USSR STATE COMMITTEE FOB UTILIZATION OF ATOMIC ENERGY INSTITUTE FOR HIGH ENERGY PHYSICS

Transcription

1 USSR STATE COMMITTEE FOB UTILIZATION OF ATOMIC ENERGY INSTITUTE FOR HIGH ENERGY PHYSICS И Ф В Э ОТФ V.V.Bazhanov, Yu.G.Stroganov HIDDEN SYMMETRY OF THE FREE FERMION MODEL. II. PARTITION FUNCTION Serpukhov 1984

2 USSR STATE COMMITTEE FOR UTILIZATION OF ATOMIC ENERGY INSTITUTE FOR HIGH ENERGY PHYSICS И Ф В Э ОТФ V.V.Bazhanov, Yu.G.Stroganov HIDDEN SYMMETRY OF THE FREE FERMION MODEL. II. PARTITION FUNCTION Submitted to TOTh Serpukhov 1984

3 УДК М - 24 Abstract Bazhanov V.V., Stroganov Yu.O. Hidden Symmetry of the Tree Fermlon Model. II. Partition Function: IHBP Preprint Serpukhov, p. 16, fig*. 3, refs.: 7. We «bow that the partition function of the free fermion model on a plane lattice haa some hidden symmetry properties. Using a specific elliptic parametrlzation for Boltzmann vertex weights, we calculate the partition function per aite and «how that it is а шегошогрыс function of three complex variablea. Баженов BJB.. Строганов Ю.Г. Скрытая симметрия модели свободных фермионов. II. Статистическая сумма: Препринт ИФВЭ Серпухов, с, 3 рис., бшблжогр.: 7. Мы показываем, что статистическая сумма модели свободных фермионов на плоской решетке обладает некоторой скрытой симметрией. Используя специальную эллшггжческую параметризацию для больцмановсих вершинных весов, мы вычисляем статистическую сумму аа узел решетки и показываем, что она является мероморфной функцией трех комплексных переменных. Институт физики высоких энергий, 1984.

4 INTRODUCTION This paper is the second one in the series of papers devoted to the study of the eight-vertex free fermion model on a plane lattice. Let us recall in brief the definition of the model. Consider a square M by N lattice. Every link of the lattice may occur in one of the two states. Eight types of the vertices allowed, as well as the notations for the corresponding Boltzmann weights, are given in Fig. 1. The vanishing weight is assigned to the remaining types of the vertices. The weights b, c, d satisfy the so-called free fermion condition: b o b a + b 2 b 3 - Q2 + d2 «it St. i t ГЦ. 1. The partition function is defined as a sum over all lattice states: Z = 2IIw(k) Ic where k=l,...,mn numbers the vertices of the lattice, and w(k) is the Boltzmann weight of the k-th vertex. In the thermodynamic limit, where z is called a partition function per site (or a speci fic partition function). For a more detailed description of

5 the free fermion model and the related references see In paper' (below we refer to it as I) the elliptic parametrization for the vertex weights was obtained. This parametrization includes five parameters: p, к, ф л, ф^, < Here are the required formulae: b o =PO - e^eg) where b k =p(e k - e^ege^1) (2) H = 4c 2 d 2 = 16p 4 k 2 (e 1 e 2 e 3 s 1 s 2 s 3 ) 2 e j = с з + is J = sn( ф., к), Cj = cn( ф, k), dj =dn(0j, k) (3) are Jacobi elliptic functions of the modulus k. As usual, symmetry transformations of the lattice lead to the symmetry relations for the partition function, which may be obtained on the level of the definition of Z. In section 2 we show that for the free fermion model there exist additional symmetry properties (hidden symmetry). In particular, we generalize the Kramers-Wannie duality transformation of Ising's model to the general free fermion model. In section 3 we calculate the specific partition function z in parametrization (2). It turns out that z is a symmetric and meromorphic function of three variables: ф^, ф, 0g-The technical details are given in the Appendices. When referring to a formula from refу, we put "i" before the formula number. 1. SYMMETRY PROPERTIES OF THE PARTITION FUNCTION 1.1. Let z(b, b.,, b, Ь, с, d) be a partition function per site considered as a function of the weights. There is a number of symmetry properties that may be obtained on the level of the definition of the partition function' 2 ': Z O123 (C ' Z O123 (C ' z 0123 <c> d) d) d) = Z ol23 = Z 2301 (d,c) (c,d) = z 1023 (c,d) where z jhk»( c;» d ) = z (b.,b.,b k,b,c,d). Besides, z is independent of the signs of с and d. Together with the first formula from (4), this means that z is an even and symmetric function of the variables с and d. Therefore, it is convenient to use the combinations c^ + d 2 and c^d^ instead of с and d. Using

6 condition (1), the quantity с 2 + d 2 may be expressed via b.. Hence, one may consider z as a function of five variables;- Ъ^ and c 2 d 2. Relations (4) are valid for any eight-vertex model.it turns out that for the free fermion model there exist additional symmetry relations. Let us show, for instance, that z is fully symmetric in the variables b Q, bj, b 2, b 3. The exact expression for z was obtained in ref. using the dimer techniques: 1 2n 2n 1 n z = - { f 6ф йф n{2a + 2B cos ф % о о 2C cos ф + 2D cos(f - ф 2 ) + 2E соз( ф^ + ф 2 )} (5) by where A = b Q + b][ + b 2, b 3 C - V3 " V 2 (6) D = c 2 - b Q b 3 E = d 2 - b b о 1 It may be readily checked that eq. (5) really has symmetry properties (4). Integrating over ф, one obtains: 1 n I 2 2"" 1 nz = f йф ln(a+bcos^ + y(f+g cos^ ) +H sin ф ) (7) о where P = b o, b 2 - b j - b 3 G = b Q b 2 + b l b 3 (8) H = 4c 2 d 2 Obviously, eq. (7) does not change under the permutation of bj and b 3. With the account for relations (4), this results in the full symmetry of z in the variables b, b,, b o and b,: о Л о z(b o,bj,b 2,b 3,H) - unchanged by permuting b^bj.bg^g. (9) There is another type of symmetry transformations for z. Consider the transformation b^ b A defined by the formulae: 4 " b o + b l + b 2 + b 3

7 2bl = b o - b;i + b 2 - b 3? (10) 2b 3 - b - b l - b 2 + b 3 Clearly under such a transformation the quantities A and В do not change, while F and G are interchanged. To compensate for this change, it is sufficient to replace H in integral (7) by: H' = H - J (13) where J = G - P =4 f o f l f 2 f 3 < 12) The quantities f, f, f o and f_ are defined by f ± = b o + Ь г +- b 2 + b 3-2b ±, i = 0,1,2,3. (13) Combining eqs. (10) and (11) with symmetry (9),one obtains some set of transformations, which do not change the partition function z and have the following interesting property.the fixed points of these transformations in the parameter space satisfy the relation: J = \ f o f l f 2 f 3 = < 14 > which exactly coincides with the phase transition condition of the model' 2 /. Consider as an illustration the two-dimensional Ising model, which is a particular case of the free fermion model. In our notations, b Q = x 2, bj =1, bj = bj = с = d = x (15) where x = th.(j8j), J i s t h e energy of the interaction of neighbouring spins and /3 is the inverse temperature. Performing transformation (10), (31) and interchanging b and b-p one obtains: z(x) = -(x + 1) z(- ) (16) Formula (16) coincides with the Kramers-Wannie duality relation. Thus, transformation (10), (31) may be treated as a generalization of the duality transformation for the free fermion model Let us now consider symmetry relations (9) and (30) (33) in terms of the parameters p,k, ф~, ф, ф 3 entering parametrization (2). Start with relations (9). Obviously, the permutations including the weights b, b, b only, are reduced to the permutation of the parameters ф^, ф 2, ф. The remaining permutation, ie those including the weight b, may be realized in two ways. In fact, using parametrization (2), one may easily show that both transformations:

8 Ф ± --» ф ±, ф^-*ж-фу ф к -*2К-ф к, р-*-е^е к р, (17) as well as (К and К' are complete elliptic integrals of the first kind of the moduli к and k f =(l-k 2 )z) leave Я unchanged and result in the same transformation-'*' for the weights b,b-.b o,b Q : О 1 л >J b o b i ; b j " b (19) k where (i,j,k) is the permutation of (1,2,3). In the present paper we adopt the following normalization of the weight: 9 = jl 3 / 2 з п l exp <- -ir-> 0( - 2 )} V~ )^<-V~ (2k'k*) 3/2 K 3 j=l 4K (20) with 0(x) and <Э^(х) being the Jacobi theta-functions, q = s exp(- ffk'/b;); Some properties of the theta-f unctions used below are given in Appendix C. The constant factor in eq. (20) is chosen to simplify the subsequent formulae. The new parameter p o does not change under transformations (17), and under transformations (18) it behaves as follows: Po - Vik^o fi- k = -exp[i»r( «^ ] (21) One may easily verify this, using the representation of е(ф ) through the theta-functions (C3). Symmetry relations (9) thus become: z(0j, ф 2, ф 3 ) = г(ф ±, ф j,< k ) (22) ^Ф г, ф 2, ф 3 ) = 2( ф ±, 2К- фу 2К-0 к ) (23) г{ф 1,Ф 2, Ф 3 ) =qm jk z(0 i, ^+2iK','f k +2iK') (24) where z( ф г ф^,ф 3 )=г(р к>, к, Ф^,Ф' 2> Ф$Ъ and is the (i'j»^ permutation, of (1,2,3>.-. ' / Moreover, from relation (1-32) one has: z( ф л, ф 2, 0 3 )=1 Ч 4ехр(-1>7^ /2K)z(2K+2iK' - ф ±, ф^, 0 к ) (25) Consider now transformation (10), (11). Let us prove that it ia reduced to the Jacobi imaginary transfromation for ellip- The difference between (17) and (18) lie* in the transformation lav for the weight» с and d (aee eq». (1-24).).

9 tic functions (upto the common factor). Indeed, let us substitute in eqs. (2) <fy - i0j. (j=l,2,3), к-** =(l-k 2 )2. Using standard relations (C4), we may represent (10), (11) as: b' = Ab i (p o, к', 1ф v 1ф 2, 1ф 3) H 1 = A 4 H( f> o, k\ 1ф г ±ф 2, 1ф 3 ) where ~ (26) % % }. (27) Since transformation (10), (11) doe?> not change the partition function, one obtains from (26): z(p o,k', 1ф г хф 2.1ф 3 )=A" 2 z(p o,k, ^, 0 2> ^3). (28) And, finally, combining (23), (24) and (28), one obtains symmetry relations of the form: г(ф, Ф, Ф ) = -г(ф, 0j + 2K, Ф +2К) (29) 3 г 3 ± к z(0 r <?i 2, ^ 3) = "^M Jk z(^i,2ik t -^j,2ik I - «5& k ). (30) Here ^ ij is defined by eq. (21). 3. PARTITION FUNCTION IN ELLIPTIC PARAMETRIZATION In this section we calculate integral (6) and obtain an explicit expression for the specific partition function as a function of the parameters p o»k, Фл, ф 2 > Фз entering into parametrization (2). Let us assume the weights b.b-.b^.bg to be real and H >0. As is shown in Appendix A, due to the symmetry relations for the partition function, it is sufficient to consider the following cases 2 *): 0< k 2 < 1, 0 < <^i< 2K, Jm ф ± =0 (31) k 2 >l, Ф ±(^1 к 2 <0, ф. D 2 (33) where i=1,2,3, and the regions Dj and D 2 are shown in Figures 2 and 3. We perform calculations for the case (31); the same formulae remain true for the case of (32). The case of (33) is reduced to (32) with the help of duality transformation (28). Substituting (2) into (7), one obtains: In z = ln(-4 p 2 e e e ) + L a*) 2 о For k 45 > 1 we choose Jmk'< 0, J*K> 0; and «or 1Г<0, Лк< О, ЛК'>0. 8

10 2тт О +S1 S 3 > 2 * «4" 3 (34) where compact notations (3) for the elliptic functions are used. The calculation of integral (34) is carried out in Appendix B. The result is expressed through the function: (35) where Z(fi ) is the logarithmic derivative of the Jacobi thetafunctions 8(/3 ) The function F(/3 ) is meromorphic in any finite part of the complex plane. It has the poles of the n-th order at the points /3 = -2nK + 2imK'; /8 =(2n+l)K+(2m+l)iK f, where ra,n are arbitrary integers (the zeros are treated as negative order poles). -а К Fig. 3. The function F( fi ) satisfies a number of functional equations: (36) F(/S)F(-/S) = (f^ F(/3 +2K) = (37) (38) ~- (39) These relations are obtained in Appendix C.

11 Define the quantities 2 0 о» ^-ф г ф 2 -ф 3 20 ± = Ф г +ф 2 + Ф 3-2ф ±, i = 1,2,3 satisfying the relation 18,,+ 18,+ ls, + )3 = 2K. It is shown in Appendix В that the partition function per site (34) has the form: z z (p o,k, ф 1г ф 2, ф 3 ) = С p o P( / S o )F( i 8 1 )F(j8 2 )F( Э 3 ), (41) С = 2(кк')"*ехр(-КК'А) As has already been noted, formula (41) obtained for region (31), is also valid for region (32). In order to obtain the expression of the partition function for region (33) (which we denote below through Zjj), we use duality transformation (28): zii^o' k) ^1* ^2' ^3* - Az i<p o ' k '» -1Ф г > - i 0 2 '~ i^3 ) =» \ф о т-г ^o)f(-ij5 1 )f(-i^2)l i (-ij3 3) (42) where \< is defined in (27), /3^ i =1,2,3 in i40), and /9 O = + = 2iK'-( 0 + ^ 2 03-)/ 2 ' T h e function F(/8 ) is defined by relation (35J, where the modulus of the elliptic functions is replaced by k 1, and К and K' are interchanged. The RHS of eqs. (41) and (42) are meromorphic functions of the variables ф-, ф, ф. They coincide with the value of integral (34) in regions (31), (32) and (33), respectively. Symmetries of partition function (34) are not, generally speaking, symmetries of these functions. However, using eqs. (36-39), one may verify by immediate' calculations that Zj satisfies symmetry relations (22^, (23), (24), (25), and z n satisfies relations (22), (25), ("29), (30). The degenerate case (14) (the critical free fermion model) is connected with the modulus k -.<». The expression for the partition function in this case may be obtained from (41) with the help of the corresponding limiting procedure (see paper I, Appendix B). In our next paper' 4 ' we shall show thatzj and ZJJ satisfy the inversion relations' 5 "?/. 10

12 REFERENCES 1. Bazhanov V.V., Stroganov Yu.G. - IHEP Preprint , Serpukhov, Fan C, Wu F.Y. - Phys. Rev., 1970, v. B2, p Whittaker E.T., Watson G.N. A Course of Modern Analysis. Cambridge University Press, Bazhanov V.V., Stroganov Yu.G. IHEP Preprint , Serpukhov, Stroganov Yu.G. - Phys. Lett., 1979, v. 74A, p Zamolodchikov A.B. -Sov.Sci.Rev.,[Phys.Rev Л,1980,v.2,p Baxter R.J. - In Fundamental Problems in Statistical Mechanics, 1980, v. 5, Amsterdam, North-Holland. Received 17 April, Appendix A In this Appendix we show that in the case of real weights it is sufficient to calculate the partition function in regions (31), (32), (33). Li paper I (Appendix B) the inversion of parametrization (2) was obtained, ie the parameters p,k, ф. were expressed through the weights b,b.pb 2,b 3,c,d. Remember the basic form u l a e :? 2 2 k 2 = 16c"! dv(f o f 1 f 2 f 3 ) (Al) where f^ are defined by (13). Introduce the quantities: x k = (f o f 1 f 2 f 3 )*/(f o f k ), к = 1,2,3 (A2) Then the parameters p and ф. may be found from the relations: (A4) о It follows from (Al) that к is real. The eliptic functions sn(0, k) and cn(<, k) have the following properties for real k 2. When 0^ k 2^l, the complete elliptic integrals К and K 1 are real and positive. In the range 0 <: ф^. 2К the function вп(ф,к) takes the values from 0 to 1, while сп(ф,к) varies from +1 to -1. For k 2 > 1 we choose k>0, Jm k f <0. Then K'> 0, К is complex, and Jm K=K', ReK> 0. The function sn(<, k) varies from 0 to 1 along the polygonal lines ABC and EDC (see Fig. 2).The function сп(ф, к) varies from 3 to -1 along the polygonal line ABODE. 11

13 For к < 0 we choose k'> 0, Jm k< 0. Then K> 0, JmK' = K, Re K*> 0. In this case the function sn(<,k) on the polygonal line ABGDJt is purely imaginary (Fig.3).It varies from 0 to along ABC, and from -i«> to 0 along CDE. The function cn(<, k) varies from 1 to «along ABC and from -«> to -1 along CDE Now consider the partition function per site z(b o,bj,b2, b 3,H). With the account for relations (2),(Al),(A3),(A4),one may treat it as a function of f, k 2, x-^x^xg. With the help of expression (7) for z one may easily show that z(f o,k 2,x lf x 3.x 3 ) = z(f Q,k,±х 1,±х 2> ±х 3 ) (А5) for an arbitrary choice of signs in RHS. Eqs. (Al), (A2) entail the fact that the variables x^ are real, provided к > 0, and are purely imaginary when к < 0. In the first case, using relation (A5), one may choose all x* nonnegative. When k 2 < 0, ie for purely imaginary x i,using eq. (A5), we can make Jm x.. 0. Then formulae (A4), with the account for the abovs properties of the elliptic functions ап(ф, k) and сп(у>,к), lead to the cases (31),(32),(33) given in the text. Appendix В In this Appendix we calculate the partition function in the elliptic parametrization for the case (31). Considering the singularities of the integrand in eq. (34), one may verify that the integral is an analytical function in region (31). Therefore, for conveniency, we confine ourselves to the case: 0 < ф г < ф 2 < 0з < К (Bl) Performing in eq. (34) the substitution of the variable: v - с cos ф = (B2) 1 - vc 2 and passing from ф~ and ф 3 to the combinations ф +1 =( Ф -,+ _+ < 3 )/2, one obtains: ln[ 2 x (l-k 2 s 2 sf) (B3> where s + = sn^+, с ± = en ф ±. The expression under the logarithm contains the factor without a root and two factors of the form A+y*B".For further 12

14 calculations it seems convenient to divide the integral into two parts, using the identity: In [C(A +VB)(A-+VB)1 = B)] <VVB)(A 2 +VB) ] (B4) Consider first that part of the integral which is connected with the second item. Substituting the integration variable: v = 1/cn z (B5) and differentiating over 0_, one obtains: is 2 sn2<fc. Я1Ж' dzdn2 z + r + <0.-*)<B6) 4 * о (cnz-c 2 )(cnz-cn2«^) where the integration contour goes along the imaginary axis. Using eq. (B17), we obtain: Here C t = cn/3 if Sj^ = зп^1? D ± = 6np ±, i = 0,1,2,3, f (/ 3) ^- (- + Z(/S)), (B8) 2sn (8 dnj8 JS. are defined by formulae (4$ K Z(/3 ) = d/d/3 1пв(/3 ) Consider now the part of the integral associated with the first item in formula (B4). Differentiating over ф. and using an elementary formula; 1 * dv 3 one obtains -l» and - Z(/3 2 +K)-Z(y3 3 +K)] (BIO) In deriving (BIO), we systematically express all the elliptic functions via Z-function with the help of the Liouville theorem. Combining (B7) and (BIO) and using (C5), we obtain the following expression for the derivative of integral L over ф = [? ( * } + f </ 3 ) + f ( / 3 i ) - f (^2 ) " r ( ' 8 3 ) < B n ) 13

15 where ^ - f(/3 ) = f(/3 ) + ^(0 + К) (В12) The function f(/3 ) is meromorphic in any finite part of the complex plane. It has simple poles at the points: /3 = 2nK + 2imK', residue = n, )8 = (2n+l)K+(2m+l)iK', residue = -n. Therefore, the function F(/3) defined by the relation: В F(j8 ) = exp[/ f(s)ds] (B13) о is also meromorphic. In the main text we give functional equa tions (36)-(39) for the functions F(/3 ). Let us show now that the required integral (34) equals: L = lnf( j3 J Ф,) в (ф ) в (ф 2 3 )- -in<-2_ _) 2 1 * 8k'K 3 n Indeed, it may be easily seen that eq. (B14) has a correct derivative over ф _, which is given by eq. (BID. Next, putting in eq. (B14) ф =C and using equations (37), (38), one obtain;,: 2 LI «5 ln >,=0 2 2к&в(ф )в(ф ) which exactly coincides with the expression for L ф _Q, directly obtained from eq. (34) with the help of eq. (6). When deriving the second equality in (B15) we use the Liouville theorem. Substituting eqs. (20), (СЗ), (В14) into eq. (34) and using the duplication formula for the в-function (see ге±у^, & ): 1 / 4 Т 1 / / 2 1 ^ ^ ' (B16) we come to result (41) given in the main text. What is left is to derive the formula: ^ i sn 2(9, sn 2 0, ) 3 2 2iK ' ; dz dn 2 z (cnz-cn20 )(enz-cn20_) 1 л 14

16 sn(0 i +ff 0< 0J, 0 2 < К which has been used in calculating the integrals in RHS of eq. (B6). Decomposing the integrans in eq. (B17) into simple fractions over the variable en z, we get: 0 2 ) ~ S n 2в 1 S n 2в 2 +[dn 20 a J(2<9 1 )-dn 20 2 J(20 2 )]/(cn гв^сп 20 2 )J (B18) where J(a) 2iK' =1 / Hnfen^' 0<a<2K < B19) о Since cn(z+2ik f )=-cnz, we may rewrite (B18) as follows: i 2iK> / d \ 1 d z[ z [ ] 2n 1 Q enz-ena enz+ena icna 2 i K ' dz T f 2 T i ( B 2 0 ) о sn a -sn z This is an elliptic integral of the third kind which can be expressed through theta-functions in a standard way (see ref. /3/, 22.74): J <«> " Г-Чг <* " IT ~ z (a)) Stia an а К " 0< a < 2K (B21) Substituting eq. (B21) into (B18), replacing the parameters <?, and 0 2 by $- + в о a n d и5^п^ t h e addition theorems for elliptic functions, one obtains formula (B17). Appendix С In the present Appendix we give some properties of the elliptic functions and obtain functional equations (36)-(39). The Jacobi theta-functions в( х), в (х), Н(х), НЛх) (see ref.^3/, ^ 21.62) are related to the elliptic functions sn x, dn x, en x through the following relations:

17 H(x) V^ H1 ( X ) л-т в 1, 1Ч _ ; en x s * -; dn х«=ук ' - (Cl) Vk 600 Vk~ 8 (x) в (x) List some properties of the theta-functions: в(х) в(-х) ш e(x+2k) 0(x+2iK') = -q^expi-iax/k) в(х) H(x) = iq 1 / 4 exp( - 1*х/К)в(х+1К') e t (x) == в(х+к), H^x) ж H(x+K) в(0) ж (2k'K/ff)i, q = exp(nrk'/k) ej(o) = <2К/я)*, H 2 (0) = <2kK/rf)* <С2) where К and K' are complete elliptic integrals of the first kind of the moduli k and k'=(l-k 2 )5, respectively. Here and henceforth, if a modulus is not indicated explicitly, it is meant to equal k. Using the Liouville theorem for elliptic function^3/, it is not difficult to show, that е(ф ) = спф + 1зпф - ±пф exp(--2) 2 (C3) The transformation to the complementary modulus k'=(l-k ) is performed with the help of the Jacob! imaginary transformation (see ref. /3/, 21.51; (22.4): k') = i sn(<^, Ю/спСф, k) k') = l/cn(0, k) (С4) Let us prove now relations (36)-(39). Consider the ratio of RHS and LHS of eq. (36) as a function of fi. Involving eqs. (Cl), (C2) and the relations (see ref.' 3 /, ): Z(/3 + K) = Z(/3 ) - k 2 sn/3cn0/dn/8 Z(/3 +ik')= Z(j8 ) - cn/3dnj8/sn 0 -W 2 K (C5) we may easily prove that this ratio is a constant. Assuming then /S = -(K+iK')/2, one gets convinced that this constant equals unity. Similarly, one proves that the ratios of RHS and LHS of eqs. (37)-(39) are constants. In order to shew that they are equal to unity, it is sufficient to put /3=0 in e,q. (37), /3 = -K+1K 1 in eq. (38), /3 = KTiK' in eq. (39) and apply relation (36). 16

18 ВЗ.Бажиюв, ЮЛЧСтроганов, Скрытая симметрия модели свободных фермионов. П. Статистическая сумма. Редактор А.А.Антипова. Технический редактор Л.П.Тимкива. Коррактор ЕЛ Jonang, Подписано к печати Т Формат 60/00x16. Офсетная печать. Печ.л. 1,00. Уч.-шэд.п. 1,24. Тираж 260. Заказ 636. Индекс Цена 18 коп. Институт физики высоких иерги1,142284,св{11ухов Московской обл.

19 Цена 18 коп. Индекс 3624 ПРЕПРИНТ , ИФВЭ, 1984