We received the good news this week that our paper “Emergent topological properties in interacting 1D systems with spin-orbit coupling” by Nikolaos Kainaris and myself has been accepted for publication in Physical Review B. You can read the preprint version of this work on the arXiv here.
Niko is a PhD student in the group of Prof. Mirlin in the theory of condensed matter group in the Karlsruhe Institute of Technology. This is the group I used to work in before I moved to Kent, and I had previously worked with Niko during his diploma project on sources of resistance in helical edges of quantum spin-Hall insulators (which you can read about here if you are interested!) This new work was planned for a while, but most of the calculations were completed by Niko when he had an extended visit to Canterbury last November/December.
The project can be motivated from a number of directions. From the currently fashionable field of topological insulators, the novel property of this work is that the interactions are necessary for the topological properties of the model – even for opening a gap in the spectrum giving it the “insulating” properties. I write “insulating” in quotes because actual the wire remains a metal, and it is only spin excitations that are gapped or insulating, but to begin with, that is an unimportant technical point. In the textbook picture of topological insulators, the topology is a property of the non-interacting band structure, and it is a generally accepted picture that the “topological protection” such systems enjoy (such as perfect conductance) break down in the presence of interactions. There are relatively few studies which are the other way around — where the interactions induce the topology in a strongly correlated system — which is why this particular model is so fascinating.
Another context in which to view this work is in the context of one-dimensional systems with a spin gap (described above, and known in the biz as a “Luther-Emery liquid”). Basically, our emergent state (with the topology) is a state known as a spin-density-wave (SDW for short). This is basically like an antiferromagnet – spins next to each other are in opposite directions, except that the spins aren’t localised on lattice sites, there is only quasi-long-range order (rather than a conventional ordered state), and the electrons (with both charge and spin) can still move, carrying currents through the system as if it were a metal. This SDW state has long been studied in the literature — however there are problems coming up with realistic models where this is expected to be the ground state. The trouble is that the realistic models have spin-rotational symmetry (SU(2) for those who know their groups!) and there is a rigorous argument that unless the SU(2) symmetry is broken, the SDW cannot form. The spin-orbit interaction does break this symmetry, but even then, conventional wisdom suggested that the system cannot be in the SDW state unless a magnetic field is also added. Now, for reasons relating to our desired topological properties of the state, we wanted a model with time-reversal symmetry, so we didn’t want a magnetic field (under time reversal, the magnetic field points in the opposite direction so the symmetry is broken). Fortunately, we didn’t read this previously literature before doing our own calculations – because it turned out that there was a term in the calculation that had been previously missed (technically, this term is the change in the spinon propagator due to the spin-orbit coupling to the charge plasmon modes), and when this is taken into account (the detailed calculation is in the appendix of our paper), there is indeed a realistic parameter regime where one can get a SDW in a time-reversal invariant system.
There is a third way to look at this work, which is from a much more practical point of view. From a technological point of view, quantum wires (which are one atom thick) are the ultimate goal of miniaturisation of normal wires – and the main goal of this is to carry current. The trouble is that impurities in the wire cause resistance, and this becomes more and more of a problem as wires get thinner and thinner – for the ultimate case of a single channel wire (one atom thick), even a single impurity in the wire can turn it from a metal into an insulator (something known as the Kane-Fisher effect), which is clearly not ideal for technological applications. One then has two possibilities – either improve the manufacturing process until you get rid of the impurities; or utilise the cool strongly-correlated quantum physics that goes on inside these ultra-narrow wires to drive the system into a state in which the impurities don’t matter. The best known example of such a state is a superconductor. However as we show, thanks to the topological properties of the emergent SDW state in our model of a quantum wire, we obtain a similar effect, where the current just cannot scatter off the impurities, and so one gets perfect conductance (strictly speaking this is only true at zero temperature, but the resistance remains very low and the system remains metallic at finite temperatures too). While this effect isn’t going to be beating superconductors any time soon, it is a fascinating alternative direction with potential applications in quantum devices.
So it isn’t just about the topology. I’m not that fashionable!