- Kramers-Wannier duality
The Kramers-Wannier duality is a
symmetry instatistical physics . It relates the free energy of a two-dimensionalsquare-lattice Ising model at a low temperature to that of another Ising model at a high temperature. It was discovered by Hendrik Kramers andGregory Wannier in 1941. With the aid of this duality Kramers and Wannier found the exact location of thecritical 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.
Derivation
The low temperature expansion for K^*,L^*) is
:Z_N(K^*,L^*) = 2 e^{N(K^*+L^*)} sum_{ P subset Lambda_D} (e^{-2L^*})^r(e^{-2K^*})^s
which by using the transformation
:anh K = e^{-2L*}, anh L = e^{-2K*}
gives
:Z_N(K^*,L^*) = 2( anh K ; anh L)^{-N/2} sum_{P} v^r w^s :2(sinh 2K ; sinh 2L)^{-N/2} Z_N(K,L)
where v = anh K and w = anh L , yielding a relation with the high-temperature expansion.
The relations can 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) = lim_{N ightarrow infty} f_N(K,L) = -kT lim_{N ightarrow infty} 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_c = K_c^* , implying sinh 2K_c = 1 , yielding kT_c = 2.2692J
ee also
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Ising model
*S-duality References
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