The Kosterlitz-Thouless TransitionHenrik Jeldtoft JensenDepartment of MathamticsImperial CollegeKeywords: Generalised rigidity, Topological defects, Two DimensionalXY-model, Superconductivity, Two dimensional Melting, Renormalization1 IntroductionMatter in the universe is organised in a hierarchical structure. At the bottom we haveelementary particles: quarks, gluons, electrons etc. At present we don’t know what theseparticles are made off. But we believe we know that quarks, gluons etc. make up protons,neutrons, etc. These then go together to make atoms, that are the building blocks ofmolecules. Atoms and molecules gives us gas, liquids and solids from which we get starsand planets which are grouped together in galaxies, that then form clusters and eventuallywe arrive at the entire universe. Or from atoms and molecules we get macro molecules likeproteins and DNA, that are the building blocks of organelles, which together form the cells.From cells we get organs, that put together form organisms: animals and plants of a greatvariety of species. The totality of individuals and species constitute the entire ecology.One branch of science is concerned with the breaking-up of systems into smaller andsmaller parts. The behaviour and properties are studied at each respective level. StatisticalMechanics is concerned with the opposite quest. Namely, from the interactions between thecomponents, say atoms, at one given level the aim is to understand the collective coherentbehaviour which emerges as many atoms are but together and the next level if formed.Often the microscopic details of the properties of the individual building blocks are not socrucial. Rather it happens that the collective behaviour is controlled by general propertiesof the interaction between the building “atoms”.In these lectures we shall discuss a particular case, where it is possible to follow indetail, how components at one level go together and form certain collective coherent struc-tures: topological defects or topological charges. These charges can be Coulomb chargesin two dimensions, dislocations in two dimensional crystals, vortices in two dimensionalsuperconductors and more. The interaction between the topological charges depends in allcases logarithmically on the spatial separation and this leads to some very general collectivebehaviour, most spectacular it causes a certain type of phase transition: the Kosterlitz-Thouless transition [1].12 The Two Dimensional XY-ModelWe will use the 2d XY-model as our reference model. The model consists of planar rotors ofunit length arranged on a two dimensional square lattice. The Hamilatonian of the systemis given byH = −JXhi,jiSi· Sj= −JXhi,jicos(θi− θj). (1)Here hi, ji denotes summation over all nearest neighbour sites in the lattice, and θidenotesthe angle of the rotor on site i with respect to some (arbitrary) polar direction in the twodimensional vector space containing the rotors.If we assume that the direction of the rotors varies smoothly from site to site, we canapproximate cos(θi− θj) by the first two terms 1 −12(θi− θj)2in the Taylor expansion ofcos. The sum over the nearest neighbours corresponds to the discrete Laplace operator,which we can express in terms of partial derivatives through θi− θj= ∂xθ for two site iand j which differs by one lattice spacing in the x-direction. This leads to the continuumHamiltonianH = E0+J2Zdr(∇θ)2. (2)Here E0= 2JN is the energy of the completely aligned ground state of N rotors.The thermodynamics of the system is obtained from the partition functionZ = e−βE0ZD[θ] exp{−βJ2Zdr(∇θ)2}, (3)a functional integral over all possible configurations of the director field θ(r). The integralover θ(r) can be divided into a sum over the local minima θvorof H[θ] plus fluctuations θswaround the minimaZ = e−βE0XθvorZD[θsw] exp{−β(H[θvor] +12Zdr1Zdr2θsw(r1)δ2Hδθ(r1)δθ(r2)θsw(r2))}. (4)The field configurations corresponding to local minima of H are solutions to the extremalconditionδHδθ(r)= 0 ⇒ ∇2θ(r) = 0. (5)There are two types of solutions to this equation. The first consists of the ground stateθ(r) = constant. The second type of solutions consist of vortices in the director field (seeFig. 1) and are obtained by imposing the following set of boundary conditions on the cir-culation integral of θ(r):1) For all closed curves encircling the position r0of the centre of the vortexI∇θ(r) · dl = 2πn. (6)2) For all paths that don’t encircle the vortex position r0I∇θ(r) · dl = 0. (7)2Condition 1) imposes a singularity in the director field. Note the circulation integralmust be equal to an integer times 2π since we circle a closed path and therefore θ(r) has topoint in the same direction after traversing the path as it did when we started.We can estimate the energy of a vortex in the following way. The problem is sphericalsymmetric, hence the vortex field θvormust be of the form θ(r) = θ(r). The dependence onr can be found from Eq. 6. We calculate the circulation integral along a circle of radius rcentred at the position r0of the vortex2πn =I∇θ(r) · dl = 2πr|∇θ|. (8)We solve and obtain |∇θ(r)| = n/r. Substitute this result into the Hamiltonian Eq. 2Evor− E0=J2Zdr[∇θ(r)]2(9)=Jn22Z2π0ZLardr1r2(10)= πn2J ln(La). (11)The circulation condition Eq. 6 creates a distortion in the phase field θ(r) that persistsinfinitely far from the centre of the vortex. |∇θ| decays only as 1/r leading to a logarithmicdivergence of the energy. Hence we need to take into account that the integral over r inEq. 10 is cut-off for large r-values by the finite system size L and for small r-values by thelattice spacing a. We recall that our continuum Hamiltonian is an approximation to thelattice Hamiltonian in Eq. 1. A vortex with the factor n in Eq. 6 larger than one is calledmultiple charged. We notice that the energy of the vortex is quadratic in the charge. In anmacroscopically large system even the energy of a single charge vortex will be large.Consider now a pair of single charged vortex and an anti-vortex. When we encirclethe vortex we pick upHdl · ∇θ = 2π and when we encircle the anti-vortex we pick upHdl · ∇θ = −2π. Hence, if we choose a path large enough to enclose both vortices we pickup a circulation of the phase equal to 2π + (−2π) = 0. I.e. the distortion of the phase fieldθ(r) from the vortex–anti-vortex pair is able to cancel out at