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The Potts and Ising Models of Statistical Mechanics The Potts Model The model uses a lattice, a regular, repeating graph, where each vertex is assigned a spin. This was designed to model behavior in ferromagnets; the spins interact with their nearest neighbors align in a low energy state, but entropy causes misalignment. The edges represent pairs of particles which interact. The q-state Potts Model The spins of the model can be represented as vectors, colors, or angles depending on the application. q=2 q=3 q=4 Unit vectors point in the q orthogonal directions n 2 n / q, n 0,1, . . . , q 1. q = 2 is the special case Ising Model Phase Transitions The Potts Model is used to analyze behavior in magnets, liquids and gases, neural networks, and even social behavior. Phase transitions are failures of analycity in the infinite volume limit. For ferromagnetics, this is the point where a metal gains or loses magnetism. Magnetism Magnetism N Temperature Temperature The Hamiltonian In mechanics, the Hamiltonian corresponds to the energy of a system. The lowest energy states exist where all spins are in the same direction. h( ) J ( i , j ) {i , j }E ( G ) Where ω is a state of graph G, σi is the spin at vertex i, δ is the Kronecker delta function, and J is the interaction energy This measures the number of pairings between vertices with the same spins, weighted by the interaction energy. Ising model Hamiltonian For the Ising model, the Kronecker delta may be defined as the dot product of adjacent spins where each spin is valued as the vector <1,0> or <0,1>. 1 0 1 h( ) J i j 0 1 0 {i , j}E ( G ) 0 1 0 1 Where J is uniform throughout the graph. 1 1 0 1 0 0 1 An edge between two like spins has a value of 1, while an edge between 1 1 unlike spins has a value of 0. 1 1 0 0 1 For this state, h(ω) = -14J The Partition Function The probability of a particular state: 1 exp( h ( )) Pr(i T kT 1 exp( h( )) all kT k=1.38x10-23 joules/Kelvin, the Boltzmann constant The denominator is the same for any state, and is called the partition function. 1 J Z exp( h( )) Z exp( i j ) all kT all {i , j } kT 4-cycle example Recall the 1 partition exp( all kT h( )) function: H 4J H 2J H 2J H 2J In this example, the partition function is: H 2J H 2J H 2J H 0 4J 2J 2exp( ) 12exp( ) 2 kT kT H 0 H 2J H 2J H 2J The probability of an all blue spin state: J H 2J H 2J H 2J H 4J exp(4 ) kT 4J 2J 2 exp( ) 12 exp( )2 kT kT Why use another approach? The partition function is difficult to calculate for graphs of useful size, as the number of possible states increases rapidly with the number of vertices. |i | is the number of states |i | 2 N 2 is number of possible spins, and N is the number of vertices An effective approach for estimating the partition function for large lattices is through a transfer matrix. This estimation is exact for infinite lattices. Ising Model Transfer Matrix Define a matrix P such that 1 i P j T exp( J ( i j )) kT For like spins: 1 0 J 1,0 P 0,1 P exp( ) Unlike spins: 0 1 kT 0 1 1,0 P 0,1 P exp(0) 1 1 0 This is a symmetric 2x2 transfer matrix, describing all possible combinations of spins between two vertices J exp( ) 1 kT 1 J exp( ) kT The exact partition function using a transfer matrix Using the transfer matrix for a cycle, the partition function is: 1 all exp( kT h( )) P P ... P P all 1 2 T 2 3 T N 1 N T N 1 T Which can be simplified to become: Z 1 P N 1T As I the identity matrix n T n all i =Tr (P N ) Since σ1 = σ1 in all possible states = i N i Where λ1 and λ2 are the eigenvalues of =1N 2 N the transfer matrix: J J 1 exp( ) 1 2 exp( ) 1 kT kT Lets check if this works: We calculated the partition function for a 4-cycle earlier: 4J 2J 2exp( ) 12exp( )2 kT kT Using the transfer matrix approach, Z 14 2 4 J J Z (exp( ) 1) (exp( ) 1) 4 4 kT kT 4J 2J =2exp( ) 12exp( ) 2 kT kT Phase Transitions As the partition function is positive, the eigenvalue with the largest absolute value must also be positive. And, as N Z i 1 N N i Failure of analycity occurs where two or more eigenvalues have the greatest absolute value. This is proven not occur in the one dimensional Ising model, where exp( J ) 1 and exp( J ) 1 1 2 kT kT but does occur in greater dimensions, or in Potts Models with more spin possibilities. Thank You

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