THE ONSAGER SOLUTION in Java Receive QR Code in Java THE ONSAGER SOLUTION THE ONSAGER SOLUTIONJava denso qr bar code writer in javagenerate, create qr code none for java projectsThe Matrix P Barcode barcode library in javausing java toaccess barcode for asp.net web,windows applicationFrom (15.8) and (15.3) we may obtain the matrix elements of P in the form Java barcode recognizer with javaUsing Barcode scanner for Java Control to read, scan read, scan image in Java applications.( sl' ... , Sn IPls' ... , s') l' n Quick Response Code implement in .net c#using visual .net toget qr-code on asp.net web,windows applicatione{3Hs k e{3, 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 = (15.25XY= 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){15.26Za==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 1relation~ 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 where2 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 (15.401For 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)(15.41The following properties of {f,,} are stated without proof. *