Doing Algebra over an Associative Algebra

A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to $\mathbb{R}$ and imposing the relation $i^2 = -1$. In this paper, we examine some of the elementary algebraic properties of such algebras, how they break-down when compared to standard grade-school algebra, and discuss how such properties are relevant to other areas of our research regarding algebras, such as the $\mathcal{A}$-calculus and the theory of $\mathcal{A}$-ODEs.Comment: Finished draft of work primarily done during the summer of 201

Logarithms Over a Real Associative Algebra

Extending the work of Freese and Cook, which develop the basic theory of calculus and power series over real associative algebras, we examine what can be said about the logarithmic functions over an algebra. In particular, we find that for any multiplicative unital nil algebra the exponential function is injective, and hence the algebra has a unique logarithm on the image of the exponential. We extend this result to show that for a large class of algebras, the logarithms behave incredibly similarly to the logarithms over the real and complex numbers depending on if they are "Type-R" or "Type-C" algebras.Comment: 15 pages. Final draft based on research that was primarily conducted in the summer of 201

Generalized Trigonometric Functions over Associative Algebras

Extending the work of Freese, we further develop the theory of generalized trigonometric functions. In particular, we study to what extent the notion of polar form for the complex numbers may be generalized to arbitrary associative algebras, and how the general trigonometric functions may be used to give particularly elegant formulas for the logarithm over an algebra. Finally, we close with an array of open questions relating to this line of inquiry.Comment: 15 pages. Final draft based primarily on work that was done during the summer of 201

Quantum spin transport and dynamics through a novel F/N junction

We study the spin transport in the low temperature regime (often referred to as the precession-dominated regime) between a ferromagnetic Fermi liquid (FFL) and a normal metal metallic Fermi liquid (NFL), also known as the F/N junction, which is considered as one of the most basic spintronic devices. In particular, we explore the propagation of spin waves and transport of magnetization through the interface of the F/N junction where nonequilibrium spin polarization is created on the normal metal side of the junction by electrical spin injection. We calculate the probable spin wave modes in the precession-dominated regime on both sides of the junction especially on the NFL side where the system is out of equilibrium. Proper boundary conditions at the interface are introduced to establish the transport of the spin properties through the F/N junction. A possible transmission conduction electron spin resonance (CESR) experiment is suggested on the F/N junction to see if the predicted spin wave modes could indeed propagate through the junction. Potential applications based on this novel spin transport feature of the F/N junction are proposed in the end.Comment: 7 pages, 2 figure

Spin Orbit Magnetism and Unconventional Superconductivity

We find an exotic spin excitation in a magnetically ordered system with spin orbit magnetism in 2D, where the order parameter has a net spin current and no net magnetization. Starting from a Fermi liquid theory, similar to that for a weak ferromagnet, we show that this excitation emerges from an exotic magnetic Fermi liquid state that is protected by a generalized Pomeranchuck condition. We derive the propagating mode using the Landau kinetic equation, and find that the dispersion of the mode has a $\sqrt q$ behavior in leading order in 2D. We find an instability toward superconductivity induced by this exotic mode, and a further analysis based on the forward scattering sum rule strongly suggests that this superconductivity has p-wave pairing symmetry. We perform similar studies in the 3D case, with a slightly different magnetic system and find that the mode leads to a Lifshitz-like instability most likely toward an inhomogeneous magnetic state in one of the phases.Comment: 5 pages, 3 figure

Non-Analytic Contributions to the Self-Energy and the Thermodynamics of Two-Dimensional Fermi Liquids

We calculate the entropy of a two-dimensional Fermi Liquid(FL) using a model with a contact interaction between fermions. We find that there are $T^2$ contributions to the entropy from interactions separate from those due to the collective modes. These $T^2$ contributions arise from non-analytic corrections to the real part of the self-energy which may be calculated from the leading log dependence of the imaginary part of the self-energy through the Kramers-Kronig relation. We find no evidence of a breakdown in Fermi Liquid theory in 2D and conclude that FL in 2D are similar to 3D FL's.Comment: 12 pages (RexTex, no figures

Introduction to the Theory of A\mathcal{A}-ODEs

We study the theory of ordinary differential equations over a commutative finite dimensional real associative unital algebra $\mathcal{A}$. We call such problems $\mathcal{A}$-ODEs. If a function is real differentiable and its differential is in the regular representation of $\mathcal{A}$ then we say the function is $\mathcal{A}$-differentiable. In this paper, we prove an existence and uniqueness theorem, derive Abel's formula for the Wronskian and establish the existence of a fundamental solution set for many $\mathcal{A}$-ODEs. We show the Wronskian of a fundamental solution set cannot be a divisor of zero. Three methods to solve nondegenerate constant coefficient $\mathcal{A}$-ODE are given. First, we show how zero-divisors complicate solution by factorization of operators. Second, isomorphisms to direct product are shown to produce interesting solutions. Third, our extension technique is shown to solve any nondegenerate $\mathcal{A}$-ODE; we find a fundamental solution set by selecting the component functions of the exponential on the characteristic extension algebra. The extension technique produces all of the elementary functions seen in the usual analysis by a bit of abstract algebra applied to the appropriate exponential function. On the other hand, we show how zero-divisors destroy both existence and uniqueness in degenerate $\mathcal{A}$-ODEs. We also study the Cauchy Euler problem for $\mathcal{A}$-Calculus and indicate how we may solve first order $\mathcal{A}$-ODEs.Comment: 33 page

Approaching Pomeranchuk Instabilities from Ordered Phase: A Crossing-symmetric Equation Method

We explore features of a 3D Fermi liquid near generalized Pomeranchuk instabilities using a tractable crossing symmetric equation method. We approach the instabilities from the ordered ferromagnetic phase. We find quantum multi-criticality as approach to the ferromagnetic instability drives instability in other channel(s). It is found that a charge nematic instability precedes and is driven by Pomeranchuk instabilities in both the l = 0 spin and density channels.Comment: 16 pages, 7 figure

Exotic quantum statistics and thermodynamics from a number-conserving theory of Majorana fermions

We propose a closed form for the statistical distribution of non-interacting Majorana fermions at low temperature. Majorana particles often appear in the contemporary many-body literature in the Kitaev, Fu-Kane, or Sachdev-Ye-Kitaev models, where the Majorana condition of self-conjugacy immediately results in nonconserved particle number, non-trivial braiding statistics, and the absence of a noninteracting limit. We deviate from this description and instead consider a gas of noninteracting, spin-1/2 Majorana fermions that obey the spin-statistics theorem via imposing a condensed matter analog of momentum conservation. This allows us to build a quantum statistical theory of the Majorana system in the low temperature, low density limit without the need to account for strong fluctuations in the particle number. A combinatorial analysis leads to a configurational entropy which deviates from the fermionic result with an increasing number of available microstates. A number-conserving Majorana distribution function is derived which shows signatures of a sharply-defined Fermi surface at finite temperatures. Such a distribution is then re-derived from a microscopic model in the form of a modified Kitaev chain with a bosonic pair interaction. The thermodynamics of this free Majorana system is found to be nearly identical to that of a free Fermi gas, except now distinguished by a two-fold ground state degeneracy and, subsequently, a residual entropy at zero temperature. Despite clear differences with the anyonic or Sachdev-Ye-Kitaev models, we nevertheless find surprising agreement between our theory and experimental signatures of Majorana excitations in several materials. Experimental realization of our exactly solvable model is also discussed in the realm of astrophysical and high-energy phenomena.Comment: 66 pages, 7 figures, 5 table

No Evidence for Lunar Transit in New Analysis of Hubble Space Telescope Observations of the Kepler-1625 System

Observations of the Kepler-1625 system with the Kepler and Hubble Space Telescopes have suggested the presence of a candidate exomoon, Kepler-1625b I, a Neptune-radius satellite orbiting a long-period Jovian planet. Here we present a new analysis of the Hubble observations, using an independent data reduction pipeline. We find that the transit light curve is well fit with a planet-only model, with a best-fit $\chi^2_\nu$ equal to 1.01. The addition of a moon does not significantly improve the fit quality. We compare our results directly with the original light curve from Teachey & Kipping (2018), and find that we obtain a better fit to the data using a model with fewer free parameters (no moon). We discuss possible sources for the discrepancy in our results, and conclude that the lunar transit signal found by Teachey & Kipping (2018) was likely an artifact of the data reduction. This finding highlights the need to develop independent pipelines to confirm results that push the limits of measurement precision.Comment: 7 pages, 5 figures, accepted to ApJ