Precision Thermodynamics of the Fermi polaron at strong coupling

TL;DR

This work tackles finite-temperature thermodynamics of the Fermi polaron in the strong-coupling regime using a canonical-ensemble auxiliary-field quantum Monte Carlo (AFMC) approach on a lattice with a single impurity ($N_\downarrow=1$) and fixed $N_\uparrow$. By extrapolating to zero time-step and to the continuum limit, the authors obtain precise predictions for Tan's contact $C$ and the thermal energy gap $\Delta E$ across temperature and interaction strength, finding good agreement with a one-particle-hole variational theory at unitarity and revealing discrepancies with some experiments away from unitarity. The study demonstrates that the sign problem is manageable over substantial parameter ranges, enabling controlled thermodynamic benchmarks for the polaron problem and guiding comparisons with experiments and other theories. It also highlights limitations in accessing the polaron-molecule transition and points toward future work on spectral properties and quasiparticle dynamics of the polaron.

Abstract

The Fermi polaron problem, which describes a mobile impurity that interacts with a spin-polarized Fermi sea, is a paradigmatic system in quantum many-body physics and has been challenging to address quantitatively in its strong coupling regime. We present the first controlled thermodynamic calculations for the Fermi polaron at strong coupling using finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods in the framework of the canonical ensemble. Modeled as a spin-imbalanced system, the Fermi polaron has a Monte Carlo sign problem, but we show that it is moderate over a wide range of temperatures and coupling strengths beyond the unitary limit of the BCS-BEC crossover. We calculate the contact, a quantity which measures the strength of the short-range correlations, as a function of temperature at unitarity and as a function of the coupling strength at fixed temperature and find good agreement with a variational approach based on one particle-hole excitation of the Fermi sea. We compare our results for the contact with recent experiments and find good agreement at unitarity (within error bars) but discrepancies away from unitarity on the BEC side of the crossover. We also calculate the thermal energy gap at unitarity as a function of temperature.

Figures (7)

• Figure 1: (a) Average Monte Carlo sign as a function of temperature at unitarity for 20+1 particles on lattices of size $7^3$, $9^3$, $11^3$, and $13^3$. (b) As in (a) but for the average Monte Carlo sign as a function of $(k_F a)^{-1}$ at $T=0.2\ T_F$.
• Figure 2: Contact $C$ (in units of $k_F$) vs. temperature $T$ (in units of the Fermi temperature $T_F$) at unitarity. The AFMC results for the $7+1$ system (red diamonds) and the $20+1$ system (green circles) are compared the variational results of Ref. Liu2020A (dashed blue line). The $T=0$ diffusion Monte Carlo result of Ref. Pessoa2021 is shown by the blue triangle. We also show the experimental results of Ref. Yan2019 (purple x's with error bars).
• Figure 3: (a) Contact $C$ (in units of $k_F$) as a function of coupling strength $1/(k_F a)$ at $T = 0.2\,T_F$. The AFMC results for the $7+1$ system (red diamonds) and the $20+1$ system (green circles) are compared the variational results of Ref. Parish2021 (dashed blue line), the $T=0$ functional renormalization group results of Ref. Punk2009 (dash-dotted orange line), and the experimental results of Ref. Ness2020 (purple x's with error bars). (b) As in panel (a) but for the contact $C/k_F$ shifted by the contribution $8\pi/ k_Fa$ of the two-particle binding energy.
• Figure 4: The thermal energy gap $\Delta E$ (in units of the Fermi energy $E_F$) as a function of temperature at unitarity. Our low-temperature results are consistent with the $T=0$ Chevy's ansatz of $\Delta E =-0.61 T_F$Chevy2006.
• Figure 5: Examples of extrapolations to the continuous time limit for $N=20+1$ particles at unitarity on a $9^3$ lattice. For sufficiently small time slices, the dependence on $\Delta\beta$ is less significant than the dependance on filling factor.