Physica 142A (1987) 555-562 North-Holland, Amsterdam

PERTURBATION THEORY FOR THE MULTIDIMENSIONAL POLARON

O.V. S E L JUGIN and M.A. S M O N D Y R E V

Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR

Received 1 September 1986

The optical polaron in N-dimensional space is investigated within the scheme of diagrammatic approach. The first two terms of perturbation series for the ground-state energy and effective mass are obtained. The perspectives of the 1/N-expansion are discussed which provide us with a new method valid in both weak- and strong-coupling regimes.

Intensive theoret ical and exper imenta l investigations in condensed mat te r

physics of recent years s t imulated the revival of a t tent ion to various po la ron

models descr ibed by the wel l -known Fr6hl ich Hamil tonian:

2 P ~ + ~ V x~(Ak eik'rak '}- Ak e ak)" H = ~ + wkakak _~ . -ik.r +, (J)

The u tmos t physical interest is roused in optical and acoustical polarons , not

only in the bulk, 3-dimensional ones, but also in the surface polarons in two spatial dimensions~). In this paper we shall confine ourselves to optical

polarons when the p h o n o n f requency does not depend on its m o m e n t u m :

w k = w. We shall not fix the n u m b e r N of spatial dimensions. Such a multi-

d imensional po la ron has recent ly been investigated by Devreese et al. 2). Here

we accept their parametr iza t ion:

A k = - i / k N-1 , g= oL2N--3/Z'B'(N--1)/ZF (2)

In the representa t ion with the total m o m e n t u m of the system P being a c -number , Hami l ton ian (1) can be writ ten as follows:

( + H = W - ka k a k

k

W = P/~c~-Ixw.

g21/4 ~ * + + Z a~ak + ~ (akak + Akak) ,

k (3)

0378-4371/87/$03.50 © Elsevier Science Publishers B.V. (Nor th -Ho l l and Physics Publishing Division)

5 5 6 O . V . S E L J U G I N A N D M . A . S M O N D Y R E V

Note that the energy and mass in eqs. (2) and (3) are expressed in units of w and Fz, respectively.

The goal of this paper is to construct the perturbat ion theory for such characteristics of the N-dimensional polaron as the ground-state energy, effec-

tive mass and average number of phonons. Results for the bulk and surface polarons will be obtained as particular cases. Besides, we discuss here the propert ies of the 1/N-expansion which can provide us with a completely new approach to the polaron problem both in the weak-coupling and in the strong-coupling regimes.

In ref. 3 a special diagrammatic technique has been developed which may easily be generalized to a multidimensional case. The Feynman rules are as follows:

,w , = 1 / k N 1. ~___~ CI~/~ 2 , / r ( N + l ) / 2 .

- n - 2 W k i 4- k i ; - - - - 1 .

i = l i = 1

(4)

Here k i are phonon momenta , and n is the number of phonons in a given virtual state. Let us briefly describe the main distinctions between our diagrams and the conventional ones. In addition to the ordinary strongly connected diagrams we have also the disconnected ones. On the other hand, there are no weakly connected diagrams which can be separated into two parts by cutting an electron propagator . Fur thermore , together with the e lec t ron-phonon vertices there exists another type of vertices whose role is to change the powers of the corresponding electron propagators . For more details we refer to ref. 3.

The sum of all diagrams defines the energy of a moving polaron:

= + + 2 2(w) + . . .

at small momen tum taking the form

~ g ( W ) ~- E + W 2 / m .

In the zeroth order in coupling constant a , ~0(W) = W 2, so that the required expansions for the ground-state energy E and the polaron effective mass m are as follows:

E = a E 1 + a 2E 2 + ' ' " ,

m = 1 + a m 1 + ol2m2 + " " " .

(5)

PERTURBATION THEORY FOR MULTIDIMENSIONAL POLARON 557

In the first order in a, in accordance with rules (4), we obtain

~ , ( w ) =

- - O /

F[(N- 1)/2] f dUk 1 ~&-;TTg k-~-=i (k 2 - 2W. k + 1)

F [ ( N - 1 ) / 2 ] ( W 2 ]

2 F-(5~/Z 1 + ~-~ } ,

from which there follow the first coefficients of expansions (5):

x/--ff F [ ( N - 1)/21 ~ F [ ( N - 1)/21 E 1 - 2 F(N/2) ' mx - 4N F(N/2) (6)

In the second order in a, the polaron energy is defined by a sum of three diagrams. The disconnected diagram contribution can be obtained in the analytical form

zZ2£Z~= _ o / 2 ~ l ( W ) F[(_NN --1) /___2] f dNk 1 27r(N+ 1)/2 k N-1 ( k 2 - 2 W . k + 1 ) 2

Of 2 [ ~ F[(N-1) /a]]2(1q_W2 (7)

Unfortunately, this cannot be performed for the remaining two diagrams. However, we succeeded in representing their contributions by one-dimensional integrals of elementary functions for which numerical calculations can be carried out by means of a simplest programmable calculator such as HP-34C:

e.~=_a2[F[(N--1)/2]]2f dUkdNq 1 L 27r (N+1)/2 kN-lqN-1 (q2 __ 2Wq + 1) 2

1 = G1 + B1W2 X ( q q- k) 2 _ 2W(q + k) + 2

1 G1 = __if2 ]P[(N --1) /2] ! dx x N 2

~ N ~ ~/-i- _ x 2 2 - x 2

( l-x: X x + 2 ~ : x - ~ arccos

S 1 ~-- -0~

1

r[(U- 1 ) / 2 ] dx x 2--N-~-F~72-) ~/~-- X 2 ( i - - X2) 3

0

(8)

558 O.V. SELJUGIN AND M.A. SMONDYREV

F x Ix(7 - 14x 2 + 9x 4) ÷

2,/r (N+ t),'2

f dNk d'~q 1 × kW=~Y=-~ (q2 _ 2W" q + 1)(k 2 - 2W" k + 1)

1 x = G~ + B2W 2

( q + k ) 2 _ 2 W . ( q + k ) + 2 - ,

1

6 2 = _ o 2 F [ ( N - 1 ) / 2 ] f dx ~ xN 2 ~-~F-(N--~ ~/ i ~ x ~

o

× ~/~= x~ arccos X/2 2~/1 _ x2 arccos x , (9)

B =_oe2 ~ [ F [ ( N - 1 ) / 2 ] ] 2

1

F [ ( N - 1)/2] ( dx x N 2 Ol 2 +

- N ~ F ( 7 ~ / ~ J ~ / ~ X 2 ( i -~ X2) 2 0

× ~ ( l + x 2)+ X / 2 - x 2 arccos .

By means of eqs. (7 ) - (9) we can calculate the coefficients E? and m 2 for arbitrary N. As to the E2, we obtained the same values as Devreese et al. 2) did. They used the path-integral technique and expressed their final result in the form of double integral. Thus, we confirm their numerical calculations in the scope of a completely different method. For this reason, here we shall only present our calculations for the multidimensional polaron mass for the same values of N, for which the energy has been calculated in ref. 2. See table I for first and second order expansion coefficients.

As expected, for the bulk, 3-dimensional polaron we have obtained the well-known results which can be expressed in an analytical form4). For another interesting case of a surface polaron (N = 2) integrals in eqs. (8) and (9) can be simplified. We will take the pleasure to cite our results for the N = 2 case:

2 ~ '7"/'2 ~-~ 1 E 2 - 7r 8-~" 323 7r F 2 ( 1 / 4 ) + _ ~ - F2( 1 / 4 ) - + G

~-/2

1 f 0 1 + 2 sin 0 0.063973966 4 dO sv~S~-0 l + s i n 0 -

o

2 x ] x5 ( 7 - llx2 + 8x 4 - 4x 6) arccos ;

PERTURBATION THEORY FOR MULTIDIMENSIONAL POLARON

TABLE I Expansion coefficients of the exact perturbation series for the

N-dimensional polaron effective mass.

N m 1 m 2

2 0.392 699 08 0.127 234 83 3 0.166 666 67 0.023 627 63 4 0.098 174 77 0.008 332 33 5 0.066 666 67 0.003 880 44 6 0.049 087 39 0.002 117 73 7 0.038 095 24 0.001 281 48 8 0.030 679 62 0.000 834 05 9 0.025 396 83 0.000 573 05

10 0.021 475 73 0.000 410 67 20 0.007 283 41 0.000 047 68 30 0.003 912 49 0.000 013 80

559

m 2 m 3~ -2 25~r 5 5 ]-~ 3 7 r 2 ~ 1 64- + 12~ 8 + l - ~ F 2 ( 1 / 4 ) - -i-6- F 2 ( 1 / 4 )

,n-/2

1 f 0 l + 6 s i n 0 - 6 s i n 2 0 + 4 s i n 30 + g d 0 ~ ( l + s i n 0 ) 3

0

= 0.127 234 835 .

H e r e G = 0.915 965 594 is C a t a l a n ' s cons tan t . No te tha t in the r ecen t p a p e r 5)

by Das S a r m a and M a s o n , d e v o t e d to the op t ica l p o l a r o n in two d imens ions ,

the coeff ic ients E 2 and m 2 a re exp res sed t h rough t r ip le in tegra ls and the i r

numer i ca l va lues a re ca l cu la t ed with a much lesser accuracy: E 2 = - 0 . 0 6 2 and

m 2 = 0.13. M o r e prec i se ly , the va lue for E 2 has b e e n ca l cu la t ed by D e v r e e s e et

al. in ref. 6.

In ref. 3 the r e l a t i on b e t w e e n the ave rage n u m b e r of p h o n o n s J£ and the

p o l a r o n ene rgy <g(W) has been de r ived :

3 0 W 0 ) 3 c = 1 - ~ c ~ 0 a 2 ~ ~ ( W ) . (10)

W i t h the he lp of eq. (10) we can cons t ruc t the p e r t u r b a t i o n ser ies for 3/'. In

pa r t i cu la r , for the sur face p o l a r o n with smal l m o m e n t u m we have

77" O/2 N = ce ~- + x 0.127 947 93 + ff(ce 3)

37r ~ - W 2 a ~ - - - a 2 X 0.080 933 20 + G ( a 3 ) ) .

No te tha t in ref. 3 the ene rgy of a bu lk p o l a r o n in the th i rd o r d e r in power s of

560 O.V. S E L J U G I N A N D M . A . S M O N D Y R E V

o~ has been obtained: E 3 = -0 .806 x 10 -3. In the same paper one can also find

the first three terms of perturbat ion series for N. Let us now revert to the case of the multidimensional polaron. Calculations

of the polaron energy and effective mass demonstra te that the polaron effects weaken with the increase of the number of dimensions. Eqs. (6 ) - (9 ) give us an opportuni ty to obtain the asymptotic behaviour of the expansion coefficients at

large N:

e I ~ - 2N

m l ~

E 2 - N 2 - ,

- 1 (11)

It follows from eq. (11) that a nontrivial limit at large N can be obtained through redefinition of the coupling constant, e.g., through scaling of the type a---~ a N 3/z. Such a redefinition of the coupling constant is not an innovation. For example, we used an analogous scaling while constructing the 1~N- expansions for the anharmonic oscillatorY), which allowed us to reproduce the results of perturbat ion theory as well as the strong coupling limit. The 1/N-expansion could provide us with a new method of description of the polaron propert ies at all possible values of coupling constant a. We still have no regular procedure of 1/N-expansion, but the results just obtained can help

us to guess its leading term. Indeed, it has been noticed by Devreese et al. 2) that in the Feynman

variational approximation there exist scaling laws

N E~) ( 3x/-~ F [ ( N - 1)/21 E~U)(a) = ~- a 2N ~]V/ -2)

m(FN}(a) = m(r7 ~) (a 3~-~ F [ ( N - 1 ) / 2 ] ] 2N F-(/V/~ / '

(12)

where the index in parentheses indicates the space dimensionality. For exact solutions these scaling laws (12) are valid only in the first per turbat ion order, but nevertheless they show a right way. Really, to obtain a nontrivial 1~N- expansion, e.g., for the bulk polaron, we redefine the coupling constant in accordance with eq. (12) and consider expressions

~(3) lim 3 E(N) ( 2N F(N/2) = N ~ -N oz 3x/~ F[(N-C--~/2]/ '

( 2N F(N/2) ) m(3)(a) = 1Nim ~ m 'N) a 3x/-~ F [ ~ - - 1 ) - / 2 I "

(13)

PERTURBATION THEORY FOR MULTIDIMENSIONAL POLARON 561

By definition, the functions under the limit sign at N = 3 are nothing else than the exact energy and mass of the bulk polaron. Evidently, they can be expanded in inverse powers of N. So, eq. (13) gives us the leading approxima- tion of the 1/N-expansion. What are the properties of/~(3)(a) and th(3)(a)? With the help of eq. (11) it is easy to find the first terms of perturbation series:

+

a 2 4 ( 1 1 ) rh(3)(a) = 1 + g + a ~ ~ 3~ -I- ~ ( a 3 ) .

(14)

In the theory of bulk polaron such expansions have been obtained by Haga 8) for the energy and by H6hler and Miillensiefen 9) for the effective mass. It has been shown by Saitoh l°) that both these expansions arise from the Feynman- type variational approximation with arbitrary quadratic trial action. The coinci- dence of the first terms of perturbation series for/~(3)(a), rh (3)(a) and approxi- mate expressions by Saitoh can be scarcely accidental. It seems, we have the reason to suggest that /~°)(a) and rh(3)(a) coincide with the Saitoh results at any value of a. In other words, we believe the Saitoh result to be leading approximation of the 1/N-expansion. If it is really the case, then the 1~N-

expansion is indeed suitable to describe the polaron properties in the whole range of the coupling constant. In particular, at large a when the Saitoh approximation gives the well-known Feynman results, we have

2 1 6 a 4

3"n" ' 81-./7 -2 '

rather close to the exact results. It follows then that the multidimensional polaron characteristics in the strong-coupling regime:

E ( N ) ( a ) = - A N a 2 m ( N ) ( a ) ~__ MNOt 4

should behave at large N as follows:

1 4 A N - 2 N 2 , M N = - ~ g . (15)

Predictions (15) are straightforward consequences of our analysis of the perspectives for the 1/N-expansion.

It is clear that in a similar way one can construct the first approximation of the 1/N-expansion for the polaron of arbitrary dimensionality: one has only to introduce appropriate factors into arguments and functions under the limit sign

562 o.v. SELJUGIN AND M.A. SMONDYREV

in eq. (13). Functions /~(N)(o/) and Fh (N)(oL) thus o b t a i n e d will satisfy the same

scal ing laws (12). T h e r e f o r e it is easy to ob ta in an ana logous expans ion for the

p o l a r o n at any given N di rec t ly f rom eq. (13).

O u r analysis p rov ides the answer to the ques t ion why the va r i a t iona l

m e t h o d s by F e y n m a n and Sa i toh are so good . I t t u rned ou t that the la t te r is

a sympto t i ca l ly exact at large N.

To conc lude , we would l ike to s tress once m o r e tha t it will be very useful and

in te res t ing to invent a r egu la r p r o c e d u r e for the 1 / N - e x p a n s i o n which would

give us an o p p o r t u n i t y to ca lcu la te co r rec t ions to the l ead ing a p p r o x i m a t i o n .

References

1) J. Sak, Phys. Rev. B 6 (1972) 3981. E. Evans and D.L. Mills, Solid State Commun. 11 (1972) 1093; Phys. Rev. B 8 (1973) 4004.

2) F.M. Peeters, X. Wu and J.T. Devreese, Phys. Rev. B33 (1986) 3926. 3) M.A. Smondyrev, Theor. and Math. Phys. 68 (1986) 29. 4) J. R6seler, Phys. Stat. Sol. 25 (1968) 311. 5) S. Das Sarma and B.A. Mason, Ann. Phys. 163 (1985) 78. 6) F.M. Peeters, X. Wu and J.T. Devreese, Phys. Rev. B31 (1985) 342(/. 7) A.V. Koudinov and M.A. Smondyrev, Theor. and Math. Phys. 56 (1983) 357; preprint J1NR

E2-83-412 (Dubna, 1983). 8) F. Haga, Prog. Theor. Phys. 11 (1954) 449. 9) G. H6hler and A. Miillensiefen, Z. Phys. 157 (1959) 159.

10) M. Saitoh, J. Phys. Soc. Japan 49 (1980) 878.