(15.15)

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

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

(15.16)

' IV2 ISl' ... 'Sn')

{3<Sk Sk+\

(15.17) (15.18)

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~)

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

V3V2V{

(15.19)

X ( sl, ,sn "

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

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:

qij'

(15.21)

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

D) = (AC)

(BD)

(15.22)

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.:::

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:

X 2 = 1,

y 2 = 1,

XY + YX = 0,

YZ + ZY = 0,

(15.25

XY= iZ, YZ = iX,

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

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

{15.26

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

{15.27 1

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 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!

L - xn = L -

n even n!

n odd n!

= cosh () + X sinh ()

1, ... , n).

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') =

e{3 ss'

e- 2 {3

(15.33) (15.34)

VI = [2 sinh (2/1 )]

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

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

(15.35)

VI = [2 sinh (2/1 )]

(15.36)

e- 2 {3

tanh(} ==

(15.37)

(15.38) (15.39)

(15.401

= 1.

This completes the fonnulation of the two-dimen-

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. *