Simulating Exotic Quantum States of Matter

Summary form only given. Strong quantum effects in correlated materials can give rise to unusual and exotic quantum states of matter. Examples are the so-called supersolid phase (which combines the seemingly contradictory properties of being solid and at the same time superfluid), high-temperature superconductors with exotic superconductivity driven by strong quantum fluctuations, or anyonic quantum liquids. Numerical methods are essential for our understanding of these exotic phases since the strong quantum fluctuations make approximate analytical treatments such as mean-field or weak-coupling approximations unreliable. In this talk I will review the tremendous progress in quantum Monte Carlo algorithms in the past decade, allowing the simulation of millions of quantum spins or bosons with high accuracy and the accurate understanding of phenomena such as supersolidity. I will also explain the origin of the infamous "negative sign problem" which prevents simulations of similar accuracy for electronic system and will show that this problem is NP-hard. The simulation of electronic systems will thus remain a big challenge for the foreseeable future.