Generalization of t4e Lee-Yang Theorem

Transcription

1 1328 Progress of Theoretical Physics, Vol. 40, No.6, December Generalization of t4e Lee-Yang Theorem Taro ASANO Institute of Physics, College of General Education University of Tokyo, Tokyo (Received April 3, 1968) The Lee-Yang theorem which states that all the roots of the partltion function of the. Ising model lie on the unit circle in the complex" fugacity" plane is proved for the case of spin Introduction The Lee-Yang theorem!) which states that all the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in the complex "fugacity" plane. has been proved, as far as we know, only for S=i, no' proof is given for larger magnitudes of spins than t. Recen,tly, Kawabata others2) have executed computer experiments on this problem have found that this theorem holds for S= 1 3/2. Here we prove this theorem for S = 1. Our proof is a natural extension of the original theorem. 2. The basic equations The partition function of the system of n classical spms with ferromagnetic interaction is defined by Z = Tr exp ( - (3H) n. H= - ~ JijSiSj-h ~ Sl, (2 1) n';2i>f:2,l where Jij>O the spin variable Si (i = 1, 2, "', n) takes a value equal to 1, -lor 0, h corresponds to the external magnetic field. We rewrite (2 1) as i=l (2 2) wher z = exp ({3h). Let us divide the integers 1, 2, ';', n into three groups A, B C, which correspond to the groups of spin variables with 1, respectively, so that there are nl integers in group A, n2 integers in group B n3 integers in group C, nl, n2 ns satisfying nl + n2 + n3 = n, I.e,

2 Generalization of the Le.e- Yang Theorem 1329 Si=l as iea, Si= -1 as ieb Si=O as iec. (2 3) Z (z) can be written as <Permutat;ion) exp [ - :E (3JijJ Z = :E (n1! Jl2! ns!)-1 :E nl+n2+na=n (2 4) where ab "', ani' bb "', b n2, CI, "', Cna is any permutation of the integers 1, 2, "', n, a's, b's c's belonging to A, B C respectively, the summation extends over all such permutations. x (i, j) (often denoted by Xii> i~ j, i, j= 1, 2, "', 71) is defined by x (i, j) = exp ( - /3Jij) '. x (i, j) has such properties as O<x(i,j)<l (2 5)' x(i~j) =x(j, i). (2 6) Hereafter we assume that none of x's are equal to ze:(e or one. The proof can be easily generalized to include the cases when one or more x's either vanish or equal to one. To derive (2. 4}, we have used the following relations: exp [(3Jij (SiSj - 1) ] = exp ( - (3Jij) = Xi} for iec, exp [(3Jij (SiSj -1) ] = 1 for i, jea or B, (2 7) { exp[(3jij (SiSj-1) ] =x;j for iea, jeb.. Corresponding to the partition function III an inhomogeneous magnetic field, the function Fn (Zb Z2, "', zn) is defined as <Perm) nl n3 Fn (Zb Z2, "', Zn) =:E (n1! JZ 2! JZs 1)-1 :E II II X (ai, C j) n 2 ng n3 nl 1t2 1tl n3 X II II X (bi" Cj') II x (Ci'" Cr) II IIx 2 (ak, bt) IIzukil z'b/. (2 8) It is evident that Fn(z,z, "', z) =exp(-/3:ejij)z(z). The most essential character of the Ising model is revealed in the reduction formula stated in Theorem 1. <Theorem 1) Reduction formula. + F n- 1 (Z2' "', zn) + Zl- l F n_1 (X12 Z2,.. 'X1nZn) J. (Proof) Evidently' Fn has the form Fn (Zl",zn) = f1 (Z2' "zn) Zl + f3 (Z2",zn) + f2 (Z2" 'Zn) Zl-l. (2 9) (2 10) The coefficient of Z1 8 in (2 10) comes from the terms in which 1 belongs. to A, B C when s takes a value equal to 1, -1 0, respectively. Hence,

3 1330 T. Asano (Perm) (A) /1 = ~ (n11 n 2 1 n3 1)-1 ~ 1tl+'Jt:l+7t3=n lfa n3 "1 "2 X II xci, Cj) II x(1, ale) II xci, b~) nt' ns = X12" 'Xl,,[ ~ (121,1 n2! 123!)-1 ~ II II X (ai, Cj) ("1'+n2+ n 3=n-1) (B) n2 ng 11.3 nt' n 2 X II II X (b'l" cr) II X (Ci", cr) II II x 2 (ale' b~) nt' n2 X II X-I (1, ale) zale II (x- 1 (1, b~) Zb~)-1J = X12'" x 1nFn- 1 (Xi:21Z2'" xl;zn). In the same way, X II x (ale, 1) zale II X-I (1, b t ) zr;/ X IIx (1, Cj) IIx (1, b~) II X (1, ale) X II II x 2 (ale, b~) II zak' II z~~ X II x (1, ai) II x (bi" 1) II X (1, cj") (2 11) (2 12) (2 13) (Q. E. D.) 3. Generalization of the Lee-Yang method From (2 4) we get exp(-p>~jij)z(z) =F,,(z "', z) =Fn(z-r,... Z-I), so that, to prove the Lee-Yang theorem it is only necessary to prove the following theorem: <Theorem 2> If ZI, Z2,.. 'Zn satisfy (3 1)

4 Generalization of the Lee-Yang Theorem 1331 (k=i,2,'''n), (3 2) then (k=l, 2, n). (3 3) We prove this theorem by mathematical induction. We can easily show that Theorem 2 (denoted by <T2» holds for n = 1 n = 2. The proof is given in the Appendix. Assume that the theorem holds for n = m -1 m - 2, does not hold for n = m, i.e. we set two hypotheses <HI) <H2), then we shall show that <H2) leads to self-contradiction. <HI) For n=m-2 m-l, <T2) holds. <H2) There are ZI, Z2,. "Zm which satisfy, (3 4) By the aid of <Tl) from (3 4), we ha"\,e (3 5) We get - F m-l (ZI, ZS,... ) ± ylf~-l (zt, ZS,... ) - 4F m-'l (Xl/Zt,... ) F m-l (X12Zt,... ) 2F m-l (X~lZt, x;;/zs,... ) (3 6) Referring to <Tl) agam, we have _ - zlf m-2 (X13 1Z S' ) ± Zl v'jj; + E' Z2- (3 7) 2X-';/ZlF m-2 (x;lxllzs,... ) + E where (3 8) { lim E/Zl = 0, IZll--->oo lim E'/Zl=O. IZll---'>oo (3 9) We now prove that keeping Zs, Z4,... Zm fixed, regarding Z2 as a function of Zt, one obtains a limit (2 for Z2 as Zl-----,) 00, that 1 (21 < 1. Since XI31X231/ZS/'... Xl:;;:/X2;~I/Zm/>I, the coefficient of Zl in the denominator of (3 7) cannot vanish because of <HI). Hence (2 satisfies (3.10)

5 1332 T. Asano It follows that -If' F (-1-1 ) [ -If' F (-1-1 ) X12 '-,,2 m-2 X23 XI3 Zg,.. XI2 '-,,2 m-2 X23 XI3 ZS,'" (3 11) Taking into account' (TI), we immediately get from (3 11). -If' F (-1-1 ) F (-If' -1 ) 0 XI2 '-,,2 m-2 X23 X13 ZS,'" ~ m-1 1:12 '-,,2, X13 Zs,'" =. (3 12) Because of <HI) (3 13) we get (3 14) so that (3 15) Then we can proceed in just the same manner as Lee Yang. Keeping Zs, Z4,'" Zm fixed, increasig [ZI[, then [Z2[ starts to be > 1 according to <H2) continuously changes its value since the denominator of (3.6) will never vanish because of <HI), [Z2[ tends to a limit <1 as [ZI[~OO. There must be a value of [ZI[ equal to [zt'[>1 so that [Z2[ assumes a value equal to 1. There exist z/, z/, Zg,... Zm satisfying (3 16) _ (3 17) We can fix z/, Z4,... Zm, regard Zs as a function of z/, follow the same procedure by increasing [zt' [ till [zs[ assumes a value. equal to 1. Continuing this way, we finally have a set of values {Zk} such that. Fm(zt, Z2, '"zm) =0, j [zi[>i, [Z2[ = [zs[ =... [Zm[ =1. (3 18) (3 19) -Our proof is completed by showing that a set of values {Zk}, which satisfy (3 18). (3 19) cannot exist. 4. The final stage of the proof In this section, we show that the set of {Zk} which satisfy (3 18) (3 19) can never exist so long as <HI) holds. In order to show this, first we prove the following Lemma, <Lemma I) If there existr>1 real Ot, O 2,.. f}m that satisfy

6 Generalization of the Lee-Yang Theorem 1333 (4 1) then (4 2) <Proof) From (4 1) <T1) + It IS easy to see that -1 -ib1f ( ib2.. r e m-1 XUe,,XIme ibm) t',.' (Zb... Zm') = F.<, (Z, -',... z;;;j), F': (ZI, "'Zm') =Fm' (Zl*' " Z!,). From these we obtain Then (4 3) IS reduced to Z2+ Fm_1(ei02, "')~z+l=o -1 io., I F ( ) ' m--l X]2 e -,.,. I where Z = r exp [ I 'f) 1 + I. arg F m-i (-1 X12 e io., -,... ) ]. 1 or (4 3) (4 4) (4 5) (4 6)' (4 7) (4 8) (4 9) Let the roots of (4 8) be a (3, then these roots must satisfy a = (3-1 (4 10) (3=a* (4 11) a=a*. If a=a*>l IS the case, (4;2) should.be fulfilled. If (3=a* IS the case, from (4 10) lal=lf3i=l (4 12) is derived. This is impossible for 1'>1. At this stage, we prove the last theorem that completes our theory. <Theorem 3) If Fm' (ZI,.,,zm') satisfies <T2), then, (4 13) for any t k >l real {h(k=l, 2,.. m/). <Proof) Keeping t 2, t 3,. tin';> 1 f)2,.. 8 m / fixed, we regard F m'(zi, t 2 e i02,., tm,e iom,)

7 1334 T. Asano as a function of Z1. Then, Z1F m' is a quadratic function of Zt, whose roots both lie inside or on the unit circle because F m' satisfies <T2) IZ21 >1, lzm,1 >1. (4 14) w here A, a, b, r/h r/h are functions of t 2, t m, ()2,.,. ()m" a b both lie in the interval [0, 1]. We prove the following inequality, for (4 15) However, it is very easy to show this, because the absolute value of (4 14) can be written as while 2 (4.16~ 1 J~ - a_eicf> 1 2 = t1 + _0_ - 2a cos J t1 t1 >1 + a 2-2a cos = 11- aeicf>12 (4 17) for t1>1 l>a>o. Therefore we have shown that (4 15) IS true that (4 18) Repeating this procedure for other t's, we obtain IF m,(t1e i0 t, tm,e iom ') 1>IFm,(ei0t,.. e iom ') I, (4 19) so long as F m' satisfies <T2). (Q. E. D.) According to <HI), F m-i should satisfy <T2), so that the inequality (4 13) must hold for m' = m -1. On the other h, in this case the Lemma 1 asserts that there is no set of {r, ()h ()2, "'()mlr>l} that satisfy F m (rei0t, e'loz,... e iom ) =. (4 20) This means that our first hypothesis <H2) leads to self-contradiction. (Q. E. D.) Appendix (A l) In order to show that zeros of!(z2) both lie inside of the unit circle for

8 Generalization of the Lee-Yang Theorem 1335! Zl! > 1, we make use of the following theorem. 3 ) <Theorem AI> Zeros of a polynomial f(x) =aox n ' an' (A 2) all lie within a unit circle, if only if Bezout's matrix B defined below is positive definite. Here, bin"... b J ll B-.. [ - bn'~'".. ~n'l. b12 = b 21 = (1- x 2 ) (r2-1) [(1 + x 2 ) (r2 + 1) + 2xr cos 8J, b ll = b~ = x (1- x) (r2-1) [(x + 1) (r2 + 1) + (e io + xe- io ) r J, where zl=re io X=X12 r>l, l>x>o. (A 3) b'l/s are defined as follows: ( aox n' an' ) ( an,y * n' ao *) - (n' aoy an' ) ( an' * x n' ao *) (A 4) In our case, bi/s are given as (A 5) (A 6) It is apparent that b12 is always positive. Also, we can easily show that det B IS positive if (A 6) is fulfilled. det B is written as det B/r 2 (1- X)2 (r2 _1)2x 3 = (u + 2) (u 2-1) w (u + 2) (2u -1) cos 8w + 4 (u + 1) cos (u - 2), here u=x+x- 1 >2, w=r+r- I >2. (A 7) (A S) It is easily shown that (A 7) is a monotonously increasing function of w as w>2, when u takes a value larger than two. Hence (A 7) is minimized, when w takes a value equal to two, then it is sufficient to prove the following Inequality for our purpose. 4(u+l)cos 2 8+4(u+2) (2u-l).cos 8+4(u+2) (u 2-1) - (u-2) >0, as u>2. (A 9) The discriminant or the left-h side of (A 9) as a function of cos 8 IS D=-16(2u+5) (u-2)<0, (A:I0) so that the left-h side of (A 9) is always positive. It is well known that a hermitian matrix is positive definite if only if all of its principal subdeterminants are positive.

9 1336 T. Asano Acknowledgements The author expresses his hearty thanks to Professor H. Kanazawa for his instructive suggestions encouragement. The author also thanks Professor S. Ono, Dr. M. Suzuki, Dr. C. Kawabata Mr. Y. Karaki for their valuable suggestions. References 1) T. D. Lee C. N. Yang, Phys. Rev. 87 (1952), ) C. Kawabata, M. Suzuki, S. Ono Y. K~raki, private communication. 3) T. Takagi, The Lecture on Algebra (Kyoritsu, Tokyo, 1946) (in Japanese), p. 461.