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The sign problem poses a significant challenge in Quantum Monte Carlo (QMC) simulations of strongly correlated quantum systems. In principle, QMC is capable of yielding exact results for a given quantum Hamiltonian. However, the sign problem introduces an exponential increase in computational time with system size and inverse temperature. Consequently, its impact confines the practical application of QMC algorithms to systems where the sign problem is either absent or mild enough.

 

After a short review of current approaches to the sign problem, I will concentrate on the recent advancements in the Lefschetz thimbles method. It is based on the idea of manipulating the sign problem by exploiting the Cauchy theorem and shifting the auxiliary fields in complex space. This method naturally expands the mean field approach by starting from exact saddle points and adding more and more fluctuations around them. Thus the method can provide important analytical insights into the physics of the investigated system in parallel to the alleviating the sign problem. I will demonstrate the application of the method to the Hubbard model on bipartite lattice away of half-filling. In particular, I will show how a simple analytical model can describe the exact saddle points of the Hubbard model at half-filling and away of it. I will also discuss the physical meaning of these saddle points and the strategies of QMC sampling of auxiliary fields fluctuating around them. 

 

References:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.045143  
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.014508