Where Two Dimensions Don’t Fall Flat
The intuitive classification of matter as solid, liquid, or gas, and the transition between these states, is a simple comfort of classical physics. At temperatures far from absolute zero (0 degrees Kelvin), thermal fluctuations, caused by random atomic movement, overpower quantum effects. As a result, the familiar solid, liquid, and gas phases behave according to the laws of classical physics. However, at temperatures near absolute zero, the previously subdued quantum effects emerge and the rules change, giving rise to strange properties like superconductivity and superfluidity.
Based on earlier work, scientists had held that ordered phases and, thus, phase transitions, were impossible in two-dimensional systems. Still, though, below a threshold temperature, thin liquid films would suddenly become superfluids and flat magnetic sheets would become superconductors. In response, David J. Thouless and J. Michael Kosterlitz, two condensed matter physicists, imagined these two-dimensional phase transitions in terms of vector fields with vortices, or spirals with specific properties.
A vortex (pl. vortices) is a structure that describes the spinning or rotation of a vector field. In a fluid, it is a region, like a tornado or whirlpool, in which the flow rotates around an axis located in the middle of the vortex. An analogous vortex can be formulated in a magnetic film. The atoms in the film each have an intrinsic angular momentum, or spin, which can point in any direction depending on the properties of the material. In a vortex, these spins form a spiral, where the axis of the vortex is a point around which the spins are oriented, shown in Figure 1(a).
In both systems, the complementary structure to a vortex is an antivortex in which the spins are oriented in a more complex pattern. As can be seen in Figure 1(b), there are several ways to visualize an antivortex, but the most precise is according to its winding number.
That is, for a vortex at any concentric level, if you follow the spins along the level clockwise, the spins will rotate clockwise, yielding a winding number of 1. Antivortices have winding numbers of -1 so that when you follow the spins in a clockwise direction along a concentric level, the spins will rotate counterclockwise. When thought of this way, it becomes apparent that the superposition of a vortex and antivortex cancels out and yields aligned spins, shown in Figure 1(c). All together, this annihilation can be seen in the animation in Figure 2, below.
To put all this into context, let us turn to the concept of temperature. Temperature is a quantity that expresses the average kinetic energy of the material with which it is associated. The lowest energy state for a substance occurs when all of its spins are aligned, so when the material is very cold, the energy is low, and it will tend toward this aligned state. However, as the two-dimensional film warms up, the energy of the system increases, and vortices and antivortices form. Each vortex and antivortex has a very high energy, but because the two complement one another, pairs of vortices and antivortices have much lower energy than the individuals. Thus, as the film is heated, the spins will form these vortex-antivortex pairs. The hotter the film, the more of these pairs form until a breaking point called the phase transition. Kosterlitz and Thouless explained what has previously been regarded as a bizarre sudden transition into superconductivity or superfluidity by proposing that, at some critical point, the density of these pairs essentially becomes so large that the pairings are no longer distinguishable and the vortices and antivortices unbind. So, the vortices initially couple up until the film heats to the the critical temperature, at which they suddenly separate. Technically, this critical point is the one at which the dissociation of the pairings becomes more favorable than the generation of additional pairings. This occurs because the system favors low free energy. However, as explained previously, vortex-antivortex pairings have lower energy than the individuals, but they also have lower entropy.
Entropy is a thermodynamic quantity that describes the disorder of a given state, and specifically, the number of different configurations of that state. This becomes clear if we imagine a ballroom with four pairs of dancers versus eight individuals. There are clearly more arrangements of the eight individual dancers in the room than of the four pairs. Thus, the entropy of the ballroom when the dancers are paired is lower than when they are not bound to their partners. In the same way, the vortex-antivortex pairings have lower entropy than the individual structures.
The most incredible and elegant part of this theory is that it is universal for any two dimensional system regardless of the material.
Because systems tend toward the lowest possible energy and increasing entropy, there is a balance between the favorability of paired vortices and antivortices versus individual structures. Thus, the phase transition occurs at the critical point of this tradeoff, so that above the critical temperature, there are free vortices and below, there are bound pairings.
This phenomenon is called the Kosterlitz-Thouless transition and opened the door to a completely different construction of the phases of matter. The most incredible and elegant part of this theory is that it is universal for any two dimensional system regardless of the material, and, specifically, explains both superfluidity and superconductivity. As such, this work contributed to the general understanding of topological phases, which won Kosterlitz and Thouless the 2016 Nobel Prize in Physics along with F. Duncan Haldane for their subsequent work and the explanation of a related concept: the quantum Hall effect.