A Monte Carlo algorithm for simulating fermions on Lefschetz thimbles

TL;DR

This work tackles the sign problem by employing Lefschetz thimbles to deform the integration domain to regions where the integrand is real and positive on each thimble. The authors introduce a simple Monte Carlo algorithm that samples a dominant thimble via a contraction map, using only stable upward flow to map far points to a Gaussian near-region and computing a Jacobian determinant to include the residual phase in observables. Applied to a solvable 0+1D fermionic model, the method shows strong agreement with exact results in parameter regimes where a single thimble dominates, and reveals non-negligible contributions from subleading thimbles in other regimes, including in the continuum limit. The results highlight both the potential and the limitations of single-thimble sampling, with clear indications that multi-thimble effects persist in certain parameter regions and even as one approaches the continuum, motivating further exploration in higher dimensions to assess universality and the true role of multiple thimbles.

Abstract

A possible solution of the notorious sign problem preventing direct Monte Carlo calculations for systems with non-zero chemical potential is to deform the integration region in the complex plane to a Lefschetz thimble. We investigate this approach for a simple fermionic model. We introduce an easy to implement Monte Carlo algorithm to sample the dominant thimble. Our algorithm relies only on the integration of the gradient flow in the numerically stable direction, which gives it a distinct advantage over the other proposed algorithms. We demonstrate the stability and efficiency of the algorithm by applying it to an exactly solvable fermionic model and compare our results with the analytical ones. We report a very good agreement for a certain region in the parameter space where the dominant contribution comes from a single thimble, including a region where standard methods suffer from a severe sign problem. However, we find that there are also regions in the parameter space where the contribution from multiple thimbles is important, even in the continuum limit.

Figures (7)

• Figure 1: Schematic representation of the mapping between points $z_f$ on the thimble (black points) and their images $z_n$ (blue points) in the gaussian region (shown as the light blue disk). The novelty of our algorithm is that by sampling the distribution of the blue points we can compute the integral over the whole thimble via the contraction map.
• Figure 2: Left: Probability distribution for the far (top row) and near points (bottom row) for different flow times. We plot 3000 samples from a simulation with $N=2$, $\hat{g}^2=1/6$, $\hat{m}=1$, and $\hat{\mu}=0.7$. Right: Chiral condensate as a function of flow time for the same parameters showing the convergence towards the $T\to\infty$ result. The horizontal line indicates the exact result which is expected to be very close to the one-thimble result for these parameters.
• Figure 3: Left: Near points distribution, $P(z_n)$, for a flow time $T=3$ showing the anisotropy of their distribution. The simulation uses the same parameters as in Fig. \ref{['fig:contraction']}. Middle: Simulation time evolution for the sum of the real parts $\mathop{\mathrm{Re}}\nolimits(z_1 + z_2)$. Top is the simulation using an isotropic proposal and bottom anisotropic. The step-sizes were adjusted to get the same acceptance rate. Right: Plot of the difference $\mathop{\mathrm{Re}}\nolimits(z_1 - z_2)$ which corresponds to the elongated direction in the left panel. Note that for isotropic proposals (top) the autocorrelation time is much longer.
• Figure 4: Left: Phase average in for real phase quenched simulations and the average of the residual phase for single thimble simulations. Right: the chiral condensate with and without the residual phase folded in. The solid line represents the exact result that includes contribution from other thimbles.
• Figure 5: Condensate as a function of chemical potential $\mu$ for the parameters for weak coupling (left) and strong coupling (right). In the top plots the solid lines indicate the exact result. The bottom plots indicate the difference between Monte Carlo and exact results. No discrepancy with the exact result is seen in the weak coupling case but a small statistically-significant difference is seen in the strong coupling case.