Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Ground states of spin systems

The constraints discussed in the main text are similar to those that arise in the physics of 2D spin systems. An example of such a system is the so-called Ising model discussed on page 981. The idea in all such systems is to have an array of spins, each of which can be either up or down. The energy associated with each spin is then given by some function which depends on the configuration of neighboring spins. The ground state of the system corresponds to an arrangement of spins with the smallest total energy. In the ordinary Ising model, this ground state is simply all spins up or all spins down. But in generalizations of the Ising model with more complicated energy functions, the conditions to get a state of the lowest possible energy can correspond exactly to the constraints discussed in the main text. And from the results shown one sees that in some cases random-looking ground states should occur. Note that a rather different way to get a somewhat similar ground state is to consider a spin glass, in which the standard Ising model energy function is used, but multiplied by -1 or +1 at random for each spin.

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From Stephen Wolfram: A New Kind of Science [citation]