Diagrammatic Monte Carlo study of a mass-imbalanced Fermi-polaron system

We apply the diagrammatic Monte Carlo approach to three-dimensional Fermi-polaron systems with mass imbalance, where an impurity interacts resonantly with a noninteracting Fermi sea whose atoms have a different mass. This method allows us to go beyond frequently used variational techniques by stochastically summing all relevant impurity Feynman diagrams up to a maximum expansion order limited by the sign problem. Polaron energy and quasiparticle residue can be accurately determined over a broad range of impurity masses. Furthermore, the spectral function of an imbalanced polaron demonstrates the stability of the quasiparticle and allows us to locate in addition also the repulsive polaron as an excited state. The quantitative exactness of two-particle-hole wave functions is investigated, resulting in a relative lowering of polaronic energies in the mass-imbalance phase diagram. Tan's contact coefficient for the mass-balanced polaron system is found in good agreement with variational methods. Mass-imbalanced systems can be studied experimentally by ultracold atom mixtures such as ${}^6\mathrm{Li}-{}^{40}\mathrm{K}$.

Figure 1

The first-order diagram is used for normalization purposes. The (local) appearance of this diagram topology as part of the whole diagram will be used to identify reducible diagrams in partially bold diagMC in Sec. 4.

Figure 2

This (unphysical) second-order diagram connects the first-order fake diagram with higher-order diagrams.

Figure 3

Illustration of inverse updates insert-mushroom and remove-mushroom.

Figure 4

Illustration of the first and second case of update reconnect. The dotted vertical bath propagator line is connected to an arbitrary $T$ matrix in the diagram.

Figure 5

Illustration of the third case of update reconnect. The dotted and full vertical bath propagator lines are connected to arbitrary $T$ matrices in the diagram (as long as neither the full line of the left figure nor the dotted line of the right figure is connected to itself.

Figure 6

Resummation methods of type Riesz with different exponents $\delta$ are compared for unitarity in a three-dimensional setup. The plot shows polaron energies depending on maximum sampling order.

Figure 7

Resummation methods of type Riesz with different exponents $\delta$ applied to the modified bare series [see Eq. (15)] are compared at unitarity in a three-dimensional setup. The plot shows polaron energies depending on maximum sampling order.

Figure 8

Polaron energy at unitarity for different mass ratios $r$. The inset shows the flat part of the figure. Many-body effects get more pronounced for a lighter polaron.

Figure 9

Polaron residue at unitarity for different mass ratios $r$. The whole range of masses shows a clear difference between the first order result and the full diagrammatic answer. The inset shows the curve for additional values of $r$.

Figure 10

The polaron energy of two values of $r$ is plotted for various interaction parameters $k_{F}a$. Note that the binding energy $E_B={\left(2m_r{a}^2\right)}^{-1}$ is subtracted. For strong interactions, the light impurity acquires a higher effective dressing relative to the binding energy than the heavy impurity. This is eventually reversed in the BEC regime.

Figure 11

Different contributions to the full polaron energy are compared for three maximum expansion orders $N$. $E_n$ denotes contributions including up to $n$-particle-hole diagrams. Two of the curves have been offset by $\pm 0.03E_F$ for clarity. The points were measured for a mass-imbalance ratio $r=0.125$ at unitarity.

Figure 12

The polaron spectral function is plotted for a mass imbalance $r=2$ at unitarity. Note the quadratic dispersion and the repulsive polaron at positive energies.

Figure 13

Molecule energy extrapolation for $k_{F}a=0.5$ and $r=0.25$. The maximum expansion order is denoted by $N$. The solid line corresponds to a linear fit of the last five points. The crosses mark the one-particle-hole result of Ref. [4]. Riesz resummation with exponent $\delta =6$ was used.

Figure 14

Polaron energy extrapolation for $k_{F}a=0.5$ and $r=0.25$. The maximum expansion order is denoted by $N$. The solid line corresponds to a linear fit of the last five points, the dashed line is a special fitting function explained in Appendix pp2. The one-particle-hole result of Ref. [4] is identically with the point $N=2$. Riesz resummation with exponent $\delta =4$ was used.

Figure 15

The extrapolation procedure on resummed data is demonstrated for $\eta =-0.248$ and $\delta =2$. The dotted curve shows the linear part, whereas the dashed line is a fit with $f$.

Figure 16

The extrapolation procedure on resummed data is demonstrated for $\eta =-0.248$ and $\delta =4$. The dotted curve shows the linear part, whereas the dashed line is a fit with $f$. Note that the resulting error bar exceeds the corresponding $\delta =2$ error bar.

Figure 17

The extrapolation procedure on resummed data is demonstrated for $\eta =-1.02$ and $\delta =4$. The dotted curve shows the linear part, whereas the dashed line is a fit with $f$.

Figure 18

The extrapolation procedure on resummed data is demonstrated for a molecule with $\eta =-1.02$ and $\delta =6$.