Strings and things

Our PRL on the statistical mechanics of 2D polar condensates just came out this week. This is the work of my former postdoc Andrew James (now at Brookhaven), showing that a spin-1 Bose gases has an interesting phase diagram in two dimensions driven by the interplay of two different types of topological defects: vortices and strings.

In general the order parameter in a spin-1 condensate is a complex three component vector. For the polar case, which corresponds to spin-spin interactions of antiferromagnetic sign (the situation prevailing in 23Na), this vector is restricted to be a real vector multiplied by a phase \( \phi=\mathbf{n} e^{i\theta}\). This parametrization has some redundancy: \((\mathbf{n},\theta)\) and \((-\mathbf{n},\theta+\pi)\) describe the same state. An immediate consequence of this is that the elementary vortex in a polar condensate has only a \(\pi\) winding of the phase, and thus half the circulation quantum, of a vortex in a regular superfluid. It must, however, coincide with a disclination in the vector \(\mathbf{n}\).

Below you can see a picture of a pair of such half-vortex / disclination defects, where the blue arrows indicate the phase \(\theta\) and the red arrows the vector \(\mathbf{n}\)


The main point of our paper is that, once you turn on a magnetic field, the quadratic Zeeman effect creates an easy axis anisotropy that causes the \(\mathbf{n}\) to align either parallel or antiparallel to the field. Thus in the picture above the red arrows mostly lie horizontally. However, they still have to reverse going around the center of each of the defects, but now this reversal is confined to a string of a well-defined thickness and energy per length (or tension) set by the field.

The Kosterlitz-Thouless transition mediated by the vortices and the Ising transition mediated by the strings fit together in an interesting way. We’ll have more to say about systems with this kind of phase diagram soon!