# The transfer matrix method

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:

*reduced Hamiltonian*. We now rewrite the partition function in the following "symmetric" way:

*transfer matrix*such that

^{[1]}:

^{[2]}we get:

^{[3]}, 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.

## Free energy[edit | edit source]

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:

*entire*thermodynamics of the system can be obtained by only knowing the largest eigenvalue of the transfer matrix

^{[4]}. 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:

**Theorem**(Perron-Frobenius)

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

- non-degenerate

- 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^{[5]}.

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:

We begin by computing ; this is the thermodynamic limit of the quantity:

What we now want to show is that:

This way, becomes:

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:

## 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:

The isothermal susceptibility of the system is:

Considering now the correlation length, we have:

^{[6]}.

- ↑ 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.