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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 Kramers and Gregory Wannier in 1941. With the aid of this duality Kramers and Wannier found the exact location of the critical point for the Ising model on the square lattice.

Similar dualities establish relations between free energies of other statistical models. For instance, in 3 dimensions the Ising model is dual to an Ising gauge model.

Intuitive idea

The 2-dimensional Ising model exists on a lattice, which is a collection of squares in a chessboard pattern. With the finite lattice, the edges can be connected to form a torus. In theories of this kind, one constructs an involutive transform. For instance, Lars Onsager suggested that the Star-Triangle transformation could be used for the triangular lattice. Now the dual of the discrete torus is itself. Moreover, the dual of a highly disordered system (high temperature) is a well ordered system (low temperature). This is because the fourier transform takes a high bandwidth signal (more standard deviation) to a low one (less standard deviation). So one has essentially the same theory with an inverse temperature.

When one raises the temperature in one theory, one lowers the temperature in the other. If there is only one phase transition, it will be at the point at which they cross, at which the temperature is equal. Because the 2D Ising model goes from a disordered state to an ordered state, there is a near one-to-one mapping between the disordered and ordered phases.

The theory has been generalized, and is now blended with many other ideas. For instance, the square lattice is replaced by a circle, random lattice, nonhomogenous torus, triangular lattice, labyrinth, lattices with twisted boundaries, chiral potts model, and many others.

Derivation

Define these variables. The low temperature expansion for (K^*,L^*) is

$Z_N(K^{*},L^{*})=2e^{N(K^{*}+L^{*})}\sum _{P\subset {\mathrm{\Lambda }}_D}(e^{-2L^{*}}{)}^r(e^{-2K^{*}}{)}^s$

which by using the transformation

$\tanh K=e^{-2L*},\tanh L=e^{-2K*}$

gives

$Z_N(K^{*},L^{*})=2(\tanh K\tanh L{)}^{-N/2}\sum _P{v}^r{w}^s$

$=2(\sinh 2K\sinh 2L{)}^{-N/2}{Z}_N(K,L)$

where v = tanh K and w = tanh L. This yields a relation with the high-temperature expansion. The relations can be written more symmetrically as

$\sinh 2K^{*}\sinh 2L=1$

$\sinh 2L^{*}\sinh 2K=1$

With the free energy per site in the thermodynamic limit

$f(K,L)=\underset{N\to \mathrm{\infty }}{\lim }{f}_N(K,L)=-kT\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\log Z_N(K,L)$

the Kramers–Wannier duality gives

$f(K^{*},L^{*})=f(K,L)+\frac{1}{2}kT\log (\sinh 2K\sinh 2L)$

In the isotropic case where K = L, if there is a critical point at K = K_c then there is another at K = K^*_c. Hence, in the case of there being a unique critical point, it would be located at K = K^* = K^*_c, implying sinh 2K_c = 1, yielding kT_c = 2.2692J.

See also

• Ising model
• S-duality

External links

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang