Editing Potts model (section)

=== Interaction energy ===
The interaction between the spins is then given by a [[continuous function (topology)|continuous function]] ''V'' : ''Q''<sup>'''Z'''</sup> → '''R''' on this topology. ''Any'' continuous function will do; for example
: <math>V(s) = -J\delta(s_0,s_1)</math>
will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of ''s''<sub>0</sub>,  ''s''<sub>1</sub> and ''s''<sub>2</sub> will describe a next-nearest neighbor interaction.  A function ''V'' gives interaction energy between a set of spins; it is ''not'' the Hamiltonian, but is used to build it. The argument to the function ''V'' is an element ''s'' ∈ ''Q''<sup>'''Z'''</sup>, that is, an infinite string of spins. In the above example, the function ''V'' just picked out two spins out of the infinite string: the values ''s''<sub>0</sub> and ''s''<sub>1</sub>.  In general, the function ''V'' may depend on some or all of the spins; currently, only those that depend on a finite number are exactly solvable.

Define the function ''H<sub>n</sub>'' : ''Q''<sup>'''Z'''</sup> → '''R''' as
: <math>H_n(s)= \sum_{k=0}^n V(\tau^k s)</math>

This function can be seen to consist of two parts: the self-energy of a configuration [''s''<sub>0</sub>, ''s''<sub>1</sub>, ..., ''s<sub>n</sub>''] of spins, plus the interaction energy of this set and all the other spins in the lattice. The {{nowrap|''n'' → ∞}} limit of this function is the Hamiltonian of the system; for finite ''n'', these are sometimes called the '''finite state Hamiltonians'''.