ISING MODEL

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ISING MODEL SPIN REPRESENTATIONS

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email wlawton_at_math.nus.edu.sg Tel (65)
874-2749 Fax [phone]
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ONE-DIMENSIONAL MODEL
Partition Function
Energy Function
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ONE-DIMENSIONAL MODEL
Transfer Matrix
Trace Formula
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TWO-DIMENSIONAL MODEL
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TWO-DIMENSIONAL MODEL
Transfer Matrix
Trace Formula
Problem Compute the largest eigenvalue of P
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PROBLEM FORMULATION
Factorization
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PROBLEM FORMULATION
Pauli spin matrices
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PROBLEM FORMULATION
construct
For
matrices
by tensor products of n factors
For distinct subscripts everything commutes
For any subscript, the Pauli matrix relations
hold
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PROBLEM FORMULATION
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CLIFFORD ALGEBRA
Generated by
that satisfy the anticommutation rule
Example
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CLIFFORD ALGEBRA
For any orthogonal matrix
the entries below satisfy the anticommutation
rules
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SPIN REPRESENTATION
Lemma 1.There exists
such that
Proof For planar rotators
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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
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SPIN REPRESENTATION
Lemma 3 Let
where
and
are complex numbers. Then
has eigenvalues
has eigenvalues
Proof Obvious
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SOLUTION
If there is no external magnetic field (H0), then
where
is the largest eigenvalue of
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SOLUTION
implies that
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SOLUTION
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SOLUTION
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SOLUTION
The matrix
commutes with both
(however
do not commute with each other
as erroneously claimed in line 7, page 380 Huang)
therefore
and
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SOLUTION
To find the eigenvalues of
we first find the 2n x 2n rotation matrices
such that
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SOLUTION
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SOLUTION
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SOLUTION
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REFERENCES
R. Herman, Spinors, Clifford and
CayleyAlgebra, Interdisciplinary Mathematics,
Vol. 17, Math. Sci. Press, Brookline, Mass. 1974.
K. Huang, Statistical Mechanics, Wiley, 1987
N. Hurt and R. Hermann, Quantum
Statistical Mechanics and Lie Group Harmonic
Analysis, Math. Sci. Press, Brookline,
E. Ising, Z. Phys. 31(1925)
B. Kaufman, Crystal statistics, II.
Partition function evaluated by spinor analysis,
Physical Review 76(1949), 1232-1243.
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REFERENCES
L. Onsager, Crystal statistics, I. A
two-dimensional model with an order-disorder
transition, Physical Review 65, (1944), 117.
D. H. Sattinger and O. L. Weaver, Lie Groups and
Algebras with Applications to Physics, Geometry,
and Mechanics, Springer 1986.
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.