Probing the Fermi Sea Topology in a Quantum Gas

Abstract

Pauli's exclusion principle forces fermions to occupy distinct quantum states, creating a filled region of momentum space at low temperature, the Fermi sea, whose topology governs the system's response to perturbations and the nature of its correlation functions. Recent theory predicts that for non-interacting fermions, the Euler characteristic of a $D$-dimensional Fermi sea -- the topological invariant that describes its shape -- is encoded in its ($D$+1)-point density correlations. Here we experimentally demonstrate this connection in a two-dimensional degenerate gas of neutral $^{6}$Li atoms using single-atom-resolved imaging. By measuring three- and four-point connected density correlations in real space, we directly extract topological invariants of the underlying Fermi sea, including the Euler characteristic. Our results are in remarkable agreement with ideal-gas predictions, despite the presence of sizeable interactions, and establish a new pathway for probing many-body topology through correlation measurements.

Figures (11)

• Figure 1: Probing the Fermi Sea Topology via Single-Atom Imaging. (a) The Euler characteristic $\chi$ is an integer that distinguishes topologically distinct shapes. Originally formulated by Euler to characterize polyhedra (like a cube or a tetrahedron, with $\chi=2$), $\chi$ can be determined by counting the number of points, links and faces. Applied to two dimensional regions (blue), $\chi$ can be evaluated by introducing a triangulation (green mesh), defined by vectors ${\bf q}_{1}$ and ${\bf q}_{2}$, and computing $\chi$ from the triangles, lines and points included in the region (red). For a sufficiently fine mesh, $\chi$ is independent of the triangulation. (b) A dilute Fermi gas is held in a highly oblate gaussian trap ("light sheet") providing a quasi-two-dimensional geometry. A combination of a 2D pinning lattice (red arrows) and laser cooling (blue arrows) produces images of the spatial distribution of atoms with single atom resolution, as shown in (c) The three-point density correlations $\langle n({\bf r}_1)n({\bf r}_2)n({\bf r}_3)\rangle_{\rm c}$ in a Fermi gas are extracted from averages over quantum gas microscopy images. The Euler characteristic of the Fermi sea, $\chi_{\rm F}$, is encoded in the Fourier transform $s_3({\bf q}_1,{\bf q}_2)$, where ${\bf q}_{1,2}$ define the triangulation in (a).
• Figure 2: Measurement of Euler characteristic from density correlations.$s_3(\textbf{q}_1, \textbf{q}_2)$ for different configurations of $\textbf{q}_1$ and $\textbf{q}_2$ shown in the top row, with $\textbf{q}_2 \perp \textbf{q}_1$. For each column, $\textbf{q}_1$ is kept fixed, while $\textbf{q}_2$ is varied along the dotted red line. The Fermi surface is pictured as a solid black circle, with a circle of radius $2k_{\rm F}$ also shown as a dotted black curve. The shaded area corresponds to the region of validity of the topological formula, $S_{\rm topo}$ for the chosen value of $\textbf{q}_1$, as defined by Eq. \ref{['eq:validity_criterion']}. From left to right: $|\textbf{q}_1|/k_{\rm F} = 0.5$, $1.25$, $1.75$. Bottom row: Measured values of $s_3$ as a function of the amplitude $q_2$, averaged over different absolute orientations of $(\textbf{q}_1, \textbf{q}_2)$ (blue markers). The data is compared to the topological formula Eq. \ref{['eq:topological_formula']} with $\chi_{\rm F}\, =\, 1$ (black dotted line), the exact analytical formula Eq. (\ref{['eq:exact_formula_T0']}) (green solid line), and finite temperature numerical predictions (red dash-dotted line). Within $S_{\rm topo}$, indicated by the shaded areas, $s_3$ has the characteristic V-shape predicted by the topological formula and is proportional to $\chi_{\rm F} |\textbf{q}_1||\textbf{q}_2|$. Errorbars show the standard error of the mean and if not visible are smaller than the markers.
• Figure 3: Robustness of topological scaling. Zoom on the central part of $s_3$ for small values of $q_{2}$. The markers are experimental data. Solid green line: analytical prediction at $T=0$, Eq. (\ref{['eq:exact_formula_T0']}); dotted black line: topological formula Eq. (\ref{['eq:topological_formula']}); solid purple line: finite temperature prediction Eq. (\ref{['eq:finiteT_expansion']}); cyan dashed line: numerical results for a pure 2D system, which incorporate finite-size effects; red dashed line: numerical results including the effect of population of excited transverse motional levels. $S_{\rm topo}$ is shown as a shaded area. The chosen configuration for the triangulating vectors is shown as an inset, corresponding to $|\textbf{q}_1|/k_{\rm F} = 1.5$. Finite temperature and finite-size effects both contribute to smoothing the singularity predicted at $q_{2}=0$. The offset between the experimental data points and the linear branches of the topological formula is fully explained by the presence of atoms in excited $z$-levels.
• Figure 4: Volume scaling of $s_3$. Measured slope of $s_3$ (times $(2\pi)^2/|\textbf{q}_1|$) within the topological region for different values of the angle $\theta$ between $\textbf{q}_1$ and $\textbf{q}_2$ (top left inset) and different values of $|\textbf{q}_1|$. The orange circles, red squares, light blue hexagons, blue diamonds and dark blue triangles correspond to $|\textbf{q}_1|/k_{\rm F} = 0.75,\, 1,\, 1.25,\, 1.5,\, 1.75$ respectively. The prediction of the topological formula is shown in black. In insets, we show corresponding $s_3$ for different values of $\theta \mod \pi$. The markers are experimental data. Solid green line: analytical prediction at $T=0$, Eq. (\ref{['eq:exact_formula_T0']}); dotted black line: topological formula, Eq. (\ref{['eq:topological_formula']}); dashed red line: numerical results. $S_{\rm topo}$ is shown as a shaded area.
• Figure 5: Four-point connected density correlations.$s_4$ for $|{\bf q}_1|/k_{\rm F}=1.5$, $|{\bf q}_2|/k_{\rm F}=1.5$ with $\theta_{12}=2\pi/3$ and $\theta_{13}=\theta_{12}/2\ \rm{mod}\ \pi$. Blue markers: experimental data; green curve: analytical result; red curve: numerical results. $s_4$ is seen to remain nearly constant and close to zero over a large region in momentum space, which coincides with the region of validity for $s_4=0$ (grey area). (Inset) Plot of the chosen configuration $\{\textbf{q}_p \}_{p=1,2,3}$.