The Yang-Lee pictures of phase transitions in two variables

PHYSICS LETTERS

Volume 29A, number 9

with 0.3% (wt.) Mg which w e r e made by coevapor a t i o n of Mg and A1 f r o m a t u n g s t e n f i l a m e n t . C h e m i c a l a n a l y s i s p r o v i d e d the data for f i l m composition. M e a s u r e m e n t of R(T) on the A1 f i l m s shows the o c c u r r e n c e of a peak d u r i n g the r e s i s t i v e t r a n sition. The peak and, in fact, the e n t i r e t r a n s i tion c u r v e is quite r e p r o d u c i b l e when low enough p r o b e c u r r e n t s a r e used; high c u r r e n t s o b l i t e r a t e all the detail. The m i n i m a w e r e c o m p l e t e l y r e p r o d u c i b l e and s t a b l e a f t e r c y c l i n g a s a m p l e through liquid h e l i u m and a l s o allowing it to s t a n d i n a t m o s p h e r e for a week b e f o r e m a k i n g a n o t h e r m e a s u r e m e n t . Fig. 1 shows a typical c u r v e , together with R(T) for the s a m e f i l m with edges r e m o v e d ; edge r e m o v a l evidently wipes out all the high t e m p e r a t u r e detail and l e a v e s the n o r m a l r e s i s t a n c e . The b e h a v i o r of A1- 0.3% Mg f i l m s i s even m o r e m a r k e d , as i s shown i n fig. 2. H e r e it is a p p a r e n t that the peaks which o c c u r a r e l i m i t e d by a n envelope which i s i d e n t i c a i to the R(T) c u r v e for the " h i g h - c u r r e n t " (15 ~ A in this case) limit. Both t e m p e r a t u r e

THE

YANG-LEE

PICTURES

OF

PHASE

28 July 1969

and height of each peak a r e c o m p l e t e l y r e p r o ducible for any single film. Once again, edge r e m o v a l wipes out this detail, a s fig. 2 shows. Typical v a l u e s of n o r m a l sheet r e s i s t i v i t y w e r e 30 f~ for the g r a n u l a r f i l m s and 5 ~2 for the A1-Mg f i l m s . In view of the r e p r o d u c i b i l i t y and s h a r p n e s s of such detail in the R(T) curve, we s u g g e s t that it is r e s u l t of s t r u c t u r a l i n s t a b i l i t y at the f i l m edges. The a u t h o r s wish to thank I. M. P u f f e r for his v e r y c o n s i d e r a b l e aid and e c o u r a g e m e n t .

References 1. w. L. McMillan, Phys. Rev. 167 (1968) 331. 2. B. Abeles, Roger W. Cohen and W. R. Stowell, Phys. Rev. Letters 18 (1967) 902. 3. J.R. Clement and E.H.Quinnell, Rev. Sei. Inst. 23 (1952) 213.

TRANSITIONS

IN T W O V A R I A B L E S

O. S T ORIVI.ARK Department of Theoretical Physics, Royal Institute of Technology, S-10044 Stockholm 70, Sweden Received 10 June 1969

By a certain extension of the Yang-Lee theory of phase transitions to the case of two complex variables, we have been able to calculate the properties of a ferromagnetic system in a neighbourhood of the critical point.

With a p a r t i t i o n function ZN(W,Z) that is a n a lytic i n both its v a r i a b l e s (which, for example, can be the m a g n e t i c field and the t e m p e r a t u r e if we t r e a t a f e r r o m a g n e t i c s y s t e m ) for a fixed p a r t i c l e n u m b e r N, the f r e e e n e r g y - k T log Z N can be s i n g u l a r only w h e r e Z N equals zero. The z e r o s e t of ZN(W , z) is a s e t of t w o d i m e n s i o n a i a n a l y t i c s u r f a c e s (according to the p r e p a r a t i o n t h e o r e m of W i e r s t r a s s [1]) i n the four d i m e n s i o n a i (w, z) - space; when N goes to infinity, these z e r o s u r f a c e s m a y c o a l e s c e to f o r m a t h r e e d i m e n s i o n a l h y p e r s u r f a c e which divides the (w, z) - space into two p a r t s , that e v e n t u a l l y c o r r e s p o n d to different p h y s i c a l p h a s e s . 566

We will m a i n l y be i n t e r e s t e d in ZN(W,Z) when (w, z) belong to a s m a l l neighbourhood of the point w = z = 0 - which is chosen such that it p h y s i c a l l y i s the c r i t i c a l point - and in that neighbourhood we a s s u m e that it i s p o s s i b l e to a p p r o x i m a t e the z e r o s u r f a c e s by p l a n e s (observe that to each o r d i n a r y point an a n l y t i c s u r f a c e , t h e r e e x i s t s a tangent plane [1]). In the end we may continue our r e s u l t s a n a l y t i c a l l y to a l a r g e r neighbourhood which s u r e l y is big enough to allow e x p e r i m e n t a l investigations. With z e r o p l a n e s of the f o r m ZnW + wn z = w n z n and ~n w +Wn z = ~nZn (the complex conjugation is i n o r d e r to get a r e a l Z N for r e a l w and z),

Volume 29A, number 9 w h e r e n = 1, 2 , . . . ,

PHYSICS LETTERS

28 July 1969

we can write: M(I-I~O, T = Tc) =

w ~n)X

ZN(W , z ) = nIn~. w z=zn ) 0 - ~ " =1~ (1 ~-~

× exp

k-1 ~ -~----+-~---~ + ( b'=-I w w n zrt

lrkT(l+l/5) cos (~/25)

+

)v

1 / 5 I +1 when H > 0 I'//I

That is, M is a n t i s y m m e t r i c in H a s it should be. × (H= O, T--" Tc) =

w h e r e N is a fixed p a r a m e t e r a p p e a r i n g in w n and Zn; with k chosen a p p r o p r i a t e l y , this p r o duct is e a s i l y seen to be convergent. The z e r o e s of ZN(W , O) a r e w = Wn, w--n. When applicating this t h e o r y to a f e r r o m a g n e t i c s y s tem, we want a m a g n e t i s a t i o n M which is a n t i s y m m e t r i c in the field H, and this we get if the z e r o e s a r e s i t u a t e d along the i m a g i n a r y axis in the w-pllme, and t h e r e f o r e we put wn = = i ( n / N ) 5/(l+b), where 5 is s o m e positive c o n stant. C l e a r l y , the z e r o e s a c c u m u l a t e to w = 0 when N ~ ~o. Z~r(o, z) equals z e r o for z = z n , ~'n. We choose z n = ~ e x p { i ~ ( n / N ) l / ( 2 - a ) } , where a < 2 and 0 < ~ < v, that is, the z e r o e s in the z - p l a n e a r e s i t u a t e d on the l i n e s z = r . exp (~i @), 0 < r - < - co, and they c o a l e s c e into z = 0 when N ~ oo. The r e a s o n why we c o n s i d e r only z e r o e s along s t r a i g h t l i n e s and not along a r b i t r a r y c u r v e s , i s that the only z e r o e s c o n t r i b u t i n g to the s i n g u l a r i t i e s i n the c r i t i c a l point a r e those which a r e s i t u a t e d in a s u f f i c i e n t l y little neighbourhood of that point, and in this neighbourhood we s u p pose that the z e r o c u r v e m a y be a p p r o x i m a t e d by a s t r a i g h t line (we do not c o n s i d e r z e r o e s a c c u m u l a t i n g in t w o d i m e n s i o n a l r e g i o n s of the z plane, as they do not s e e m to give welldefined physical properties.[2]). Defining Z = Z N I / N we get the f r e e e n e r g y p e r p a r t i c l e a s F = - k T log Z. This b e c o m e s a n infinite s u m ; by d i f f e r e n t i a t i n g t e r m w i s e , and, when N t u r n s to infinity, l e t t i n g n / N b e c o m e a continous v a r i a b l e so that the s u m c o n v e r t s into a n i n t e g r a l , we can c a l c u l a t e the i n t e r e s t i n g p h y s i c a l p r o p e r t i e s such as the m a g n e t i s a t i o n M, the s u s c e p t i b i l i t y × and the specific heat C H. T h i s c a l c u l a t i o n i s a bit tedious; however it gives the following r e s u l t s (here we have put Im w = I m z = 0, Re z = T c - T ( T c = c r i t i c a l t e m p e r a t u r e ) and Re w = - H ) : M~

= 0, T - ~ T c) =

27rkT(2-a),,IT-Tcl ~ ~ s i n ( ~ ) when T > T c m

sin ~)

~ - s i n [~(~-~)] when T< Tc

w h e r e /~ = (2-a)/(1+5) or a + 8(1+5) = 2.

- 1 when H < 0 .

-2~kT(2-a)(~-l) iT_rci-r X s i n (y~) "

l c o s ( r ~ ) when T > Tc × {cos[r0r-~)] when T < Tc w h e r e ~ - ( 2 - a ) ( 5 - 1 ) o r a + 2 ~ + y = 2. 5+1 T h i s e x p r e s s i o n is not defined for y = 1; howe v e r , then we g e t : ×(H=O, T " " T c) = when T > =2kT.

(2-ol) c o s ~ " ] T - T c 1 - 1 . 1 +1 -lwhenT

CH(H= O, T--* T c) - 2~kT2(2- ~)(1-rv) IT- TC] s i n (a~)

Tc,

< Te; -or

×

~COS[~(2-a)] when T > T c , × c o s [ a ~ + ~(2- a)] when T < Tc .

This is c o r r e c t if o t ¢ i n t e g e r . Thus f a r we supposed that t h e r e i s j u s t one kind of z e r o d i s t r i b u t i o n ( n a m e l y wn = = i(n/N) ~/(1+5), zn = exp{i~(n/N)lT(2-~)}, but g e n e r a l l y we could have Z N ( w , z) built up f r o m s e v e r a l d i s t r i b u t i o n s (like. w~ = i ( n / N ) h v / ( l + h v ) , ~ = exp{i,v(n/N)i/(~-~v~} w h e r e v -- 1 , 2 , . . . ) which g i v e s M = ~ v M , ×= ~ , , ×V and CH = v p = ~vCR, w h e r e M , ×v and C ~ a r e e x p r e s s i o n s of the kind just d e s c r i b e d . E s p e c i a l l y , we can for given a and 5 c o m b i n e two d i s t r i b u t i o n s with + ~ and - ~ , b e c a u s e then the m a g n e t i s a t i o n at H = 0 equals z e r o for T > Tc; this can also be achieved by choosing ~ and ~ such that s i n ( ~ ) = 0. The i m p o r t a n t r e s u l t of all these f o r m u l a s is, that the s o - c a l l e d s c a l i n g laws (a+~(l+5) = 2, a+~+y=2, Ot(T T c ) , y ( T < T c) = y ( T > Tc) [3]) a r e c o r r e c t at l e a s t as long as none of the c o s i n e - f a c t o r s a p p e a r i n g in X and C H equals zero, which happens if a , 5 and ~ a r e r e l a t e d to each o t h e r in c e r t a i n m a n n e r s . As a concluding r e m a r k , we o b s e r v e that p h y s i c a l l y t h e r e should be no s i n g u l a r i t y in M(Ho, T) 567

Volume 29A, number 9

PHYSICS LETTERS

28 July 1969

References 1. Behnke-Thullen, Theorie der Funktionen mehrerer komplexer VerHnderlichen, Erg. d. Math: HI, 3. 2. S. Grousmann, Phys. Letters 28A (1968) 162. 3. M.E. Fisher, Rep.Progr. Phys. 30 (1967) 615.

at T = T e if H o ¢ 0; it is e a s y to see that this is fulfilled in our c a s e if 2 - a > 1 + 1 / 5 . P u t t i n g in the experimental values for ~ and 5, we see that this inequality is indeed valid. *****

THEORY

OF

THE

MAGNETIZATION

OF

PURE

TYPE-II

SUPERCONDUCTORS

U. BRANDT A bteiltmg fl~*- Theo*-etische F estk~rpe*-physik am lnstitut fa*Angewandte Physik de," Unive*-siRlt Hambu*-g, Germany Received 17 June 1969

The magnetization M of a bulk pure type-II superconductor is calculated for magnetic fields Ha below Hc2 and all temperatures T. Comparison is made with experimental data in Nb. In the limit T = 0 and Ha near Hc2 the magnetization has the nonanalytic form Mac [Hc2-Ha]/ln [(ttc2_Ha)/Hc2].

The difference of the t h e r m o d y n a m i c potentials of the superconducting and the n o r m a l state has been e x p r e s s e d in t e r m s of the t h e r m o d y n a m i c Green function G00(r , r ' ) in analogy to ref. 1. A s s u m i n g the o r d e r p a r a m e t e r A (r) is proportional to A b r i k o s o v ' s v o r t e x solution, neglecting the spatial variation of the magnetic field H(r), and using the solution of the F o u r i e r t r a n s f o r m of the Green function of ref. 2 and the BCS-identity, one obtains

ns ~2N V

1 - 8,r (Ha- B)2 + A2N°

in

r Tc

-4~'T

/>~0 N o _os i n o d O

\

2A

Xo

w/

- ~/"

.

(1)

H e r e is V the volume of the superconductor, B the spatial a v e r a g e of H(r), A2 Athe averba~e of IA(r) 12, N o the n o r m a l density of s t a t e s at the F e r m i surface, v F the F e r m i velocity, = (2eB)- , and co/= = (2/+ 1)~rT; Xo(W) (>0) is the solution of the equation

2 ,A

VFSinO

xo+

# ( _ ,,A \ v F sin O /

w0x o)--0

(2)

where the function w(z) is defined as w ( z ) = exp ( - z 2) [ 1 + 2i,t a~ / dt exp (t2)] . o

(3)

Minimization of eq. (1) with r e s p e c t to a 2 yields the relation *r [ a V(Xo)l 0 /2WAHc2, ] O=~.~r ~ f, dOsinO ~ - ~ V ~ o ) Wl ' l >~0 o wl which d e t e r m i n e s A2 as a function of A and T; h e r e V(x) is defined by 1 dV0~)/dx = w(ix); AHc 2 = (2eHc2)-~.

568

(4)