elucidated by considering a continuous version of the XY model. Here, the discrete spins are replaced by a field representing the spin's angle at any point in space. In this case the angle of the spins must vary smoothly over changes in position. Expanding the original cosine as a Taylor series, the Hamiltonian can be expressed in the continuum approximation as The continuous version of the XY model is often used to model systems that possess order parameters with the same kinds of symmetry, e.g. superfluid helium, hexatic liquid crystals. This is what makes them peculiar from other phase transitions which are always accompanied with a symmetry breaking. Topological defects in the XY model lead to a vortex-unbinding transition from the low-temperature phase to the high-temperature disordered phase. Indeed, the fact that at high temperature correlations decay exponentially fast, while at low temperatures decay with power law, even though in both regimes , is called Kosterlitz–Thouless transition. Kosterlitz and Thouless provided a simple argument of why this would be the case: this considers the ground state consisting of all spins in the same orientation, with the addition then of a single vortex. The presence of these contributes an entropy of roughly , where is an effective length scale (for example, the lattice size for a discrete lattice) Meanwhile, the energy of the system increases due to the vortex, by an amount . Putting these together, the free energy of a system would change due to the spontaneous formation of a vortex by an amount In the thermodynamic limit, the system does not favor the formation of vortices at low temperatures, but does favor them at high temperatures, above the critical temperature . This indicates that at low temperatures, any vortices that arise will want to annihilate with antivortices to lower the system energy. Indeed, this will be the case qualitatively if one watches 'snapshots' of the spin system at low temperatures, where vortices and antivortices gradually come together to annihilate. Thus, the low-temperature state will consist of bound vortex-antivortex pairs. Meanwhile at high temperatures, there will be a collection of unbound vortices and antivortices that are free to move about the plane. To visualize the Ising model, one can use an arrow pointing up or down, or represented as a point colored black/white to indicate its state. To visualize the XY spin system, the spins can be represented as an