An exact chiral spin liquid with non-Abelian anyons

by Hong Yao, Steven A. Kivelson

This is another interesting development in the field of topological quantum computing that came out during the Aspen work shop on the subject this summer. In the Kitaev model, a bizarre but exactly solvable spin model on the honeycombe lattice, a topological phase with non-abelian topological defects (vortices) can arise in the presence of a magnetic field. Unfortunately, a magnetic field spoils the exact solvability of the model leaving us only with a perturbative understanding of this interesting phase.

Here, Yao and Kivelson extend the Kitaev model to a lattice that is somewhere in between a honeycombe and a kagome lattice. They show that the topological phase with non-abelian excitations can arise spontaneously in this extended model. Furthermore, they were able to analyse the phase exactly. In addition, they make use of a recent discovery by Feng, Zhang and Xiang who solved the Kitaev model by a simple Jordan-Wigner transformation which converts the model into spinless fermions that in turn reduces to a model of free Marajona (charge neutral) fermions. As a result, Yao and Kivelson are able to explain their results in a very simple and intuitive language.

Engineering exotic phases for topologically-protected quantum computation by emulating quantum dimer models

by A. Fabricio Albuquerque, Helmut G. Katzgraber, Matthias Troyer, Gianni Blatter

While at Aspen this summer, I had the opportunity to hear Matthias Troyer introduce this interesting study on the possibility of simply engineering a topological quantum computer. In this paper, they consider the quantum dimer model on a triangular lattice (see Moessner and Sondhi, 2000) which is known to have a gapped resonating dimer (valence bond) phase with topologically protected degenerate ground states (that could be used to store a 0 or a 1, for example). They study a "microscopic" model of a system of Josephson junctions engineered in such a way as to produce the mentioned quantum dimer model at low energies. Their central result, however, is negative: under the best circumstances, this system needs to be at or below 1 mK in order to successfully operate as a storage device of a quantum bit. The lesson to be drawn is then quite simple: to engineer such a phase, one needs to start from a system with very large energy scales so that its low energy sector can still be studied at accessible temperatures.

I find in particular that I am drawn to their numerical technique called CORE. This technique projects the lowest lying energy states of an exactly diagonalized hamiltonian onto a hoped for low energy subspace. The quality of the projection is then analyzed and if successful, provides strong evidence that a particular low energy effective model truly captures the low lying excitations of a particular system. This seems to be a particularly effective analysis of an exact diagonalization study.