The Standard Model (SM) of the particle physics, based on the gauge group SU(3) ×SU(2)L × U(1)Y , has been a successful theory which provides consistent description of all phenomena ranging from the nuclear beta decay to known processes at the high energy colliders like the LHC which operates at the TeV scale. Nevertheless, the SM is considered to be only a low energy (weak scale) theory and not a theory that is valid up to an energy scale (∼ 1019 GeV) where the effects of gravity are expected to be strong. The reasons for this view include the sensitivity of the higgs mass to the high energy scale (the hierarchy and the fine tuning problems), lack of explanation, within the SM, of the observation that the matter in the Universe dominates the anti-matter by orders of magnitude, lack of explanation for the number of fermion generations etc. Many extensions of the SM have been proposed so far which come with their own phenomenology to be tested at the high energy particle colliders like the LHC. Many of these extensions offer a special role to the heavy fermions of the SM, viz., the third generation leptons and quarks, the top quark in particular. An example of such a model is the Minimal Supersymmetric Standard Model. The special role given to top quarks is because of the closeness of the mass of the top quark mt (∼ 173 GeV) to the scale of the electroweak √ symmetry breaking (v/ 2 ∼ 175 GeV, where v is the vacuum expectation value of the Higgs field). This also means that the coupling of the top quark to the Higgs boson is large O(1) which makes the top quark loops major contributors to the fine tuning and hierarchy problems of the Higgs sector. Moreover, the interactions of the third generation fermions are the places where some room is left for new physics to appear as the experimental measurements of the properties of the first two generation fermions are very precise. Hence, the third generation particles, and the top quark in particular are expected to have new non-SM couplings to particles that are expected in this Beyond the Standard Model(BSM) physics. These particles can be either fermions or bosons. We focus first on a simple model that has a new fermion generation with the same quantum numbers as the corresponding SM fermions. This model is called the fourth generation Standard Model. Note that the Standard Model has no explanation for the number of fermion generations. The number of neutrinos extracted from the invisible Z-boson decay width at LEP is consistent with three. But this constrain can be evaded when the fourth generation neutrino is sufficiently heavy: mν′ ≳ mZ /2, where mZ is the mass of the Z-boson. Direct search constraints on the charged lepton of the fourth generation put it’s mass above ∼ 100 GeV. The lower bounds on the masses of fourth generation quarks t′ and b′ (mt′ , mb′ , respectively) have changed very much since the beginning of our work. We had used a model independent lower bound mt′,b′ > 290 GeV that was available at the time of our work. One can easily see that the fourth generation fermions were necessarily heavy, heavier than the top quark, at the time of our work. Since then the lower bounds only moved up. The present limits are mt′ > 700 GeV and mb′ > 675 GeV, if they decay through charged current processes. One important aspect of the fourth generation fermions is that they do not decouple when they are heavy. This affects the precision EW observables (see Introduction) through the loops. Earlier works focusing mainly on low Higgs mass (mh) suggested that the precision EW constraints imply a mass splitting |mt′ −mb′ | ≲ mW , where mW is the mass of the W boson. Another important effect of the heavy fourth generation fermions is that some of the tree-level scattering amplitudes like t′t′ → t′t′ at high energies, grow as GF m2f′ , where GF is the Fermi’s constant and m f ′ is the mass of the fourth generation fermion f ′ = t′,b′,ν′,τ′, which could be of O(1), potentially violating the tree level perturbative unitarity of the S-matrix. We combined the constraints from the precision EW observables -the S,T,U parameters, and the perturbative unitarity constraints to find available fourth generation SM parameter space in the light of a heavy Higgs as required by the then available LHC Higgs exclusion limits. We allowed for a small mixing between the third and fourth generation fermions: sin θ34 ≲ 0.3, where θ34 is the mixing angle of the third and the fourth generation quarks. This necessitated inclusion of amplitudes involving the top quark along with those of t′ and b′ in the perturbative unitarity analysis which had not been done before. We found that a heavy higgs with mass mh ≳ 800 GeV allowed large mass splitting between t′ and b′ and τ′ and ν′: |mt′ − mb′ | and mτ′ − mν′ could be greater than mW as long as sin θ34 ≤ 0.3. This meant that there was a non-negligible possibility that t′ → b′W /b′ → t′W and τ′ → ν′W could be open. Further we showed that the branching ratios for t′ → b′W or b′ → t′W could be close to unity (100%) for sin θ34 ≲ 0.05. The implication for the direct search experiments, which till then had not considered such decay modes, was that the search strategies should be altered to include these decay modes. Another important aspect of our result is that the large mass splittings mentioned above could be achieved even with one Higgs doublet, in contrast to earlier works which obtained such mass splittings only with two Higgs doublets. An epilogue is necessary here. The main point of our work was to show that a heavy Higgs could be allowed when a fourth generation of heavy fermions were present. At the time of the publication of this work, hints, but not a discovery, for a light Higgs appeared at the LHC. We did not take these hints to constitute as an evidence at that time. The discovery of a 125 GeV higgs boson at the LHC rules out the simple picture we had considered in our work. This was due to the huge suppression of the B.R of h → γγ channel by two orders of magnitude relative to it’s SM value despite a factor ten enhancement relative to the SM of the production channel gg → h. This results in a net suppression of the gg → h process relative to it’s SM value. Even after the Higgs discovery, a fourth generation model with a two Higgs doublet model could, however, still be viable. The top quark has an important property which is not shared by any other known quarks: Once produced, it decays before it can form any hadron. Hence, information about it’s spin state is transferred to the kinematical distributions of it’s decay products. One of the forms in which the spin information is revealed is via the kinematical distributions of decay products of the top which are sensitive to the polarization of the top quark. Different distributions have different sensitivity to the top polarization. The polarization of the top gives information about the chiral structure of the interaction responsible for the production. In the SM, the main mode of top production is the tt¯ pair production through QCD interactions. Due to the parity conserving nature of the QCD interactions (in other words, purely vector interactions), the polarization of the top quarks along their direction of motion is very small-less than about a percent. On the other hand, the single top production process which involves vector-axial-vector (maximally parity violating) weak interactions, produces highly polarized top quarks. Any possible chiral new physics interaction in the top production could affect the polarization of the produced top quarks. Hence, the top polarization can be a probe of new physics in top production. However, when any possible new physics effects appear in the top decay vertex, such as the W tb vertex associated with t → bW , measurement of top polarization is affected. This is because of the new Lorentz structures induced by the new physics which affect the kinematic distributions of the decay products. These additional couplings can be induced by higher order SM loops also. The possible deviations of these coefficients from the SM value are called anomalous couplings. Different kinematic distributions have different sensitivities to the anomalous couplings. In the second work, we constructed asymmetries from four kinematic distributions: θℓlab, xℓ = 2Eℓlab/mt , u = Eℓlab/(Eℓlab + Eblab) and z = Eℓlab/Etlab; ℓ and b denote the charged lepton and the b-quark from the top decay. The superscript lab denotes that our asymmetries are evaluated in the lab frame. Lab frame asymmetries do not need full reconstruction of a top event. We compare the four asymmetries for their sensitivity to the top polarization and the anomalous coupling f2R (The anomalous couplings of the W tb vertex are denoted as f1R, f2L, f2R (we set f1L = 1). Due to the strong indirect constraints from the measured branching ratio of b → sγ, we considered only one anomalous coupling, i.e f2R). We focused on a particular scenario where the top is highly boosted in the lab frame. Since a typical new physics process is expected to be in the TeV scale, the top produced through such processes would be highly boosted in the lab frame. Since effects of a possible chiral new physics in the top production appear in the top polarization and in the top decay, through the anomalous couplings like f2R of W tb, a simultaneous constraint on the top polarization and anomalous couplings is very useful, as it does not rely upon any specific assumptions on the decay or production. We combined asymmetries in a χ2 analysis to determine how much they can constrain the longitudinal top polarization (polarization along the direction of motion) and the anomalous W tb coupling f2R simultaneously. We also studied the effects of systematic uncertainties in the asymmetries and found that our asymmetries were sensitive to both P and f2R at a level of O(10−2) −O(10−1), for systematic uncertainties upto 5%. The top quarks are produced at the LHC dominantly as tt¯ pairs through QCD interactions. The other modes of production that have been observed include the single top (t-channel), associated production with a electroweak gauge boson etc. But the top can also be produced through possible new physics processes such as the one where a heavy new physics particle decays into a top quark. The couplings of the top with the heavy particle determine it’s polarization, in the rest frame of the heavy particle, for given masses of the parent and the daughter particles that are produced along with the top. The polarization of the top is a frame dependent quantity. For example, if we define the top polarization in the helicity basis, i.e. taking the direction of motion of the top in a given frame as it’s spin quantization axis, the polarization of the top in the rest frame of the heavy particle is not the same as it’s value in the lab frame. This is because the helicity states of the top are not invariant under the Lorentz transformations which are not along the direction of motion of the top. The probes of top polarization defined in the lab frame, do not require a full reconstruction of the event which is complicated by the possible presence of missing energy at the detectors. To probe the mechanism of the top production through the measured top polarization in the lab frame, a prediction of the polarization of the top in the lab frame as a function of the dynamical parameters of a theory like the couplings, mixing angles etc. is needed. In the third work, we studied how the top polarization in the rest frame of the heavy particle can be related to it’s value in the lab frame. In particular, we provide a simple procedure of calculation of top polarization in the lab frame given the dynamical parameters of the theory and the masses of the particles involved in the decay. We show that this can be achieved by the convolution of the velocity distribution of the heavy particle in the lab frame with a formula for top polarization in the lab frame. This formula depends only on the velocity of the heavy particle in the lab frame and not it’s direction of motion. We derive the formula and provide a simple explanation for the absence of the dependence on the direction of motion of the heavy particle. We illustrate our formula with two examples: the top produced from the decay of a gluino, and the top produced in the decay of stop. The analytical expression which we have derived gives the value of top polarization in any boosted frame. We establish the validity of our formula through a Monte Carlo simulation. We also give how finite width effects can be included. We find that a simple approach of folding the expression for the top polarization (after convoluting with the velocity distribution of the heavy particle) with a Breit-Wigner form for the distribution of mass of the heavy particle around it’s on-shell mass is sufficient in most of the cases. To summarize, we explored some aspects of the phenomenology of heavy quarks and leptons which are currently known or which are hypothetical. The first work focuses on the fourth generation Standard Model in the light of an LHC exclusion limit on Higgs boson. Taking into consideration all the indirect constraints, including the precision electroweak tests, we found that a heavy Higgs boson allowed a large mass splitting between the fourth generation fermions which implied that the direct search strategies need to include some more decays of fourth generation fermions. In the second work, we constructed observables which are sensitive to top polarization and used them to constrain possible anomalous couplings associated with the W tb vertex. We studied these observables for their potential to constrain both the top polarization and the possible anomalous couplings of W tb vertex. In the third work, we gave a simple procedure to calculate the top polarization in the lab frame, when the top quarks are produced in the decays of heavy particle. We showed that the lab frame polarization of the top could be obtained simply by convoluting the velocity distribution of the heavy particle in the lab frame with an expression for top polarization. We derived the expression and gave reasons for why the analytical expression does not depend on the direction of motion of the heavy particle. We demonstrated use of a simple procedure to include the effects of finite width of the heavy particle.

Orientador: Pascoal José Giglio Pagliuso / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-14T11:11:25Z (GMT). No. of bitstreams: 1 HolandaJunior_LinoMartinsde_M.pdf: 2170608 bytes, checksum: 3abed7533be911b023d90094d4aa28f4 (MD5) Previous issue date: 2009 / Resumo: Nesta Dissertação desenvolvemos estudos de Ressonância de Spin Eletrônico (ESR) em monocristais dos compostos tipo férmions pesados YbR h2Si2 e YbAlB4. No caso do sistem YbRh2Si2, exploraramos os experimentos de ESR para três bandas de frequência existentes em nosso laboratório (n = 4,1 GHz (Banda-S), n = 9,4 GHz (Banda-X) e n = 34,0 GHz (Banda-Q)), o que situa o campo de ressonância da linha de ESR em diferentes regimes do diagrama de fase do YbRh2Si2, onde as fases antiferromagnética (AFM), líquido de Fermi (FL) e não-líquido de Fermi (NFL) podem ser encontradas. Foram investigados efeitos de campo cristalino e mudanças da dinâmica de spin dos elétrons 4¦ do Yb, para as diferentes bandas, e também para monocristais de YbR h2Si2 dopados com Lu e crescidos em uxo de Zn. Nossos resultados sugerem que o sinal de ESR observado em YbRh2Si2 consiste em um modo acoplado entre os 4f do Yb3+ e os elétrons de condução, em um regime tipo bottleneck com presença de interações antiferromagnéticas. Para o sistema YbAlB4, realizamos experimentos de ESR em banda-X nas duas fases iso-estequiométricas b -YbAl B4 e - a -YbAlB4. Para as amostras de b - YbAlB4 encontramos um comportamento interessante do sinal de ESR que apresenta características de uma ressonância de elétrons de condução (CESR) a altas temperaturas e adquire propriedades do íon de Yb3+ a baixas temperaturas. Tal dualidade não foi observada na fase a - YbAlB4. Nós discutiremos uma possível correlação entre o espectro de ESR observado nos dois sistemas férmions pesados que se situam em lados opostos de um ponto crítico quântico (Quantum Critical Point - QCP) em seus diagramas de fase. Essa correlação permitiu uma especulação sobre origem desse sinal de ESR em compostos férmions pesados, na qual a proximidade ao QCP desenvolve um papel crucial para o comportamento do espectro de ESR observado nessa classe de compostos. / Abstract: In this work we have performed Electron Spin Resonance (ESR) experiments on single crystals of YbR h2Si2 and YbAlB4 heavy fermion compounds. For YbRh2Si2, we explore the ESR measurements at three frequency bands (n = 4,1 GHz (S-Band), n = 9,4 GHz (X-Band) e n = 34,0 GHz (Q-Band)) which places the ESR resonance field within different regimes in the phase diagram of YbRh2Si2 where antiferromagnetic (AFM), Fermi liquid (FL) and non Fermi liquid (NFL) phases could be found. We have also explored the effects of crystal field and spin dynamics of Yb3+ for these different bands and also as a function of Lu-doping and single crystal growth methods. Our results indicate that the ESR signal found YbRh2Si2 behave such a Kondo coupled mode in a bottleneck-like regime with the presence of antiferromagnetic interactions. For YbAlB4, we have performed X-band experiments for single crystals the two different phases b -YbAl B4 and a -YbAlB4. For b -YbAlB4, we found a remarkable ESR signal that behaves as a conduction electron spin resonance (CESR) at high temperatures and acquires characteristics of the Yb3+ local moment ESR at low temperature. This behavior was not found in the a -YbAlB4. The striking and unique dual behavior observed in the same ESR spectra of b -YbAl B4 - a -YbAlB4 associated to the ESR results found for a -YbAlB4, YbRh2Si2 allow us to propose a qualitative scenario that may explain the origin of the ESR signal in HF systems. We speculate that for HF systems a strongly coupled 4¦ and ce give origin to Kondo coupled ESR modes with may behave as CESR or LM ESR, depending on the strength of Jfs. Moreover, HF systems near a QCP may have propitious conditions to present such a signal. / Mestrado / Física da Matéria Condensada / Mestre em Física

L'objectif principal de cette thèse est de répondre à la question: étant donné une matrice aléatoire avec spectre réel, combien de valeurs propres tomber entre A et B? Ceci est une question fondamentale dans la théorie des matrices aléatoires et toutes ses applications, autant de problèmes peuvent être traduits en comptant les valeurs propres à l'intérieur des régions du spectre. Nous appliquons la méthode de gaz Coulomb à ce problème général dans le cadre de différents ensembles de matrice aléatoire et l'on obtient de résultats pour intervalles générales [a, b]. Ces résultats sont particulièrement intéressants dans l'étude des variations des systèmes fermioniques unidimensionnelles de particules confinées non-interaction à la température zéro. / The main goal of this thesis is to answer the question: given a random matrix with real spectrum, how many eigenvalues fall between a and b? This is a fundamental question in random matrix theory and all of its applications, as many problems can be translated into counting eigenvalues inside regions of the spectrum. We apply the Coulomb gas method to this general problem in the context of different random matrix ensembles and we obtain many results for general intervals [a,b]. These results are particularly interesting in the study of fermionic fluctuations for one-dimensional systems of confined non-interacting particles at zero temperature.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22238号 / 理博第4552号 / 新制||理||1654(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 松田 祐司, 教授 石田 憲二, 教授 寺嶋 孝仁 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22986号 / 理博第4663号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 松田 祐司, 教授 石田 憲二, 准教授 笠原 裕一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Patterning graphene with a spatially periodic potential provides a powerful means to modify its electronic properties. In particular, in twisted bilayers, coupling to the resulting moiré superlattice yields an isolated flat band that hosts correlated many-body phases. However, both the symmetry and strength of the effective moiré potential are constrained by the constituent crystals, limiting its tunability. Here, we have exploited the technique of dielectric patterning⁶ to subject graphene to a one-dimensional electrostatic superlattice (SL). We observed the emergence of multiple Dirac cones and found evidence that with increasing SL potential the main and satellite Dirac cones are sequentially flattened in the direction parallel to the SL basis vector, behavior resulting from the interaction between the one-dimensional SL electric potential and the massless Dirac fermions hosted by graphene. Our results demonstrate the ability to induce tunable anisotropy in high-mobility two-dimensional materials, a long-desired property for novel electronic and optical applications. Moreover, these findings offer a new approach to engineering flat energy bands where electron interactions can lead to emergent properties. The photon analog of electronic superlattice is photonic crystals. Efficient control of photons is enabled by hybridizing light with matter. The resulting light-matter quasi-particles can be readily programmed by manipulating either their photonic or matter constituents. Here, we hybridized infrared photons with graphene Dirac electrons to form surface plasmon polaritons (SPPs) and uncovered a previously unexplored means to control SPPs in structures with periodically modulated carrier density. In these photonic crystal structures, common SPPs with continuous dispersion are transformed into Bloch polaritons with attendant discrete bands separated by bandgaps. We explored directional Bloch polaritons and steered their propagation by dialing the proper gate voltage. Fourier analysis of the near-field images corroborates that this on-demand nano-optics functionality is rooted in the polaritonic band structure. Our programmable polaritonic platform paves the way for the much-sought benefits of on-the-chip photonic circuits.

In this work, we present a lattice QCD calculation of two closely related quantities: 1). The 𝜋𝜋 scattering phase shift for both 𝑰=0 and 𝑰=2 channels at seven energies in total, and 2). The 𝜟𝑰=1/2, 𝛫 → 𝜋𝜋 decay amplitude 𝐴₀ and 𝜖′, the measure of direct CP violation. These two results improve our earlier calculation presented in 2015 [1]. The calculation is performed on an ensemble of 32³ × 64 lattice with 𝛼⁻¹=1.3784(68)GeV. This is a physical calculation, where the chiral symmetry breaking is controlled by the 2+1 flavor Möbius Domain Wall Fermion, and we take the physical value for both kaon and pion. The G-parity boundary condition is used and carefully tuned so that the ground state energy of the 𝜋𝜋₁₌₀ state matches the kaon mass. Three sets of 𝜋𝜋 interpolating operators are used, including a scalar bilinear ``σ" operator and paired single-pion bilinear operators with the constituent pions carrying various relative momenta. Several techniques, including correlated fits and a bootstrap determination of the 𝑝-value have been used, and a detailed analysis of all major systematic error is performed. The 𝜋𝜋 scattering phase shift results are presented in Fig. 5.10 and Tab. 5.12. For the Kaon decay amplitude, we finally get Re(𝐴₀) = 2.99(0.32)(0.59) × 10⁻⁷GeV, which is consistent with the experimental value of Re(𝐴₀) = 3.3201(18) × 10⁻⁷GeV, and Im(𝐴₀) = -6.98(0.62)(1.44) × 10⁻¹¹GeV. Combined with our earlier lattice calculation of 𝐴₂ [2], we obtained Re(𝜖′/𝜖) = 21.7(2.6)(6.2)(5.0) × 10⁻⁴, which agrees well with the experimental value of Re(𝜖′/𝜖) = 16.6(2.3) × 10⁻⁴, and Re(𝐴₀)/Re(𝐴₂) = 19.9(2.3)(4.4), consistent with the experimental value of Re(𝐴₀)/Re(𝐴₂) = 22.45(6), known as the 𝜟𝑰=1/2 rule.

Lattice gauge theory is a special regularization of continuum gauge theories and the numerical simulation of lattice quantum chromodynamics (QCD) remains as the only first principle method to study non-perturbative QCD at low energy. The lattice spacing a, which serves as the ultraviolet cut off, plays a significant role in determining error on any lattice simulation results. Physical results come from extrapolating a series of simulations with different values for a to a=0. Reducing the size of these errors for non-zero a improves the extrapolation and minimizes the error. In the strong coupling limit the coarse lattice spacing pushes the analysis of the finite lattice spacing error to its limit. Section 4 measures two renormalized physical observables, the neutral kaon mixing parameter BK and the Delta I=3/2 K pi pi decay amplitude A2 on a lattice with coarse lattice spacing of a ~ 1GeV and explores the a^2 scaling properties at this scale. In the weak coupling limit the lattice simulations suffer from critical slowing down where for the Monte Carlo Markov evolution the cost of generating decorrelated samples increases significantly as the lattice spacing decreases, which makes reliable error analysis on the results expensive. Among the observables the topological charge of the configurations appears to have the longest integrated autocorrelation time. Based on a previous work where a diffusion model is proposed to describe the evolution of the topological charge, section 2 extends this model to lattices with dynamical fermions using a new numerical method that captures the behavior for different Fourier modes. Section 3 describes our effort to find a practical renormalization group transformation to transform lattice QCD between two different scales, whose knowledge could ultimately leads to a multi-scale evolution algorithm that solves the problem of critical slowing down. For a particular choice of action, we have found that doubling the lattice spacing of a fine lattice yields observables that agree at the few precent level with direct simulations on the coarser lattice. Section 5 aims at speeding up the lattice simulations in the weak coupling limit from the numerical method and hardware perspective. It proposes a preconditioner for solving the Dirac equation targeting the ensemble generation phase and details its implementation on currently the fastest supercomputer in the world.

No description available.

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.