142
21 Symmetry Breaking in the Ising Model
By the same argument as above, one can show that all the correlation functions
at T = 0 satisfy the cluster property and therefore their symmetry proves that there
is no spontaneous symmetry breaking at non-zero temperature.
2. Periodic and Cyclic Boundary Conditions
A commonly used choice is that of periodic boundary conditions, mainly because
they have the virtue of preserving translational invariance in finite volume. But, being
invariant under internal symmetries, they also lead to a mixed phase, when there is
symmetry breaking.
The computation of the correlation functions with periodic boundary conditions
is instructive also because it allows the use of the transfer matrix, which has become
a powerful tool in statistical mechanics and in lattice quantum field theory.
118 To this
purpose, the exponential
T (i, i + 1) ≡ e
β J σ i σ i+1 +βh(σ i +σ i+1 )/2
= T (i + 1, i)
(21.4)
can be viewed as the matrix element < σ i |T |σ i+1 > of an operator T , called the
transfer matrix, between vectors |σ i > labelled (only) by the value (±1) taken by
the spin σ i , e.g. |σ i >= |+ >= |σ i+1 >, if σ i = 1 = σ i+1 . Thus, T is effectively
acting on a two-dimensional space and is given by
T =
T ++ T +−
T −+ T −−
=
e
β J +βh e
−β J
e
−β J
e
β J −βh
.
(21.5)
Its eigenvalues are
λ ± (h) = e
β J cosh βh ± [e
2β J sinh
2
(βh) + e
−2β J
]
1/2
.
(21.6)
Then, the partition function becomes
Z N =
σ 1 ,σ N
< σ 1 |T
N −1
|σ N > e
βh(σ 1 +σ N )/2
.
(21.7)
Z N is easily computed for periodic boundary conditions, σ 1 = σ N , if h = 0, since it
is given by the trace of T
N −1
Z N = λ
N −1
+
+ λ
N −1
− , λ + = 2 cosh β J, λ − = 2 sinh β J.
(21.8)
118 See T. D. Schulz, D. C. Mattis and E. H. Lieb, Rev. Mod. Phys. 36, 856 (1964) and references
therein; E. Lieb, in Boulder Lectures in Theoretical Physics, Vol. XI D, K. T. Mahantappa and
W. E. Brittin eds., Gordon and Breach 1969, p. 329; J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
21 Symmetry Breaking in the Ising Model
By the same argument as above, one can show that all the correlation functions
at T = 0 satisfy the cluster property and therefore their symmetry proves that there
is no spontaneous symmetry breaking at non-zero temperature.
2. Periodic and Cyclic Boundary Conditions
A commonly used choice is that of periodic boundary conditions, mainly because
they have the virtue of preserving translational invariance in finite volume. But, being
invariant under internal symmetries, they also lead to a mixed phase, when there is
symmetry breaking.
The computation of the correlation functions with periodic boundary conditions
is instructive also because it allows the use of the transfer matrix, which has become
a powerful tool in statistical mechanics and in lattice quantum field theory.
118 To this
purpose, the exponential
T (i, i + 1) ≡ e
β J σ i σ i+1 +βh(σ i +σ i+1 )/2
= T (i + 1, i)
(21.4)
can be viewed as the matrix element < σ i |T |σ i+1 > of an operator T , called the
transfer matrix, between vectors |σ i > labelled (only) by the value (±1) taken by
the spin σ i , e.g. |σ i >= |+ >= |σ i+1 >, if σ i = 1 = σ i+1 . Thus, T is effectively
acting on a two-dimensional space and is given by
T =
T ++ T +−
T −+ T −−
=
e
β J +βh e
−β J
e
−β J
e
β J −βh
.
(21.5)
Its eigenvalues are
λ ± (h) = e
β J cosh βh ± [e
2β J sinh
2
(βh) + e
−2β J
]
1/2
.
(21.6)
Then, the partition function becomes
Z N =
σ 1 ,σ N
< σ 1 |T
N −1
|σ N > e
βh(σ 1 +σ N )/2
.
(21.7)
Z N is easily computed for periodic boundary conditions, σ 1 = σ N , if h = 0, since it
is given by the trace of T
N −1
Z N = λ
N −1
+
+ λ
N −1
− , λ + = 2 cosh β J, λ − = 2 sinh β J.
(21.8)
118 See T. D. Schulz, D. C. Mattis and E. H. Lieb, Rev. Mod. Phys. 36, 856 (1964) and references
therein; E. Lieb, in Boulder Lectures in Theoretical Physics, Vol. XI D, K. T. Mahantappa and
W. E. Brittin eds., Gordon and Breach 1969, p. 329; J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).