The method we have just seen is an ad hoc solution that works only in this case, and it would be rather difficult to extend it in the presence of an external field. A more general method that allows us to extend our considerations also when and to compute other interesting properties is the so called transfer matrix method, which basically consists in defining an appropriate matrix related to the model such that all the thermodynamic properties of the system can be extracted from the eigenvalues of this matrix.
We are going to see this method applied to the one-dimensional Ising model, but its validity is completely general; we will stress every time if we are stating general properties of the transfer matrix method or restricting to particular cases.
The Hamiltonian of a one-dimensional Ising model with periodic boundary conditions when an external field is present is such that:
is sometimes called reduced Hamiltonian
We now rewrite the partition function in the following "symmetric" way:
If we therefore define the transfer matrix
we can write
as a product of the matrix elements of
If we now choose
so that they are orthonormal, i.e.:
then an explicit representation of
Note that the matrix elements of
are in one-to-one correspondence with the spin variables, and that the dimension of the transfer matrix depends on the number of possible values that they can assume.
Now, since the vectors
are orthonormal we have:
is the identity matrix, and
This is the general purpose of the transfer matrix method: being able to write the partition function of a system as the trace of the
-th power of an appropriately defined matrix (the transfer matrix).
Now, the trace can be easily computed if we diagonalize
; if we call
the diagonalization of the transfer matrix we will have:
is an invertible matrix whose columns are the eigenvectors of
, the partition function becomes:
and using the cyclic property of the trace
In the case of the one-dimensional Ising model we are considering
matrix so it will have two eigenvalues which we call
, with the convention
(we could in principle also consider the case
, but we will shortly see why this is not necessary). We will therefore have:
In general, if
matrix whose eigenvalues are
we will have:
Let us note that the dimension of the transfer matrix can increase if we consider interactions with longer ranges or if we allow the spin variables to assume more than one value
, and clearly the larger the matrix the harder its eigenvalues are to be computed, but the principle is always the same.
We will now use the transfer matrix in order to compute some interesting properties of a generic system.
Considering a general situation, the partition function of a model can be written with the use of the transfer matrix as . Therefore, in the thermodynamic limit the free energy of the system will be:
This is an extremely important result, since this means that the entire
thermodynamics of the system can be obtained by only knowing the largest eigenvalue of the transfer matrix
Furthermore, the fact that only
is involved in the expression of the free energy in the thermodynamic limit has a very important consequence on the possibility for phase transitions to occur. In fact, there exists a theorem called Perron-Frobenius theorem
in linear algebra which states the following:
If is an square matrix (with finite) such that all its elements are positive, namely , then the eigenvalue with largest magnitude is:
- real and positive
- an analytic function of the elements
We omit the proof of this theorem.
This means that if the transfer matrix of a model satisfies such properties, since is an analytic function the system will never exhibit phase transitions because also will be analytic.
For the one-dimensional Ising model with nearest neighbour interaction that we are considering, these properties are satisfied and so we have:
- , so that is well defined
- (this justifies a posteriori why we have considered from the beginning)
- is analytic, and therefore so is
From the last fact we deduce that no phase transition can occur for ; if some of the elements of diverge and Perron-Frobenius theorem can't be applied.
In general, in higher dimensions or with different kinds of interactions the transfer matrix can become infinite-dimensional in the thermodynamic limit: in this case the assumptions of Perron-Frobenius theorem don't hold and so the system can actually exhibit phase transitions (since is not necessarily an analytic function any more).
Correlation function and correlation length[edit | edit source]
The transfer matrix method can also be used to compute the correlation function and so the correlation length of a system.
As we know (see Long range correlations) in order to do that we first have to compute the two-point correlation function.
The connected correlation function of two spins which are at sites of distance is defined as:
where we have considered the first and the
-th spins because we are assuming periodic boundary conditions (so our choice is equivalent to considering two generic spins at sites
). For very large distances, we know that the correlation function decays exponentially, namely
is the correlation length. Therefore we can define the correlation length
of the system as:
We begin by computing ; this is the thermodynamic limit of the quantity:
Using the same factorization of the Hamiltonian that we have previously seen and that led to the expression of
in terms of the eigenvalues
, we have:
Now, we can write:
are the eigenvectors of
their relative eigenvalues (again ordered so that
); this way since these eigenvectors are orthonormal we also have:
Now, we introduce the matrices:
which are diagonal matrices such that on their diagonal there are all the possible spin values at the
-th site. This way, moving
at the beginning of
(since it is simply a number) and summing over
and using the expression of
given in terms of the
Multiplying and dividing by
In the thermodynamic limit the surviving terms are those containing
, and also all the terms with
(because they are not affected by the limit in
where we have used the symbolic notation "
" in the sum to indicate that we are excluding the case
What we now want to show is that:
and proceeding like we have done for
, we get to:
and again, using the expression of
and multiplying and dividing by
Again, the only surviving term in the thermodynamic limit is that with
, so indeed:
This way, becomes:
and thus the connected correlation function is:
If we now take the limit the leading term will be that with the largest possible eigenvalue , i.e. and all the other will vanish. Therefore:
and thus the correlation length will be such that:
are just numbers). Therefore:
Explicit computations for the one-dimensional Ising model[edit | edit source]
We now want to apply what we have just shown in order to do some explicit computations on the one-dimensional Ising model.
We have seen that the explicit expression of the transfer matrix for the one-dimensional Ising model is:
Its eigenvalues can be determined, as usual, solving the equation
, which yields:
Thus, the free energy of the system in the thermodynamic limit is:
while its magnetization (remembering that
Let us note that for
vanishes: this is again the expression that the Ising model in one dimension does not exhibit phase transitions. Furthermore we can see that in the limit
, we have
: again,the only value of the temperature for which the unidimensional Ising model with nearest-neighbour interactions exhibits a spontaneous magnetization is
The isothermal susceptibility of the system is:
Instead of explicitly calculating
generic, since we are interested in the behaviour of
when there are no external fields let us see what happens for small values of
. Since in this case
For high and low temperatures, we get:
and as we can see
diverges exponentially for
; this agrees with the fact that (as already previously stated) for
some elements of the transfer matrix diverge and thus the Perron-Frobenius theorem can't be applied.
Considering now the correlation length, we have:
In the particular case
Now, in the limit
, the following asymptotic expansion holds:
and so again we find an exponential divergence for
On the other hand, for
- ↑ We symbolically use Dirac's bra-ket notation.
- ↑ Namely, .
- ↑ In this case (which will be studied later on, see Bragg-Williams approximation for the Potts model) the model is called Potts model; if the spin variables can assume different values, then the transfer matrix of a one-dimensional Potts model will be .
- ↑ This is a blessing also from a computational point of view: it often happens, in fact, that the exact expression of the transfer matrix can be obtained but it is too big or complicated to diagonalize completely. There are however several algorithms that allows one to compute only the largest eigenvalue of the matrix in a rather efficient way.
- ↑ This agrees with what we have noted in Bulk free energy, thermodynamic limit and absence of phase transitions about the fact that a "phase transition" occurs in the one-dimensional Ising model for .
- ↑ Remember that in general correlation lengths are negligible for large temperatures and become relevant near a critical point.