q originates from the screened Coulomb interaction and its value is a measure of the impor- tance of the Coulomb contribution. We have calculated its temperature dependence using (65) and the minimization procedure described in the next section for the determination of the ampli tude C. The result, given in Fig. 1, shows that Q is only negligible for temperatures just below T c.

For Q = 0, (95) represents the standard expression for the free energy of incommensurate-commensurate systems in the phase-only approximat ion [13]. It pre- dicts the formation of a soliton lattice at sufficiently low temperatures where the solitons repel each other exponentially. The transition to the commensurate phase occurs continuously with the lattice constant of the soliton lattice tending to infinity. The presence of the Coulomb interaction modifies this picture in a twofold way: First, a finite Q leads to an enhance- ment of the elastic energy of the charge-density wave corresponding to an increase of the repulsion between solitons at short distances. Secondly, there is an addi- tional term ~ - Q p 2 in the free energy. It describes

6

5

4

(2 3

2

1

0

0 .2 ,4 .6 .B 1.0

T / T c

Fig. 1. Coulomb coupling constant Q as function of T/Tc for h=0

2O

A OJ 1 5

10

(3 5

~3

q~ 0 L.

..J 5 d )

-10

-15 Q

- 2 0 I I I I [ I I I I

0 .2 .4 .6 ,8 0

L a g L i c e c o n s L a n L x ~ [ p ~ / 2 r r )

F i g . 2 . I n t e r a c t i o n e n e r g y o f t h e s o l i t o n l a t t i c e ( i n u n i t s o f F o 6 2,

see Eq. (96)) along the transition line as function of the lattice constant. The lines and the Q-values have the same vertical order

a long-ranged and attractive interaction between solitons which always overwhelms the exponential repulsion at sufficiently large distances. Figure 2 shows the calculated effective soliton-soliton interaction energy for different values of Q as a function of the distance. The dashed line corresponds to the case of vanishing Coulomb forces. The diagram illustrates that even very small values of Q lead to an attraction at large distances and thus to a first-order transition where the soliton density jumps from zero in the commensurate phase to a finite value in the incommensurate phase.

The minimization of the functional in (95) can be reduced to the solution of a transcendental equation. Varying F' with respect to 4~ leads to the usual sine- Gordon equation which can be solved in terms of elliptic functions and one constant of mot ion which can be expressed by p. It is convenient to introduce


the variable

~z _ / w Y = P K(y2) V2(1 + Q) (100)

where K is the complete elliptic integral of the first kind. The variation of F' with respect to y yields the equation of state

7 : 2 W - - f f y2(1 +Q)[E(y2)+Q(E(y 2) r~z \ ] -2 4 K -y2))J .(101)

E denotes the complete elliptic integral of the second kind. The difference of the free energies of the commensurate and incommensurate phase becomes

2YZ--1 Fic-- F'c= b2 Fo ( ~ f T - W+ Q pZ). (102)

The phase boundary between the two phases obeys the condition that F;c = F~, i.e.,

7r2 Q K-2(y2). (103) y2=l 4 I + Q

The above equations reduce to well-known expressions of the sine-Gordon case if Q is set equal to zero.

The solutions of (101) and (103) are illustrated in Fig. 3. The left part of this figure shows the phase boundary using Q and W as independent variables. With increasing Q the commensurate phase grows at the expense of the incommensurate phase. The right part of Fig. 3 shows the soliton density p along the transition line. p is equal to zero at Q = 0, approaching, however, very rapidly the value one for increasing Q. The transition line is therefore strongly first-order except at Q=0. Figure 4 shows the spatial dependence of 4~ at the phase boundary. The dashed line corresponds to Q = 0 showing the squaring up of the modulation in the extreme soliton limit. In contrast

1 . 5

w

1 .O

. 5

C

f c

1 . 5

p

1.O

. 5

/

0 i i i 0 . ) ) I I

0 1 2 3 4 5 l 2 3 4

s 0

Fig. 3. Left part: Transition line between the commensurate (C) and incommensurate (IC) phase in the W-Q plane (see Eqs. (97) and (98)). Right part: Soliton density p along the transition line as a function of Q

1 . 0

c 3 O = D. [ d a s h e d l

.a 050 / / / / ~O~ 0 . 7 5 / / / / "

. 7 / # . "

. 6

. 5

. 3

. 2

. 1

0 . ~ i f i i i r i

o .2 .4 .6 .a l.a x *(pO/2nl

Fig. 4. Phase nfb/2r~ as a function of x(pt/27r) along the transition line. The Q-values and the curves have the same vertical order

to that the phase @ differs only little from a straight line if Q is finite. This can be understood easily: The charge density induced by the order parameter is proportional to 9~b/Ox. Every variation in the slope of ~b causes inhomogeneous charge densities which interact via the Coulomb interaction and thus drive up the free energy. This occurs in spite of the fact that thermally activated electrons above the Peierls gap screen the Coulomb interaction.

The results of this section rely on the phase-only approximation. Its validity however is problematic in our case: The first-order transition line lies inside the Q = 0 incommensurate region so that this line may be reached long before the phase-only approximation becomes valid. The small deviation of the phase from a straight line even at the transition line suggests a different approximation, namely to expand the whole order parameter in plane waves and to keep a few hamonics. This will be done in the next section.

d) Phase Diagram and Properties of the Order Parameter

The most general Ansatz for the order parameter in the incommensurate phase is

zl(x) = ziei~176 + ~(x)), (104)

~b(x)= Z f ~b-k ei~"k~ (105) ~=_+Ik=l

Motivated by the considerations and results of section c) we insert (104) and (105) into F and keep only terms up to second order in 4. Without loss of gener- ality, A can be chosen to be positive and the coefficients @~k to be real. The free energy becomes then


F=F(O)(3, p)+ ~ (l) 3 o~,k

+ �89 - F'~k,&i(A , p) eb~g ~aj + ... (106) ~,k /LJ

with the coefficients

) L f l ~ F(~ p = ~ ( ~ ~ dp log((1 + p ) t z)

0

+ B(3) ~-2" p)2 62}, (107)

( l t - - z n + l F~k(A , p ) = 2 n M h A" L6~1 6k 1 , (108)

tudes of C I and C i i vanishes at the critical point CP which exists for a finite temperature if T c exceeds the value 2 exp( -5 .25)~0 .00105. For a finite mismatch 6, the original region of C~ is divided up into an incommensurate phase at small values of u and the commensurate phase CI. Both are separated by a first- order line which ends in a new triple point TP2 below T c. The position of the new IC - C~ boundary is close to the continuous I C - C line for Q--0. The presence of Coulomb forces makes it, however, always first- order. For increasing 6, T P a shifts to lower temperatures, the region of the IC-phase grows and that of the Ci-phase shrinks. We obtained very similar phase

FifSAZ ' B6

B62 Q[(1 - p ) 2 1 1l (109)

In (109), we used the fact that h is a small quantity which can be set to zero in evaluating F (2). Variation of F with respect to p yields

p = l - - � 8 9 ~ {6~p(2+Q)(nk-cQ)

-5~, _~ nkQ} @~k ~ek. (110)

Keeping only terms up to O(h2), it is sufficient to set p equal to 1 in F for the variation with respect to ~b~k. We obtain then

( 2 ) - i - - q)~k=~ F~k.ak(A, 1) F~)(3, 1). (111) fl

Because F (2) is diagonal in k the inverse of F (2) can be formed trivially. F rom (108), it follows that only the two harmonic with k = 1 are nonzero.

Figure 5 shows the phase diagram calculated in the above approximation. The independent variables are T/Tc and u Tc with h = u "-2 and we use n = 4, 6 ---0.003, and Tc--1/90. N, IC, and Cr, Cn denote the normal, incommensurate, and the two commensurate phases, respectively. The solid and dashed lines represent first- and second-order transition lines, respectively. The dotted line indicates the incommensurate- commensurate transition line in the absence of Cou- lomb interactions. Figure 6 shows the same phase diagram for 3--0.01 and the same values for the other parameters. Figs. 5 and 6, as well as our discussion in Sect. 3, lead to the following picture. For 6 = 0 there are a normal phase N and two commensurate phases C I and C~I. The continuous transition line N - C I at T= Tc ends in a triple point TP 1 = TP2. Above and below TP1, a first-order line separates N and Cn, and C I and CH, respectively. The finite jump in the ampli-

diagrams for the case n = 3. For n = 2 3 becomes real so that a as well as Fc vanish. This means that there are no long-range forces in the case n = 2. This is in agreement with [-141 and [15] where it is shown that exchange interactions may also lead to an attraction between solitons.

Figure 7 shows the effective mismatch p as function of TIT c for n =4 , u T c =0.03. The solid and dashed lines have been calculated with the approximat ion of Sect. 3d) with and without Coulomb forces, respectively. The dotted line corresponds to the phase-only approximat ion without Coulomb forces. For 0.7 ~-T/Tc< 1 the three curves are close: The Peierls gap is small and the screening good so that Q can be approximated by zero. The two cases, however, differ

t . 6

t . 4

N 1 . 2

1 . 0 . . . . . . . . . . . . . . . . . . . . . . . .

.G

.4

.2

O. i i if: i I ~ I P I l i t i i I

0 . 0 2 . 0 4 . 0 6 , 0 8 . t O . t 2 . 1 4 . t 6

u ~ T c

Fig. 5. Phase diagram in the T--u plane for 5=0.003, Tc = 1/90, and n=4; u is equal to ]/h


1 . 6

1 . 4

2

I C y T~2 . S

42 /It 0 . I I I I I f ~ I I I F I I I I I I

. 0 5 . 1 0 , 1 5

u ~ T c

Fig. 6. Phase diagram in the T - u plane for 6=0.01, Tc= 1/90, and

n=4 ; u is equal to ~/h

C:k

t3

r~

E t~

~3

ka 03

1 . 0 0

. 9 9

. 9 8

. 9 7 , ' ' ' / / �9

�9 9 6 / / , " / - " ~

�9 9 5 / / /

/ �9 9 4 / / , ~

�9 9 3 i I / /

/ / ,~ �9 9 2

.91 , '~176

.913 q " I I q

. 3 . 4 5 .G . 7 . 8 . 9 t . O

T I T c

Fig. 7. Effective mismatch p as a function of T/Tc for uTc=0.03, Tc=1/90, and n=4. - - Q@0, - - Q=0, calculated to O(h2); ..... Q = 0 in phase-only approximation

.I0

,08

.06

.04 4 m .02

N ~ ' - . 0 2

- . 0 4

- . 0 6

- . 0 / 3

. 1 0 i i I i i r I

.3 .4 .5 .6 .7 .8 .9

T/% Fig. 8. Amplitudes ~bla and 421 for uTc=O.03, Tc = 1/90, and n=4. - - Q@0, Q=0, calculated to O(h z)

fundamentally towards lower temperatures: With Coulomb forces, p shows a minimum around T/Tc

0.6 and then increases towards lower temperatures approaching essentially one again at very low temperatures. In contrast to that, the dotted line decreases monotonically and rapidly with decreasing TIT c approaching zero at very low temperatures for our parameters. Looking at absolute values, we find that p is practically equal to one over the whole incommensurate phase if Coulomb forces are taken into account. The dashed line in Fig. 7 demonstrates that the different behavior of the solid and dotted lines is not due to the approximation of keeping only the two lowest harmonics in calculating the solid curve.

Figure 8 shows the amplitudes 4~1t, 4~21 corresponding to the two lowest harmonics as a function of TIT c for u T c = 0.03, T c - 1/90, and n = 4. The solid and dashed lines correspond to Q 4= 0 and Q = 0, respectively, and both have been calculated in the approximation of Sect. 3d). The amplitudes are similar in the two cases for 0.7---T/Tc < 1, but behave quite differently towards lower temperatures: For Q=~0, ~ 1 decrease and assumes rather small values at very low tempertures.

Finally Fig. 9 shows the amplitude C and the phase q~ of the order parameter in the periodicity interval for uTc=0.03, Tc = 1/90, n = 4 and three different temperatures: a) T/Tc=0.8; b) T/Tc=0.4; c) T/Tc=0.2. The three diagrams illustrate the single- plane wave character of the order parameter over the whole incommensurate phase. Deviations from a single plane wave are always small and, in particular, do not increase towards lower temperatures. Without Coulomb forces one finds a quite different behavior: For medium temperatures both C and �9 show strong spatial oscillations and at low the temperatures C becomes constant but 4~ develops a staircase-like shape.

In conclusion, we have investigated the influence of only partially screened Coulomb interactions on the phase diagram and the order parameter of quasi- one dimensional metals. Starting from a standard microscopic model we have derived the free energy functional using the heat kernel method. This technique is very convenient because the usual Landau expansion of the free energy in powers of the order parameter does not work in our case. Our main results are: a) The incommensurate-commensurate transition is always first-order. Our theory may therefore explain the discontinuous lock-in transition observed in NbTe 2,3; b) The order parameter consists essentially of one plane wave over the entire incommensurate phase and the admixture of higher harmonics is very small. This may explain the experimental fact that higher harmonics are very difficult to observe in qua-

C

C

1 . 2

1 . 0

./3

. 6

. 4

. 2

O.

c j

~ T / T c 0 . 8 0 i i I r I [ I i

. 2 . 4 . S . 8

X

i

1 . 0

( p d / 2 r r )

r r / 2

I / 4

c

1 . 2

1 . 0

.B

. 6

. 4

. 2

I.


. 2 . 4 . 6 . 8 1 . 0

b x ~ ( p O / 2 F )

r r / 2

r r / 4

1 . 2

c 1 . 0 r f / 2

g . 8

. 6

7 r / 4

. 4

. 2

T/T c O. 28 B . i I I i I i I i i

. 2 . 4 , g . 8 1 . O

C • ~ { p d / 2 r r }

Fig. 9a-c. Amplitude C and phase q~ of the order parameter as a function of x(p6/27z), for uTc.=0.03, Tc=l/90, and n-4 and a) r/rc=0.8; b) T/Tc-0.4; e) T/Tc=0.2

si-one dimensional metals because of their small amplitudes (~0.01 of the amplitude of the fundamental E16]). This is probably also related to the small pinning energies found in the nonlinear conductance of many quasi-one dimensional metals E2] since pinning

is only possible in the presence of higher harmonics. Our findings rule out intrinsic pinning mechanisms for systems described by our Hamiltonian because they need a large admixture of high harmonic in the order parameter [13, 17]; c) The wavelength of the modulation is nearly independent of the temperature over the entire incommensurate phase which agrees, for instance, with the experimental data in NbTe 3.

Appendix

Calculation of the Coefficients c i

Inserting the Ansatz (38) into (34) and making use of the differential equation satisfied by U, we obtain

OU OU - v ~ vc,,~-'= (Df) oT +(Df*) OT ,

V = I

+(DAo)~-~oo-I-1.Uffo-i0- 3 Uff 1 cvz v (A1) v

D is the differential operator

D = 1 "0o-- i0- 3 ffl (A2)

and o- 3 the Pauli matrix

(A3)

Let us introduce the functions g, ~, and 7

0U = Ug, (A4)

of 0U

- U ~ , (A5) Of* 0- 3 U = U h. (A 6)

From (A4)-(A6) follow the relations

0U a 3 0 f = c % U g = U h g ,

0U 0-3 0f~=0-3 U~.

Noting that

(A7)

(18)

0U izU

&lo

we obtain, by inserting (A4)-(19) into (A 1),

(A9)


v c~ z ~-1 + {(6of) g-- i(J1 f ) h g + (0of*) V=I

--i(Jlf*) h g - i'c(~?o Ao)-'c(J1 Ao)h oo

(A 10)

For the higher coefficients we can put x' equal to x immediately and obtain

1/iD_ A o D _ f * ] (A19) c2(x,x)=~ D+f iD+ Ao]'

and

1 (iD 2_ A o - (D + f ) f * + (D_ f * ) f - 2 i (J l f ) f* D 2_ f* + 4 (J1 A o)f* ~ (A 20) c3(x, x )= - -~ D2+ f --4i(~l Ao)f iDZ+ Ao +(D + f ) f * - ( D _ f * ) f + 2iJlf*)f]

It follows from (A4)-(A6) that g, g, and h depend only on x 1 -X'l, A, A*, and "c. Taking the limit x] --*xl at once and using power expansions in "c, one obtains in a straightforward way

g = -- 'co-_ -- ~ f * o - 3 "C2 _~ " ' ' , (A 11)

~ : - - 'CO'+ ~- 21--a3 T2-~ - . . . , (A12)

h = o- 3 + 2"c (fo-_ - f * o%) + . . . (A 13)

where we have used the abbreviations

a :(01 00) (A14)

o-+ :(00 ~). (A15)

Inserting (A 1 1)-(A 13) into (A 10) yields equations for the c i if the sum of all contributions with the same power in z is put to zero separately. It this way, one obtains for general arguments

C 1 (X, X') : (~0 - - i0"3 J1) Co (x , x ' ) . (A 16)

From the Ansatz as well as the initial condition (34.5), it is clear that

Co(X, x ' )= 1 (A 17)

can be chosen. Equation (A 16) then yields

C 1 (X, X) = 0. (A 18)


D+ = ~0__+ iJ1. (A21)

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(1985) 9. Faddeev, L.D., Slavnov, A.A.: Gauge fields, introduction to

quantum theory, Chap. 2. London: Benjamin/Cummings 1980 10. Popov, V.N.: Functional integrals in quantum field theory and

statistical physics. Dordrecht: Reidel 1983 11. Kleinert, H.: Fortschr. Phys. 30, 187 (1982) 12. Ramond, P.: Field theory, a modern primer, p. 115. London:

Benjamin/Cummings 1981 13. Bak, P.: Rep. Progr. 45, 587 (1982) 14. Grabowski, M., Subbaswamy, K.R., Horovitz, B.: Solid State

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R. Brendle R. Zeyher Max-Planck-Institut fiir Festk6rperforschung Heisenbergstrasse 1 D-7000 Stuttgart 80 Federal Republic of Germany

influence of coulomb interactions on the phase diagram of quasi-one dimensional metals

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