Experimental Discovery of a Topological Weyl Semimetal State in Tap
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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. -
Novel Electronic Properties of Monoclinic
www.nature.com/scientificreports OPEN Novel electronic properties of monoclinic M P4 (M = Cr, Mo, W) compounds with or without topological nodal line Muhammad Rizwan Khan1,2, Kun Bu1,2, Jun‑Shuai Chai1,2 & Jian‑Tao Wang1,2,3* Transition metal phosphides hold novel metallic, semimetallic, and semiconducting behaviors. Here we report by ab initio calculations a systematical study on the structural and electronic properties of 6 MP4 (M = Cr, Mo, W) phosphides in monoclinic C2/c (C 2h ) symmetry. Their dynamical stabilities have been confrmed by phonon modes calculations. Detailed analysis of the electronic band structures and density of states reveal that CrP4 is a semiconductor with an indirect band gap of 0.47 eV in association with the p orbital of P atoms, while MoP4 is a Dirac semimetal with an isolated nodal point at the Ŵ point and WP4 is a topological nodal line semimetal with a closed nodal ring inside the frst Brillouin zone relative to the d orbital of Mo and W atoms, respectively. Comparison of the phosphides with group VB, VIB and VIIB transition metals shows a trend of change from metallic to semiconducting behavior from VB-MP4 to VIIB-MP4 compounds. These results provide a systematical understandings on the distinct electronic properties of these compounds. Transition metal phosphides (TMPs) have been attracted considerable research interest due to their structural and compositional diversity that results in a broad range of novel electronic, magnetic and catalytic properties1–4. Tis family consists of large number of materials, having distinct crystallographic structures and morpholo- gies because of choices of diferent TMs and phosphorus atoms 5. -
Holographic Topological Semimetals Arxiv:1911.07978V1 [Hep-Th]
Holographic Topological Semimetals Karl Landsteiner Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera 13-15, Campus Cantoblanco, 28049, Spain E-mail: [email protected] Yan Liu Center for Gravitational Physics, Department of Space Science, Beihang University, Beijing 100191, China Key Laboratory of Space Environment Monitoring and Information Processing, Ministry of Industry and Information Technology, Beijing, China E-mail: [email protected] Ya-Wen Sun School of physics & CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China E-mail: [email protected] Abstract The holographic duality allows to construct and study models of strongly cou- pled quantum matter via dual gravitational theories. In general such models are characterized by the absence of quasiparticles, hydrodynamic behavior and Planck- ian dissipation times. One particular interesting class of quantum materials are ungapped topological semimetals which have many interesting properties from Hall transport to topologically protected edge states. We review the application of the holographic duality to this type of quantum matter including the construction of holographic Weyl semimetals, nodal line semimetals, quantum phase transition to arXiv:1911.07978v1 [hep-th] 18 Nov 2019 trivial states (ungapped and gapped), the holographic dual of Fermi arcs and how new unexpected transport properties, -
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. -
Hear the Sound of Weyl Fermions
PHYSICAL REVIEW X 9, 021053 (2019) Featured in Physics Hear the Sound of Weyl Fermions Zhida Song1,2 and Xi Dai1,* 1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA (Received 4 February 2019; revised manuscript received 10 April 2019; published 17 June 2019) Quasiparticles and collective modes are two fundamental aspects that characterize quantum matter in addition to its ground-state features. For example, the low-energy physics for Fermi-liquid phase in He-III is featured not only by fermionic quasiparticles near the chemical potential but also by fruitful collective modes in the long-wave limit, including several different sound waves that can propagate through it under different circumstances. On the other hand, it is very difficult for sound waves to be carried by electron liquid in ordinary metals due to the fact that long-range Coulomb interaction among electrons will generate a plasmon gap for ordinary electron density oscillation and thus prohibits the propagation of sound waves through it. In the present paper, we propose a unique type of acoustic collective mode in Weyl semimetals under magnetic field called chiral zero sound. Chiral zero sound can be stabilized under the so-called “chiral limit,” where the intravalley scattering time is much shorter than the intervalley one and propagates only along an external magnetic field for Weyl semimetals with multiple pairs of Weyl points. The sound velocity of chiral zero sound is proportional to the field strength in the weak field limit, whereas it oscillates dramatically in the strong field limit, generating an entirely new mechanism for quantum oscillations through the dynamics of neutral bosonic excitation, which may manifest itself in the thermal conductivity measurements under magnetic field. -
Arxiv:2008.10628V3 [Cond-Mat.Str-El] 14 Jan 2021
Prediction of Spin Polarized Fermi Arcs in Quasiparticle Interference of CeBi Zhao Huang,1 Christopher Lane,1, 2 Chao Cao,3 Guo-Xiang Zhi,4 Yu Liu,5 Christian E. Matt,5 Brinda Kuthanazhi,6, 7 Paul C. Canfield,6, 7 Dmitry Yarotski,2 A. J. Taylor,2 and Jian-Xin Zhu1, 2, * 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Center for Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3Department of Physics, Hangzhou Normal University, Hangzhou 310036, China 4Department of Physics, Zhejiang University, Hangzhou 310013, China 5Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 6Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA 7Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Dated: January 15, 2021) We predict that CeBi in the ferromagnetic state is a Weyl semimetal. Our calculations within density func- tional theory show the existence of two pairs of Weyl nodes on the momentum path (0,0,kz) at 15 meV above and 100 meV below the Fermi level. Two corresponding Fermi arcs are obtained on surfaces of mirror- symmetric (010)-oriented slabs at E = 15 meV and both arcs are interrupted into three segments due to hy- bridization with a set of trivial surface bands. By studying the spin texture of surface states, we find the two Fermi arcs are strongly spin-polarized but in opposite directions, which can be detected by spin-polarized ARPES measurements. Our theoretical study of quasiparticle interference (QPI) for a nonmagnetic impurity at the Bi site also reveals several features related to the Fermi arcs. -
Nonsaturating Large Magnetoresistance in Semimetals
This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. Nonsaturating large magnetoresistance in semimetals Leahy, Ian A.; Lin, Yu‑Ping; Siegfried, Peter E.; Treglia, Andrew C.; Song, Justin Chien Wen; Nandkishore, Rahul M.; Lee, Minhyea 2018 Leahy, I. A., Lin, Y.‑P., Siegfried, P. E., Treglia, A. C., Song, J. C. W., Nandkishore, R. M., & Lee, M. (2018). Nonsaturating large magnetoresistance in semimeta. Proceedings of the National Academy of Sciences of the United States of America, 115 (42), 10570‑10575. doi:10.1073/pnas.1808747115 https://hdl.handle.net/10356/137998 https://doi.org/10.1073/pnas.1808747115 © 2018 The Author(s). All rights reserved. This paper was published by National Academy of Sciences in Proceedings of the National Academy of Sciences of the United States of America and is made available with permission of The Author(s). Downloaded on 01 Oct 2021 03:16:44 SGT Non-saturating large magnetoresistance in semimetals Ian A. Leahy,1 Yu-Ping Lin,1 Peter E. Siegfried,1 Andrew C. Treglia,1 Justin C. W. Song,2 Rahul M. Nandkishore,1, 3 and Minhyea Lee1, 4, ∗ 1Department of Physics, University of Colorado, Boulder, CO 80309, USA 2Division of Physics and Applied Physics, Nanyang Technological University, Singapore 637371 3Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA 4Center for Experiments on Quantum Materials, University of Colorado, Boulder, CO 80309, USA (Dated: September 6, 2018) The rapidly expanding class of quantum materials known as topological semimetals (TSM) dis- play unique transport properties, including a striking dependence of resistivity on applied magnetic field, that are of great interest for both scientific and technological reasons. -
Interaction and Temperature Effects on the Magneto-Optical Conductivity Of
Interaction and temperature effects on the magneto-optical conductivity of Weyl liquids S. Acheche, R. Nourafkan, J. Padayasi, N. Martin, and A.-M. S. Tremblay D´epartement de physique; Institut quantique; and Regroupement qu´eb´ecois sur les mat´eriauxde pointe; Universit´ede Sherbrooke; Sherbrooke; Qu´ebec; Canada J1K 2R1 (Dated: July 29, 2020) Negative magnetoresistance is one of the manifestations of the chiral anomaly in Weyl semimetals. The magneto-optical conductivity also shows transitions between Landau levels that are not spaced as in an ordinary electron gas. How are such topological properties modified by interactions and temperature? We answer this question by studying a lattice model of Weyl semimetals with an on-site Hubbard interaction. Such an interacting Weyl semimetal, dubbed as Weyl liquid, may be realized in Mn3Sn. We solve that model with single-site dynamical mean-field theory. We find that in a Weyl liquid, quasiparticles can be characterized by a quasiparticle spectral weight Z, although their lifetime increases much more rapidly as frequency approaches zero than in an ordinary Fermi liquid. The negative magnetoresistance still exists, even though the slope of the linear dependence of the DC conductivity with respect to the magnetic field is decreased by the interaction. At elevated temperatures, a Weyl liquid crosses over to bad metallic behavior where the Drude peak becomes flat and featureless. We comment on the effect of a Zeeman term. I. INTRODUCTION nodes.11,12 This is a consequence of the parabolic density of states. It has been observed in the low-temperature, low-frequency optical spectroscopy of the known Weyl Weyl semimetals are three-dimensional (3D) analogs of 13 graphene with topologically protected band crossings and semimetal TaAs. -
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) × . -
It's Been a Weyl Coming
CONDENSED MATTER It’s been a Weyl coming Condensed-matter physics brings us quasiparticles that behave as massless fermions. B. Andrei Bernevig “Mathematizing may well be a creative activity of man, like language or music”1 — so said Hermann Weyl, the German physicist whose penchant for mathematical elegance prompted his prediction that a new particle would arise when the fermionic mass in the Dirac equation vanished2. Such a particle could carry charge but, unlike all known fermions, would be massless. During the course of his career, Weyl actually fell out of love with his prediction, largely because it implied the breaking of a particular symmetry, known as parity, which at the time was thought to be obeyed. More to the point, no such particle was observed during his lifetime. After his death, the Weyl fermion was proposed to describe neutrinos, which are now known to have mass. For some time, it seemed that the Weyl fermion was destined to be just an abstract concept from another beautiful mind. That was until the Weyl fermion entered the realm of condensed-matter physics. For several years this field has been considered fertile ground for finding the Weyl fermion. Now, three papers in Nature Physics3–5 have cemented earlier findings6,7 to confirm the predictions8,9 of Weyl physics in a family of nonmagnetic materials with broken inversion symmetry. In condensed-matter physics, specifically in solid-state band structures, Weyl fermions appear when two electronic bands cross. The crossing point is called a Weyl node, away from which the bands disperse linearly in the lattice momentum, giving rise to a special kind of semimetal. -
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† .