Dynamic structure factor for ${}^3$He in two dimensions

Recent neutron scattering experiments on ${}^3$He films have observed a zero-sound mode, its dispersion relation, and its merging with—and possibly emerging from—the particle-hole continuum [H. Godfrin et al., Nature 483, 576 (2012)]. Here we address the study of excitations in the system via quantum Monte Carlo methods: we suggest a practical scheme to calculate imaginary time correlation functions for moderate-size fermionic systems. Combined with an efficient method for analytic continuation, this scheme affords an extremely convincing description of the experimental findings.

Figure 1

Comparison between the spectral functions $S_1(q,\omega )$ (shaded curve down to the baseline) obtained with the GIFT algorithm from $F_1(q,\tau )$ and $S_2(q,\omega )$ (unshaded curve) obtained from $F_2(q,\tau )$ with GIFT for some wave vectors $q$ at density $\rho =0.047$ Å${}^{-2}$. The two spectral functions have a compatible shape, with a shift in energy compatible with $E_0^F-E_0^B$.

Figure 2

From left to right the coherent dynamic structure factor, obtained as an average of several independently extracted $S_1(q,\omega )$, for increasing wave vectors at $\rho =0.047$ Å${}^{-2}$. Orange shading represents statistical uncertainties and filled (blue) circles are the available experimental data from Refs. 3 and 4. The wave vectors shown are those accessible from our simulation; the experimental wave vectors are $q=0.55$ Å${}^{-1}$ (b), $q=1.15$ Å${}^{-1}$ (d), $q=1.25$ Å${}^{-1}$ (e), and $q=1.65$ Å${}^{-1}$ (f). We have used different scales in the panels to make the comparison with experimental data more easily visible. The dashed (green) line shows the dynamic structure factor of a fictitious system of bosons of mass $m_3$. The bosonic peaks in the roton region are five to nine times higher than the fermionic ones.

Figure 3

Color map of normalized $S_2(q,\omega )$ for many wave vectors $q$. For better visibility, each $S_2(q,\omega )$ for different $q$ has been normalized in order to have their maximum value equal to 1. The vertical scale has been shifted by a quantity $E_0^B-E_0^F$, so that the excitation energies are measured with respect to the fermionic ground state.

Figure 4

Static response function of ${}^3$He obtained from ${\chi }_q=-2\rho \int d\omega \frac{S\left(q,\omega \right)}{\omega }$. Circles, $\rho =0.038$ Å${}^{-2}$; squares, $\rho =0.047$ Å${}^{-2}$; triangles, $\rho =0.060$ Å${}^{-2}$. Filled circles show the results for a fictitious system of bosons of mass $m_3$ at $\rho =0.047$ Å${}^{-2}$.