Susceptibility and fluctuations in the Ising ferromagnet

Volume 43A, number 2

PHYSICS LETI’ERS

26 February 1973

SUSCEPTIBILITY AND FLUCTUATIONS iN THE ISING FERROMAGNET D.B. ABRAHAM Department of Theoretical Chemistry, Oxford OIl 3TG, UK Received 5 January 1973 A rigorous proof is given of a formula relating the susceptibility and local magnetisation fluctuations for the ferromagnetic Ising model in the thermodynamic limit. This completes the derivation of the exponent ~ = 7/4 for two dimensions.

>~$,II) = urn

Consider a ferromagnetic Ising spin system on a hypercubical lattice A described by the configurational energy

E

1—hEa~,

(1)

ieA

/aA where J> 0, the first sum is over neighbouring spins in the interior A’, and the spins {~} on the boundary ~A are fixed. The canonical partition function is ZA = Tr exp(—I3EA) and the magnetic susceptibility is definedby 2 lim ~J~logZ~ H*0

a

lim

x03, Ii),

(2)

where H 13h. The pair Ursell function, which is defined in terms of spin correlations by U(i, /113, H, A, bA) = (UjO/)A,H, bA

~i~A,H,bA

=-j-J-I E

Then for H * 0

4/cA

of even length: then lim lim S(13, H, A, +) = lim S(J3, 0, A, +). (5)

h-”O+ A-’°~

From the GHS inequality [2, 31, which also holds forbA=+,wehaveforH~O UQ,jI13,H,A,+)~U(i,jI13,0,A,+). (6) h oo the sites on the double cross, we note that U(i, / 113, H, A, +) = 0 if i,/ are in different sublattices A 1. Hence from GHS we have S(j3, H, A, +)> S(J3, H, A1, +) which, with (6), (5). at H = 0, but 13c, the derivative in gives [2] exists For 13 < it is also continuous [3] so that no ambiguity arises. -~

The spectrum of the transfer matrix with + boundary conditions has been obtained [4] establishing the (G/)A,H, bA

‘

_______

describes fluctuations in local magnetisation for the boundary condition bA. It is related to the susceptibilityby Lemma 1: (Lee-Yang [1]). Let S(J3,H, A, +)

functions: Lemma 2: Let A be a hypercubical lattice with sides

The proof is illustrated ford = 2 by fi~1; letung

~H2 IAI~ I~

x03, 0) =

(4)

The next lemma relates x(I3~0) to the zero field Ursell

a~cr,—JE o~ff ieA’

S(J3, H, A, +).

IAI—~

U(i,1113,H,

A, +).

(3)

—

A,

A, -

______

-

________

—

_________

A _________

_______

A,

FIg. 1. The lattice A partitioned into 4 sublattices Aj.

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Volume 43A, number 2

PHYSICS LE’l’TERS

identity of (2) and (5) with (1) of ref [5] which refers to toroidal boundary conditions. This completes the rigorous derivation of eexponent ‘y = 7/4. Incidentally, ref. [4] also justifies the Onsager-Yang formula for the spontaneous magnetisationfor all 13 [6]. This result has been obtained independently by Benettin et al. using a different method. I thank J. Lebowitz for a useful correspondence.

164

26 February 1973

References [1] T.D. Lee and C.N. Yang, Phys. Rev. 87 (1952) 404, 410. [2] R.B.Griffiths, C.A. Hurst and S. Sherm~i,J. Math. Phys. [3] J.L.Lebowitz,Commun. Math. Phys. 28 (1972) 313. [4] D.B. Abraham and A. Martin-Löf, to be published. (5] D.B. Abraham, Phys. Lett. 39A (1972) 357. [6] C.N. Yang, Phys. Rev. 85 (1952) 808; A. Martin-Lof, Phys. 24(1972) 253;Steila, G. Benettin, G.Commun. Gallavotti,Math. G. Jona-Lasimo and A.L. preprint.