Surmounting the sign problem in nonrelativistic calculations: A case study with mass-imbalanced fermions

The calculation of the ground state and thermodynamics of mass-imbalanced Fermi systems is a challenging many-body problem. Even in one spatial dimension, analytic solutions are limited to special configurations and numerical progress with standard Monte Carlo approaches is hindered by the sign problem. The focus of the present work is on the further development of methods to study imbalanced systems in a fully nonperturbative fashion. We report our calculations of the ground-state energy of mass-imbalanced fermions using two different approaches which are also very popular in the context of the theory of the strong interaction (quantum chromodynamics, QCD): (a) the hybrid Monte Carlo algorithm with imaginary mass imbalance, followed by an analytic continuation to the real axis; and (b) the complex Langevin algorithm. We cover a range of on-site interaction strengths that includes strongly attractive as well as strongly repulsive cases which we verify with nonperturbative renormalization group methods and perturbation theory. Our findings indicate that, for strong repulsive couplings, the energy starts to flatten out, implying interesting consequences for short-range and high-frequency correlation functions. Overall, our results clearly indicate that the complex Langevin approach is very versatile and works very well for imbalanced Fermi gases with both attractive and repulsive interactions.

Figure 1

Ground-state energy of $N=3+3$ fermions as a function of imaginary (top) and real (bottom) mass imbalance for various couplings $\gamma$ from weak to strong attractive interaction (lines ordered from top to bottom). Top: iHMC results for imaginary $\overline{m}$ (black diamonds, statistical error bars are of the size of the symbols) with Padé approximations according to Eq. (20) (solid colored lines). The black solid line shows the noninteracting result on the lattice. Bottom: analytically continued ground-state energies as a function of real mass imbalance (solid lines). Although the fits as a function of imaginary mass imbalance are precise, small uncertainties result in wide confidence bands (shaded areas) when displayed as a function of real mass imbalance. The plot range in the bottom panel was limited to $\overline{m}=0.65$ due to large uncertainties beyond that point.

Figure 2

Ground-state energy of $N=5+5$ fermions of equal mass ($\overline{m}=0$) as a function of interaction strength computed with iHMC (red error bars), CL (blue squares), and DFT-RG (dash-dotted line).

Figure 3

Ground-state energy of $N=5+5$ fermions computed using iHMC (solid lines) and CL (symbols). The shaded areas represent the 95%-confidence interval of iHMC data, the uncertainties in the CL data are smaller than the symbol sizes. We find agreement in the ground-state energies at low imbalances up to $\overline{m}\sim 0.5$. Beyond that point the higher-order terms in the Padé approximants reduce the curvature of the iHMC lines. The energies calculated with CL increase monotonically as $\overline{m}$ approaches unity as naively expected for a species with diverging kinetic energy (zero mass).

Figure 4

Ground-state energy of $N=5+5$ fermions as a function of the dimensionless coupling $\gamma$ for several mass imbalances $\overline{m}$ as obtained from our CL approach. Error bars are of the size of the symbols and below. The dashed-dotted lines show the first-order perturbative result of Eq. (23).

Figure 5

Real part of the logarithm of the fermion determinant (see Sec. 3 for our conventions) as a function of the magnitude of the auxiliary field (at a randomly chosen point on the lattice) for an exemplaric choice of parameters (random seed, time of CL evolution, and coupling) for an unregulated (i.e. $.\xi =0$; dashed lines) and an action regulated by choosing $\xi =0.1$ (solid lines).

Figure 6

Top panels: Ground-state energy of a system with $N=5+5$ fermions with mass imbalance $\overline{m}=0.3$ (top left panel) and $\overline{m}=0.6$ (top right panel) as a function of the target CL step $h_0$ for different strengths $\xi$ of the regulating term appearing in the CL equations. The interaction strength is set to $\gamma =-1.0$ in both cases. Dashed lines represent linear fits which have been used to extrapolate to the limit $h_0\to 0$. The latter are marked by diamonds. Bottom panels: Ground-state energy of a system with $N=5+5$ fermions with mass imbalance $\overline{m}=0.3$ (bottom left panel) and $\overline{m}=0.6$ (bottom right panel) as obtained from an extrapolation to $h_0=0$ is shown as a function of the regulator strength $\xi$. The interaction strength is set to $\gamma =-1.0$ in both cases. Dashed lines represent linear fits of the data. For comparison, we also show the results obtained from our iHMC approach via analytic continuation. Note that, whereas the error bars on the CL data points originate from statistical errors, the error bars on all extrapolated CL values as well as the iHMC values refer to errors from associated fits.

Figure 7

Left panels: (logarithmic $y$-scale) of the measured ground-state energies for an attractively (top) and a repulsively (bottom) interacting system. While the former follows a Gaussian distribution, the latter exhibits so-called “fat tails” as a consequence of a signal-to-noise problem. Right panel: distributions of the real part of the path integral measure, i.e. the real part of the negative logarithm of the fermion determinant for various systems from strongly attractive to strongly repulsive (from left to right).