Transport of interacting electrons in 1d: nonperturbative RG approach Dmitry Aristov 1,3 and Peter Wölfle 1,2 Universität Karlsruhe (TH) 1) Institut für.

Slides:



Advertisements
Similar presentations
Feynman Diagrams Feynman diagrams are pictorial representations of
Advertisements

Anderson localization: from single particle to many body problems.
Summing planar diagrams
Theory of the pairbreaking superconductor-metal transition in nanowires Talk online: sachdev.physics.harvard.edu Talk online: sachdev.physics.harvard.edu.
From weak to strong correlation: A new renormalization group approach to strongly correlated Fermi liquids Alex Hewson, Khan Edwards, Daniel Crow, Imperial.
Does instruction lead to learning?. A mini-quiz – 5 minutes 1.Write down the ground state wavefunction of the hydrogen atom? 2.What is the radius of the.
A two-loop calculation in quantum field theory on orbifolds Nobuhiro Uekusa.
Solving non-perturbative renormalization group equation without field operator expansion and its application to the dynamical chiral symmetry breaking.
QCD-2004 Lesson 1 : Field Theory and Perturbative QCD I 1)Preliminaries: Basic quantities in field theory 2)Preliminaries: COLOUR 3) The QCD Lagrangian.
Non-equilibrium physics Non-equilibrium physics in one dimension Igor Gornyi Москва Сентябрь 2012 Karlsruhe Institute of Technology.
Green’s function of a dressed particle (today: Holstein polaron) Mona Berciu, UBC Collaborators: Glen Goodvin, George Sawaztky, Alexandru Macridin More.
January 16, 2001Physics 8411 Introduction to Feynman Diagrams and Dynamics of Interactions All known interactions can be described in terms of forces forces:
Green’s Functions From Heisenberg to Interaction Picture Useful once we have it, but since we need to know the full ground state and the field operators.
Functional renormalization – concepts and prospects.
Gerard ’t Hooft Spinoza Institute Yukawa – Tomonaga Workshop, Kyoto, December 11, 2006 Utrecht University.
QCD – from the vacuum to high temperature an analytical approach an analytical approach.
Renormalised Perturbation Theory ● Motivation ● Illustration with the Anderson impurity model ● Ways of calculating the renormalised parameters ● Range.
Universality in ultra-cold fermionic atom gases. with S. Diehl, H.Gies, J.Pawlowski S. Diehl, H.Gies, J.Pawlowski.
Modified Coulomb potential of QED in a strong magnetic field Neda Sadooghi Sharif University of Technology (SUT) and Institute for Theoretical Physics.
Physics of fusion power Lecture 10 : Running a discharge / diagnostics.
Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.
An Introduction to Field and Gauge Theories
Many-body Green’s Functions
Vibrational Spectroscopy
Monte Carlo Simulation of Interacting Electron Models by a New Determinant Approach Mucheng Zhang (Under the direction of Robert W. Robinson and Heinz-Bernd.
System and definitions In harmonic trap (ideal): er.
Spin and Charge Pumping in an Interacting Quantum Wire R. C., N. Andrei (Rutgers University, NJ), Q. Niu (The University of Texas, Texas) Quantum Pumping.
The World Particle content. Interactions Schrodinger Wave Equation He started with the energy-momentum relation for a particle he made the quantum.
What do we know about the Standard Model? Sally Dawson Lecture 2 SLAC Summer Institute.
Cross section for potential scattering
Lecture 20: More on the deuteron 18/11/ Analysis so far: (N.B., see Krane, Chapter 4) Quantum numbers: (J , T) = (1 +, 0) favor a 3 S 1 configuration.
QED at Finite Temperature and Constant Magnetic Field: The Standard Model of Electroweak Interaction at Finite Temperature and Strong Magnetic Field Neda.
Finite Temperature Field Theory Joe Schindler 2015.
Chiral condensate in nuclear matter beyond linear density using chiral Ward identity S.Goda (Kyoto Univ.) D.Jido ( YITP ) 12th International Workshop on.
Exact bosonization for interacting fermions in arbitrary dimensions. (New route to numerical and analytical calculations) K.B. Efetov, C. Pepin, H. Meier.
Quantum pumping and rectification effects in interacting quantum dots Francesco Romeo In collaboration with : Dr Roberta Citro Prof. Maria Marinaro University.
Disordered Electron Systems II Roberto Raimondi Perturbative thermodynamics Renormalized Fermi liquid RG equation at one-loop Beyond one-loop Workshop.
Quantum Mechanical Cross Sections In a practical scattering experiment the observables we have on hand are momenta, spins, masses, etc.. We do not directly.
Introduction to Time dependent Time-independent methods: Kap. 7-lect2 Methods to obtain an approximate eigen energy, E and wave function: perturbation.
1 Methods of Experimental Particle Physics Alexei Safonov Lecture #4.
Interacting Molecules in a Dense Fluid
Current noise in 1D electron systems ISSP International Summer School August 2003 Björn Trauzettel Albert-Ludwigs-Universität Freiburg, Germany [Chung.
The Importance of the TeV Scale Sally Dawson Lecture 3 FNAL LHC Workshop, 2006.
The nonperturbative analyses for lower dimensional non-linear sigma models Etsuko Itou (Osaka University) 1.Introduction 2.The WRG equation for NLσM 3.Fixed.
THERMODYNAMICS OF THE HIGH TEMPERATURE QUARK-GLUON PLASMA Jean-Paul Blaizot, CNRS and ECT* Komaba - Tokyo November 25, 2005.
Electronic transport in one-dimensional wires Akira Furusaki (RIKEN)
LORENTZ AND GAUGE INVARIANT SELF-LOCALIZED SOLUTION OF THE QED EQUATIONS I.D.Feranchuk and S.I.Feranchuk Belarusian University, Minsk 10 th International.
Quantum Two 1. 2 Time-Dependent Perturbations 3.
Particle Physics Particle Physics Chris Parkes Feynman Graphs of QFT QED Standard model vertices Amplitudes and Probabilities Forces from particle exchange.
Non-Linear Effects in Strong EM Field Alexander Titov Bogoliubov Lab. of Theoretical Physics, JINR, Dubna International.
Nonlocal Condensate Model for QCD Sum Rules Ron-Chou Hsieh Collaborator: Hsiang-nan Li arxiv:
Lecture 2 - Feynman Diagrams & Experimental Measurements
Lecture 4 – Quantum Electrodynamics (QED)
Fundamental principles of particle physics Our description of the fundamental interactions and particles rests on two fundamental structures :
Kondo Effect Ljubljana, Author: Lara Ulčakar
THE ISOTOPIC FOLDY-WOUTHUYSEN REPRESENTATION
Quantum Mechanics for Applied Physics
Schrödinger Representation – Schrödinger Equation
Vivek Sinha (09MS 066) Amit Kumar (09 MS 086)
NGB and their parameters
Concept test 15.1 Suppose at time
Derivation of Electro-Weak Unification and Final Form of Standard Model with QCD and Gluons  1W1+  2W2 +  3W3.
RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS
The Rendering Equation
Chapter V Interacting Fields Lecture 7 Books Recommended:
Adnan Bashir, UMSNH, Mexico
Scattering Theory: Revised and corrected
It means anything not quadratic in fields and derivatives.
QCD at very high density
Review for Int 3 C10 Test.
Presentation transcript:

Transport of interacting electrons in 1d: nonperturbative RG approach Dmitry Aristov 1,3 and Peter Wölfle 1,2 Universität Karlsruhe (TH) 1) Institut für Theorie der Kondensierten Materie (TKM), Universität Karlsruhe, KIT DFG-Center for Functional Nanostructures (CFN) 2) Institut für Nanotechnologie (INT), Forschungszentrum Karlsruhe Karlsruhe Institute of Technology 3) Petersburg Nuclear Physics Institute

Outline Introduction: conductance of electrons in Luttinger liquids Renormalization group approach: nonperturbative β-function; d.c. conductance for any Luttinger parameter K and barrier reflection R. Universität Karlsruhe (TH) Barrier in a strongly interacting Luttinger liquid; current algebra representation Perturbation theory in the interaction g 2 : summing up the principal terms linear in log(T) to all orders in g 2 Demonstration of agreement with known results at R~0 and R~1 Role of non-universal terms beyond the ladder series.

Introduction: Transport in clean Luttinger liquids Universität Karlsruhe (TH) Early results for conductance G of clean infinite spinless Luttinger liquids: forward scattering, interaction parameters: in units of W. Apel and T. M. Rice, 1982 C. Kane and M.P.A. Fisher, 1992 A. Furukawa and N. Nagaosa, 1993 Two-terminal conductance: Luttinger liquid attached to ideal leads (T=0) Proper sequence of limits (1) ω → 0, (2) L → ∞ D. Maslov and M. Stone, 1995 I. Safi and H. Schulz, 1995 A. Kawabata, 1996 Y. Oreg, A. Finkelstein, 1995 Screened internal electric field Later interpreted as four-terminal conductance

Introduction: Quantum wire with single barrier I Universität Karlsruhe (TH) Landauer conductance in non-interacting limit: Transmission amplitude Effect of interaction: Friedel oscillations of charge density around impurity lead to dynamically generated extended effective potential : The spatial extent L of the Friedel oscillations is determined by phase relaxing inelastic processes: Where T is the temperature and is the excitation energy of a fermion

Introduction: Poor man’s scaling at weak interaction Universität Karlsruhe (TH) : Yue, Matveev and Glazman, 1995 Integrating out high momentum states, reducing the band width, one finds a renormalization group equation for the transmission amplitude as a function of the bandwidth D The transmission coefficient as a function of energy follows as

Single particle scattering states Universität Karlsruhe (TH) Scattering of spinless fermions by potential barrier: S-matrix: Single particle scattering states for right (left) moving particles (k>0) : Neglect k-dependence of t, r in the following

Fermion operators in scattering state representation Universität Karlsruhe (TH) Creation ops. for right (left) moving fermions in scattering states k : Creation ops. for R, L fermions at position x: Creation operators for fermions at position x::

Interaction in scattering state representation Universität Karlsruhe (TH)

Current algebra representation Universität Karlsruhe (TH) Definition of current operators: Scattering by the potential barrier does not affect the isocharge component but rotates the isospin vector Affleck, 1990 Comps. of R: Current operators obey Kac-Moody algebra.

Interaction in chiral (current algebra) representation Universität Karlsruhe (TH) In this representation the potential barrier may be viewed as a local magnetic field rotating the isospin vector of a wave packet, when it passes through the field. “Nonlocal” interaction:

Physical currents and conductance Universität Karlsruhe (TH) Electron density: Electron current obtained from continuity equation: Applied voltage:, couples only to isospin component Linear response two-terminal conductance (in units of ) :

Perturbation theory in g 2 : Feynman diagrams Universität Karlsruhe (TH) Diagram rules for nth order contributions in (energy-position)-representation: (1)Draw n vertical wavy lines representing interaction (-2g 2 ), the ith line connecting the upper point -x i with vertex and the lower point x i with vertex (2) Connect all points with two propagator lines entering and leaving the point: (4) In each fermion loop take trace of product of vertex matrices (5) Take limit of external frequency (3) Integrate over internal position variables from a to L

Conductance in 0 th order Universität Karlsruhe (TH) Define : -x-y y -x

Conductance in 1st order Universität Karlsruhe (TH) Logarithmic correction: Agrees with Yue, Matveev, and Glazman, 1995

Summation of linear log-terms Universität Karlsruhe (TH) Each diagram of n th order has a leading scale dependent contribution Principal diagrams with linear logarithmic dependence are those with the maximum number of loops; they are independent of the cutoff scheme The sum of these diagrams is obtained from a ladder summation:

Conductance up to linear log-terms Universität Karlsruhe (TH) Substituting L in place of the bare interaction into the first order diagrams, one finds the conductance Taking the limits and where we defined: and put v F =1.

Renormalization group approach Universität Karlsruhe (TH) In perturbation theory the n-th order contribution is a polynomial in of degree n If the theory is renormalizable, all terms of higher powers in should be generated by a renormalization group equation for the scaled conductance We will use instead. The beta-function is given by the prefactor of in the perturbation expansion of G:

Renormalization group equation Universität Karlsruhe (TH) The RG-equation may be expressed in the symmetric form: where is the Luttinger parameter. The RG-equation is invariant under

Solution of the renormalization group equation Universität Karlsruhe (TH) The RG-equation may be integrated from to give Agrees with Kane a. Fisher ( cases |t|→1,0 ) except that G→1 for K>1 Repulsive int., K<1 Attractive int., K>1 Low T: Limiting cases: High T:

Check of renormalizability Universität Karlsruhe (TH) The expansion gives in 2 nd and 3 rd order: By using computer algebra to evaluate Feynman diagrams up to third order ( more than 4000 diagrams) one finds agreement with the RG result, except for additional terms ~ (1-Y 2 ) 2 g 3 Λ within the hard cutoff (T=0) scheme The renormalizability of the theory may be checked by comparing the terms with higher powers of generated by expanding the solution of the RG equation with perturbation theory.

β-function beyond ladder series Universität Karlsruhe (TH) Feynman graphs, leading to lowest order linear logarithmic contribution in third order in g, beyond ladder series. Our evaluation at T=0,  = ln(L/a) : c 3 =  2 /12 Exact solution in S. Lukyanov, Ph. Werner (2007) : L=1,  = ln(v F /  aT) (?) => c 3 =1/4 β-function is not universal, but depends on cutoff scheme

Summary Universität Karlsruhe (TH) Calculated d.c. linear conductance G of Luttinger liquid with barrier in perturbation theory in g 2 as a function of length of interacting region and of temperature T Analyzed scale dependent terms of G(T): powers of, and summed up the principal terms linear in Assuming (and checking to third order) renormalizability of the theory extracted the beta-function of the renormalization group equation for G Integrated the RG-equation to give G(T) for any g 2 or equivalently, Luttinger liquid parameter K, and any (narrow) potential barrier Comparing with exact solutions known from Thermodynamic Bethe Ansatz, clarified the meaning of the obtained solution as a robust, “universal”, part of Beta-function independent of the RG cutoff scheme

Thank you

Luttinger Hamiltonian Universität Karlsruhe (TH) Define partial densities for incoming and outgoing particles: Hamiltonian (free H in bosonized form): Hard cutoff scheme: a<x<L

Nonuniversality of β-function Universität Karlsruhe (TH) Usual reasoning: higher order coefficients in beta-function depend on the cutoff scheme in (third loop and higher order) 1st diagram is linear in  => universal 2nd and 3rd diagrams contain both    and  : at finite T the coefficient in front of  is different ! Overall result : is replaced by “Non-universality” for observable quantity: dependence on the problem setup! In our problem, we used a hard cutoff scheme at T=0 and a soft cutoff at T ≠0   = ln(L/a) vs.