Quantum liquid crystal phases and unconventional magnetism in electronic and atomic Fermi systems

This thesis is devoted to the study of quantum liquid crystal phases in metallic systems and itinerant Fermi systems with dipole-dipole interactions. It is based on published works of Refs. [65, 193, 68, 67, 66]. In the first part of the Thesis, I present the construction of a generalization of the McMillan-de Gennes theory of nematic-smectic thermal phase transition of classical liquid crystals to describe an analogous quantum phase transition in metallic systems. This theory qualitatively can also describe similar quantum phase transitions in high Tc superconductors. Recent experimental advances in cooling atoms with large electric or magnetic moment[125] make possible the study of novel quantum phases that arise due to the anisotropic and long-range nature of the dipole-dipole interactions. In the next part of the Thesis I present a Landau Fermi liquid theory of one-component Fermi systems with electric or magnetic dipolar interactions. I use this theory to construct an order parameter theory and predict elongated Fermi surface (FS) along the polarization axis. I calculate the temperature and interaction dependence of FS distortions. For a two-component system with magnetic dipole-dipole interactions I predict the existence of "spin textures" in momentum space and of prolate/oblate deformed FS's. This constitutes a new phase of matter I called ferronematic. Possible observation of this phase in ultracold Fermi gases with large magnetic moments such as 163Dy are discussed. The stability of Fermi gases under attractive interactions at unitarity was also studied.;I now briefly summarize the content of this thesis. Chapter 2 is based on Ref. [65]. I present a unified overview, from the mean-field to the unitarity regime, of the stability of a trapped Fermi gas with short range attractive interactions. Unlike in a system of bosons, a Fermi gas is always stable in these regimes, no matter how large the particle number. However, when the interparticle spacing becomes comparable to the range of the interatomic interactions, instability is not precluded.;Chapter 3 is based on Ref. [193]. I discuss the quantum phase transition between a quantum nematic metallic state to an electron metallic smectic state in terms of an order-parameter theory coupled to fermionic quasiparticles. Both commensurate and incommensurate smectic (or stripe) cases are studied. Close to the quantum critical point (QCP), the spectrum of fluctuations of the nematic phase has low-energy fluctuating stripes. I study the quantum critical behavior and find evidence that, contrary to the classical case, the gauge-type of coupling between the nematic and smectic is irrelevant at this QCP. The collective modes of the electron smectic (or stripe) phase are also investigated. The effects of the low-energy bosonic modes on the fermionic quasiparticles are studied perturbatively. I find that at the nematic-smectic critical point, due to the critical smectic fluctuations, the dynamics of the fermionic quasiparticles near several points on the Fermi surface, around which it is reconstructed, are not governed by a Landau Fermi liquid theory. On the other hand, the quasiparticles in the smectic phase exhibit Fermi liquid behavior. I also present a detailed analysis of the dynamical susceptibilities in the electron nematic phase close to this QCP (the fluctuating stripe regime) and in the electronic smectic phase.;Chapter 4 is based on Ref. [68]. I demonstrate the possibility of a spontaneous symmetry breaking biaxial phase in these systems, which may be realized in, e.g., gases of ultracold polar molecules or strongly magnetic atoms. This biaxial nematic phase is manifest in a spontaneous distortion of the Fermi surface perpendicular to the axis of polarization. I describe these dipolar interaction induced phases using Landau Fermi liquid theory.;Chapter 5 is based on Ref. [67]. I show that a homogeneous two-component Fermi gas with long range dipolar and short-range isotropic interactions has a ferronematic phase for suitable values of the dipolar and short-range coupling constants. The ferronematic phase is characterized by having a non-zero magnetization and long range orientational uniaxial order. The Fermi surface of majority component is elongated while the minority component is compressed along the direction of the magnetization.;Chapter 6 is based on Ref. [66]. I study the magnetic structure of the ground state of a magnetic dipolar Fermi system of spin-1/2 particles in an imbalanced, partially polarized, state. I determine the distribution of quasiparticle spins states in momentum space. The k-dependent spin quantization axis is not along the spin magnetization axis but develops instead spin textures in momentum space. I compute the shape of the Fermi surfaces, and the effective magnetic moment of the quasiparticles, at zero temperature and weak coupling. I discuss realizations of this state with ultracold magnetic atoms.