A large part of our activities is the design and implementation of new numerical methods. In particular we work on `diagrammatic’ or `continuous-time’ methods for quantum impurity and lattice models. These algorithms are based on a diagrammatic expansion of the system’s partition function, and they are by now the methods of choice for solving quantum impurity models.

**Example 1: Continuous-time Auxiliary Field Algorithm**

The continuous-time auxiliary field algorithm is based on the partition function expansion of an impurity model into a `weak coupling’ series, coupled with an auxiliary field decomposition. The algorithm can be applied to any Hamiltonian with density – density interactions. Similarly to the stochastic ‘flipping of spins’ in a simple Ising simulation, the algorithm is based on the stochastic insertion and removal of diagrams’ in a diagrammatic configuration space.

Because it is numerically exact and because it is substantially faster than other algorithms, CT-AUX is ideally suited to the simulation of large impurity clusters.

**Example 2: Inchworm Monte Carlo out of equilibrium**

Out of equilibrium, on the Keldysh contour, most standard numerically exact algorithms fail. With a combination of many-body physics and computer science we were able to construct iterative numerically exact algorithms that can avoid the exponential slowing down of standard Monte Carlo methods and propagate quantum systems to almost arbitrary times.

**Example 3: Diagrammatic Monte Carlo for the Dual Fermion method**

Most standard numerical methods are good at getting local or short-distance correlations. However, many physical phenomena are based on long wavelength fluctuations. A way of combining local physics with long wavelength fluctuations is given by the Dual Fermion method, for which we have developed a diagrammatic Monte Carlo method.

**Relevant Publications**

Taming the Dynamical Sign Problem in Real-Time Evolution of Quantum Many-Body Problems

Guy Cohen, Emanuel Gull, David R. Reichman, and Andrew J. Millis

Phys. Rev. Lett. 115, 266802 (2015)

Continuous-time Monte Carlo methods for quantum impurity models,

Emanuel Gull, Andrew J. Millis, Alexander I. Lichtenstein, Alexey N. Rubtsov, Matthias Troyer, Philipp Werner.

Rev. Mod. Phys. 83, 349 (2011).

Diagrammatic Monte Carlo for Dual Fermions,

Sergei Iskakov, Andrey E. Antipov, Emanuel Gull.

Phys. Rev. B 94, 035102 (2016).