The concept of the Fermi sea is central to understanding the electronic properties of many-body quantum systems. In one-dimensional systems, the Fermi sea is characterized by a simpler topology, with electrons occupying states along a linear momentum axis. In contrast, in three dimensions, the free Fermi sea takes on a more complex geometry, forming a spherical surface in momentum space. This difference profoundly influences the density of states, interactions between particles, and the emergent phenomena in these systems. Three-dimensional fermionic systems are governed by a rich spectrum of quantum states and exhibit different collective behaviors.
The concept of the Fermi sea is central to understanding the electronic properties of many-body quantum systems. In one-dimensional systems, the Fermi sea is characterized by a simpler topology, with electrons occupying states along a linear momentum axis. In contrast, in three dimensions, the free Fermi sea takes on a more complex geometry, forming a spherical surface in momentum space. This difference profoundly influences the density of states, interactions between particles, and the emergent phenomena in these systems. Three-dimensional fermionic systems are governed by a rich spectrum of quantum states and exhibit different collective behaviors.
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 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.
Most of the visible matter around us is made of fermions, e.g. neutrons, protons and electrons. Their properties are thus relevant to understand the behavior of matter in different situations and at very different scales. Fermions, in contrast to bosons, are described by antisymmetric wavefunctions, as explained in Chapter 1.3. This has an important consequence already for non-interacting fermionic systems. Namely, the way a set of non-interacting identical fermions builds its ground state is quite peculiar as compared to bosons and gives rise to the so-called Fermi sea.
Interacting Fermi systems in two dimensions display intriguing phenomena such as high temperature superfluidity and pseudogap physics. These features emerge from the collective behavior of the constituent particles but are extremely hard to predict from microscopic theory. I will report on experiments that realize microscopic two-dimensional systems with ultracold fermionic lithium atoms. With the ability to deterministically prepare few-body ground states and to observe individual particles in momentum space, they enable a microscopic view of strongly interacting two-dimensional Fermi systems. Surprisingly, we find that characteristic features of many-body Fermi gases can already be found in systems of no more than a dozen particles: In spectroscopy, we observe collective excitations that are the few-body precursor of the Higgs amplitude mode of a superfluid. Moreover, in spin-resolved momentum space probes, we can directly image individual ‘Cooper pairs’ and show the presence of fermionic pairing even in a microscopic setting. These findings may broaden our understanding of finite-size Fermi systems, such as atomic nuclei and superconducting grains.
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.
At finite temperatures, the sharp distinction between occupied and unoccupied states near the Fermi surface becomes blurred, as the filling of the states is ruled by the Fermi distribution. This leads to significant changes in the physical properties of the system. Analyzing these effects is crucial for understanding phenomena like electrical and thermal conductivity, as well as the response of materials to external perturbations. Moreover, working at nonzero temperatures allows us to study transitions between quantum and classical regimes, shedding light on the statistical mechanics that govern systems from condensed matter to nuclear physics and astrophysics.
The properties of fermions in one dimension are markedly different from those in higher dimensions. In two or three dimensions, the many-body behaviour of fermions is described by the Fermi liquid model, whose most prominent result is that, instead of studying the behaviour of individual particles, it is better to describe the emergent phenomena of the whole system, known as quasiparticles. These are collective modes of excitations born out of particle-particle interactions, which resemble the constituent single particles, albeit with renormalised mass. In one dimension, however, the formation of such a correlated state is unfavourable and the quasiparticles are completely absent. Instead, 1D fermions are described by a model known as the Tomonaga-Luttinger liquid. Central to this model is the process of bosonisation, a mathematical technique which maps the excitation of fermions to bosonic modes of spin-density and charge-density waves. As a prerequisite of the bosonisation technique, the fermionic single-particle dispersion is approximated as a linear function. Consequently, the Luttinger liquid model is limited in its scope of applicability. Specifically, it is valid only in the vicinity of the Fermi points, where the excitation energy is low. Recent advancement in the theoretical field has expanded our understanding of 1D interacting fermions beyond the limit of linearity. The mobile-impurity model, proposed by L. Glazman and his co-authors, is a generalised model of 1D fermions which extends the scope of applicability of the Tomanaga-Luttinger model, accounting for the curvature of single-particle dispersion relations. It extends the region of validity of the tradition TLL theory from the neighbourhood of the Fermi points to a band which follows the single-particle dispersion. This thesis presents the results of an experimental work which provides direct evidence of non-linear behaviour predicted by the new model. It revolves around the measurement of the spectral function of electrons in surface-gate-defined quantum wires on a double-quantum-well, GaAs/AlGaAs device, and the comparison and fitting of the result to the non-linear theory. The spectral function is a theoretical concept frequently used in discussions on many-body problems. Its physical interpretation makes it closely related to experimental observables, such as the tunnelling current between the double quantum wells in this work. This thesis consists of three parts. Part one concerns the theories of the various many-body models and the experimental technique: Chapter 1 briefly introduces the concept of Fermi liquid, before focusing on a detailed account of the Luttinger-liquid model. The mobile-impurity model will be introduced in the framework of the Luttinger liquid model afterwards, followed by a review of experimental works reported in the literature on the study of TLL physics. In Chapter 2, the principles of the experimental tunnelling spectroscopy technique are described. The second part of the thesis details the structure of the test device used in the experiment and its fabrication. In particular, we will discuss the intricacies of the three-dimensional EBL fabrication process developed specifically for this project. The setup and running of the measurement circuit will also be discussed in this part of the thesis, in the context of a prerequisite sample characterisation measurement. In the final part of the work, data from the main experiment will be presented and analysed, in search for the electron-electron interaction effects which are predicted by the theoretical discussions in the first chapter.