Kramers–Wannier duality

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.Somendra M. Bhattacharjee, and Avinash Khare, Fifty Years of the Exact Solution of the Two-Dimensional Ising Model by Onsager (1995), {{arxiv|cond-mat/9511003}} 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 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. For instance, the square lattice is replaced by a circle,{{arXiv|cond-mat/9805301}}, Self-dual property of the Potts model in one dimension, F. Y. Wu random lattice,{{arXiv|hep-lat/0110063}}, Dirac operator and Ising model on a compact 2D random lattice, L.Bogacz, Z.Burda, J.Jurkiewicz, A.Krzywicki, C.Petersen, B.Petersson nonhomogeneous torus,{{arXiv|hep-th/9703037}}, Duality of the 2D Nonhomogeneous Ising Model on the Torus, A.I. Bugrij, V.N. Shadura triangular lattice,{{arXiv|cond-mat/0402420}}, Selfduality for coupled Potts models on the triangular lattice, Jean-Francois Richard, Jesper Lykke Jacobsen, Marco Picco labyrinth,{{arXiv|solv-int/9902009}}, A critical Ising model on the Labyrinth, M. Baake, U. Grimm, R. J. Baxter lattices with twisted boundaries,{{arXiv|hep-th/0209048}}, Duality and conformal twisted boundaries in the Ising model, Uwe Grimm chiral Potts model,{{arXiv|0905.1924}}, Duality and Symmetry in Chiral Potts Model, Shi-shyr Roan and many others.

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.Morris, C. M., et al. "Duality and domain wall dynamics in a twisted Kitaev chain." Nature Physics 17.7 (2021): 832-836.

We define first the variables. In the two-dimensional square lattice Ising model the number of horizontal and vertical links are taken to be equal. The couplings $J,\; J\text{'}$ of the spins $\backslash sigma\_i$ in the two directions are different, and one sets $K^*=\backslash beta\; J$ and $L^*\; =\backslash beta\; J\text{'}$ with $\backslash beta\; =\; 1/kT$.

The low temperature expansion of the $N$ spin partition function $Z\_N$ for (K^*,L^*)

obtained from the standard expansion

:::$Z\_N(K^*,L^*)\; =\; 2\backslash sum\_\{P\; \backslash subset\; \backslash Lambda\_D\}\; e^\{K^*(N-2s)\}e^\{L^*(N-2r)\}$

is

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

the factor 2 originating from a spin-flip symmetry for each $P$.

Here the sum over $P$ stands for summation over closed polygons on the lattice resulting in the graphical correspondence from the sum over spins with values $\backslash pm\; 1$.

By using the following transformation to variables $(K,\; L)$, i.e.

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

one obtains

:::$Z\_N(K^*,L^*)\; =\; 2(\backslash tanh\; K\; \backslash ;\; \backslash tanh\; L)^\{-N/2\}\; \backslash sum\_\{P\}\; v^r\; w^s$

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

where $v\; =\; \backslash tanh\; K$ and $w\; =\backslash tanh\; L$ . This yields a mapping relation between the low temperature expansion $Z\_N(K^*,\; L^*)$ and the high-temperature expansion $Z\_N(K,L)$ described as duality (here Kramers-Wannier duality). With the help of the relations

:::$\backslash tanh\; 2x\; =\; \backslash frac\{2\backslash tanh\; x\}\{1+\backslash tanh^2x\},\; \backslash ;\; \backslash sinh\; 2x\; =\; 2\backslash sinh\; x\backslash cosh\; x$

the above hyperbolic tangent relations defining $K$ and $L$ can be written more symmetrically as

:::$\backslash ,\; \backslash sinh\; 2K^*\; \backslash sinh\; 2L\; =\; 1,\; \backslash ;\backslash ;\; \backslash sinh\; 2L^*\; \backslash sinh\; 2K\; =\; 1.$

With the free energy per site in the thermodynamic limit

:::$f(K,L)\; =\; \backslash lim\_\{N\; \backslash rightarrow\; \backslash infty\}\; f\_N(K,L)\; =\; -kT\; \backslash lim\_\{N\backslash rightarrow\; \backslash infty\}\; \backslash frac\{1\}\{N\}\; \backslash log\; Z\_N(K,L)$

the Kramers–Wannier duality gives

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

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

:::$kT\_c\; =\; 2.2692J$.

The result can also be written and is obtained below as

:::$e^\{2K\_c\}=\; 1+\backslash sqrt\{2\}.$

The Kramers-Wannier duality appears also in other contexts. P. Severa, Quantum Kramers-Wannier duality and its topology, hep-th/9803201P. Severa, (Non-)Abelian Kramers-Wannier duality and topological field theory, hep-th/0206162B.N. Shalaev, S.A. Antonenko and A.I. Sokolov, Five-loop $\backslash sqrt\{\backslash epsilon\}$ expansions for random Ising model and marginal spin dimensionality for cubic systems, cond-mat/9803388 We consider here particularly the two-dimensional theory of a scalar field $\backslash Phi.$B.N. Shalaev, Kramers-Wannier symmetry and strong-weak coupling duality in the two-dimensional $\backslash Phi^4$ field model, cond.mat/0110205G. Jug and B.N. Shalaev, Duality symmetry, strong coupling expansion and universal critical amplitudes in two-dimensional $\backslash Phi^4$field models, J. Phys. A32 (1999) 7249, cond-mat/9908068 In this case a more convenient variable than $\backslash sinh\; (2K)$is

:::$s(K):=\; \backslash exp(2K)\backslash tanh(K).$

With this expression one can construct the self-dual quantity

:::$\backslash xi^2:=\backslash frac\{s(K)\}\{(1-s(K))^2\}=\backslash frac\{s(K^*)\}\{(1-s(K^*))^2\}.$

In field theory contexts the quantity $\backslash xi$ is called correlation length. Next set

:::$\backslash beta(K):=\; \backslash xi\backslash frac\{dK\}\{d\backslash xi\}=\backslash frac\{2s(K)(1-s(K))\}\{(1+s(K))[ds(K)/dK]\}.$

This function is the beta function of renormalization theory. Now suppose there is a value $K^*$ of $K$ for which $\backslash beta(K^*)\; =0$, i.e. $s(K^*)=1$. The zero of the beta function is usually related to a symmetry - but only if the zero is unique. The solution of $s(K^*)=+1$ yields (obtained with MAPLE)

:::$K^*=\; i\backslash pi\; +\; \backslash frac\{1\}\{2\}\backslash ln(1+\backslash sqrt\{2\}),\; \backslash ;\backslash ;\; \backslash frac\{1\}\{2\}\backslash ln(1+\backslash sqrt\{2\}),\; \backslash ;\backslash ;\; \backslash ln\backslash sqrt\{\{1-\backslash sqrt2\}\}$.

Only the second solution is real and gives the critical value of Kramers and Wannier as

:::$\backslash exp(2K^*)\; =\; 1\; +\backslash sqrt\{2\}$.

• Ising model
• S-duality
• Z N model

• {{cite journal | author=H. A. Kramers and G. H. Wannier | title=Statistics of the two-dimensional ferromagnet| journal = Physical Review | volume=60 | pages=252–262 | year=1941 | issue=3| doi=10.1103/PhysRev.60.252|bibcode = 1941PhRv...60..252K }}
• {{cite journal | author=J. B. Kogut | title=An introduction to lattice gauge theory and spin systems| journal = Reviews of Modern Physics | volume=51 | pages=659–713 | year=1979 | issue=4| doi=10.1103/RevModPhys.51.659 | bibcode=1979RvMP...51..659K}}

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Category:Statistical mechanics

Category:Exactly solvable models

Category:Lattice models