2010 IUPAP Young Scientist Prize in Computational Physics

By Prof. Philipp Werner, ETHZ

We have developed a new class of algorithms, 'Continuous time Quantum Monte Carlo methods for fermions', which have opened new directions in the theoretical study of materials with strong electron correlations. Strongly correlated materials such as transition metal and actinide compounds, exhibit remarkable physical properties including superconductivity at unprecedentedly high temperatures (cuprates and pnictides), colossal magentoresistance (manganites), high temperature ferromagnetism (double perovskites), metal-insulator transitions (vanadates), and volume collapse transitions (Cerium and Plutonium).
The striking properties of these materials arise from the combination of quantum mechanics and lattice structure, making  the theoretical description and numerical simulation of these materials very challenging.

In fact, even the simplest models for correlated fermions cannot be solved numerically on a large lattice.  In order to make progress, simplifications are needed, and one such simplified description of correlated lattice models is dynamical mean field theory (DMFT) [1].
This theoretical framework, which has been developed in the 1990s, replaces the unsolvable lattice model by a computationally more tractable quantum impurity model and a self-consistency condition. One can think of the impurity as representing a single site of the lattice, while the self-consistency condition fixes the hoppings between the impurity and the uncorrelated bath in such a way that the local dynamics of the lattice problem is reproduced. More recently, DMFT has been combined with input from band structure calculations to enable "ab-initio" simulations of strongly correlated materials (LDA +DMFT) [2].

A DMFT calculation requires the repeated numerical solution of a quantum impurity model. This step is the bottleneck in the iterative calculation and the efficiency and flexibility of the so-called impurity solver therefore determines what kind of problems can be addressed. Over the past few years, our work [3] and that of others [4] has led to significant advances, opening new avenues for progress in the study of correlated electron systems.Our work is based on an insight, also dating back to the early 1990s, that  the diagrammatic expansions of a partition function can be sampled to all relevant orders by Monte Carlo techniques.

These ideas led to the development of efficient quantum Monte Carlo techniques for models involving bosons on a lattice [5]. However, as formulated these techniques could not be applied to study electrons, which are fermions with different quantum mechanical behavior than bosons.

We have shown how to adapt the Monte Carlo techniques to solve the fermion quantum impurity models arising in dynamical mean field theory (Fig 1). The continuous-time Monte Carlo techniques are not only efficient [6], which means that they enable investigations of the low- temperature physics, they are also very flexible and free of systematic errors, such as truncation or time discretization errors.
Calculations which required access to supercomputer facilities a few years ago can now be run on desktop machines. The new methods enable the study of multi-orbital models with general interactions [7], as required in LDA+DMFT calculations, or large impurity clusters in the single-band case [8]. Extensions to models with phonon couplings [9] or dynamically screened interactions [10] are possible and computationally cheap. The generalization to nonequilibrium systems [11] enables the study of transport through nano-structures [12] or simulations of the relaxation dynamics of bulk systems [13]. Diagrammatic impurity solvers are meanwhile used by most DMFT groups worldwide and will play an important role in the further development and improvement of the DMFT formalism.

The considerable impact of these methodological innovations was recognized by the Commission C20 of the International Union of Pure and Applied Physics, which awarded the 2010 Young Scientist Prize in Computational Physics to Philipp Werner, SNF Professor at the Institute for Theoretical Physics, ETH Zürich,  "For the development and implementation of quantum Monte Carlo methods which have transformed the study of interacting electrons in solids".

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