Condensed Matter

Topological aspects of physical systems, including the called topological states of matter, have become hot topics in the frontiers of physics in recent years. Here I would like to present a mathematically "popular" talk for professional physicists for a highlight or overview of how one can systematize knowledge of topological aspects of quantum field theories. Our starting points are Descent Equations and Gauge Structure in Configuration Space in Field Theory. (The audience needs only to know the meaning of "differential forms".)

In this talk, I will present our latest results on the development of algorithms for computer simulations based on tensor network states (TNS). The major part of this talk is concerned with finite projected entangled pair states (PEPS) in 2D: I will discuss their algorithmic properties as well as their performance for typical benchmark Hamiltonians [1, 2]. A minor part of this talk deals with the question whether TNS methods can be useful for density functional theory (DFT).

[1] M. Lubasch, J. I. Cirac, and M.-C. Ba\~{n}uls, New J. Phys. \textbf{16}, 033014 (2014).

[2] M. Lubasch, J. I. Cirac, and M.-C. Ba\~{n}uls, Phys. Rev. B \textbf{90}, 064425 (2014).

Modern numerical methods have revolutionized the practice of science, creating a third discipline between traditional theory and experiment. Perhaps the most widely known and successful technique has been the Monte Carlo method in general, and the Metropolis algorithm in particular. In this talk, I will present a new way of performing unbiased Monte Carlo simulations based on highly-accurate tensor network contractions. The resulting technique inherits the legendary precision of tensor networks without any of the variational bias. From a Monte Carlo point-of-view, the method can be seen as an aggressive multi-sampling technique where each sample may account for the vast majority of the entire partition function resulting a a drastic reduction in sample-to-sample variance (in contrast to standard
configuration-based Monte Carlo, where only a small subset of possible configurations are sampled). The presented results are all classical, though applications to quantum systems and the sign problem will be discussed.