My watch list
my.chemeurope.com  

my.chemeurope.com

With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.

  • My watch list
  • My saved searches
  • My saved topics
  • My newsletter
Register free of charge
Login  
eng  
DeutschEnglishFrançaisEspañol

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 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.

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

\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 = tanhK and w = tanhL, yielding a relation with the high-temperature expansion.

The relations can written more symmetrically as

sinh2K * sinh2L = 1
sinh2L * sinh2K = 1

With the free energy per site in the thermodynamic limit

f(K,L) = \lim_{N \rightarrow \infty} f_N(K,L) = -kT \lim_{N\rightarrow \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 = Kc, 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 sinh2Kc = 1, yielding kTc = 2.2692J

See also

References

  • H. A. Kramers and G. H. Wannier (1941). "Statistics of the two-dimensional ferromagnet". Physical Review 60: 252-262.
  • J. B. Kogut (1979). "An introduction to lattice gauge theory and spin systems". Reviews of Modern Physics 51: 659-713.
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Kramers-Wannier_duality". A list of authors is available in Wikipedia.
Your browser is not current. Microsoft Internet Explorer 6.0 does not support some functions on Chemie.DE