Numerical challenges in Quantum Monte Carlo simulations in condensed matter physics

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We will address two outstanding problems in quantum Monte Carlo (QMC) simulations of condensed matter systems. The first challenge is to reach lattices sizes that allow direct comparison with experiments. These large lattice simulations will allow us to study various graphene-based problems including, superlattice structures hosting flat bands and subject to strong long-range Coulomb repulsion, and the study of hydrodynamic flow. The second challenge addresses the sign problem, which often prevents the study of many experimentally relevant systems. A unified approach will be taken for both challenges. We will exploit the special representation of the partition function of the quantum many-body system with continuous auxiliary fields. This approach works to our advantage, since we can use the formalism developed to deal with extremely large systems in lattice quantum chromodynamics, by adopting a stochastic representation of the fermionic determinant. In some cases, it leads to a superior scaling in system size as compared to the standard auxiliary field QMC. As we demonstrate on some examples below, this algorithm, when implemented efficiently on Graphic Processing Units (GPUs) can reach lattices as large as 102x102. For instance, we already have been able to observe the logarithmic divergence of the Fermi velocity for the first time in non-perturbative QMC calculations. The approach using continuous auxiliary fields also allows us to address the sign problem in a novel and exciting way. Cauchy's theorem, which is valid for continuous fields, allows us to shift the integration domain of the functional integral into complex space. It was shown that in some cases we can construct an optimal contour with substantially reduced sign problem. It consists of an ensemble of Lefschetz thimbles, each corresponding to the set of points that role down to a saddle within a steepest descent scheme. Due to the fact that the complex phase of the integrand is constant along each thimble, the sign problem is often substantially weaker along this optimal contour. Thus, the shift into complex space towards this contour, can give us an efficient algorithm to suppress the sign problem. We intend to study various algorithmic approaches to efficiently sample field configurations over curved manifolds in complex space. An additional benefit is that we gain insight on exact saddle points, without any assumptions like uniformity in space or Euclidean time. This allows us to systematically build more accurate quasi-classical approximations.