Antiferromagnetic interaction

An anti-ferromagnetic interaction acts to anti-align neighboring spins. If the energy is expressed as the sum of all pairs, i, j, over an interaction term J(i,j), times the spin of atom i times the spin of atom j, J<0 is a ferromagnetic interaction and J>0 is an antiferromagnetic interaction. The combination of both can lead to spin glass behavior.

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Unlike ferromagnetism, anti-ferromagnetic interactions can lead to multiple optimal states (ground states—states of minimal energy). In one dimension, the anti-ferromagnetic ground state is an alternating series of spins: up, down, up, down, etc. Yet in two dimensions, multiple ground states can occur.

Consider an equilateral triangle with three spins, one on each vertex. If each spin can take on only two values (up or down), there are 2^3=8 possible states of the system, six of which are ground states! The two situations which are not ground states are when all three spins are up or are all down. In any of the other six states, there will be two favorable interactions and one unfavorable one. This illustrates frustration: the inability of the system to find a single ground state. This is type of magnetic behavior has been found in minerals that have a crystal stacking structure called a Kagome lattice.