Exact diagonalization at the petaflop scale

Exact diagonalization is an unbiased and versatile method to study a large variety of quantum many-body systems, ranging from quantum chemistry and nuclear structure calculations to correlated systems in condensed matter physics and ultracold atomic gases. This project aims at developing and applying a parallel exact diagonalization code for current open problems in the fields of frustrated quantum magnetism and SU(N) magnetism of ultracold atomic gases. This code will be able to solve eigenvalue problems with linear dimensions beyond 1012 states, corresponding - for S=1/2 quantum spins - to systems consisting of about 50 sites. Using this technology we will address the low-energy physics and the entanglement structure of the S=1/2 kagome antiferromagnet, the anomalous excitation spectrum of the S=1/2 triangular lattice antiferromagnet, and their experimental response functions relevant for inelastic neutron scattering, Raman and resonant X-ray scattering. In the field of SU(N) quantum magnetism with alkaline-earth atoms, we will focus on systems with N>3 in the fundamental representation on various lattices, where exotic spin liquid phases have been predicted theoretically in large-N studies. The technology developed in this project is expected to be useful in the future to push the limits in exact diagonalizations of fractional quantum Hall effects at various fractions under debate, and to contribute to the exploration of the rapidly growing field of interacting topological insulators.

Quantum mechanical effects as well as strong interactions between the electrons play a prominent role in solid state materials at lowest temperatures. These two ingredients can lead to fascinating new physical effects, yet the equations describing these systems pose major problems for theoretical physicists. Massive computer simulations on supercomputers are often necessary to gain insight into the behavior of these systems. The goal of our project has been to predict novel states of matter in these materials and to develop simulation software therefore. Chiral spin liquids are an example of such exotic states of matter. We know from particle physics that electrons are elementary particles, i.e. they are not composed of smaller particles. Yet if many strongly correlated electrons are in the state of a chiral spin liquid they astonishingly split up into smaller constituents, called spinons. Although this state of matter has already been proposed in the eighties it remained an open question whether there are systems that realize such a state. In the course of our project we now have finally provided evidence for the emergence of such a state in two dimensional quantum magnets. This provides a first step to the experimental discovery of this new physics. Applications could lie in the field of fault-tolerant quantum computing.These discoveries were made possible by a technical breakthrough in the simulation software we developed. The computational effort for this kind of computations grows exponentially with the number of simulated particles, i.e. for every additional particle we have to use twice the computational resources. Since we are interested in the collective behavior of many particles it is important to simulate as much particles as possible. Our software is now for the first time worldwide able to simulate 50 Spin 1/2 particles. It thus gives us a powerful tool to discover further interesting physical phenomena.