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arXiv:2310.03737 · Physical review. B./Physical review. B

Topological density correlations in a Fermi gas

Pok Man Tam, C. L. Kane

Published Jan 11, 2024• 13 citations
arXivHTMLPDF

Abstract

A Fermi gas of noninteracting electrons, or ultracold fermionic atoms, has a quantum ground state defined by a region of occupancy in momentum space known as the Fermi sea. The Euler characteristic ${\ensuremath{\chi}}_{F}$ of the Fermi sea serves to topologically classify these gapless fermionic states. The topology of a $D$-dimensional Fermi sea is physically encoded in the $D+1$ point equal time density correlation function. In this work, we first present a simple proof of this fact by showing that the evaluation of the correlation function can be formulated in terms of a triangulation of the Fermi sea with a collection of points, links, and triangles and their higher dimensional analogs. We then make use of the topological $D+1$ point density correlation to reveal universal structures of the more general $M$ point density correlation functions in a $D$-dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in $D=2$. In cold atomic gases imaged by quantum gas microscopy, our analysis supports the feasibility of measuring the third-order density correlation, from which ${\ensuremath{\chi}}_{F}$ can be reliably extracted in systems with as few as around 100 atoms. For solid-state electron gases, we propose measuring correlations in the speckle pattern of intensity fluctuations in nonlinear x-ray scattering experiments.

DOI
10.1103/physrevb.109.035413

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A Fermi gas of noninteracting electrons, or ultracold fermionic atoms, has a quantum ground state defined by a region of occupancy in momentum space known as the Fermi sea. The Euler characteristic \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ of the Fermi sea serves to topologically classify these gapless fermionic states. The topology of a DDD-dimensional Fermi sea is physically encoded in the D+1D+1D+1 point equal time density correlation function. In this work, we first present a simple proof of this fact by showing that the evaluation of the correlation function can be formulated in terms of a triangulation of the Fermi sea with a collection of points, links, and triangles and their higher dimensional analogs. We then make use of the topological D+1D+1D+1 point density correlation to reveal universal structures of the more general MMM point density correlation functions in a DDD-dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in D=2D=2D=2. In cold atomic gases imaged by quantum gas microscopy, our analysis supports the feasibility of measuring the third-order density correlation, from which \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ can be reliably extracted in systems with as few as around 100 atoms. For solid-state electron gases, we propose measuring correlations in the speckle pattern of intensity fluctuations in nonlinear x-ray scattering experiments.
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 DDD-dimensional Fermi sea -- the topological invariant that describes its shape -- is encoded in its (DDD+1)-point density correlations. Here we experimentally demonstrate this connection in a two-dimensional degenerate gas of neutral 6^{6}6Li 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.
We characterize a singularity in the equal-time three-point density correlations that is generic to two-dimensional interacting Fermi liquids. In momentum space where the three-point correlation is determined by two wave vectors, q_{1} and q_{2}, the singularity takes the form |q_{1}×q_{2}|. We explain how this singularity is sharply defined in a long-wavelength collinear limit. For a noninteracting Fermi gas, the coefficient of this singularity is given by the quantized Euler characteristic of the Fermi sea, and it implies a long-range real space correlation favoring collinear configurations. We show that this singularity persists in interacting Fermi liquids, and express the renormalization of the coefficient of singularity in terms of Landau parameters, for both spinless and spinful Fermi liquids. Our results offer an independent method for measuring Landau parameters. Implications for quantum gas experiments are discussed.
We characterize a singularity in the equal-time three-point density correlations that is generic to two-dimensional interacting Fermi liquids. In momentum space where the three-point correlation is determined by two wavevectors q1\mathbf{q}_1q1​ and q2\mathbf{q}_2q2​, the singularity takes the form ∣q1×q2∣|\mathbf{q}_1\times\mathbf{q}_2|∣q1​×q2​∣. We explain how this singularity is sharply defined in a long-wavelength collinear limit. For a non-interacting Fermi gas, the coefficient of this singularity is given by the quantized Euler characteristic of the Fermi sea, and it implies a long-range real space correlation favoring collinear configurations. We show that this singularity persists in interacting Fermi liquids, and express the renormalization of the coefficient of singularity in terms of Landau parameters, for both spinless and spinful Fermi liquids. Implications for quantum gas experiments are discussed.
Density three-point correlations are known to probe the topology of the Fermi sea in two-dimensional noninteracting systems. Here, we study how these correlations are modified by interactions using the coadjoint-orbit effective field theory. A key advantage of the coadjoint-orbit formulation is that it provides a systematic way to incorporate generalized Landau interactions in terms of bosonized degrees of freedom, mapping fermionic loop contributions onto simpler tree-level diagrams. We show that, for a general isotropic dispersion ε(p)ε(p)ε(p), even at linear order in the generalized Landau interaction, O(F(2,0))\mathcal{O}(\mathcal{F}^{(2,0)})O(F(2,0)), there exists a contribution proportional to the band curvature ε′′(pF)ε''(p_F)ε′′(pF​) that changes the nonanalytic structure of the free density three-point correlation function. This contribution introduces a distinct nonanalytic structure beyond that found in either the noninteracting case or an interacting Galilean-invariant system, showing that interaction effects can modify the topology-detecting density three-point correlation.
This thesis presents a series of experimental studies that leverage atom-resolved imaging to probe fundamental properties of ultracold Fermi gases in continuous space. We demonstrate the viability of this continuum microscopy by reliably observing the expansion of a Gaussian wave packet and the Fermi hole of a two-dimensional ideal Fermi gas, measuring its second- and third-order correlation functions. Building upon this foundation, we introduce a novel local thermometer based on the relationship between fluctuations and correlations. This method enables accurate global and local temperature measurement without requiring trap calibration. Further investigations reveal sub-extensive fluctuations absent from the well-known fluctuation-dissipation theorem. We describe how these fluctuations are linked to entanglement entropy and observe a violation of the area law. Following this we connect the spatial organization of non-interacting fermions to random matrix theory via the Fredholm determinant. Measurements of the counting statistics and nearest-neighbor spacing are compared with numerical predictions, showing excellent agreement. Finally, we explore the physics of two-dimensional interacting fermions in the BEC-BCS crossover. Our measurements of two- and three-point correlation functions show significant discrepancies with mean-field BCS theory, but are corroborated by exact quantum Monte Carlo calculations. We also perform a precise measurement of Tan's contact. As a whole, this work provides a unique and unprecedented microscopic perspective on the fundamentals of Fermi gases, and establishes quantum gas microscopy as a powerful and versatile tool for exploring quantum gases in the continuum.
We show that the topology of the Fermi sea of a D-dimensional Fermi gas is reflected in the multipartite entanglement characterizing D þ 1 regions that meet at a point.For odd D we introduce the multipartite mutual information and show that it exhibits a log D L divergence as a function of system size L with a universal coefficient that is proportional to the Euler characteristic χ F of the Fermi sea.This provides a generalization, for a Fermi gas, of the well-known result for D ¼ 1 that expresses the log L divergence of the bipartite entanglement entropy in terms of the central charge c characterizing a conformal field theory.For even D we introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation.We show that the corresponding charge-weighted mutual information exhibits a similar log D L divergence proportional to χ F .Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the D þ 1 order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to χ F .Our analytic results are based on the replica method.In addition, we perform a numerical study of the charge-weighted mutual information for D ¼ 2 that confirms several aspects of the analytic theory.Finally, we consider the effect of interactions perturbatively within the replica theory.We show that for D ¼ 3 the log 3 L divergence of the topological mutual information is not perturbed by weak short-ranged interactions, though for D ¼ 2 the charge-weighted mutual information is perturbed.Thus, for D ¼ 3 the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.