Summing up

The table lists the energy and magnetization for both free ends and periodic boundary conditions.

State Energy (FE) Energy (PBC) Magnetization
\( 1= \uparrow\uparrow \) \( -J \) \( -2J \) 2
\( 2=\uparrow\downarrow \) \( J \) \( 2J \) 0
$ 3=\downarrow\uparrow$ \( J \) \( 2J \) 0
$ 4=\downarrow\downarrow$ \( -J \) \( -2J \) -2