Explorations in Dirac Fermions and Spin Liquids
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Effective Dirac Equations in Honeycomb Structures
Effective Dirac equations in honeycomb structures Effective Dirac equations in honeycomb structures Young Researchers Seminar, CERMICS, Ecole des Ponts ParisTech William Borrelli CEREMADE, Universit´eParis Dauphine 11 April 2018 It is self-adjoint on L2(R2; C2) and the spectrum is given by σ(D0) = R; σ(D) = (−∞; −m] [ [m; +1) The domain of the operator and form domain are H1(R2; C2) and 1 2 2 H 2 (R ; C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories. Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1@1 + σ2@2) + mσ3: (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C2-valued spinors. The domain of the operator and form domain are H1(R2; C2) and 1 2 2 H 2 (R ; C ), respectively. Remark The negative spectrum is associated with antiparticles, in relativistic theories. Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1@1 + σ2@2) + mσ3: (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the particle. It acts on C2-valued spinors. It is self-adjoint on L2(R2; C2) and the spectrum is given by σ(D0) = R; σ(D) = (−∞; −m] [ [m; +1) Effective Dirac equations in honeycomb structures Dirac in 2D The 2D Dirac operator The 2D Dirac operator is defined as D = D0 +mσ3 = −i(σ1@1 + σ2@2) + mσ3: (1) where σk are the Pauli matrices and m ≥ 0 is the mass of the particle. -
2018 March Meeting Program Guide
MARCHMEETING2018 LOS ANGELES MARCH 5-9 PROGRAM GUIDE #apsmarch aps.org/meetingapp aps.org/meetings/march Senior Editor: Arup Chakraborty Robert T. Haslam Professor of Chemical Engineering; Professor of Chemistry, Physics, and Institute for Medical Engineering and Science, MIT Now welcoming submissions in the Physics of Living Systems Submit your best work at elifesci.org/physics-living-systems Image: D. Bonazzi (CC BY 2.0) Led by Senior Editor Arup Chakraborty, this dedicated new section of the open-access journal eLife welcomes studies in which experimental, theoretical, and computational approaches rooted in the physical sciences are developed and/or applied to provide deep insights into the collective properties and function of multicomponent biological systems and processes. eLife publishes groundbreaking research in the life and biomedical sciences. All decisions are made by working scientists. WELCOME t is a pleasure to welcome you to Los Angeles and to the APS March I Meeting 2018. As has become a tradition, the March Meeting is a spectacular gathering of an enthusiastic group of scientists from diverse organizations and backgrounds who have broad interests in physics. This meeting provides us an opportunity to present exciting new work as well as to learn from others, and to meet up with colleagues and make new friends. While you are here, I encourage you to take every opportunity to experience the amazing science that envelops us at the meeting, and to enjoy the many additional professional and social gatherings offered. Additionally, this is a year for Strategic Planning for APS, when the membership will consider the evolving mission of APS and where we want to go as a society. -
2 Lecture 1: Spinors, Their Properties and Spinor Prodcuts
2 Lecture 1: spinors, their properties and spinor prodcuts Consider a theory of a single massless Dirac fermion . The Lagrangian is = ¯ i@ˆ . (2.1) L ⇣ ⌘ The Dirac equation is i@ˆ =0, (2.2) which, in momentum space becomes pUˆ (p)=0, pVˆ (p)=0, (2.3) depending on whether we take positive-energy(particle) or negative-energy (anti-particle) solutions of the Dirac equation. Therefore, in the massless case no di↵erence appears in equations for paprticles and anti-particles. Finding one solution is therefore sufficient. The algebra is simplified if we take γ matrices in Weyl repreentation where µ µ 0 σ γ = µ . (2.4) " σ¯ 0 # and σµ =(1,~σ) andσ ¯µ =(1, ~ ). The Pauli matrices are − 01 0 i 10 σ = ,σ= − ,σ= . (2.5) 1 10 2 i 0 3 0 1 " # " # " − # The matrix γ5 is taken to be 10 γ5 = − . (2.6) " 01# We can use the matrix γ5 to construct projection operators on to upper and lower parts of the four-component spinors U and V . The projection operators are 1 γ 1+γ Pˆ = − 5 , Pˆ = 5 . (2.7) L 2 R 2 Let us write u (p) U(p)= L , (2.8) uR(p) ! where uL(p) and uR(p) are two-component spinors. Since µ 0 pµσ pˆ = µ , (2.9) " pµσ¯ 0(p) # andpU ˆ (p) = 0, the two-component spinors satisfy the following (Weyl) equations µ µ pµσ uR(p)=0,pµσ¯ uL(p)=0. (2.10) –3– Suppose that we have a left handed spinor uL(p) that satisfies the Weyl equation. -
The Bound States of Dirac Equation with a Scalar Potential
THE BOUND STATES OF DIRAC EQUATION WITH A SCALAR POTENTIAL BY VATSAL DWIVEDI THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2015 Urbana, Illinois Adviser: Professor Jared Bronski Abstract We study the bound states of the 1 + 1 dimensional Dirac equation with a scalar potential, which can also be interpreted as a position dependent \mass", analytically as well as numerically. We derive a Pr¨ufer-like representation for the Dirac equation, which can be used to derive a condition for the existence of bound states in terms of the fixed point of the nonlinear Pr¨uferequation for the angle variable. Another condition was derived by interpreting the Dirac equation as a Hamiltonian flow on R4 and a shooting argument for the induced flow on the space of Lagrangian planes of R4, following a similar calculation by Jones (Ergodic Theor Dyn Syst, 8 (1988) 119-138). The two conditions are shown to be equivalent, and used to compute the bound states analytically and numerically, as well as to derive a Calogero-like upper bound on the number of bound states. The analytic computations are also compared to the bound states computed using techniques from supersymmetric quantum mechanics. ii Acknowledgments In the eternity that is the grad school, this project has been what one might call an impulsive endeavor. In the 6 months from its inception to its conclusion, it has, without doubt, greatly benefited from quite a few individuals, as well as entities, around me, to whom I owe my sincere regards and gratitude. -
Introduction to Supersymmetry
Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × . -
Parkin CV and Publication List 11-20-2020
Curriculum Vitae - Stuart Stephen Papworth PARKIN Nationality Joint United Kingdom and United States Birthdate December 9, 1955 Current address Trothaer Strasse 17c, 06118 Halle (Saale), Germany Education and Employment The Edinburgh Academy (1971-1973) 1973 A levels- Physics, Chemistry and Maths (Grade A); S levels- Chemistry and Maths (Grade 1) 1973 Gold Medal and Academical Club Prize for Dux of School Trinity College, Cambridge (1974- ) 1974 Entrance Scholarship; 1975 Senior Scholarship; 1976 Science Essay Prize; 1977 Research Scholarship 1977 B.A. in Physics and Theoretical Physics (Theoretical Physics Option), class I, comprising class I in parts Ia (1975), Ib (1976) and II (1977) 1979 Research Fellow 2014 Honorary Fellow The Cavendish Laboratory, Cambridge (1977-1980) 1977 Research Student in the Physics and Chemistry of Solids Group, headed by Dr. A.D. Yoffe 1980 Ph.D. awarded (April) Laboratoire de Physique des Solides, Orsay, Paris (1980-1981) 1980 Royal Society European Exchange Fellowship, Laboratoire de Physique des Solides, Université Paris-Sud IBM Almaden Research Center, San Jose, California (formerly IBM San Jose Research Laboratory) 1982 IBM World Trade Fellowship 1983 Adjunct Research Staff Member (January); 1984 Research Staff Member (October) 1999 IBM Fellow (June) 2004-2014 Director, IBM-Stanford Spintronic Science and Applications Center (SpinAps); co-directed by Shoucheng Zhang (Physics) and James Harris (Electrical Engineering), Stanford University Max Planck Institute of Microstructure Physics and Martin Luther University Halle-Wittenberg (2014- ) 2014 Alexander von Humboldt Professor, Martin Luther University Halle-Wittenberg, Halle, Germany 2015 Director, Max Planck Institute of Microstructure Physics, Halle, Germany, April 1, 2015. 2016-2019 Managing Director, Max Planck Institute of Microstructure Physics, Halle, Germany Degrees B.A. -
Spin-1/2 Fermions in Quantum Field Theory
Spin-1/2 fermions in quantum field theory µ First, recall that 4-vectors transform under Lorentz transformations, Λ ν, as p′ µ = Λµ pν, where Λ SO(3,1) satisfies Λµ g Λρ = g .∗ A Lorentz ν ∈ ν µρ λ νλ transformation corresponds to a rotation by θ about an axis nˆ [θ~ θnˆ] and ≡ a boost, ζ~ = vˆ tanh−1 ~v , where ~v is the corresponding velocity. Under the | | same Lorentz transformation, a generic field transforms as: ′ ′ Φ (x )= MR(Λ)Φ(x) , where M exp iθ~·J~ iζ~·K~ are N N representation matrices of R ≡ − − × 1 1 the Lorentz group. Defining J~+ (J~ + iK~ ) and J~− (J~ iK~ ), ≡ 2 ≡ 2 − i j ijk k i j [J± , J±]= iǫ J± , [J± , J∓] = 0 . Thus, the representations of the Lorentz algebra are characterized by (j1,j2), 1 1 where the ji are half-integers. (0, 0) is a scalar and (2, 2) is a four-vector. Of 1 1 interest to us here are the spinor representations (2, 0) and (0, 2). ∗ In our conventions, gµν = diag(1 , −1 , −1 , −1). (1, 0): M = exp i θ~·~σ 1ζ~·~σ , butalso (M −1)T = iσ2M(iσ2)−1 2 −2 − 2 (0, 1): [M −1]† = exp i θ~·~σ + 1ζ~·~σ , butalso M ∗ = iσ2[M −1]†(iσ2)−1 2 −2 2 since (iσ2)~σ(iσ2)−1 = ~σ∗ = ~σT − − Transformation laws of 2-component fields ′ β ξα = Mα ξβ , ′ α −1 T α β ξ = [(M ) ] β ξ , ′† α˙ −1 † α˙ † β˙ ξ = [(M ) ] β˙ ξ , ˙ ξ′† =[M ∗] βξ† . -
![Arxiv:1003.1912V2 [Hep-Ph] 30 Jun 2010 Fermion and Complex Vector Boson Dark Matter Are Also Disfavored, Except for Very Specific Choices of Quantum Numbers](https://docslib.b-cdn.net/cover/0927/arxiv-1003-1912v2-hep-ph-30-jun-2010-fermion-and-complex-vector-boson-dark-matter-are-also-disfavored-except-for-very-speci-c-choices-of-quantum-numbers-1290927.webp)
Arxiv:1003.1912V2 [Hep-Ph] 30 Jun 2010 Fermion and Complex Vector Boson Dark Matter Are Also Disfavored, Except for Very Specific Choices of Quantum Numbers
Preprint typeset in JHEP style - HYPER VERSION UMD-PP-10-004 RUNHETC-2010-07 A Classification of Dark Matter Candidates with Primarily Spin-Dependent Interactions with Matter Prateek Agrawal Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742 Zackaria Chacko Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742 Can Kilic Department of Physics and Astronomy, Rutgers University, Piscataway NJ 08854 Rashmish K. Mishra Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742 Abstract: We perform a model-independent classification of Weakly Interacting Massive Particle (WIMP) dark matter candidates that have the property that their scattering off nucleons is dominated by spin-dependent interactions. We study renormalizable theories where the scattering of dark matter is elastic and arises at tree-level. We show that if the WIMP-nucleon cross section is dominated by spin-dependent interactions the natural dark matter candidates are either Majorana fermions or real vector bosons, so that the dark matter particle is its own anti-particle. In such a scenario, scalar dark matter is disfavored. Dirac arXiv:1003.1912v2 [hep-ph] 30 Jun 2010 fermion and complex vector boson dark matter are also disfavored, except for very specific choices of quantum numbers. We further establish that any such theory must contain either new particles close to the weak scale with Standard Model quantum numbers, or alternatively, a Z0 gauge boson with mass at or below the TeV scale. In the region of parameter space that is of interest to current direct detection experiments, these particles naturally lie in a mass range that is kinematically accessible to the Large Hadron Collider (LHC). -
![Arxiv:1702.04624V3 [Cond-Mat.Str-El]](https://docslib.b-cdn.net/cover/0661/arxiv-1702-04624v3-cond-mat-str-el-1420661.webp)
Arxiv:1702.04624V3 [Cond-Mat.Str-El]
Type-III and IV interacting Weyl points J. Nissinen1 and G.E. Volovik1, 2 1Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland 2Landau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka, Russia (Dated: July 7, 2017) 3+1-dimensional Weyl fermions in interacting systems are described by effective quasi-relativistic µ Green’s functions parametrized by a 16 element matrix eα in an expansion around the Weyl point. µ The matrix eα can be naturally identified as an effective tetrad field for the fermions. The corre- spondence between the tetrad field and an effective quasi-relativistic metric gµν governing the Weyl fermions allows for the possibility to simulate different classes of metric fields emerging in general relativity in interacting Weyl semimetals. According to this correspondence, there can be four types 00 of Weyl fermions, depending on the signs of the components g and g00 of the effective metric. In addition to the conventional type-I fermions with a tilted Weyl cone and type-II fermions with an 00 overtilted Weyl cone for g > 0 and respectively g00 > 0 or g00 < 0, we find additional “type-III” 00 and “type-IV” Weyl fermions with instabilities (complex frequencies) for g < 0 and g00 > 0 or g00 < 0, respectively. While the type-I and type-II Weyl points allow us to simulate the black hole event horizon at an interface where g00 changes sign, the type-III Weyl point leads to effective spacetimes with closed timelike curves. PACS numbers: I. INTRODUCTION Weyl fermions1 are massless fermions whose masslessness (gaplessness) is topologically protected2–5. -
An Effective Theory of Dirac Dark Matter
SLAC-PUB-13482 An Effective Theory of Dirac Dark Matter Roni Harnik1 and Graham D. Kribs2 1SITP, Physics Department, Stanford University, Stanford, CA 94305 and SLAC, Stanford University, Menlo Park, CA 94025 2Department of Physics and Institute of Theoretical Science, University of Oregon, Eugene, OR 97403 (Dated: October 31, 2008) A stable Dirac fermion with four-fermion interactions to leptons suppressed by a scale Λ ∼ 1 TeV is shown to provide a viable candidate for dark matter. The thermal relic abundance matches cosmology, while nuclear recoil direct detection bounds are automatically avoided in the absence of (large) couplings to quarks. The annihilation cross section in the early Universe is the same as the annihilation in our galactic neighborhood. This allows Dirac fermion dark matter to naturally explain the positron ratio excess observed by PAMELA with a minimal boost factor, given present astrophysical uncertainties. We use the Galprop program for propagation of signal and background; we discuss in detail the uncertainties resulting from the propagation parameters and, more impor- tantly, the injected spectra. Fermi/GLAST has an opportunity to see a feature in the gamma-ray spectrum at the mass of the Dirac fermion. The excess observed by ATIC/PPB-BETS may also be explained with Dirac dark matter that is heavy. A supersymmetric model with a Dirac bino provides a viable UV model of the effective theory. The dominance of the leptonic operators, and thus the observation of an excess in positrons and not in anti-protons, is naturally explained by the large hypercharge and low mass of sleptons as compared with squarks. -
UNIVERSITY of CALIFORNIA RIVERSIDE Explore Spin
UNIVERSITY OF CALIFORNIA RIVERSIDE Explore Spin Dependent Phenomenon in Topological Insulator and Magnetic Insulator A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics by Zilong Jiang December 2015 Dissertation Committee: Dr. Jing Shi, Chairperson Dr. Ward Beyermann Dr. Jeanie Lau Copyright by Zilong Jiang 2015 The Dissertation of Zilong Jiang is approved: Committee Chairperson University of California, Riverside Acknowledgment Accomplishment of Ph.D. study is the most meaningful and proud moment for me. The life at University of California, Riverside endows me so much memory, capability, painful experience, happy time, active mind and strong work ethics, which will add significant value to my future. Writing this dissertation is not only an essential requirement for pursuing a Ph.D. degree, also a very good chance to take a look back at what I have achieved and who I should appreciate in the past five years. First and foremost I want to make a sincere acknowledgment to my supervisor, Prof. Jing Shi who gives me extensive research opportunities and professional academic advices. I joined the Shi’s group since the second year of Ph.D. study. Dr. Shi helps me build up a long term research proposal focusing on the emergent material topological insulator and related heterostructures. Dr.Shi encourages me to have the aggressive attitude towards the research work with great passion. He teaches me the scientific thinking and working mode and enjoys discussing about insightful ideas and challenging problems with me. With his support and delicate supervision, I also set up a successful collaboration with people inside and outside campus. -
Majorana Fermions As Emergent Quasiparticles
Majorana fermions as emergent quasiparticles Abhisek Sahu1 1Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., V6T 1Z1, Canada (Dated: December 19, 2020) Majorana fermions are a special type of fermion predicted by the Dirac’s equation that are their own antiparticles. Currently they have rapidly gained interest in condensed matter physics as emergent quasiparticles in certain systems like topological superconductors. In this article, we review the theory of Majorana fermions starting from the Dirac equation. Then we discuss using Bogoliubov-deGennes formalism how Superconductors form ideal hunting grounds for Majorana particles and introduce the notion of Majorana zero modes. Finally we discuss the Kitaev model- a paradigmatic model to look for unpaired Majorana zero modes. I. INTRODUCTION II. WHAT ARE MAJORANA FERMIONS? In this section, we briefly review how Majorana In 1928 developed the wave equation that describes fermions come about from Dirac’s equations. The relativistic spin 1=2 particles. The solutions of this Dirac equation for a free particle 1is equation are complex valued four component spinors µ which can be interpreted as a spin 1=2 particle- (iγ @µ − m)Ψ(x) = 0 (1) antiparticle pair. It was Etorre Majorana’s insight to T look for a completely real set of solutions for the Dirac Where Ψ(x) = ( 1; 2; 3; 4) is a four-component µ equation in order to create a symmetric theory of par- spinor field and γ are 4 × 4 matrices satisfying the ticles and anti-particles. As a result in the year 1937 following algebra: he introduced the notion of fermions which are their µ ν µν y own anti-particles; known today as Majorana fermions fγ ; γ g = 2η ; γ0γµγ0 = γµ: (2) [1].