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The tempered Lefschetz thimble method (TLTM) is a parallel-temperingalgorithm towards solving the numerical sign problem. It tames both the signand ergodicity problems simultaneously by tempering the system with the flowtime of continuous deformations of the integration region. In this article,after reviewing the basics of the TLTM, we explain a new algorithm within theTLTM that enables us to estimate the expectation values precisely with acriterion ensuring global equilibrium and the sufficiency of the sample size.To demonstrate the effectiveness of the algorithm, we apply the TLTM to thequantum Monte Carlo simulation of the Hubbard model away from half filling on atwo-dimensional lattice of small size, and show that the obtained numericalresults agree nicely with exact values.

Tempered Lefschetz thimble method and its application to the Hubbard model away from half filling