Duality in generalized Ising models Franz J. Wegner Institut f¨ ur Theoretische Physik Universit¨ at Heidelberg August 21, 2014 Contribution to the summer school ’Topological aspects of condensed matter physics’ Les Houches August 2014 preliminary Abstract This paper rests to a large extend on a paper I wrote quite awhile ago on Duality in generalized Ising models and phase transitions without local or- der parameter . It deals with Ising models with interactions containing products of more than two spins. In contrast to this old paper I will first give examples before I come to the general statements. Finally I will shortly mention a few, but very important developments, which have some relation to this paper, (i) the basic idea by Ken Wilson of a theory for quarks and gluons, and (ii) the idea by Thomas Weiland to discretize Maxwell’s equations on lattices used here. 1 Introduction In this contribution I consider a number of Ising models, which arose out of the question, whether there is duality for Ising models in dimensions larger than two. Indeed the idea of duality can be used to construct a whole class of such systems, which however, differ from conventional Ising models in some properties. First these models contain products of more than two Ising spins, Secondly they have no longer local order parameters, but they can still have two phases. For a number of these systems the order appears in the expectation value of the product of the spins along a loop, called Wilson-loop. It shows in the limit of large loops an area law at high temperatures and a perimeter law at low temperatures. Such models, where the ising spins are replaced by members of groups, typi- cally by the groups U, SU(2) and SU(3) have become important in high-energy physics in the description of quarks and gluons. In sect. 2 I review the Kramers-Wannier duality for two-dimensional Ising models. In sect. 3 I introduce the model dual to the conventional three- dimensional Ising model. In sect. 4 the general concept of Ising models and 1
duality is introduced. In sect. 5 this is applied to general lattices and in sect. 6 to models on hypercubic lattices. The correlation functions are considered in sect. 7. The basic idea of lattice gauge theory is given in sect. 8 and a useful lattice for the discretization of Maxwell’s equations is mentioned in sect. 9. 2 Kramers-Wannier Duality Kramers and Wannier[6, 9] predicted in 1941 the exact critical temperature of the two-dimensional Ising model on a square lattice. They did this by comparing the high- and the low-temperature expansion for the partition function of this model. Consider a square lattice with N s = N 1 × N 2 lattice points and periodic boundary conditions. There is an Ising spin S i,j = ± 1 at each lattice site. The Hamiltonian reads N 1 N 2 � � H = − J ( S i,j S i,j +1 + S i,j S i +1 ,j ) . (1) i =1 j =1 High temperature expansion ( HTE ) We may rewrite � e − βH = (cosh K + sinh KS i,j S i,j +1 )(cosh K + sinh KS i,j S i +1 ,j ) i,j (cosh K ) N b � = (1 + tanh KS i,j S i,j +1 )(1 + tanh KS i,j S i +1 ,j ) (2) i,j In order to determine the partition function, we may expand this expression in powers of tanh KSS ′ and sum over all spin configurations. This summation yields zero unless all spins appear with even powers. In this latter case the sum is 2 N s . This is the case, when the interaction bonds form closed loops. That is at each lattice site meet an even number of bonds as shown in the upper figs. of fig. 1. The partition function can be expanded Z ( K ) = (cosh K ) N b f (tanh K ) , (3) � c l a l , f ( a ) = (4) l where K = βJ . The coefficients c l count the number of closed loops of length l , c 0 = 1, c 2 = 0, c 4 = N s , c 6 = 2 N s , c 8 = N s ( N s + 9) / 2, etc. and c l = 0 for odd l . Low temperature expansion ( LTE ) We now consider the low temperature expansion on the dual lattice. The dual lattice is obtained by placing a spin S ∗ ( r ∗ ) inside each of the squares (in general polygons) of the original lattice. We muliply spins S ∗ in polygons with a common edge and sum over these products, which in the case of the square lattice writes H ∗ = − J ∗ � ( S ∗ i − 1 / 2 ,j − 1 / 2 S ∗ i − 1 / 2 ,j +1 / 2 + S ∗ i − 1 / 2 ,j − 1 / 2 S ∗ i +1 / 2 ,j − 1 / 2 ) . (5) i,j 2
Figure 1: Examples for closed loops in the HTE and Bloch walls in the LTE on the dual lattice Assuming positive J ∗ the states lowest in energy are those where all S ∗ are equal. There energy is E ∗ min = − N b J ∗ , (6) where N b = 2 N ∗ s is the number of bonds. Excited states are found by flipping some spins. Flipping one spin over costs an excitation energy 2 lJ , if the spin interacts with l other spins. Quite general the excitation energy is given by 2 lJ , if the flipped spins are surrounded by Bloch walls of a total number of l edges. In the case of the square lattice one obtains Z ∗ ( K ∗ ) = 2e N b K ∗ f (e − 2 K ∗ ) (7) with f defined in (4). Comparison Kramers and Wannier argued: If the partition function or equiv- alently the free energy has a singularity at the critical point and no other sin- gularity, then it must be determined by e − 2 K c = tanh K c , (8) which yields √ K c = 1 2 ln(1 + 2) = 0 . 4407 , (9) which indeed turned out to be correct from Onsager’s exact solution[7]. Thus there is a relation between the partition function and similarly the free energy at high ( K < K c ) and low ( K ∗ > K c ) temperatures for ↔ tanh K ∗ = e − 2 K tanh K = e − 2 K ∗ → sinh(2 K ) sinh(2 K ∗ ) = 1 . (10) 3
The square lattice is called self-dual, since the HTE and the LTE are per- formed on the same lattice. This is different for the triangular lattice, where the HTE is performed on the triangular lattice and the LTE on the honeycomb lattice. See fig. 2. Then however the HTE of the triangular lattice and the LTE of the honeycomb lattice are given by the same sum f ( a ), Z hte ( K ) = 2 N s 3 (cosh K ) N b f 3 (tanh K ) , Z lte 6 ( K ) = 2e N b K f 3 (e − 2 K ) , (11) 3 Z hte ( K ) = 2 N s 6 (cosh K ) N b f 6 (tanh K ) , Z lte 3 ( K ) = 2e N b K f 6 (e − 2 K ) , (12) 6 where the number N b of bonds are equal in both lattices and N s 3 = N b / 3 and N s 6 = 2 N b / 3. The coefficients c l in f 3 and f 6 count the number of closed loops on the triangular and the honeycomb lattice, resp. Figure 2: Triangular and dual hexagonal lattice. The thick black triangle indi- cates a product of 3 interactions on the triangular lattice contributing to HTE and the Bloch wall for a flipped spin on the hexagonal lattice. Similarly the thick red hexagon indicates a product of 6 interactions on the hexagonal lattice contributiong to HTE and the Bloch wall of a flipped spin on the triangular lattice. As a consequence the partition functions Z 3 ( K ) and Z 6 ( K ∗ ) are directly related for K and K ∗ given by (10). One cannot directly read of the critical values K c for these lattices. However, the Ising model on the honeycomb lattice can be related to that on the triangular lattice by means of the star-triangle transformation[9]. To do this one eliminates every other spin of the hexago- S 0 e KS 0 ( S 1 + S 2 + S 3 ) one obtains C e K ′ ( S 1 S 2 + S 1 S 3 + S 2 S 3 ) , nal lattice by summing � which yields the Ising model on the triangular lattice. 3 Duality in 3 dimensions The basic question I asked myself, when I started my paper[10] on duality in generalized Ising models was: Does there exist a dual model to the three- 4
dimensional Ising model? It turned out, that there is such a model, but of a different kind of interaction. (Compare also [1]) In order to see this, I consider the low-temperature expansion of the 3d-Ising model on a cubic lattice. I start out from the ordered state and then flip single spins. These single spins are surrounded by closed Bloch walls. The expansion of the partition function is again of the form (4), but now with c 2 = 0, c 4 = 0, c 6 = N s , c 8 = 0, c 10 = 3 N s , c 12 = N s ( N s − 7) / 2, etc. The HTE of the dual model must be given by an interaction such that only closed surfaces yield a contribution. Thus locate a spin at each edge and introduce the interaction as a product of the spins surrounding an elementary square called plaquette. Thus the interaction of the dual model reads � H = − J ( S i +1 / 2 ,j,k +1 / 2 S i +1 / 2 ,j +1 / 2 ,k S i +1 / 2 ,j,k − 1 / 2 S i +1 / 2 ,j − 1 / 2 ,k i,j,k + S i +1 / 2 ,j +1 / 2 ,k S i,j +1 / 2 ,k +1 / 2 S i − 1 / 2 ,j +1 / 2 ,k S i,j +1 / 2 ,k − 1 / 2 + S i +1 / 2 ,j,k +1 / 2 S i,j +1 / 2 ,k +1 / 2 S i − 1 / 2 ,j,k +1 / 2 S i,j − 1 / 2 ,k +1 / 2 ) . (13) It is the sum over three differently oriented plaquettes. They are shown in fig. 3. Figure 3: Elementary cube with spins. The red circles (ellipses) indicate, which four spins are multiplied in the interaction S -independent products of R ( b ) From fig. 3 it is obvious that the product of the six R ( b ) around the cube does not depend on the spin configuration, since each spin appears twice in the product. Gauge invariance This model has a local gauge invariance. Flipping all spins around the corner of a cube does not change the energy of the configuration. As an example in fig. 3 the three spins around the corner close to the center are flipped from the state, in which all spins are aligned upwards. 5
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