Topological phases of matter : beyond two dimensions – TopO

A computer exploiting quantum-mechanical properties such as superposition and entanglement would run more efficient algorithms than a classical machine. It would also be able to simulate many-body quantum systems much faster. The greatest challenge posed by quantum computers is decoherence. Certain zero-temperature phases of matter have anyons, quasiparticles that are neither bosons nor fermions. These phases exhibit topological order and have recently received a lot of attention as a promising approach to fault-tolerant quantum computation. They can store quantum information non-locally in a way that is intrinsically robust to local decoherence.

Topological phases of matter is an extremely active and highly competitive field of research. Dozens of preprints appear on the arXiv every day, and many workshops are held all year long. Indeed, after the 2016 Nobel Prize in physics was awarded for the theoretical discoveries of topological phase transitions and topological phases of matter, interest in this line of research will certainly continue to grow in the foreseeable future.

Still, the lack of a general framework for topological order has proven to be a stumbling block and many challenging questions remain unanswered. Despite promising candidates among fractional quantum Hall states, to date no experiment has conclusively proven the existence of non-Abelian anyons. It is therefore of crucial importance to look for other experimental realizations of these quasiparticles.

In particular the prospect of three dimensional topological phases of matter supporting anyonic excitations is extremely exciting. Such systems could potentially transcend the fragile 2D FQHE by providing a more practical and robust physical implementation of topological quantum computing, using bulk materials instead of layered ones. While non-Abelian statistics in three or more spatial dimensions are not possible for point-like excitations, collective excitations need not be point-like : defect loops and membranes are known to occur in condensed matter physics. For instance, superfluids and superconductors exhibit vortex lines. Topological phases in 3D are expected to be of a variety not yet known, supporting strongly correlated surface states (just like FQH states exhibit Luttinger liquid behavior at their edge) and hosting exotic collective excitations : both point-like quasiparticles and extended defect lines.

This proposal’s objective is to explore the emergence of topological order in two and three dimensions and to propose new physical realizations in solid state materials and ultracold atomic gases. This proposal also addresses fundamental questions about the nature of topological order, such as its holographic properties in two dimensions and the nature of its (extended) collective excitations in higher dimensions. A better understanding of these questions will ultimately lead to the proposal of new, potentially more robust experimental realizations of anyonic systems.

The time is ripe to address these questions. On the one hand the recent discovery of time-reversal invariant topological insulators in two and three dimensions has provided a natural hunting ground for anyonic systems. Moreover recent advances in the control of ultracold quantum gases have opened the way for quantum simulations of many-body systems, and they will thus provide direct simulations to complement condensed matter experiments. On the other hand promising new analytical and numerical methods, based on quantum information, have been developed in the last couple of years. These methods - dubbed “Tensor Network” - have just begun to be applied to strongly correlated systems with unprecedented success. Such methods provide for the first time the correct framework for understanding topological phases of matter, offering a way to overcome the technical and conceptual difficulties posed by topological order.