A possible solution of the notorious sign problem preventing direct MonteCarlo calculations for systems with non-zero chemical potential is to deformthe integration region in the complex plane to a Lefschetz thimble. Weinvestigate this approach for a simple fermionic model. We introduce an easy toimplement Monte Carlo algorithm to sample the dominant thimble. Our algorithmrelies only on the integration of the gradient flow in the numerically stabledirection, which gives it a distinct advantage over the other proposedalgorithms. We demonstrate the stability and efficiency of the algorithm byapplying it to an exactly solvable fermionic model and compare our results withthe analytical ones. We report a very good agreement for a certain region inthe parameter space where the dominant contribution comes from a singlethimble, including a region where standard methods suffer from a severe signproblem. However, we find that there are also regions in the parameter spacewhere the contribution from multiple thimbles is important, even in thecontinuum limit.
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