466
8 Electric and Magnetic Phenomena
that even though the spins are distinguishable by virtue of their locations at distinct
lattice sites, because they interact with one another we are no longer able to express
the canonical partition function Z N (T ) for our assembly of N spins as the canonical
partition function z(T ) for a single spin particle raised to the power N. We are
thus reduced in principle to explicitly performing the sum over all 2 N microstates
accessible to such a one-dimensional collection of N spins. However, as the 2 N
energies involved in the partition function sum will be too numerous to take into
account directly for N 1, we must either be able to obtain Z N (T ) from its
defining expression via some mathematical trick, or we shall be reduced to deducing
the mathematical form for Z N (T ) by an essentially inductive procedure.
We note also that while boundary conditions are irrelevant for the thermodynamic
limit in which N → ∞, V → ∞, and the number density N/V is held constant,
they do matter for finite N. There are two obvious choices for boundary conditions
for a one-dimensional system. These are: either (a) allow the ends to be free, in
which case two of the spins (one located at each end of the chain) will have only
a single nearest-neighbour, while the remaining N −2 spins each have two nearestneighbours, or (b) place the N spins equidistant from one another on a circle,
with the N th spin next to the first spin, so that all spins are equivalent, thereby
eliminating the boundary. We shall examine the one-dimensional Ising model firstly
in the absence, thereafter in the presence, of an external constant and homogeneous
magnetic field. The first type of boundary condition (referred to as free boundary
conditions) will prove useful in the former case, while the elimination of the
boundary via the second condition (referred to as the toroidal boundary condition)
will aid us in working through the latter case.
8.6.2 One-Dimensional Ising Chain in the Absence of an
External Magnetic Field
We shall find it to be helpful to choose free boundary conditions when there is no
external magnetic field, so that the energy of our Ising spin-chain is given simply by
E N = −J
N −1
k=1
s k s k+1 .
(8.6.3)
In order to obtain a closed-form expression for the partition function Z N (T )
associated with this Hamiltonian, we note firstly that the 2 N energy values, E r N ,
associated with the eigenstates of H N arise from the product s k s k+1 when s k and
s k+1 are independently assigned their allowed values ±1. The 2 N values for r N may
thus be determined via the multiple two-term summations of s 1 , s 2 · · · s N , so that
the summation over r N from 1 to 2 N that appears in the defining relation
8 Electric and Magnetic Phenomena
that even though the spins are distinguishable by virtue of their locations at distinct
lattice sites, because they interact with one another we are no longer able to express
the canonical partition function Z N (T ) for our assembly of N spins as the canonical
partition function z(T ) for a single spin particle raised to the power N. We are
thus reduced in principle to explicitly performing the sum over all 2 N microstates
accessible to such a one-dimensional collection of N spins. However, as the 2 N
energies involved in the partition function sum will be too numerous to take into
account directly for N 1, we must either be able to obtain Z N (T ) from its
defining expression via some mathematical trick, or we shall be reduced to deducing
the mathematical form for Z N (T ) by an essentially inductive procedure.
We note also that while boundary conditions are irrelevant for the thermodynamic
limit in which N → ∞, V → ∞, and the number density N/V is held constant,
they do matter for finite N. There are two obvious choices for boundary conditions
for a one-dimensional system. These are: either (a) allow the ends to be free, in
which case two of the spins (one located at each end of the chain) will have only
a single nearest-neighbour, while the remaining N −2 spins each have two nearestneighbours, or (b) place the N spins equidistant from one another on a circle,
with the N th spin next to the first spin, so that all spins are equivalent, thereby
eliminating the boundary. We shall examine the one-dimensional Ising model firstly
in the absence, thereafter in the presence, of an external constant and homogeneous
magnetic field. The first type of boundary condition (referred to as free boundary
conditions) will prove useful in the former case, while the elimination of the
boundary via the second condition (referred to as the toroidal boundary condition)
will aid us in working through the latter case.
8.6.2 One-Dimensional Ising Chain in the Absence of an
External Magnetic Field
We shall find it to be helpful to choose free boundary conditions when there is no
external magnetic field, so that the energy of our Ising spin-chain is given simply by
E N = −J
N −1
k=1
s k s k+1 .
(8.6.3)
In order to obtain a closed-form expression for the partition function Z N (T )
associated with this Hamiltonian, we note firstly that the 2 N energy values, E r N ,
associated with the eigenstates of H N arise from the product s k s k+1 when s k and
s k+1 are independently assigned their allowed values ±1. The 2 N values for r N may
thus be determined via the multiple two-term summations of s 1 , s 2 · · · s N , so that
the summation over r N from 1 to 2 N that appears in the defining relation