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.
[1] 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.
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.
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.
If there is only one phase transition, it will be at the point at which they cross, at which the temperatures are 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.
One of the consequences of Kramers–Wannier duality is an exact correspondence in the spectrum of excitations on each side of the critical point.
This was recently demonstrated via THz spectroscopy in Kitaev chains.
In the two-dimensional square lattice Ising model the number of horizontal and vertical links are taken to be equal.
for (K*,L*) obtained from the standard expansion is the factor 2 originating from a spin-flip symmetry for each
stands for summation over closed polygons on the lattice resulting in the graphical correspondence from the sum over spins with values
This yields a mapping relation between the low temperature expansion
can be written more symmetrically as With the free energy per site in the thermodynamic limit the Kramers–Wannier duality gives 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* = K*c, implying sinh 2Kc = 1, yielding The result can also be written and is obtained below as The Kramers-Wannier duality appears also in other contexts.
The zero of the beta function is usually related to a symmetry - but only if the zero is unique.
yields (obtained with MAPLE) Only the second solution is real and gives the critical value of Kramers and Wannier as