An Ising chain

We have solved the one-dimensional Ising chain with free boundary conditions for either J = 0 or h = 0. We also solved the one-dimensional Ising chain with periodic boundary conditions with both J ≠ 0 and h ≠ 0. For this assignment we will return to the one-dimensional Ising chain with J ≠ 0, but h = 0.

1. Use the same matrix methods we used in class to solve the general one dimensional chain to find the free energy with J ≠ 0, but h = 0. (Do not simply copy the solution and set h equal to zero! Repeat the derivation.)

2. Find the free energy per site FN /N in the limit of an infinite system for both free boundary conditions and periodic boundary conditions. Compare the results.

3. The following problem is rather challenging, but it is worth the effort. A curious problem related to these calculations caused some controversy in the late 1990s. Consider the following facts.

(a) Free boundary conditions. The lowest energy level above the ground state corresponds to having all spins equal to +1 to the left of some point and all spins equal to -1 to the right. Or vice versa. The energy of these states is clearly 2J, so that the thermal probability of the lowest state should be proportional to exp(-2βJ), and the lowest-order term in an expansion of the free energy should be proportional to this factor.

(b) Periodic boundary conditions. The ground state still has all spins equal to 1 ( por -1), but the deviation from perfect ordering must now occur in two places. One of them will look like
··· ++++ ----···
and the other will look like
···---- ++++ ···

The energy of these states is clearly 4J, so the thermal probability of the lowest state will be exp(-4βJ) for all N. In this case, the lowestorder term in an expansion of the free energy should be proportional to exp(-4βJ).

In the limit of an infinite system, this difference is not seen in the free energies per site. Can you find the source of this apparent contradiction?