spinor condensates
Strings and things
09/04/11 10:32
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!
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!
Spin-1 microcondensates paper out
09/03/11 22:29
Just out in PRA today is my paper on spin-1 Bose condensates in what I’ve dubbed the microcondensate regime, meaning that there is no variation in the spin state across the system. In some ways this is analogous to studying a single magnetic domain of a ferromagnet, but considerably richer because of the larger phase space afforded by spin-1.
This project turned into a terrific excuse to learn some fascinating things about classical integrable systems -- notably the idea of Hamiltonian monodromy, which relates to the global existence of action-angle variables -- semiclassical quantization, hyperbolic geometry, and the Bethe ansatz.
This system can be reduced to two coupled hyperbolic (or SU(1,1)) spins and the resulting phase space analyzed in some neat ways:
Here the black line is a classical trajectory of one of the hyperbolic spins, and the red lines are projections onto a hemisphere below the hyperbola, and then from the hemisphere to a half plane. The latter is the celebrated Poincaré half-plane model of hyperbolic geometry.
This project turned into a terrific excuse to learn some fascinating things about classical integrable systems -- notably the idea of Hamiltonian monodromy, which relates to the global existence of action-angle variables -- semiclassical quantization, hyperbolic geometry, and the Bethe ansatz.
This system can be reduced to two coupled hyperbolic (or SU(1,1)) spins and the resulting phase space analyzed in some neat ways:
Here the black line is a classical trajectory of one of the hyperbolic spins, and the red lines are projections onto a hemisphere below the hyperbola, and then from the hemisphere to a half plane. The latter is the celebrated Poincaré half-plane model of hyperbolic geometry.