High-precision numerical solution of the Fermi polaron problem and large-order behavior of its diagrammatic series

TL;DR

This paper develops a determinant diagrammatic Monte Carlo method to accurately compute the Fermi polaron ground state, revealing the divergence pattern of its diagrammatic series and achieving high-precision energy estimates.

Contribution

The authors introduce a simple, effective Monte Carlo algorithm that handles high-order diagrams without sign problems, and analyze the divergence of the series for the Fermi polaron problem.

Findings

Diagrammatic series diverges exponentially with order N.

Radius of convergence R is less than 1, determined by three-body diagrams.

Resummation yields highly accurate polaron energy estimates.

Abstract

We introduce a simple determinant diagrammatic Monte Carlo algorithm to compute the ground-state properties of a particle interacting with a Fermi sea through a zero-range interaction. The fermionic sign does not cause any fundamental problem when going to high diagram orders, and we reach order $N=30$. The data reveal that the diagrammatic series diverges exponentially as $(-1/R)^{N}$ with a radius of convergence $R<1$. Furthermore, on the polaron side of the polaron-dimeron transition, the value of $R$ is determined by a special class of three-body diagrams, corresponding to repeated scattering of the impurity between two particles of the Fermi sea. A power-counting argument explains why finite $R$ is possible for zero-range interactions in three dimensions. Resumming the divergent series through a conformal mapping yields the polaron energy with record accuracy.