Quantum many-body systems can give rise to remarkable collective states of matter that have no counterpart in their classical analogs. Archetypal examples include superfluids, superconductors, and insulating quantum liquids in the context of condensed matter physics and materials science, but collective quantum phenomena are also ubiquitous in nuclear physics, quantum chemistry, high-energy physics, traditional atomic physics as well as ultracold atoms.
In connecting such complex emergent behavior to a microscopic picture in terms of short-ranged interactions between the elementary quantum mechanical degrees of freedom analytical approaches on the level of mean-field theory or effective field theory descriptions can often provide qualitative guidance and occasionally understanding. However, the validity of such analytical approaches and their underlying abstractions are often a matter of debate and unbiased numerical simulations play a crucial role in verifying these assumptions. Often quantitative guidance in mapping out phase diagrams (in terms of the microscopic interactions) and the respective phase transitions is obtained solely by numerical methods. In the interplay between theory and experiment, computational physics has finally established itself as a vital discipline for quantum many-body physics this past decade.
The continuing demand for new methods has motivated the development of a plethora of novel algorithms over recent years, which significantly extend the applicability of well-established techniques such as quantum Monte Carlo (QMC) methods and the density matrix renormalization group (DMRG) approach. In a remarkable interplay between quantum information theory and computational condensed matter physics time-dependent renormalization group techniques have been developed that allow to study one-dimensional quantum systems far out of equilibrium. Recent extensions made possible by thorough reformulation of DMRG in terms of matrix product states (MPS) and the introduction of novel variational algorithms based on tensor product state bear the intriguing potential to simulate the two-dimensional fermion problem or frustrated quantum spin systems which both harbor the infamous sign problem making them intractable for QMC approaches. In classical statistical mechanics it has been realized that extended ensemble sampling techniques can overcome the equilibration problem caused by competing interactions or phases and the resulting rough energy landscapes even when there are no non-local update techniques available. The applicability of these ideas to quantum systems still needs to be fully explored. Finally, a new class of continuous-time impurity solvers has recently been developed for dynamical mean-field theory (DMFT) calculations that will allow to study the many-fermion problem in this approximation at considerably lower temperatures.
Development of methods has been rapid and there are no text books available for these new methods. The motivation for the tutorial is to present the fundamentals of simulations of strongly correlated systems, to the participants, and to introduce them to existing open-source codes and recent developments in the field.
Contents and objectives
This tutorial will cover this broad range of modern numerical approaches to strongly correlated quantum many-body systems and materials. In particular, there will be a series of tutorials that introduce the established numerical techniques in more detail as well as open-source implementations of these, such as the ALPS project (see alps.comp-phys.org), which has been founded amongst others by the organizers of the proposed workshop - and followed by hands-on sessions in the afternoon.
The tutorial shall expose the students to the wide variety of methods in the field as well as recent developments, and teach them how to use the ALPS open source simulation codes.
It will cover the following topics:
classical Monte Carlo
quantum Monte Carlo for lattice models
density matrix renormalization group (DMRG)
introduction to the ALPS software
hands-on sessions with ALPS
Plan of lectures
Day 1, morning: Introduction to ALPS and classical Monte Carlo
Day 1, afternoon: hands-on session with classical Monte Carlo in ALPS
Day 2, morning: exact diagonalization and DMRG
Day 2, afternoon: hands-on session with exact diagonalization in ALPS
Day 3, morning: dynamics in DMRG and recent developments
Day 3, afternoon: hands-on session with DMRG in ALPS
Day 4, morning: introduction to quantum Monte Carlo and the loop algorithm
Day 4, afternoon: hands-on session with the QMC loop algorithm in ALPS
Day 5, morning: the QMC worm and directed loop algorithms, and extended ensembles
Day 5, afternoon: hands-on session with the other QMC algorithm in ALPS