Diagrammatic Monte Carlo study of quasi-two-dimensional Fermi polarons

We apply a diagrammatic Monte Carlo method to the problem of an impurity interacting resonantly with a homogeneous Fermi bath for a quasi-two-dimensional setup. Notwithstanding the series divergence, we can show numerically that the three particle-hole diagrammatic contributions are not contributing significantly to the final answer, thus demonstrating a nearly perfect destructive interference of contributions in subspaces with higher-order particle-hole lines. Consequently, for strong enough confinement in the third direction, the transition between the polaron and the molecule ground state is found to be in good agreement with the pure two-dimensional case and agrees very well with the one found by the wave-function approach in the two-particle-hole subspace.

Figure 1

Examples of (top) “exchange-hole” and (bottom) “direct-hole” contributions to the 2-ph diagrams are shown. This demonstrates that every order $N>2$ has at least two diagrams counting as 2-ph.

Figure 2

Difference in energy between the 2-ph and 3-ph contributions as a function of the number of $T$ matrices $N$ in the Feynman diagrams. Since this difference is essentially vanishing within the error bars, rapid convergence in the ph expansion order is seen. Note that two of the data sets have been offset by $\pm 0.08E_F$ for clarity.

Figure 3

Polaron and molecule energies as obtained by the diagrammatic Monte Carlo method. For low values of the interaction parameter $ln\left(k_{F}a\right)$, the molecule is the stable ground state, while the polaron (green triangles) dominates in the weak-coupling regime $ln\left(k_{F}a\right)\gtrsim -1$. The first-order quasi-two-dimensional energy is also shown for comparison, from which we see that the many-body modification is nearly independent of the interaction strength. These data were produced for ${\omega }_z=5000E_F$.

Figure 4

The influence of ${\omega }_z$ on both polaron energy and binding energy is demonstrated. Using ${\omega }_z=5000E_F$, the results are well saturated, justifying the assumption of neglecting transitions between the lowest and higher harmonic oscillator levels. The data in the plot were measured at $ln\left(k_{F}a\right)=-1.4$. Blue diamonds mark the value of ${\omega }_z$ we use in our simulations. Riesz resummation was applied to the bare data.