ISING MODEL & SPIN REPRESENTATIONS Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore Tel (65) Fax (65)

ONE-DIMENSIONAL MODEL Partition Function Energy Function

ONE-DIMENSIONAL MODEL Transfer Matrix Trace Formula

TWO-DIMENSIONAL MODEL

Transfer Matrix Trace Formula Problem : Compute the largest eigenvalue of P

PROBLEM FORMULATION Factorization

PROBLEM FORMULATION Pauli spin matrices

PROBLEM FORMULATION construct For distinct subscripts everything commutes For any subscript, the Pauli matrix relations hold by tensor products of n factors Formatrices

PROBLEM FORMULATION

CLIFFORD ALGEBRA Generated by that satisfy the anticommutation rule Example

CLIFFORD ALGEBRA For any orthogonal matrix the entries below satisfy the anticommutation rules

SPIN REPRESENTATION Lemma 1.There exists such that Proof For planar rotators

SPIN REPRESENTATION Lemma 2.The eigenvalues of are 1 with multiplicity (2n-2) and The eigenvalues of are each with multiplicity Proof First part is trivial. For the second, choose

SPIN REPRESENTATION Lemma 3 Let where andare complex numbers. Then has eigenvalues Proof Obvious

SOLUTION If there is no external magnetic field (H=0), then whereis the largest eigenvalue of

SOLUTION implies that

SOLUTION

The matrixcommutes with both (howeverdo not commute with each other as erroneously claimed in line 7, page 380 Huang) therefore and

SOLUTION To find the eigenvalues of we first find the 2n x 2n rotation matrices such that

SOLUTION

REFERENCES K. Huang, Statistical Mechanics, Wiley, 1987 N. Hurt and R. Hermann, Quantum Statistical Mechanics and Lie Group Harmonic Analysis, Math. Sci. Press, Brookline, B. Kaufman, “Crystal statistics, II. Partition function evaluated by spinor analysis”, Physical Review 76(1949), E. Ising, Z. Phys. 31(1925) R. Herman, Spinors, Clifford and CayleyAlgebra, Interdisciplinary Mathematics, Vol. 17, Math. Sci. Press, Brookline, Mass

REFERENCES D. H. Sattinger and O. L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, Springer L. Onsager, Crystal statistics, I. “A two-dimensional model with an order-disorder transition”, Physical Review 65, (1944), 117. T. D. Schultz, Mattis, D. C. and E. H. Lieb, “Two dimensional Ising model as a soluble problem of many fermions”, Reviews of Modern Physics, 36 (1964), C. Thompson, Mathematical Statistical Mechanics, MacMillan, New York, 1972.