468
8 Electric and Magnetic Phenomena
and
C spin (N, T ) = Nk B [βJ sech βJ ]
2 .
(8.6.9)
8.6.3 One-Dimensional Ising Chain in the Presence of an
External Magnetic Field
Free boundary conditions do not permit a straightforward solution of the onedimensional Ising chain in the presence of an external magnetic field. In this case,
it is convenient to employ the toroidal boundary condition, for which the nearestneighbours for the spin ˆ
s N are ˆ
s N −1 and ˆ
s 1 . For the toroidal topology, we are thus
able to re-express our Hamiltonian (8.6.1b) in a symmetric manner as
H N = −J
N −1
k=1
ˆ
s k ˆ
s k+1 −
1
2 μB
N −1
k=1
(ˆ s k + ˆ
s k+1 ).
(8.6.10)
The corresponding energy expression is
E N = −J
N −1
k=1
s k s k+1 −
1
2 μB
N −1
k=1
(s k + s k+1 ),
(8.6.11)
in which the s k take values ± 1 in the multiple summation version of the sum-overstates that then defines Z N (B, T ).
In order to obtain a closed analytic form for the summation that appears in the
expression for Z N (B, T ) when the energy of the spin chain is given by Eq. (8.6.11),
we shall employ a generalization [25] of the procedure that we employed in the
previous subsection for Z N (T ) in the absence of an external magnetic field B. 10
As the presence of the external field term in the energy expression (8.6.11) greatly
complicates the form for the final result obtained from carrying out the summations
over the spin values s k , we shall draw upon our knowledge of matrix algebra to
simplify the mathematics.
Interlude: Transfer Matrices
To introduce this procedure, let us consider briefly the essentially trivial two-spin
case, in which s 1 and s 2 are one another’s nearest-neighbours (i.e., s N +1 ≡ s 3 is
10 This procedure was introduced by Kramers and Wannier [25] in order to study the twodimensional Ising model. Although they referred to their procedure as the eigenvalue method,
it is now commonly called the transfer matrix method.
8 Electric and Magnetic Phenomena
and
C spin (N, T ) = Nk B [βJ sech βJ ]
2 .
(8.6.9)
8.6.3 One-Dimensional Ising Chain in the Presence of an
External Magnetic Field
Free boundary conditions do not permit a straightforward solution of the onedimensional Ising chain in the presence of an external magnetic field. In this case,
it is convenient to employ the toroidal boundary condition, for which the nearestneighbours for the spin ˆ
s N are ˆ
s N −1 and ˆ
s 1 . For the toroidal topology, we are thus
able to re-express our Hamiltonian (8.6.1b) in a symmetric manner as
H N = −J
N −1
k=1
ˆ
s k ˆ
s k+1 −
1
2 μB
N −1
k=1
(ˆ s k + ˆ
s k+1 ).
(8.6.10)
The corresponding energy expression is
E N = −J
N −1
k=1
s k s k+1 −
1
2 μB
N −1
k=1
(s k + s k+1 ),
(8.6.11)
in which the s k take values ± 1 in the multiple summation version of the sum-overstates that then defines Z N (B, T ).
In order to obtain a closed analytic form for the summation that appears in the
expression for Z N (B, T ) when the energy of the spin chain is given by Eq. (8.6.11),
we shall employ a generalization [25] of the procedure that we employed in the
previous subsection for Z N (T ) in the absence of an external magnetic field B. 10
As the presence of the external field term in the energy expression (8.6.11) greatly
complicates the form for the final result obtained from carrying out the summations
over the spin values s k , we shall draw upon our knowledge of matrix algebra to
simplify the mathematics.
Interlude: Transfer Matrices
To introduce this procedure, let us consider briefly the essentially trivial two-spin
case, in which s 1 and s 2 are one another’s nearest-neighbours (i.e., s N +1 ≡ s 3 is
10 This procedure was introduced by Kramers and Wannier [25] in order to study the twodimensional Ising model. Although they referred to their procedure as the eigenvalue method,
it is now commonly called the transfer matrix method.