Question

In Problem 8.15 you manually computed the energy of a particular state of a 4 x 4 square lattice. Repeat that computation, but this time apply periodic boundary conditions.

Short answer

Therefore, the energy is 4J.

Step-by-step solution

Given information

In Problem 8.15 you manually computed the energy of a particular state of a 4 x 4 square lattice. Apply periodic boundary conditions.

Explanation

In the Ising model, a 4x4 square lattice has 16 sites. When periodic boundary conditions are applied, the energy of a certain state is given by:

Where spins are equal to:

$$\begin{gathered} S_k^{\nu }=+1 \\ {S}_k^{\mu }=-1 \\ {S}_i^{\nu }=S_i^{\mu } \end{gathered}$$

Energy is equal to:

role="math" localid="1647835907601" $$\begin{gathered} E=-J·\sum _{i,j}{S}_i^{\nu }·S_k^{\nu }+J·\sum _{i,j}{S}_i^{\nu }·S_k^{\mu }=-J·(-2)+J·(+2) \\ E=4J \end{gathered}$$