Lectures on Theoretical Physics Linear Response Theory Peter Hertel University of OsnabrÄuck, Germany The theory of linear response to perturbations of the equilibrium state, or linear response theory, is the subject of this series of lectures. Ordinary matter, if left alone, will sooner or later attain an equilibrium state. This equilibrium state depends on the temperature of the environment and on external parameters. External parameters may be the region of space within which a certain number of particles are con¯ned, mechanical stress, or the strength of an external electric or magnetic ¯eld. If temperature or the external parameters change slowly enough, the system can attain the new equilibrium state practically instantaneously, and we speak of a reversible process. On the other hand, if the external parameters vary so rapidly that the system has no chance to adapt, it remains away from equilibrium, and we speak of irreversibility. The most important application is optics. There is a medium which is exposed to an electromagnetic wave. The electric ¯eld changes so rapidly that matter within a region of micrometer dimensions cannot react instantaneously, it responds with retardation. We shall work out the retarded response in linear approximation. There are quite a few general and important results which hold irrespective of a particular Hamiltonian, such as the Kramers-Kronig relations, the fluctuation- dissipation theorem, the second law of thermodynamics, and Onsager's relation. We discuss various electro- and magnetooptic e®ects, such as the Pockels e®ect, the Faraday e®ect, the Kerr e®ect, and the Cotton-Mouton e®ect. We also treat spatial dispersion, or optical activity and indicate how the theory is to be developed further in order to handle the non-linear response as well. lrt-10.12.01 2 CONTENTS Contents 1 Maxwell equations 4 1.1 The electromagnetic ¯eld . .................... 4 1.2 Potentials . ............................ 4 1.3 Field energy . ............................ 5 1.4 Polarization and magnetization .................... 6 2 A simple model 8 2.1 Equation of motion . ........................ 8 2.2 Green's function ............................ 8 2.3 Susceptibility . ............................ 9 3 Thermodynamic equilibrium 11 3.1 Observables and states ........................ 11 3.2 The ¯rst law of thermodynamics . ................ 11 3.3 Entropy ................................ 12 3.4 The second law of thermodynamics . ................ 12 3.5 Irreversible processes . ........................ 13 4 Perturbing the equilibrium 15 4.1 Time evolution . ............................ 15 4.2 Interaction picture . ........................ 16 4.3 Perturbing the Gibbs state . .................... 16 4.4 Time dependent external parameter . ................ 17 5 Dielectric susceptibility 18 5.1 Polarization of matter . ........................ 18 5.2 Dielectric susceptibility ........................ 19 5.3 Susceptibility proper and optical activity . ............ 20 6 Dispersion relations 21 6.1 Retarded Green function . .................... 21 6.2 Kramers-Kronig relations . .................... 23 6.3 Refraction and absorption . .................... 23 6.4 Oscillator strength . ........................ 24 7 Dissipation-fluctuation theorem 26 7.1 The Wiener-Khinchin theorem .................... 26 7.2 Kubo-Martin-Schwinger formula . ................ 27 7.3 Response and correlation . .................... 28 7.4 The Callen-Welton theorem . .................... 29 7.5 Energy dissipation . ........................ 29 CONTENTS 3 8 Onsager relations 32 8.1 Symmetry of static susceptibilities . ................ 32 8.2 Time reversal . ............................ 32 8.3 Onsager theorem . ........................ 34 8.4 Onsager relation for kinetic coe±cients ................ 35 8.5 Electrical conductivity and Hall e®ect ................ 36 9 Electro- and magnetooptic e®ects 37 9.1 Crystal optics . ............................ 37 9.2 Pockels e®ect . ............................ 39 9.3 Faraday e®ect . ............................ 39 9.4 Kerr e®ect . ............................ 40 9.5 Magneto-electric e®ect ........................ 41 9.6 Cotton-Mouton e®ect . ........................ 41 10 Spatial dispersion 42 10.1 Dispersion relation . ........................ 42 10.2 Optical activity ............................ 43 11 Non-linear response 45 11.1 Higher order response . ........................ 45 11.2 Susceptibilities . ............................ 46 11.3 Second harmonic generation . .................... 47 A Causal functions 49 B Crystal symmetry 51 B.1 Spontaneous symmetry breaking . ................ 51 B.2 Symmetry groups . ........................ 52 B.3 A case study . ............................ 52 C Glossary 55 4 1 MAXWELL EQUATIONS 1 Maxwell equations In this serious of lectures we will study the interaction of a rapidly oscillating electromagnetic ¯eld with matter. Therefore, a recollection of basic electrody- namics seems to be appropriate. 1.1 The electromagnetic ¯eld The electromagnetic ¯elds E = E(t; x) and B = B(t; x) are de¯ned by their action on charged particles. The trajectory t ! x(t) of a particle with charge q is a solution of p_ = q (E + v £ B) ; (1.1) p where v = x_ and p = mv= 1 ¡ v2=c2. The electromagnetic ¯eld is to be evaluated at the current particle location t; x(t). The electromagnetic ¯elds act on charged particles, as described by the Lorentz formula (1.1), and charged particles generate the electromagnetic ¯eld. dQ = dV ½(t; x) is the amount of electric charge in a small volume element dV at x at time t. Likewise, dI = dA ¢ j(t; x) is the charge current passing the small area element dA from the back to to the front side. Maxwell's equations read 1 ²0r ¢ E = ½ and r £ B ¡ ²0E_ = j (1.2) ¹0 as well as r ¢ B = 0 and r £ E + B_ =0 : (1.3) The ¯rst group of four equations describe the e®ect of electric charge and cur- rent, the second group of likewise four equations say that there is no magnetic charge (magnetic monopoles). It is a consequence of Maxwell's equations that charge is conserved, ½_ + r ¢ j =0 : (1.4) 1.2 Potentials Stationary ¯elds decouple. ²0r ¢ E = ½ and r £ E = 0 (1.5) describe the electrostatic ¯eld, 1 r £ B = j and r ¢ B = 0 (1.6) ¹0 1.3 Field energy 5 the magnetostatic ¯eld. The electrostatic ¯eld may be derived from a scalar potential, E = ¡rÁ (1.7) which obeys Poisson's equation ¡²0¢Á = ½: (1.8) The magnetostatic ¯eld can be expressed by B = r £ A (1.9) in terms of a vector potential A. Adding the gradient of an arbitrary scalar ¯eld ¤, A 0 = A+r¤, does not change the induction ¯eld B. This ambiguity allows to subject the vector potential to a gauge, e.g. the Coulomb gauge r ¢ A =0. The three components of the vector potential the obey a Poisson equation each, 1 ¡ ¢A = j : (1.10) ¹0 The full set of Maxwell equations are solved by E = ¡rÁ ¡ A_ and B = r £ A : (1.11) If we now impose the Lorentz gauge 1 ²0Á_ + r ¢ A =0 ; (1.12) ¹0 the following equations result: 1 ²0¤ Á = ½ and ¤ A = j : (1.13) ¹0 2 ¡ The box, or wave operator is @0 ¢ where @0 is thep partial derivative with respect to time, divided by c which is de¯ned as c =1= ²0¹0. 1.3 Field energy The potential of a point charge q resting at y is the Coulomb potential q 1 ÁC(x)= : (1.14) 4¼²0 jx ¡ yj If charges q1;q2;:::are brought from in¯nity to their locations at x1; x2;:::the following work has to be done: X 1 qbqa W = : (1.15) j b ¡ aj b>a 4¼²0 x x 6 1 MAXWELL EQUATIONS For a smooth charge distribution ½ this expression may be rewritten into Z Z 3 00 3 0 00 0 1 d x d x ½(x )½(x ) 1 3 W = 00 0 = d x½(x)Á(x) : (1.16) 8¼²0 jx ¡ x j 2 By partial integration we obtain Z ² E2 W = d3x 0 : (1.17) 2 2 We interpret ²0E =2 as the energy density of the electric ¯eld. Using Maxwell's equation we may prove the following balance equation: ´_ + r ¢ S = ´¤ (1.18) where ² E2 B2 1 ´ = 0 + , S = E £ B and ´¤ = ¡j ¢ E : (1.19) 2 2¹0 ¹0 2 As explained before, ²0E =2 is the energy density of the electric ¯eld. The 2 energy density of the magnetic ¯eld is B =2¹0. The energy current density S =(E £ B)=¹0 is also called Poynting's vector. ´¤ = ¡j ¢ E describes the production of ¯eld energy per unit time per unit volume. Field energy is created if charge runs counter to the electric ¯eld (Bremsstrahlung). Field energy vanishes if charges run with the direction of the electric ¯eld (Ohm's law, Joule's heat). 1.4 Polarization and magnetization R R Q = d3x½(x) is the charge of system. By p = d3x x ½(x) we denote its electric dipole moment. The dipole moment does not depend on the choice of the coordinate system origin if its charge vanishes. Denote by P the density of a probe's dipole moments, its polarization. For ordinary matter, which is locally neutral, this is a well de¯ned quantity. One can show that ¡r ¢ P is the charge density causing the polarization and that P_ contributes to the current density. We likewise de¯ne magnetic dipole moments m and their density M, the mag- netization. r £ M also contributes to the current density. The charge and current density should therefore be split into ½ = ¡r ¢ P + ½f and j = P_ + r £ M + jf : (1.20) The remainders ½f and jf are the charge and current density of free, or mobile charges, as opposed to bound charges. We introduce auxiliary ¯elds D = ²0E + P (dielectric displacement) and H = B=¹0 ¡ M (magnetic ¯eld strength). They obey the following equations: r ¢ D = ½f and r £ H ¡ D_ = jf : (1.21) 1.4 Polarization and magnetization 7 r ¢ B = 0 and r £ E + B_ = 0 (1.22) remain unchanged. 8 2 A SIMPLE MODEL 2 A simple model For warming up, we develop a very simple model of the dielectric susceptibility. Consider an atom which is exposed to an oscillating electric ¯eld E = E(t) along the z-direction.