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One-Dimensional Ising Models: N Spins and Period Boundary

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Consider One-dimensional Ising Model with N spins and periodic boundary condition, i.e., sN+1 = S1 (no external field).
a) Compute the partition exactly using the transfer matrix approach (justify steps).
b) Compute the free energy.
c) Compute the magnetization.
d) Compute the average energy.
e) Compute the specific heat.

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Solution Preview

Please refer to the attached document for the solutions.

The Hamiltonian of a nearest neighbors pair interaction is given by:

Where each spin can attain only one of two possible values:

The systems Hamiltonian is therefore:

The transfer matrix is defined as

And:

So in our case:
...

Solution Summary

This solution calculates the partition, free energy, magnetization, energy, and heat for a one-dimensional ising model.

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