THE ONSAGER SOLUTION

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The Matrix P

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From (15.8) and (15.3) we may obtain the matrix elements of P in the form

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( sl' ... , Sn IPls' ... , s') l' n

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e{3Hs k e{3<Sk Sk+\ e{3<sk sL

(15.15)

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Let us define three 2 n X 2 n matrices Vi, V2 , and V3 whose matrix elements are respectively given by

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(sl, ... ,snlv{ls{, ... ,s~) ==

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( sl' ... , S n

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e{3<Sk

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(15.16)

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' IV2 ISl' ... 'Sn')

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= us\s' ... Us s' II - II \ n n

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{3<Sk Sk+\

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(15.17) (15.18)

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where Sss' is the Kronecker symbol. Thus V2 and V3 are diagonal matrices in the present representation. It is easily verified that

(SI' .. ' snIPls{, ... , s~)

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s{', ... s~' s{', .... s~'

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V3V2V{

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(15.19)

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in the usual sense of matrix multiplication, namely

X ( sl, ,sn "

"IV2IS1'" ' ... 'Sn"')( sl, ,sn'" IV'I Sl, ,sn') ", ' I

Direct Product of Matrices

Before describing a convenient way to represent the matrices V3 , V2 , and VI we introduce the notion of a direct product of matrices. Let A and B be two m X m matrices whose matrix elements are respectively (iIAIi) and (iIBIJ>, where i and j independently take on the values 1,2, ... , m. Then the direct product A X B is the m 2 X m 2 matrix whose matrix elements are

(ii'IA

Blii') == (iiA li)(i'IBli')

(15.20)

This definition can be immediately extended to define the direct product A X ... X C of any number of m X m matrices A, B, , C:

(ii' ... i"IA

X ... X

qij'

(15.21)

== (iiA Ij)(i'IBlj') ... (i"ICIi")

multiplication, then

If AB denotes the product of the matrices A and B under ordinary matrix

B)(C

D) = (AC)

(BD)

(15.22)

SPECIAL TOPICS IN STATISTICAL MECHANICS

To prove this, take matrix elements of the left side:

(ii'I{A X B){C X D)IiJ') = L(ii'IA X Blkk')(kk'IC X DIJJ')

= L(iIAlk)(kIClJ) L(i'IBlk')(k'IClJ')

(iIAClJ)(i'IBDlj') (ii'l (AC)

(BD )IJJ')

A generalization of (15.22) can be proved in the same way:

(A X B X ... X C){D X Ex .. XF) = (AD) X (BE) X ... X (CF) (15.:::

Spin Matrices

We now introduce some special matrices in terms of which VI' V2 , and V3 may t'C conveniently expressed. Let the three familiar 2 X 2 Pauli spin matrices t'C denoted by X, Y, and Z:

(15.2~

The following properties are easily verified

X 2 = 1,

y 2 = 1,

XY + YX = 0,

YZ + ZY = 0,

Z2 = 1 ZX + XZ =

(15.25

XY= iZ, YZ = iX,

ZX= iY

Let three sets of 2 X 2 matrices X a , Va' Za (a = 1, ... , n) be defined afollows*:

X a == 1 X 1 X

xX X

Xl xl xl

Ya == 1 X 1 X

XY X xZX

(n factors) (n factors) (n factors)

{15.26

Za==lX1X

For a

*' f3 we can easily verify that

ath factor

[Xa,X p] = [Va' Yp ] = [Za,Zp] = [X a, Yp ] = [X a, Zp] = [Va' Zp] =

{15.27 1

relation~

For any given a the 2 n X 2 n matrices X a , Va' Za formally satisfy all the (15.25).

*The matrices Xa , Y , Za are familiar in quantum mechanics. For example, for a system of ' a nonrelativistic electrons the spin matrices for the ath electron are precisely Xa , Ya , and Za'

THE ONSAGER SOLUTION

The following identity holds for any matrix X whose square is the unit matrix e 8X = cosh () + X sinh () (15.28) where () is a number. The proof is as follows. Since xn = X if n is odd, e 8X =

n=O n!

xn =

1 if n is even, and

L - xn = L -

n even n!

n odd n!

= cosh () + X sinh ()

In particular (15.28) is satisfied separately by X, Y, Z and by X a, Ya,Za (a =

1, ... , n).

The Matrices VI' V2 , and V

By inspection of (15.16) it is clear that V{ is a direct product of n 2 matrices: VI = a X a X ... Xa where

2 identical (15.29) (15.30)

(slals') =

Therefore

e{3 ss'

(15.31) Using (15.28) we obtain (15.32) where tanh(} == Hence

e- 2 {3

(15.33) (15.34)

VI = [2 sinh (2/1 )]

e 8X

n/2 e 8X

X e 8X X ... X e 8X

The following identity can be verified by a direct calculation of matrix elements:

X e 8X X ... X e 8X

e 8X1 e 8X2 e 8Xn

e 8 (X 1 +X 2 + ." +X n )

(15.35)

Applying (15.35) to (15.34) we obtain

VI = [2 sinh (2/1 )]

n/2 V1

(15.36)

e- 2 {3

tanh(} ==

(15.37)

A straightforward calculation of matrix elements shows that

e{3 Za Z a+l

(15.38) (15.39)

SPECIAL TOPICS IN STATISTICAL MECHANICS

Therefore

(15.401

For the case H = 0, V3 sional Ising model.

= 1.

This completes the fonnulation of the two-dimen-

15.2 MATHEMATICAL DIGRESSION

The following study of a general class of matrices is relevant to the solution of the two-dimensional Ising model in the absence of magnetic field (H = 0). Let 2n matrices f" (p. = 1, ... , 2n) be defined as a set of matrices satisfying the following anticommutation rule

(p.=I,

,2n) ,2n)

(15.41

The following properties of {f,,} are stated without proof. *