Square-lattice Ising Model
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In statistical mechanics, the two-dimensional square-lattice Ising model was solved by (Lars Onsager 1944) for the special case that the external field H = 0. A solution for the general case for H neq 0 has yet to be found. Construct a dual lattice D as depicted in the diagram. For every configuration { }, a polygon is associated to the lattice by drawing a line on the edge of the dual lattice if the spins separated by the edge are unlike. Since by traversing a vertex...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In statistical mechanics, the two-dimensional square-lattice Ising model was solved by (Lars Onsager 1944) for the special case that the external field H = 0. A solution for the general case for H neq 0 has yet to be found. Construct a dual lattice D as depicted in the diagram. For every configuration { }, a polygon is associated to the lattice by drawing a line on the edge of the dual lattice if the spins separated by the edge are unlike. Since by traversing a vertex of the spins need to change an even number of times so that one arrives at the starting point with the same charge, every vertex of the dual lattice is connected to an even number of lines in the configuration, defining a polygon. Spin configuration on a dual lattice This reduces the partition function to Z_N(K,L) = 2e^{N(K+L)} sum_{P subset Lambda_D} e^{-2Lr-2Ks} summing over all polygons in the dual lattice, where r and s are the number of horizontal and vertical lines in the polygon, with the factor of 2 arising from the inversion of spin configuration.
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