Ab initio study of the neutron and Fermi polarons on the lattice

TL;DR

The paper develops and applies auxiliary-field Quantum Monte Carlo on a lattice to study a single impurity (polaron) immersed in a spin-polarized Fermi sea, spanning ultra-cold atomic and nuclear physics. It combines a lattice AFQMC implementation with Lüscher’s finite-volume formalism to tune two-body interactions and introduces a parametric matrix model (PMM) emulator to accelerate parameter tuning. The work reports new ab initio results for the Fermi polaron across the BCS-BEC crossover and at unitarity, and provides the first lattice QMC calculations of the neutron polaron over a broad density range, with continuum extrapolations and careful treatment of the sign problem via the constrained path method. These results yield stringent benchmarks for theory and experiment and demonstrate a unified approach to polaron physics across disparate many-body systems, with potential implications for energy-density functionals and exotic nuclear phenomena.

Abstract

We have used the auxiliary-field Quantum Monte Carlo (AFQMC) many-body approach on the lattice to study the equation of state for a fermionic impurity interacting with a background sea of spin-polarized fermions. The impurity, or polaron, is an interesting system in both cold atomic and nuclear physics. Our approach is general, and we are able to straightforwardly study the polaron across these regimes. We first study the Fermi polaron at unitarity and for a wide range of scattering lengths, comparing against previous theoretical and experimental studies. We then explore the neutron polaron which has been shown to be an important constraint for nuclear physics. We have also employed the recently developed parametric matrix model to emulate AFQMC solutions to the two-body problem on the lattice, to accelerate the tuning of our lattice Hamiltonian parameters directly to two-body energies in a periodic box, following Lscher's formula. Our lattice Quantum Monte Carlo results for the polaron in both a cold atomic and nuclear physics context can serve as stringent benchmarks for future theoretical and experimental research.

Figures (6)

• Figure 1: Average percent error of 4x4 PMMs as a function of the number of training points given to the model. Each PMM was trained independently on a different set of AFQMC calculations for a given lattice size.
• Figure 2: Schematic description of our new approach to tuning the lattice Hamiltonian parameters (for $\alpha=1.5\ \text{fm}$) to reproduce known scattering length and effective range parameters. PMMs that have been trained on successively larger lattice sizes emulate AFQMC results, $q^2 = (Lp/2\pi)^2 = EL^2/4m\pi^2$, for any choice of $\gamma$ and $U$ until they reproduce the exact Lüscher's formula results. These parameters are then used for AFQMC many-body calculations of larger lattices, showing excellent agreement with the exact result.
• Figure 3: Energy of the Fermi polaron, in units of ${E_F = k_F^2/2m}$, as a function of $1/k_Fa$ where $a$ is the tunable scattering length of the interaction. AFQMC calculations are carried out at exactly $r_e=0$. Over the full range of interactions we compare our AFQMC calculations against previous Diagrammatic Monte Carlo calculations Prokofev_Svistunov_2008 and experimental results from MIT Schirotzek_Wu_Sommer_etal_2009. In the inset we also show our AFQMC calculations at unitarity, and our extrapolation to the continuum limit of zero filling factor. We find excellent agreement with recent experiments Schirotzek_Wu_Sommer_etal_2009Yan_Patel_Mukherjee_etal_2019, and other ab initio many-body calculations such as diagrammatic Monte Carlo and impurity lattice Monte Carlo (ILMC) Prokofev_Svistunov_2008VanHoucke_Werner_Rossi_2020Bour_Lee_Hammer_etal_2015. The horizontal scatter in the points at $\nu_{\uparrow}^{1/3}=0$ is purely to aid in legibility.
• Figure 4: Extrapolation to the continuum limit of zero filling factor, $\nu_{\uparrow} = N_{\uparrow}/M^3$ for our AFQMC calculations of the Fermi polaron. The number of spin-up particles is kept fixed at ${N_{\uparrow}=33}$, and we simultaneously increase the number of lattice sites while decreasing the lattice spacing in order to extrapolate to the continuum limit while maintaining a fixed $k_Fa$.
• Figure 5: Energy of the Fermi and neutron polaron across a range of negative $k_F a$. The dashed lines are included only to guide the eye. All the AFQMC results are extrapolated to the continuum limit, see Fig. \ref{['fig:extrap']}.