Thesis advisor: Kevin S. Bedell / We present several results based upon non-trivial extensions of Landau-Fermi liquid theory. First proposed in the mid-20th century, the Fermi liquid approach assumes an adiabatic “switching-on” of the interaction, which allows one to describe the collective excitations of the many-body system in terms of weakly-interacting quasiparticles and quasiholes. At its core, Landau-Fermi liquid theory is often considered a perturbative approach to study the equilibrium thermodynamics and out-of-equilibrium response of weakly-correlated itinerant fermions, and therefore non-trivial extensions and consequences are usually overlooked in the contemporary literature. Instead, more emphasis is often placed on the breakdown of Fermi liquid theory, either due to strong correlations, quantum critical fluctuations, or dimensional constraints. After a brief introduction to the theory of a Fermi liquid, I will first apply the Landau quasiparticle paradigm to the theory of itinerant Majorana-like fermions. Defined as fermionic particles which are their own anti-particle, traditional Majorana zero modes found in topological materials lack a coherent number operator, and therefore do not support a Fermi liquid-like ground state. To remedy this, we will apply a combinatorical approach to build a statistical theory of self-conjugate particles, explicitly showing that, under this definition, a filled Fermi surface exists at zero temperature. Landau-Fermi liquid theory is then used to describe the interacting phase of these Majorana particles, from which we find unique signatures of zero sound in addition to exotic, non-analytic contributions to the specific heat. The latter is then exploited as a “smoking-gun” signature for Majorana-like excitations in the candidate Kitaev material Ag3LiIr2O6, where experimental measurements show good agreement with a sharply-defined, “Majorana-Fermi surface” predicted in the underlying combinatorial treatment. I will then depart from Fermi liquid theory proper to tackle the necessary conditions for the applicability of Luttinger’s theorem. In a nutshell, Luttinger’s theorem is a powerful theorem which states that the volume of phase space contained in the Fermi surface is invariant with respect to interaction strength. In this way, whereas Fermi liquid only describes fermionic excitations near the Fermi surface, Luttinger’s theorem describes the fermionic degrees of freedom throughout the entire Fermi sphere. We will show that Luttinger’s theorem remains valid only for certain frequency and momentum-dependencies of the self-energy, which correlate to the exis- tence of a generalized Fermi surface. In addition, we will show that the existence of a power-law Green’s function (a unique feature of “un-particle” systems and a proposed characteristic of the pseudo-gap phase of the cuprate superconductors) forces Luttinger’s theorem and Fermi liquid theory to be mutually exclusive for any non-trivial power of the Feynman propagator. Finally, we will return to Landau-Fermi liquid theory, and close with novel out-of-equilibrium behavior and stability in unconventional Fermi liquids. First, we will consider a perfectly two- dimensional Fermi liquid. Due to the reduction in dimension, the traditional mode expansion in terms of Legendre polynomials is modified to an expansion in terms of Chebyshev polynomials. The resulting orthogonality conditions greatly modifies the stability and collective modes in the 2D system. Second, we will look at a Fermi liquid in the presence of a non-trivial gauge field. The existence of a gauge field will effectively shift the Fermi surface in momentum space, resulting in, once again, a modified stability condition for the underlying Fermi liquid. Supplemented with a modernized version of Mermin’s condition for the propagation of zero sound, we outline the full effects a spin symmetric or anti-symmetric gauge would have on a Fermi liquid ground state. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Abstract Interacting many-body problems are central to most fields of physics. In condensed matter physics, the systems of interest consists of a number of bodies on the order of Avogadro's constant, ~10²³. The precise modeling of such systems is usually impossible. Under certain circumstances however, even these problems can become tractable. One such circumstance is that of a Fermi liquid. At sufficiently low temperatures, in describing the dynamics of a system of interacting fermions, it is possible to forgo description of the fermions themselves, and instead concentrate on the collective excitations of the entire fermion system. These collective excitations are called quasiparticles. In this thesis we study two phenomena related to the motion of objects in a Fermi liquid. First, we study the transmission of transverse oscillations through a thin film of normal Fermi liquid. The dynamics of normal Fermi liquid are described by Landau's Fermi liquid theory. Landau's theory predicts the existence of new modes of sound under conditions where sound ordinarily would not propagate. Using the equations of motion for the Fermi liquid quasiparticles, we calculate the linear response of a Fermi liquid film to the transverse oscillations of a planar substrate under a wide range of conditions. We present the linear response in terms of the film's acoustic impedance and study the effects of quasiparticle collisions and of the Fermi liquid interactions. The second phenomenon we study is the supercritical motion of a wire in a superfluid Fermi liquid. The prevailing assumption is that if the velocity of an object moving in a superfluid Fermi liquid surpasses a characteristic critical velocity, the object experiences a sudden onset of viscous forces. This viscosity is caused by the escape of quasiparticles, produced by pair breaking on the surface of the object, into the surrounding superfluid. We study Andreev reflection of the quasiparticles by the surrounding superfluid flow field, and modifications to the flow caused by pair breaking, as possible mechanisms for low-dissipation motion above the critical velocity. / Original publications The original publications are not included in the electronic version of the dissertation. Kuorelahti, J. A., Tuorila, J. A., & Thuneberg, E. V. (2016). Fermi liquid theory applied to a film on an oscillating substrate. Physical Review B, 94(18). https://doi.org/10.1103/physrevb.94.184103 Kuorelahti, J. A., & Thuneberg, E. V. (2018). Two-parameter boundary condition applied to transverse acoustic impedance of a Fermi liquid. Journal of Physics: Conference Series, 969, 12010. https://doi.org/10.1088/1742-6596/969/1/012010 http://jultika.oulu.fi/Record/nbnfi-fe2018060425173 Kuorelahti, J. A., Laine, S. M., & Thuneberg, E. V. (2018). Models for supercritical motion in a superfluid Fermi liquid. Physical Review B, 98(14). https://doi.org/10.1103/physrevb.98.144512 http://jultika.oulu.fi/Record/nbnfi-fe2018112148794

Abstract The problem of describing a system of many interacting particles is one of the most fundamental questions in physics. One of the central theories used in condensed matter physics to address the problem is the Fermi liquid theory developed by L. D. Landau in the 1956. The theory describes interacting fermions, and can be used to explain transport phenomena of electrons in metals and dynamics of helium three. Even when the theory is not directly applicable, it forms a basis against which other, more sophisticated theories can be compared. this thesis the Fermi liquid theory is applied to 3He-4He-mixtures at temperatures where the bosonic 4He part is superfluid, and the mechanical properties of the system are largely determined by the 3He component, treated as a degenerate normal Fermi liquid. The dynamics of strongly interacting liquid 3He can be described as a collection of quasiparticles, elementary excitations of the system, which interact only weakly. In 3He-4He mixtures the interactions can be continuously tuned by changing the temperature and the concentration of the mixture. The scattering time of quasiparticles depends on temperature, and thus the transition from the hydrodynamic limit of continuous collisions at higher temperatures to the collisionless ballistic limit at low temperatures can be studied. This gives invaluable information on the role of the interactions in the dynamics of the system. In this work, by using the Fermi liquid theory and Boltzmann transport equation, the dynamics of helium mixture disturbed by a mechanical oscillator is described in the full temperature range. The solution necessarily is numeric, but new analytical results in the low temperature limit are obtained as well. The numerical approach enables one to study various boundary conditions thoroughly, and allows application of the theory to a specic geometry. It is shown that in order to explain the experimental observations, it is necessary to take into account the reflection of quasiparticles from the walls of the container. For suitable choice of oscillator frequency and container size, second sound resonances are observed at higher temperatures, while in the ballistic limit quasiparticle interference can be seen. The numerical results are in quantitative agreement with experiments, thus attesting the accuracy of Fermi liquid theory. In particular, the previously observed decrease of inertia of a mechanical oscillator immersed in helium at low temperatures is reproduced in the calculations, and is explained by elasticity of the fluid due to Fermi liquid interactions.

We demonstrate a remarkable property of metallic Fermi liquids: the transverse conductivity assumes a universal value in the quasi-static (ω → 0) limit for wavevectors q in the regime l −1 mfp q pF, where lmfp is the mean free path and pF is the Fermi momentum. This value is (e2/h)RFS/q in two dimensions (2D), where RFS measures the local radius of curvature of the Fermi surface (FS) in momentum space. Even more surprisingly, we find that U(1) spin liquids with a spinon FS have the same universal transverse conductivity. This means such spin liquids behave effectively as metals in this regime, even though they appear insulating in standard transport experiments. Moreover, we show that transverse current fluctuations result in a universal low-frequency magnetic noise that can be directly probed by a spin qubit, such as a nitrogen-vacancy (NV) center in diamond, placed at a distance z above of the 2D metal or spin liquid. Specifically the magnetic noise is given by CωPFS/z, where PFS is the perimeter of the FS in momentum space and C is a combination of fundamental constants of nature. Therefore these observables are controlled purely by the geometry of the FS and are independent of kinematic details of the quasi-particles, such as their effective mass and interactions. This behavior can be used as a new technique to measure the size of the FS of metals and as a smoking gun probe to pinpoint the presence of the elusive spinon FS in two-dimensional systems. We estimate that this universal regime is within reach of current NV center spectroscopic techniques for several spinon FS candidate materials.