- Square-lattice Ising model
The

**two-dimensional square-lattice**was solved byIsing model Lars Onsager in1944 for the special case that the external field "H" = 0. The general case for $H\; eq\; 0$ has yet to be found.Consider the 2D

Ising model on a square lattice $Lambda$ with "N" sites, with periodicboundary condition s in both the horizontal and vertical directions, which effectively reduces thegeometry of the model to atorus . In a general case, the horizontalcoupling "J" is not equal to the coupling in the vertical direction, "J*". With an equal number of rows and columns in the lattice, there will be "N" of each. In terms of:$K\; =\; eta\; J$:$L\; =\; eta\; J*$

where $eta\; =\; 1/(kT)$ where "T" is

absolute temperature and "k" isBoltzmann's constant , thepartition function $Z\_N(K,L)$ is given by:$Z\_N(K,L)\; =\; sum\_\{\{sigma\; exp\; left(\; K\; sum\_\{langle\; ij\; angle\_H\}\; sigma\_i\; sigma\_j\; +\; L\; sum\_\{langle\; ij\; angle\_V\}\; sigma\_i\; sigma\_j\; ight).$

**Dual lattice**Consider a configuration of spins $\{\; sigma\; \}$ on the square lattice $Lambda$. Let "r" and "s" denote the number of unlike neighbours in the vertical and horizontal directions respectively. Then the summand in $Z\_N$ corresponding to $\{\; sigma\; \}$ is given by

:$e^\{K(N-2s)\; +L(N-2r)\}$Construct a dual lattice $Lambda\_D$ as depicted in the diagram. For every configuration $\{\; sigma\; \}$, 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 $lambda$ 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. 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.

**Low-temperature expansion**At low temperatures, "K, L" approach infinity, so that as $T\; ightarrow\; 0,\; e^\{-K\},\; e^\{-L\}\; ightarrow\; 0$, so that

:$Z\_N(K,L)\; =\; 2\; e^\{N(K+L)\}\; sum\_\{\; P\; subset\; Lambda\_D\}\; e^\{-2Lr-2Ks\}$

defines a low temperature expansion of $Z\_N(K,L)$.

**High-temperature expansion**Since $sigma\; sigma\text{'}\; =\; pm\; 1$ one has

:$e^\{K\; sigma\; sigma\text{'}\}\; =\; cosh\; K\; +\; sinh\; K(sigma\; sigma\text{'})\; =\; cosh\; K(1+\; anh\; K(sigma\; sigma\text{'})).$

Therefore:$Z\_N(K,L)\; =\; (cosh\; K\; cosh\; L)^N\; sum\_\{\{\; sigma\; prod\_\{langle\; ij\; angle\_H\}\; (1+v\; sigma\_i\; sigma\_j)\; prod\_\{langle\; ij\; angle\_V\}(1+wsigma\_i\; sigma\_j)$

where $v\; =\; anh\; K$ and $w\; =\; anh\; L$. Since there are "N" horizontal and vertical edges, there are a total of $2^\{2N\}$ terms in the expansion. Every term corresponds to a configuration of lines of the lattice, by associating a line connecting "i" and "j" if the term $v\; sigma\_i\; sigma\_j$ (or $w\; sigma\_i\; sigma\_j)$ is chosen in the product. Summing over the configurations, using

:$sum\_\{sigma\_i\; =\; pm\; 1\}\; sigma\_i^n\; =\; egin\{cases\}\; 0\; mbox\{for\; \}\; n\; mbox\{\; odd\}\; \backslash \; 2\; mbox\{for\; \}\; n\; mbox\{\; even\}\; end\{cases\}$

shows that only configurations with an even number of lines at each vertex (polygons) will contribute to the partition function, giving

:$Z\_N(K,L)\; =\; 2^N(cosh\; K\; cosh\; L)^N\; sum\_\{P\; subset\; Lambda\}\; v^r\; w^s$

where the sum is over all polygons in the lattice. Since tanh "K", tanh "L" $ightarrow\; 0$ as $T\; ightarrow\; infty$, this gives the high temperature expansion of $Z\_N(K,L)$.

The two expansions can be related using the

Kramers-Wannier duality .**References*** R.J. Baxter, "Exactly solved models in statistical mechanics", London, Academic Press, 1982

*Wikimedia Foundation.
2010.*

### См. также в других словарях:

**Ising model**— The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It has since been used to model diverse phenomena in which bits of information, interacting in pairs, produce collectiveeffects.Definition… … Wikipedia**Spherical model**— The spherical model in statistical mechanics is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T.H. Berlin and M. Kac. It has the remarkable property that when applied to systems of dimension d greater than four … Wikipedia**Vertex model**— A vertex model is a type of statistical mechanics model in which the Boltzmann weights are associated with a vertex in the model (representing an atom or particle). This contrasts with a nearest neighbour model, such as the Ising model, in which… … Wikipedia**Kramers-Wannier duality**— The Kramers Wannier duality is a symmetry in statistical physics. It relates the free energy of a two dimensional square lattice Ising model at a low temperature to that of another Ising model at a high temperature. It was discovered by Hendrik… … Wikipedia**List of mathematics articles (S)**— NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia**Exchange interaction**— In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap.… … Wikipedia**Geometrically frustrated magnet**— Geometrical frustration and ice rules The word frustration was introduced to describe the situation where a system cannot simultaneously minimize the interaction energies between its components [cite journal last = Schiffer first = P. authorlink … Wikipedia**Scientific phenomena named after people**— This is a list of scientific phenomena and concepts named after people (eponymous phenomena). For other lists of eponyms, see eponym. NOTOC A* Abderhalden ninhydrin reaction Emil Abderhalden * Abney effect, Abney s law of additivity William de… … Wikipedia**Rule 184**— is a one dimensional binary cellular automaton rule, notable for solving the majority problem as well as for its ability to simultaneously describe several, seemingly quite different, particle systems:* Rule 184 can be used as a simple model for… … Wikipedia**Corner transfer matrix**— In statistical mechanics, the corner transfer matrix describes the effect of adding a quadrant to a lattice. Introduced by Rodney Baxter in 1968 as an extension of the Kramers Wannier row to row transfer matrix, it provides a powerful method of… … Wikipedia