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November 2017
by Professor Bruno Nachtergaele

At the atomic and subatomic scale, the world is governed by the laws of quantum mechanics. Conservation laws of energy and momentum hold just as firmly as we have come to expect from our experience with the classical universe at larger scales. But in quantum mechanics, a system of particles can be in an entangled state, which does not resemble anything we know in daily life. The striking phenomena that highlight how different the quantum world is from our everyday experiences can all be traced back to the property called entanglement, which in essence means that the state of the system as a whole cannot be described in terms of properties of its individual components. Said differently, we cannot deduce the complete state of a system of particles by combining observations made on each particle separately.

Indeed the notion of particle itself is ambiguous. It is not so easy to pin down properties such as a particle’s mass, charge, spin, etc. The creators of quantum mechanics realized that particles should be described as a state of a more fundamental object: a quantum field. The field may describe a given number of particles but it generally also has other meanings. This realization led to a working theory of quantum electrodynamics.

Bogoliubov first realized that interactions between the atoms in condensed matter systems such as liquid Helium may cause the quantum field that describes their collective state to present itself as a field with elementary excitations that are a different kind of particle than the atoms of the original description. These new particles are often referred to as quasiparticles. Bogliubov’s point of view led to the first successful first-principles explanation of a strange state of matter called superfluidity.

A decade later saw the first successful theory of superconductivity by Bardeen, Cooper, and Schrieffer. Superconductivity is another quantum state of matter arising from the interaction between electrons and phonons, the latter being the quantum field describing the vibrations of the atoms in a crystal lattice. Another success of the quantum field point of view of ‘elementary’ particles is the connection between spin and statistics given by the spin-statistics theorem. This theorem says that for quantum fields in 3+1 dimensional Minkowski space there is link between the transformation properties of the particle states under permutations of the particles and space rotations. As a result particles are of one of two kinds: on the one hand there are bosons with state vectors that are symmetric under permutation of the particles (coordinates) and for which the one particle states span an odd-dimensional irreducible representation of SU(2), and on the other hand we have fermions, of which the state vectors are anti-symmetric under permutations and for which the behavior under space rotations is described by an even-dimensional irreducible representation of SU(2).

In the seventies it was discovered that if space is two-dimensional, multi-particle state of identical particles are not required to be symmetric or anti-symmetric under permutations (Leinaas-Myrheim). In particular, states that form a representation of the braid group, instead of the permutation group, are consistent with all known principles of physics. The particles associated with such quantum fields are called anyons (Wilczek).

Two further developments prompted us and many other researchers to study the emergence of anyons in models of condensed matter physics. The first is the tantalizing prospect that anyons may provide a robust implementation of quantum memory useful for quantum computation (Kitaev). The second is the experimental confirmation that anyons occur in fractional quantum Hall systems. Therefore we have reason to believe that anyons are both real and useful. In order to start bridging the gap between toy models that demonstrate the theoretical possibility of a large variety of anyonic condensed matter systems and the experimental observations in systems of strongly correlated electrons, we are making a mathematical analysis of the stability of the anyonic properties of quantum lattice systems under perturbations of the interactions.

This is the topic of Matthew Cha’s dissertation (Ph.D. 2017), as well as recently published and forthcoming joint work with Matthew and Pieter Naaijkens (Marie Sklodowska-Curie Fellow at Davis 2015-17). We proved that the anyonic nature of the excitation spectrum of Kitaev’s abelian quantum double models, and the tensor category of the associated superselection sectors, is robust under a broad class of perturbations.

The mathematical quest to understand the quantum states of matter continues. Anyons and other intriguing questions about our universe are the focus of the new Center for Quantum Mathematics and Physics on campus. The possibility of helping bring quantum computers a step closer to reality only adds to the excitement.