ISING MODEL & SPIN REPRESENTATIONS

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email wlawton@math.nus.edu.sgTel (65) 874-2749Fax (65) 779-5452

ONE-DIMENSIONAL MODEL

R1}{-1, X:E N

N

1i

N

1i

)i(sH)1i(s)i(s E(s)

Partition Function

R

kT

1,E(s))exp(-) Z(

Xs

Energy Function

RR:Z

ONE-DIMENSIONAL MODEL

N1N3221

PTracePPP ) Z( ssssss

Transfer Matrix

)H(

)H(

ee

ee)P(

Trace Formula

22RR:P

42 e)H(sinh)Hcosh(e

TWO-DIMENSIONAL MODEL

)s,,s(),,,( n1n1

n

1)(E),(E )E( 112

2nN

n

1iii2 ss ),(E

n

1i

n

1ii1ii1 sHss )(E

TWO-DIMENSIONAL MODEL

nPTrace ) Z(

Transfer Matrix

Trace Formula

nn 22RR:P

)(E),(EexpP 12

Problem : Compute the largest eigenvalue of P

PROBLEM FORMULATION

123 VVVP

n

1iiiss1 ssexp )V(

Factorization

n

1issss 1iinn11ss2 ssexp )(V

n

1issss inn11ss3 Hsexp )(V

PROBLEM FORMULATION

IZYX 222

Pauli spin matrices

10

01Z,

0i

i0Y,

01

10X

0XZZXZYYZYXXY iYZX,iXYZ,iZXY

sinhXcoshe X

siniZcose Zi

PROBLEM FORMULATION

construct

IXIIX

n,1,

IYIIY

IZIIZ

For distinct subscripts everything commutes For any subscript, the Pauli matrix relations hold

by tensor products of n factors

For nn 22 matrices

PROBLEM FORMULATION

Xe)2sinh(2ee

eea

2e tanh 1

2n1 V)2sinh(2aa V

n

1XXX

1 e ee V

n

11ZZ

2 e V

n

1ZH

3 e V

CLIFFORD ALGEBRA

ijijji 2

Generated by n2,,1i, i

that satisfy the anticommutation rule

1211 Y,Z Example

214213 YX,ZX

k1k1k2k1k11k2 YXX,ZXX

CLIFFORD ALGEBRA

n2

2

1

n2,n22,n21,n2

n2,22221

n2,11211

n2

2

1

C)SO(2n, For any orthogonal matrix

the entries below satisfy the anticommutation rules

SPIN REPRESENTATION

n2,,1i),(S)(S 1ii

n2n2CC)SO(2n,:S Lemma 1.There existssuch that

Proof For planar rotators

2kj2kj2sincosexp)(S

sincos)(S)(S kj1

j

cossin)(S)(S kj1

k

)|k(j,

SPIN REPRESENTATION

ie Lemma 2.The eigenvalues of

are 1 with multiplicity (2n-2) and

)(S

,YZ,XZ 21k21j

)|(jk

The eigenvalues of are 2ie

each with multiplicity 1n2

Proof First part is trivial. For the second, choose

222kj iZYX

IIe0

0eI)(S

2i

2i

SPIN REPRESENTATION

Lemma 3 Let

)(S

where

)|()|()|( n21 }n2,,2,1{},,,,,,{

and },,{ n1 are complex numbers. Then

2n

21 expexp)(S

has eigenvalues ni1i e,,e

has eigenvalues 2/)ni2i1i(ie

Proof Obvious

SOLUTION

If there is no external magnetic field (H=0), then

21VVV

logn

1

nlim)]2sinh(2log[

2

1)(Zlog

N

1

Nlim

where is the largest eigenvalue of

n

1jjX

1 e V

n

1k1kZkZ

2 e V

SOLUTION

implies that

n

1j

n

1j1j2j2ijX

1 e e V

1ZnZ1n

1j

1jZjZ2 ee V

jjj1j2j2 iXZY

1n,,1j,ZiZYZX 1jjj1jjj21j2

)XX(ZiZ n1n1n21

SOLUTION

1n

1j

j21j2in21Ui2 e e V

n21 XXX U

n

1k

1k2k2i1n

1j

j21j2in21Ui eee V

21 etanh0,,

SOLUTION

)UI()UI(U,UI)UI(U

,I U,iU 2n221

n

n21i2

UIn21i2

UIUn21i eee

n

1k

1k2k2i1n

1j

j21j2in21i eee V

V2

UIV

2

UIV

SOLUTION

RXX U

The matrix U commutes with both

(however

I0

0IU~

RUR 1-

B0

0AV~

B0

0AV~

VV and VV and do not commute with each other

as erroneously claimed in line 7, page 380 Huang)

therefore

and

SOLUTION

To find the eigenvalues of VV and

we first find the 2n x 2n rotation matrices

such that )2i|(1,2n)S(V

1n

1j

n

1k)i2|1k2,k2()i2|j2,1j2(

and

)i|n2,1n2()i|4,3()i|2,1(

)i2|n2,1( )i2|1n2,2n2()i2|3,2(

SOLUTION

A0B

AB

0BAB

BBA

**

*

cosh2cosh2h2insicosh2

h2insicosh2-cosh2cosh2A

h2ins0.5sinh2-hcosisinh2-

hinsisinh2h2ins0.5sinh2-B

2

2

SOLUTION

1n2,,1,0k,e kk

2sinh2sinhnkcos2cosh2coshoshc k

)0.5(exp Vof ig.e 2n220

)0.5(exp Vof ig.e 1n231

)0.5(exp of ig.elargest 1n231

SOLUTION

2coth2coshD

dt

0

)t2cos211(Dlog2

1log

n

1

nlim

D

2

REFERENCES

K. Huang, Statistical Mechanics, Wiley, 1987

N. Hurt and R. Hermann, Quantum StatisticalMechanics and Lie Group Harmonic Analysis,Math. Sci. Press, Brookline,

B. Kaufman, “Crystal statistics, II. Partitionfunction evaluated by spinor analysis”, PhysicalReview 76(1949), 1232-1243.

E. Ising, Z. Phys. 31(1925)

R. Herman, Spinors, Clifford and CayleyAlgebra,Interdisciplinary Mathematics, Vol. 17, Math. Sci.Press, Brookline, Mass. 1974.

REFERENCES

D. H. Sattinger and O. L. Weaver, Lie Groupsand Algebras with Applications to Physics,Geometry, and Mechanics, Springer 1986.

L. Onsager, Crystal statistics, I. “A two-dimensionalmodel with an order-disorder transition”, PhysicalReview 65, (1944), 117.

T. D. Schultz, Mattis, D. C. and E. H. Lieb, “Twodimensional Ising model as a soluble problem ofmany fermions”, Reviews of Modern Physics,36 (1964), 856-871.C. Thompson, Mathematical Statistical Mechanics,MacMillan, New York, 1972.