Introduction to Linear Response Propagators and Green’s Functions

1. Many-Body TheoryApplication to Electrons in SolidsCharles Patterson Charles.Patterson@tcd.ieSchool of Physics, Trinity College Dublin • Introduction to Linear Response • Propagators and Green’s Functions • Green’s Function for Schrödinger Equation • Functions of a Complex Variable • Contour Integrals in the Complex Plane • Schrödinger, Heisenberg, Interaction Pictures • Occupation Number Formalism • Field Operators • Wick’s Theorem • Many-Body Green’s Functions • Equation of Motion for the Green’s Function • Evaluation of the Single Loop Bubble • The Polarisation Propagator • The GW Approximation • The Bethe-Salpeter Equation • Numerical Aspects of Many-Body Theory • The GW Approximation • Hybrid Density functionals

2. Recommended Texts • A Guide to Feynman Diagrams in the Many-Body Problem, 2nd Ed. R. D. Mattuck, Dover (1992). • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, Dover (2003). • Many-Body Theory of Solids: An Introduction, J. C. Inkson, Plenum Press (1984). • Green’s Functions and Condensed Matter, G. Rickaysen, Academic Press (1991). • Mathematical Methods for Physicists, 5th (Int’l) Ed. G. B. Arfken and H. J. Weber, Academic Press (2001) • Elements of Green’s Functions and Propagation, G. Barton, Oxford (1989).

3. Overview • Propagation of single particles or holes • Zero temperature formalism (c.f. finite temperature formalism) • Applicable in first principles, model Hamiltonian, … methods • Scattering of particles and holes from each other and external potentials • Renormalisation of particle or hole energies • Finite lifetimes for particles or hole excitations (quasiparticles) • Particle-hole bound states (excitons, plasmons, magnons, …) • New concepts such as • N-body Green’s functions particle, hole, particle-hole, particle-particle, … propagators • Self energy (energy renormalisation and excitation lifetime)

4. Overview • Approximations to propagators derived from expansion in (Feynman) diagrams or by functional derivative technique • Wick’s Theorem for evaluating time-ordered products of operators • Lehmann Representation to extract retarded functions • Integral equations which arise in the new concepts • Dyson’s equation G = Go + GoSG • Bethe-Salpeter EquationP = Po + Po v P • Effective Potential V = v + v Po V • Applications of Concepts and Methods in Many-Particle Theory • Correlation Effects and the Total Energy (No-body propagator) • Self-Energies and the GW approximation (1-body propagator) • Collective Excitations and the Bethe-Salpeter Equation (2-body propagator)

5. Classical Statistical Mechanics • Average value of variable A • Probability distribution in phase space r Probability that system is in infinitesimal region of phase space dG Elements of position p and momenta q Average value of variable A Density in phase space

6. Classical Statistical Mechanics • Average value of variable A in NVT Canonical Ensemble Probability depends on total energy of state Canonical partition function normalises r Helmholtz free energy Average value of variable A

7. Classical Statistical Mechanics • Correspondence with Quantum Mechanics Density operator Variables represented as operators Complete set of states Expectation value of A Trace of A Sum of diagonal elements with probability weights Pure state Mixed state

8. Classical Statistical Mechanics • Linear Response in Classical Mechanics Hamiltonian contains Ho and perturbation B Average value of A in absence of perturbation B b = 1/kT <A> changes in presence of perturbation B Defines the linear response

9. Classical Statistical Mechanics • Linear Response in Classical Mechanics Expansion of exponential with scalar exponent Linear response function

10. Classical Statistical Mechanics • Example of application of Static Linear Response Molecule in gas phase with permanent dipole moment -p.E term in Hamiltonian <px>o vanishes in absence of field p is magnitude of permanent moment P is polarisation c is molecular susceptibility of gas phase

11. Classical Statistical Mechanics • Time-dependent linear response • Consider a system where a steady perturbation –lB is applied from t→- • The perturbation is switched off abruptly at t = 0 • To obtain <A(t)> we must use the perturbed system density at t = 0 (full H) Relaxation after static perturbation in time domain

12. Classical Statistical Mechanics • Time-dependent linear response • Onsager regression hypothesis The way in which spontaneous fluctuations in a system relax back to equilibrium is the same as the way in which a perturbed system relaxes back to equilibrium, so long as the perturbation is small Relaxation after static perturbation in time domain

13. Classical Statistical Mechanics • Time-dependent linear response Define the response function cAB Causality requires this for t < t’ Response depends only on time difference Introduce change of variable f(t’) switches off at t’ = 0 Linear response in terms of response function

14. Classical Statistical Mechanics • Linear response function Linear response function is correlation function

15. Classical Statistical Mechanics • Example: Mobility of particle in a fluid m Phenomenological relation Force acting on charged particle in uniform field Linear response approach Need to know velocity auto-correlation function Mobility

16. Classical Statistical Mechanics • Example: Electric conductivity s Hamiltonian for charged particle in vector potential, A Omit nonlinear A2 term Define current density Hamiltonian as Ho + perturbing part

17. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon Equation of motion Set driving force to zero Equation of motion in absence of driving force Solution in absence of driving force A, d depend on initial conditions Required for EoM to be satisfied – defines w1

18. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Impulse Response Function We can view the continuous force applied to a mass on a spring as a sequence of delta function impulses. If we know the response of the system to a single impulse, provided the system is linear, we can immediately write down the solution in terms of the impulse response function. G(t-t’) is the Green’s function Dirac delta function is a unit impulse function Velocity of oscillator when given an unit impulse at t = 0

19. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Impulse Response Function Unit impulse F(t’)D = 1 Position following unit impulse Velocity following unit impulse Position following impulse series Position following continuous force

20. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Impulse Response Function Green’s function Defining relation for G Position, from G for a given F Velocity, from G for a given F

21. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Impulse Response Function Change of variable Example: impulse at t’=0 Recover original solution from impulse response function

22. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Impulse Response Function Example: Periodic forcing If we set the lower limit in the integral dt’ to 0 instead of – we obtain additional transient terms which depend on initial conditions

23. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Evaluate velocity auto-correlation function

24. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon • Parseval’s Theorem Convolution of f1 and f2 Fourier transform of f1 Product of Fourier transforms Fourier transform of convolution

25. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon

26. Classical Statistical Mechanics • Example: Electric conductivity of harmonic oscillator e.g. phonon Same result as Drude when wo tends to zero

27. Classical Statistical Mechanics • Conclusions • Linear response functions, e.g. transport coefficients, are derived from correlation functions • The correlation function is independent of the external stimulus (Onsager) • The reponse function contains the step function q(t) to satisfy causality