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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
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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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' IV2 ISl' ... 'Sn')
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= us\s' ... Us s' II - II \ n n
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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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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
Blii') == (iiA li)(i'IBli')
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
== (iiA Ij)(i'IBlj') ... (i"ICIi")
multiplication, then
If AB denotes the product of the matrices A and B under ordinary matrix
D) = (AC)
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:
The following properties are easily verified
X 2 = 1,
y 2 = 1,
XY + YX = 0,
YZ + ZY = 0,
Z2 = 1 ZX + XZ =
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
(n factors) (n factors) (n factors)
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
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!
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') =
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 )
Applying (15.35) to (15.34) we obtain
VI = [2 sinh (2/1 )]
n/2 V1
e- 2 {3
tanh(} ==
A straightforward calculation of matrix elements shows that
e{3 Za Z a+l
(15.38) (15.39)
For the case H = 0, V3 sional Ising model.
= 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
,2n) ,2n)
The following properties of {f,,} are stated without proof. *