1. 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.sg Tel (65) 874-2749 Fax (65) 779-5452

2. ONE-DIMENSIONAL MODEL Partition Function Energy Function

3. ONE-DIMENSIONAL MODEL Transfer Matrix Trace Formula

4. TWO-DIMENSIONAL MODEL

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

6. PROBLEM FORMULATION Factorization

7. PROBLEM FORMULATION Pauli spin matrices

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

9. PROBLEM FORMULATION

10. CLIFFORD ALGEBRA Generated by that satisfy the anticommutation rule Example

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

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

13. 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

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

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

16. SOLUTION implies that

17. SOLUTION

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

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

21. SOLUTION

24. 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), 1232-1243. E. Ising, Z. Phys. 31(1925) R. Herman, Spinors, Clifford and CayleyAlgebra, Interdisciplinary Mathematics, Vol. 17, Math. Sci. Press, Brookline, Mass. 1974.

25. REFERENCES D. H. Sattinger and O. L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, Springer 1986. 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), 856-871. C. Thompson, Mathematical Statistical Mechanics, MacMillan, New York, 1972.