Luttinger Liquid at the Edge of a Graphene Vacuum
|
|
- Sandra Howard
- 5 years ago
- Views:
Transcription
1 Luttinger Liquid at the Edge of a Graphene Vacuum H.A. Fertig, Indiana University Luis Brey, CSIC, Madrid I. Introduction: Graphene Edge States (Non-Interacting) II. III. Quantum Hall Ferromagnetism and a Domain Wall at the Edge Properties of the Domain Wall IV. Ecitations from Filled (Graphene) Landau Levels (with Drew Iyengar and Jianhui Wang) V. Summary Funding: NSF
2 I. Edge States for Graphene B Honeycomb lattice, two atoms per unit cell Lattice constant:.46å Nearest neighbor distance: 1.4Å A Simple tight-binding model for p z orbitals: H = t n 1 n1n = n. n. t.5-3 ev n
3 A and B sublattice sites in unit cell For each k there are eigenvalues at ± ε particle-hole symmetry Fermi energy at ε= E F
4 t a n n l 3 ), ( ± = τ ε ± = Ψ ) ( ) ( ), ( 1 l l n n k y k y ik e n K φ φ ± = Ψ ) ( ) ( ), ( 1 l l n n k y k y ik e n K φ φ = Ψ,) ( φ ik e K = Ψ,) ( φ ik e K Wavefunctions in a magnetic field: state = harmonic oscillator φ n Energies: ε Particle-hole conjugates k With valley and spin indices, each Landau level is 4-fold degenerate
5 Real samples in eperiments are very narrow (.1-1µm) edges can have a major impact on transport Can get a full description of QHE within Dirac equation Edge structure can be probed directly via STM at very small length scales. Nothing comparable is possible in standard DEG s (GaAs samples, Si MOSFET s) K K K y K y Tight-binding results, armchair edge
6 II. Quantum Hall Ferromagnetism and the Graphene Edge E F DOS Energy E F Interactions DOS Energy Echange tends to force electrons into the same level even when bare splitting between levels is small Renormalizes gap to much larger value than epected from non-interacting theory (even if bare gap is zero!)
7 This does happen in graphene (Zhang et al., 6). Plateaus at ν=?,±1. System may be a quantum Hall ferromagnet. cf. Alicea and Fisher, 6 Nomura and Macdonald, 6 Fuchs and Lederer, 6
8 Vacuum state (undoped graphene): n= n=1 K,K n= K,K EF n=-1 n=- Low-lying ecitations: (+) spin (+valley) waves EF Spin stiffness Analogy with Heisenberg ferromagnet.
9 Consequences for edge states: 4 n= states (E z =) Electron-like edge state. (X ) Include Zeeman coupling Hole-like edge state. (X ) Domain wall Spin polarized Spin unpolarized
10 Description of the domain wall: θ( X Ψ = ) + θ( X) iϕ + + cos C + sin e C C > +, X,, X,, X, X < L X θ = ; X L θ = π; ϕ = E Pseudospin stiffness = πl ρ s X < L dθ dx + X < L ( E X )) cosθ ( X ) z ( θ(x )/π Result of minimizing energy. Width of domain wall set by strength of confinement. X /l
11 III. Properties of the Domain Wall = +, y = -, 1. φ=: Broken U(1) symmetry linearly dispersing collective mode φ ~ in-plane angle of spins m ~ position of domain wall
12 . Spin-charge coupling gapless charged ecitations! Twist phase once X =k y l = weight in w/f X =k y l Fermion operator:
13 3. Tunneling from STM tip: power law IV not a Fermi liquid! Power law eponent a function of confinement potential tunneling t Domain wall STM tip y Graphene sheet I ~ t de G [ ( E) G ( E ev ) G ( E ev ) G ( E) ] adv ret ret adv tip DW tip DW G 1 τ κ = + 1/ / ; = 4π ~ ρ / ( τ ) ~ T ( ) ( ) τψ y = ; τ ψ + y = ; ~ κ ( ) Γ U(1) spin stiffness Γ ~ confinement potential Eponent sensitive to edge confinement!
14 4. Tunneling from a bulk lead: possibility of a quantum phase transition (into 3D metal). Lead Tunneling t y Model lead as non-interacting electrons in a magnetic field with Perturbative RG: dt dl = ( κ ) t Shrinking t DW a Luttinger liquid Growing t DW + lead = Fermi liquid?
15 IV. Inter-Landau Level Ecitations (Magnetoplasmons) Low-lying ecited states = particle-hole pairs Electron in empty band Standard DEG: ql q cf. Kallin and Halperin, 1984 Hole in filled band Measurable in cyclotron resonance, inelastic light scattering. This picture is largely the same for graphene, just need to be careful about spinor structure of particle and hole states. Two-Body Problem Electron Hole (ħ=ω c =l=1)
16 To diagonalize (A= -By): 1. Adopt center and relative coordinate R=(r 1 +r )/, r= r 1 -r. Apply unitary transformation H = U+ H U with r r ip ( zˆ P) iy P r = center of mass momentum U = e e with z=+iy Wavefunctions constructed from: with
17 Wavefunctions are 4-vectors n +,n - > constructed from with energies Electron Hole 3. Apply unitary transformation to interaction H 1 : 4. Compute eigenvalues of two-body eigenenergies with fied P
18 Results: γ =4 γ 1 =4 4,, 1, 1 3,, 5, 1 Energy ħ vf 1 Lowest filled LL =, 1, Energy ħ vf 1 Lowest filled LL = 1 4, 1 3, 1, P P Interaction scale: β=(e /εl)/(ħv F /l ) (c/v F ε)/137 =.73
19 γ <4 γ 1 <4,,, 1,, 4, 1, 1, 1, 1 3, 1 Energy ħ vf 1, Energy ħ vf, 1 1, P P Note negative energy
20 Comments: 1. Negative energies because we have not included loss of echange self-energy many-body approach needed. Landau level miing relatively small. Probability P Note however for β 1, LL miing becomes much more pronounced system on cusp between weakly and strongly interacting
21 Many-Body Particle-Hole Approach A generalization of spin-wave calculation Almost, but not quite.
22 Must watch out for degeneracies n= n=1 n= 1 E F n=-1 n=- K,K Ecitations characterized by S z and τ z Also: Echange energy with infinite number of filled hole levels leads to (logarithmically) divergent self-energy. Fi this with an eplicit cutoff in number of filled Landau levels.
23 µ (1, 1) (1, 1) ( 1, 1) ( 1, 1) (, ) (, 1) (, 1) ( 1, ) (1, ) γ=4, =spin, double arrows=pseudospin (m,n)=(s z,t z )
24 Energy generically involves four terms: Noninteracting Direct (Ladders) Echange (Bubbles - RPA) Echange self-energy
25 Results: N= 1 E E e Ε γ <4 Intra-Landau level Two-body result (up to constant) E e Ε P E 1 E 3 E 4 γ <4 E γ 4 n=1 Comments: 1. Change in kinetic energy and Zeeman energy must be added in. Gapless ecitations for ν=-1,1 3. Ecitation spectra identical for ν=-1,1: particle-hole symmetry P
26 E 1 γ 1 =4 Dashed lines equivalent to two-body result. E e Ε 1 E Very large many-body correction! P E e Ε 1 E 1 γ =4 E P Minima/maima may be visible in inelastic light scattering or microwave absorption.
27 Summary Graphene: a new and interesting material both for fundamental and applications reasons. Clean system is likely a quantum Hall ferromagnet. Armchair edges: oppositely dispersing spin up and down bands domain wall Domain wall supports gapless collective ecitations, and gapless charged ecitations through pseudospin teture. Domain wall supports power law IV (Luttinger liquid). Domain wall may undergo quantum phase transition when coupled to a bulk lead. Collective inter-landau level ecitations = ecitons Many-body corrections split and distort dispersions found in two-body problem
Spin Superfluidity and Graphene in a Strong Magnetic Field
Spin Superfluidity and Graphene in a Strong Magnetic Field by B. I. Halperin Nano-QT 2016 Kyiv October 11, 2016 Based on work with So Takei (CUNY), Yaroslav Tserkovnyak (UCLA), and Amir Yacoby (Harvard)
More informationGraphite, graphene and relativistic electrons
Graphite, graphene and relativistic electrons Introduction Physics of E. graphene Y. Andrei Experiments Rutgers University Transport electric field effect Quantum Hall Effect chiral fermions STM Dirac
More informationQuantum Confinement in Graphene
Quantum Confinement in Graphene from quasi-localization to chaotic billards MMM dominikus kölbl 13.10.08 1 / 27 Outline some facts about graphene quasibound states in graphene numerical calculation of
More informationSpin orbit interaction in graphene monolayers & carbon nanotubes
Spin orbit interaction in graphene monolayers & carbon nanotubes Reinhold Egger Institut für Theoretische Physik, Düsseldorf Alessandro De Martino Andreas Schulz, Artur Hütten MPI Dresden, 25.10.2011 Overview
More informationQuantum Hall effect. Quantization of Hall resistance is incredibly precise: good to 1 part in I believe. WHY?? G xy = N e2 h.
Quantum Hall effect V1 V2 R L I I x = N e2 h V y V x =0 G xy = N e2 h n.b. h/e 2 = 25 kohms Quantization of Hall resistance is incredibly precise: good to 1 part in 10 10 I believe. WHY?? Robustness Why
More informationElectron interactions in graphene in a strong magnetic field
Electron interactions in graphene in a strong magnetic field Benoit Douçot Mark O. Goerbig Roderich Moessner K = K K CNRS and ENS Paris VI+XI cond-mat/0604554 Overview Recent experiments: integer QHE in
More informationSpin-Orbit Interactions in Semiconductor Nanostructures
Spin-Orbit Interactions in Semiconductor Nanostructures Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. http://www.physics.udel.edu/~bnikolic Spin-Orbit Hamiltonians
More informationSpins and spin-orbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spin-orbit coupling in semiconductors, metals, and nanostructures Behavior of non-equilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationKondo effect in multi-level and multi-valley quantum dots. Mikio Eto Faculty of Science and Technology, Keio University, Japan
Kondo effect in multi-level and multi-valley quantum dots Mikio Eto Faculty of Science and Technology, Keio University, Japan Outline 1. Introduction: next three slides for quantum dots 2. Kondo effect
More informationSupplementary Figure S1. STM image of monolayer graphene grown on Rh (111). The lattice
Supplementary Figure S1. STM image of monolayer graphene grown on Rh (111). The lattice mismatch between graphene (0.246 nm) and Rh (111) (0.269 nm) leads to hexagonal moiré superstructures with the expected
More informationFractional quantum Hall effect and duality. Dam T. Son (University of Chicago) Canterbury Tales of hot QFTs, Oxford July 11, 2017
Fractional quantum Hall effect and duality Dam T. Son (University of Chicago) Canterbury Tales of hot QFTs, Oxford July 11, 2017 Plan Plan General prologue: Fractional Quantum Hall Effect (FQHE) Plan General
More informationThe Quantum Spin Hall Effect
The Quantum Spin Hall Effect Shou-Cheng Zhang Stanford University with Andrei Bernevig, Taylor Hughes Science, 314,1757 2006 Molenamp et al, Science, 318, 766 2007 XL Qi, T. Hughes, SCZ preprint The quantum
More informationThe ac conductivity of monolayer graphene
The ac conductivity of monolayer graphene Sergei G. Sharapov Department of Physics and Astronomy, McMaster University Talk is based on: V.P. Gusynin, S.G. Sh., J.P. Carbotte, PRL 96, 568 (6), J. Phys.:
More informationPart III: Impurities in Luttinger liquids
Functional RG for interacting fermions... Part III: Impurities in Luttinger liquids 1. Luttinger liquids 2. Impurity effects 3. Microscopic model 4. Flow equations 5. Results S. Andergassen, T. Enss (Stuttgart)
More informationQuantum Hall Effect in Graphene p-n Junctions
Quantum Hall Effect in Graphene p-n Junctions Dima Abanin (MIT) Collaboration: Leonid Levitov, Patrick Lee, Harvard and Columbia groups UIUC January 14, 2008 Electron transport in graphene monolayer New
More informationEffects of Interactions in Suspended Graphene
Effects of Interactions in Suspended Graphene Ben Feldman, Andrei Levin, Amir Yacoby, Harvard University Broken and unbroken symmetries in the lowest LL: spin and valley symmetries. FQHE Discussions with
More informationQuantum Oscillations in Graphene in the Presence of Disorder
WDS'9 Proceedings of Contributed Papers, Part III, 97, 9. ISBN 978-8-778-- MATFYZPRESS Quantum Oscillations in Graphene in the Presence of Disorder D. Iablonskyi Taras Shevchenko National University of
More informationCorrelations between spin accumulation and degree of time-inverse breaking for electron gas in solid
Correlations between spin accumulation and degree of time-inverse breaking for electron gas in solid V.Zayets * Spintronic Research Center, National Institute of Advanced Industrial Science and Technology
More informationPOEM: Physics of Emergent Materials
POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Reference: Bernevig Topological Insulators and Topological Superconductors Tutorials:
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationLecture 2 2D Electrons in Excited Landau Levels
Lecture 2 2D Electrons in Excited Landau Levels What is the Ground State of an Electron Gas? lower density Wigner Two Dimensional Electrons at High Magnetic Fields E Landau levels N=2 N=1 N= Hartree-Fock
More informationTopological insulator with time-reversal symmetry
Phys620.nb 101 7 Topological insulator with time-reversal symmetry Q: Can we get a topological insulator that preserves the time-reversal symmetry? A: Yes, with the help of the spin degree of freedom.
More informationGRAPHENE the first 2D crystal lattice
GRAPHENE the first 2D crystal lattice dimensionality of carbon diamond, graphite GRAPHENE realized in 2004 (Novoselov, Science 306, 2004) carbon nanotubes fullerenes, buckyballs what s so special about
More informationLecture 20: Semiconductor Structures Kittel Ch 17, p , extra material in the class notes
Lecture 20: Semiconductor Structures Kittel Ch 17, p 494-503, 507-511 + extra material in the class notes MOS Structure Layer Structure metal Oxide insulator Semiconductor Semiconductor Large-gap Semiconductor
More informationFractional quantum Hall effect and duality. Dam Thanh Son (University of Chicago) Strings 2017, Tel Aviv, Israel June 26, 2017
Fractional quantum Hall effect and duality Dam Thanh Son (University of Chicago) Strings 2017, Tel Aviv, Israel June 26, 2017 Plan Fractional quantum Hall effect Halperin-Lee-Read (HLR) theory Problem
More informationarxiv: v1 [cond-mat.mes-hall] 26 Sep 2013
Berry phase and the unconventional quantum Hall effect in graphene Jiamin Xue Microelectronic Research Center, The University arxiv:1309.6714v1 [cond-mat.mes-hall] 26 Sep 2013 of Texas at Austin, Austin,
More informationElectron Interactions and Nanotube Fluorescence Spectroscopy C.L. Kane & E.J. Mele
Electron Interactions and Nanotube Fluorescence Spectroscopy C.L. Kane & E.J. Mele Large radius theory of optical transitions in semiconducting nanotubes derived from low energy theory of graphene Phys.
More informationIs the composite fermion a Dirac particle?
Is the composite fermion a Dirac particle? Dam T. Son GGI conference Gauge/gravity duality 2015 Ref.: 1502.03446 Plan Plan Fractional quantum Hall effect Plan Fractional quantum Hall effect Composite fermion
More informationPhysics of Semiconductors (Problems for report)
Physics of Semiconductors (Problems for report) Shingo Katsumoto Institute for Solid State Physics, University of Tokyo July, 0 Choose two from the following eight problems and solve them. I. Fundamentals
More informationECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015
ECE 535 Theory of Semiconductors and Semiconductor Devices Fall 2015 Homework # 5 Due Date: 11/17/2015 Problem # 1 Two carbon atoms and four hydrogen atoms form and ethane molecule with the chemical formula
More informationMn in GaAs: from a single impurity to ferromagnetic layers
Mn in GaAs: from a single impurity to ferromagnetic layers Paul Koenraad Department of Applied Physics Eindhoven University of Technology Materials D e v i c e s S y s t e m s COBRA Inter-University Research
More informationWhat is a topological insulator? Ming-Che Chang Dept of Physics, NTNU
What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU A mini course on topology extrinsic curvature K vs intrinsic (Gaussian) curvature G K 0 G 0 G>0 G=0 K 0 G=0 G
More informationMinimal Update of Solid State Physics
Minimal Update of Solid State Physics It is expected that participants are acquainted with basics of solid state physics. Therefore here we will refresh only those aspects, which are absolutely necessary
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS
2753 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS TRINITY TERM 2011 Wednesday, 22 June, 9.30 am 12.30
More informationThe Dirac composite fermions in fractional quantum Hall effect. Dam Thanh Son (University of Chicago) Nambu Memorial Symposium March 12, 2016
The Dirac composite fermions in fractional quantum Hall effect Dam Thanh Son (University of Chicago) Nambu Memorial Symposium March 12, 2016 A story of a symmetry lost and recovered Dam Thanh Son (University
More informationTopological Properties of Quantum States of Condensed Matter: some recent surprises.
Topological Properties of Quantum States of Condensed Matter: some recent surprises. F. D. M. Haldane Princeton University and Instituut Lorentz 1. Berry phases, zero-field Hall effect, and one-way light
More informationRobustness of edge states in graphene quantum dots
Chapter 5 Robustness of edge states in graphene quantum dots 5.1 Introduction The experimental discovery of graphene [3, 57], a monolayer of carbon atoms, has opened room for new electronic devices (for
More informationTight binding and emergence of "Dirac" equation in graphene.
Tight binding and emergence of "Dirac" equation in graphene. A. A. Kozhevnikov 1 1 Laboratory of Theoretical Physics, S. L. Sobolev Institute for Mathematics, and Novosibirsk State University April 22,
More informationMath Questions for the 2011 PhD Qualifier Exam 1. Evaluate the following definite integral 3" 4 where! ( x) is the Dirac! - function. # " 4 [ ( )] dx x 2! cos x 2. Consider the differential equation dx
More informationELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES
ELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES D. RACOLTA, C. ANDRONACHE, D. TODORAN, R. TODORAN Technical University of Cluj Napoca, North University Center of
More informationSupersymmetry and Quantum Hall effect in graphene
Supersymmetry and Quantum Hall effect in graphene Ervand Kandelaki Lehrstuhl für Theoretische Festköperphysik Institut für Theoretische Physik IV Universität Erlangen-Nürnberg March 14, 007 1 Introduction
More informationStrongly correlated Cooper pair insulators and superfluids
Strongly correlated Cooper pair insulators and superfluids Predrag Nikolić George Mason University Acknowledgments Collaborators Subir Sachdev Eun-Gook Moon Anton Burkov Arun Paramekanti Affiliations and
More informationRelativistic magnetotransport in graphene
Relativistic magnetotransport in graphene Markus Müller in collaboration with Lars Fritz (Harvard) Subir Sachdev (Harvard) Jörg Schmalian (Iowa) Landau Memorial Conference June 6, 008 Outline Relativistic
More informationSUPPLEMENTARY INFORMATION
doi:1.138/nature12186 S1. WANNIER DIAGRAM B 1 1 a φ/φ O 1/2 1/3 1/4 1/5 1 E φ/φ O n/n O 1 FIG. S1: Left is a cartoon image of an electron subjected to both a magnetic field, and a square periodic lattice.
More informationZero-bias conductance peak in detached flakes of superconducting 2H-TaS2 probed by STS
Zero-bias conductance peak in detached flakes of superconducting 2H-TaS2 probed by STS J. A. Galvis, L. C., I. Guillamon, S. Vieira, E. Navarro-Moratalla, E. Coronado, H. Suderow, F. Guinea Laboratorio
More informationTopology of the Fermi surface wavefunctions and magnetic oscillations in metals
Topology of the Fermi surface wavefunctions and magnetic oscillations in metals A. Alexandradinata L.I. Glazman Yale University arxiv:1707.08586, arxiv:1708.09387 + in preparation Physics Next Workshop
More information3.14. The model of Haldane on a honeycomb lattice
4 Phys60.n..7. Marginal case: 4 t Dirac points at k=(,). Not an insulator. No topological index...8. case IV: 4 t All the four special points has z 0. We just use u I for the whole BZ. No singularity.
More informationNo reason one cannot have double-well structures: With MBE growth, can control well thicknesses and spacings at atomic scale.
The story so far: Can use semiconductor structures to confine free carriers electrons and holes. Can get away with writing Schroedinger-like equation for Bloch envelope function to understand, e.g., -confinement
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS
A11046W1 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS TRINITY TERM 2015 Wednesday, 17 June, 2.30
More informationSTM spectra of graphene
STM spectra of graphene K. Sengupta Theoretical Physics Division, IACS, Kolkata. Collaborators G. Baskaran, I.M.Sc Chennai, K. Saha, IACS Kolkata I. Paul, Grenoble France H. Manoharan, Stanford USA Refs:
More informationarxiv: v1 [cond-mat.mes-hall] 1 Nov 2011
V The next nearest neighbor effect on the D materials properties Maher Ahmed Department of Physics and Astronomy, University of Western Ontario, London ON N6A K7, Canada and arxiv:.v [cond-mat.mes-hall]
More informationVarious Facets of Chalker- Coddington network model
Various Facets of Chalker- Coddington network model V. Kagalovsky Sami Shamoon College of Engineering Beer-Sheva Israel Context Integer quantum Hall effect Semiclassical picture Chalker-Coddington Coddington
More informationTime Reversal Invariant Ζ 2 Topological Insulator
Time Reversal Invariant Ζ Topological Insulator D Bloch Hamiltonians subject to the T constraint 1 ( ) ΘH Θ = H( ) with Θ = 1 are classified by a Ζ topological invariant (ν =,1) Understand via Bul-Boundary
More informationElectron-electron interactions and Dirac liquid behavior in graphene bilayers
Electron-electron interactions and Dirac liquid behavior in graphene bilayers arxiv:85.35 S. Viola Kusminskiy, D. K. Campbell, A. H. Castro Neto Boston University Workshop on Correlations and Coherence
More informationElectrons in a weak periodic potential
Electrons in a weak periodic potential Assumptions: 1. Static defect-free lattice perfectly periodic potential. 2. Weak potential perturbative effect on the free electron states. Perfect periodicity of
More informationBerry s phase in Hall Effects and Topological Insulators
Lecture 6 Berry s phase in Hall Effects and Topological Insulators Given the analogs between Berry s phase and vector potentials, it is not surprising that Berry s phase can be important in the Hall effect.
More informationIs the composite fermion a Dirac particle?
Is the composite fermion a Dirac particle? Dam T. Son (University of Chicago) Cold atoms meet QFT, 2015 Ref.: 1502.03446 Plan Plan Composite fermion: quasiparticle of Fractional Quantum Hall Effect (FQHE)
More informationRotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD
Rotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD Uwe-Jens Wiese Bern University LATTICE08, Williamsburg, July 14, 008 S. Chandrasekharan (Duke University) F.-J. Jiang, F.
More informationBeyond the Quantum Hall Effect
Beyond the Quantum Hall Effect Jim Eisenstein California Institute of Technology School on Low Dimensional Nanoscopic Systems Harish-chandra Research Institute January February 2008 Outline of the Lectures
More informationTopological Kondo Insulator SmB 6. Tetsuya Takimoto
Topological Kondo Insulator SmB 6 J. Phys. Soc. Jpn. 80 123720, (2011). Tetsuya Takimoto Department of Physics, Hanyang University Collaborator: Ki-Hoon Lee (POSTECH) Content 1. Introduction of SmB 6 in-gap
More informationElectronic structure of correlated electron systems. Lecture 2
Electronic structure of correlated electron systems Lecture 2 Band Structure approach vs atomic Band structure Delocalized Bloch states Fill up states with electrons starting from the lowest energy No
More informationTunneling Spectroscopy of Disordered Two-Dimensional Electron Gas in the Quantum Hall Regime
Tunneling Spectroscopy of Disordered Two-Dimensional Electron Gas in the Quantum Hall Regime The Harvard community has made this article openly available. Please share how this access benefits you. Your
More information7.4. Why we have two different types of materials: conductors and insulators?
Phys463.nb 55 7.3.5. Folding, Reduced Brillouin zone and extended Brillouin zone for free particles without lattices In the presence of a lattice, we can also unfold the extended Brillouin zone to get
More informationThe semiclassical semiclassical model model of Electron dynamics
Solid State Theory Physics 545 The semiclassical model of The semiclassical model of lectron dynamics Fermi surfaces and lectron dynamics Band structure calculations give () () determines the dynamics
More informationPseudospin Magnetism in Graphene
Title Phys. Rev. B 77, 041407 (R) (008) Pseudospin Magnetism in Graphene Hongi Min 1, Giovanni Borghi, Marco Polini, A.H. MacDonald 1 1 Department of Physics, The University of Texas at Austin, Austin
More informationPHYSICS 250 May 4, Final Exam - Solutions
Name: PHYSICS 250 May 4, 999 Final Exam - Solutions Instructions: Work all problems. You may use a calculator and two pages of notes you may have prepared. There are problems of varying length and difficulty.
More informationTopological insulators. Pavel Buividovich (Regensburg)
Topological insulators Pavel Buividovich (Regensburg) Hall effect Classical treatment Dissipative motion for point-like particles (Drude theory) Steady motion Classical Hall effect Cyclotron frequency
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationResonating Valence Bond point of view in Graphene
Resonating Valence Bond point of view in Graphene S. A. Jafari Isfahan Univ. of Technology, Isfahan 8456, Iran Nov. 29, Kolkata S. A. Jafari, Isfahan Univ of Tech. RVB view point in graphene /2 OUTLINE
More informationManipulation of Dirac cones in artificial graphenes
Manipulation of Dirac cones in artificial graphenes Gilles Montambaux Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France - Berry phase Berry phase K K -p Graphene electronic
More informationAtkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 8: Quantum Theory: Techniques and Applications
Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas Chapter 8: Quantum Theory: Techniques and Applications TRANSLATIONAL MOTION wavefunction of free particle, ψ k = Ae ikx + Be ikx,
More informationSymmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona
Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether
More informationTopological Insulator Surface States and Electrical Transport. Alexander Pearce Intro to Topological Insulators: Week 11 February 2, / 21
Topological Insulator Surface States and Electrical Transport Alexander Pearce Intro to Topological Insulators: Week 11 February 2, 2017 1 / 21 This notes are predominately based on: J.K. Asbóth, L. Oroszlány
More informationChapter 4 (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence
V, E, Chapter (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence Potential Total Energies and Probability density
More informationTopological Defects in the Topological Insulator
Topological Defects in the Topological Insulator Ashvin Vishwanath UC Berkeley arxiv:0810.5121 YING RAN Frank YI ZHANG Quantum Hall States Exotic Band Topology Topological band Insulators (quantum spin
More informationCorrelated 2D Electron Aspects of the Quantum Hall Effect
Correlated 2D Electron Aspects of the Quantum Hall Effect Magnetic field spectrum of the correlated 2D electron system: Electron interactions lead to a range of manifestations 10? = 4? = 2 Resistance (arb.
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationLecture 4: Basic elements of band theory
Phys 769 Selected Topics in Condensed Matter Physics Summer 010 Lecture 4: Basic elements of band theory Lecturer: Anthony J. Leggett TA: Bill Coish 1 Introduction Most matter, in particular most insulating
More informationQuantum numbers and collective phases of composite fermions
Quantum numbers and collective phases of composite fermions Quantum numbers Effective magnetic field Mass Magnetic moment Charge Statistics Fermi wave vector Vorticity (vortex charge) Effective magnetic
More informationCalculating Band Structure
Calculating Band Structure Nearly free electron Assume plane wave solution for electrons Weak potential V(x) Brillouin zone edge Tight binding method Electrons in local atomic states (bound states) Interatomic
More informationZooming in on the Quantum Hall Effect
Zooming in on the Quantum Hall Effect Cristiane MORAIS SMITH Institute for Theoretical Physics, Utrecht University, The Netherlands Capri Spring School p.1/31 Experimental Motivation Historical Summary:
More informationPhysics of Semiconductors
Physics of Semiconductors 13 th 2016.7.11 Shingo Katsumoto Department of Physics and Institute for Solid State Physics University of Tokyo Outline today Laughlin s justification Spintronics Two current
More informationQuantum transport through graphene nanostructures
Quantum transport through graphene nanostructures S. Rotter, F. Libisch, L. Wirtz, C. Stampfer, F. Aigner, I. Březinová, and J. Burgdörfer Institute for Theoretical Physics/E136 December 9, 2009 Graphene
More informationTopological Physics in Band Insulators II
Topological Physics in Band Insulators II Gene Mele University of Pennsylvania Topological Insulators in Two and Three Dimensions The canonical list of electric forms of matter is actually incomplete Conductor
More informationwhere a is the lattice constant of the triangular Bravais lattice. reciprocal space is spanned by
Contents 5 Topological States of Matter 1 5.1 Intro.......................................... 1 5.2 Integer Quantum Hall Effect..................... 1 5.3 Graphene......................................
More informationBloch, Landau, and Dirac: Hofstadter s Butterfly in Graphene. Philip Kim. Physics Department, Columbia University
Bloch, Landau, and Dirac: Hofstadter s Butterfly in Graphene Philip Kim Physics Department, Columbia University Acknowledgment Prof. Cory Dean (now at CUNY) Lei Wang Patrick Maher Fereshte Ghahari Carlos
More information5.5. Representations. Phys520.nb Definition N is called the dimensions of the representations The trivial presentation
Phys50.nb 37 The rhombohedral and hexagonal lattice systems are not fully compatible with point group symmetries. Knowing the point group doesn t uniquely determine the lattice systems. Sometimes we can
More informationSpinon magnetic resonance. Oleg Starykh, University of Utah
Spinon magnetic resonance Oleg Starykh, University of Utah May 17-19, 2018 Examples of current literature 200 cm -1 = 6 THz Spinons? 4 mev = 1 THz The big question(s) What is quantum spin liquid? No broken
More informationLecture 4 (19/10/2012)
4B5: Nanotechnology & Quantum Phenomena Michaelmas term 2012 Dr C Durkan cd229@eng.cam.ac.uk www.eng.cam.ac.uk/~cd229/ Lecture 4 (19/10/2012) Boundary-value problems in Quantum Mechanics - 2 Bound states
More informationSUPPLEMENTARY INFORMATION
Dirac electron states formed at the heterointerface between a topological insulator and a conventional semiconductor 1. Surface morphology of InP substrate and the device Figure S1(a) shows a 10-μm-square
More informationarxiv:cond-mat/ v2 8 May 1998
Canted antiferromagnetic and spin singlet quantum Hall states in double-layer systems S. Das Sarma 1, Subir Sachdev 2,andLianZheng 1 1 Department of Physics, University of Maryland, College Park, Maryland
More informationOptoelectronic and Transport Properties of Gapped Graphene
0 Optoelectronic and Transport Properties of Gapped Graphene Godfrey Gumbs, Danhong Huang, Andrii Iurov, and Bo Gao CONTENTS Abstract... 489 0. Introduction... 489 0. Dirac Fermions, Chirality, and Tunable
More informationTopological insulator part I: Phenomena
Phys60.nb 5 Topological insulator part I: Phenomena (Part II and Part III discusses how to understand a topological insluator based band-structure theory and gauge theory) (Part IV discusses more complicated
More informationLecture. Ref. Ihn Ch. 3, Yu&Cardona Ch. 2
Lecture Review of quantum mechanics, statistical physics, and solid state Band structure of materials Semiconductor band structure Semiconductor nanostructures Ref. Ihn Ch. 3, Yu&Cardona Ch. 2 Reminder
More informationNematic and Magnetic orders in Fe-based Superconductors
Nematic and Magnetic orders in Fe-based Superconductors Cenke Xu Harvard University Collaborators: Markus Mueller, Yang Qi Subir Sachdev, Jiangping Hu Collaborators: Subir Sachdev Markus Mueller Yang Qi
More informationSUPPLEMENTARY INFORMATION
doi:10.1038/nature10941 1. Motivation and summary of results. In this work we combine a central tenet of condensed matter physics how electronic band structure emerges from a periodic potential in a crystal
More informationWORLD SCIENTIFIC (2014)
WORLD SCIENTIFIC (2014) LIST OF PROBLEMS Chapter 1: Magnetism of Free Electrons and Atoms 1. Orbital and spin moments of an electron: Using the theory of angular momentum, calculate the orbital
More informationLocal currents in a two-dimensional topological insulator
Local currents in a two-dimensional topological insulator Xiaoqian Dang, J. D. Burton and Evgeny Y. Tsymbal Department of Physics and Astronomy Nebraska Center for Materials and Nanoscience University
More information