Lectures on Non-Equilibrium Theory of Condensed Matter

The following sections are included:

• Preface

• Contents

Nowadays most theorists working on condensed matter or statistical mechanics are involved with problems related to equilibrium. The interest is focused on different phase transitions. Many of them are spectacular, being candidates for technological applications too. On the other hand, the situation is very clear from a conceptual point of view. The object of mathematical interest is the statistical sum in the thermodynamic limit. However, one has to face a very intricate question - how equilibrium may be reached by physical systems obeying time reversal invariant (classical-or quantum-) mechanics? This is the central problem non-equilibrium theorists have to deal with, and the issue as such is getting a promising yet not satisfactory and exhaustive answer in the last decades only. Although Boltzmann formulated his famous and very successful equation for gases at the end of the 19th century, he was desperate about the lack of any sound foundation, neither was he able to defend it against the harsh criticism of Mach…

The Lagrange-function of a classical system of point-like interacting particles

Let us consider a classical one-dimensional model [Caldeira and Leggett (1983)] for the motion of an electron (of unit mass and unit charge) of coordinate x, interacting with acoustical phonons in the presence of a constant electric field ɛ. The acoustical phonons are described by the oscillator coordinates Q_q for each wave-vector q having the oscillator frequencies ω_q = c|q|, with c being the sound velocity. The continuous wave-vectors of phonons are cut-off at the Debye wave-vector q_D. The Lagrangian of the whole system is

In this chapter we consider another solvable classical model [Bányai (1993)] - now in the three-dimensional space - describing an electron (of unit mass and charge) moving in a homogeneous electric field and interacting with LO phonons (of unit frequency). This model is nothing but the classical version of the Fröhlich interaction, and it is described by the Lagrange function

Unfortunately, in the study of real quantum-systems one cannot obtain the connection between the ballistic and dissipative behaviors without a chain of non-controllable approximations. Solvable models - as the classical models described in the previous chapters - are not known. On the other hand, it was felt more pragmatical to elaborate general equations describing the evolution towards equilibrium while incorporating elements of quantum mechanics. Indeed, a physicists in his everyday work, is confronted with a dual description of the phenomena. The basic understanding of the nature stems from the Quantum Mechanics of finite, isolated systems (finite number of particles in a finite volume), i.e. the Schrodinger equation for normalized the probability amplitude ψ(t)…

Although we discuss later in detail the quantum-mechanical derivation of the Master and rate equations, we are going to give here a simplified version for the derivation of the rate equation so often found in the literature. (The relationship in itself between the Master and rate equations will be discussed in a later chapter.) This kind of derivation (which I called here “naive”) just indicates which approximations have to be implemented in order to obtain the desired result, without any deep justification of the approximations themselves…

The question arises whether it is not better to avoid the last step (iii) in the deduction of the rate equation. The goal is to obtain an equation for the adequate description of rapid processes at short times. Thus we are looking for a pre-Markovian description including memory effects. This means going back from the delta-function…

The following sections are included:

• Properties of the Master equation

• Rate equations

• Scattered versus hopping electrons

The rate-equations (or the Master equation itself) do not allow the calculation of averages of operators, which do not commute with the unperturbed Hamiltonian of the system. For example, in the case of electrons weakly interacting with phonons, one may compute with the rate equation the average occupations of the -states, therefore also the average overall homogeneous flow, but not local charge and current densities in the case of an inhomogeneous flow. The situation is essentially different in the classical statistical mechanics. Here helps the well-known Boltzmann equation.

The following sections are included:

• Macroscopic description of charge evolution in a semiconductor

• The scale projection conjecture

• Scaling in a neutral hopping model

• Scaling in a charged hopping model

• Scaling in a charged Boltzmann model

We have seen in the previous chapters the way one may determine the kinetic coefficients within the frame of some solvable models. In this chapter we want to try a more general approach for the determination of the same coefficients…

The classical Nyquist theorem was deduced [Nyquist (1928)] within the classical thermodynamics of electric circuitry. It is a statement relating the thermal fluctuation of the potential drop U in a conductor to its resistance R and temperature T:

The following sections are included:

• The Kubo formula for the electric conductivity

• The non-mechanical kinetic coefficients

• Symmetry relations, dispersion relations and sum rules

• Implementation of the Master (rate) — approach into the Kubo formulae. Transport by scattered electrons versus transport by hopping electrons

• Ideal relaxation

• The transverse strong-field magneto-resistance

Now, in order to illustrate the application of the linear response, let us consider a simple model of a semiconductor at T = 0, where light absorption implies the excitation of Coulomb interacting electron-hole pairs from the vacuum. We admit that both the valence and conduction bands have simple isotropic parabolic dispersions with maximum (minimum) at . We consider that the valence band is p-type with threefold degeneracy, while the conduction band is nondegenerate s-type. This is a good model for many direct-band semiconductors, which are important for applications. However, we ignore the spin, which leads only to a degeneracy factor. The kinetic energies of the electrons and holes are thus

Until now we always assumed that the system under discussion is spatially homogenous. One often constructs through various approximations such models, even for crystalline solids. (Within the approximation of simple parabolic bands in the modeling of a semiconductor, for example.) At a certain macroscopic level surely even a crystal may be regarded as a homogeneous (but anisotropic) medium. Nevertheless, a better understanding of these aspects, as well as a more profound treatment of the linear response of solids, needs a very careful treatment. In this chapter we restrict ourselves to the problem of the macroscopic longitudinal dielectric function for Coulomb interacting electrons in a given periodical potential at T = 0…

The following sections are included:

• Classical microscopical theory

• Non-relativistic quantum electrodynamics

• The transverse dielectric function in the nonrelativistic quantum electrodynamics

• One-loop (RPA) susceptibility

In this Chapter we want to describe the derivation of the Master equation from the quantum mechanics of the system proper A interacting with a reservoir (thermostat) B. Our more ambitious purpose is to derive not only the already discussed Master equation for the diagonal matrix elements of the density operator (probabilities) for the system A, but also that of describing the evolution of the off-diagonal ones. This would enable the calculation of the average of an arbitrary operator, therefore essentially enlarging the domain of applications…

The earliest and also simplest application of the Master equation (including the non-diagonal matrix elements) was in the field of quantum optics of atoms. Let us consider the Hamiltonian of a two-level system (atom) in the second quantization formalism:

Until now we have discussed only the case where irreversibility comes in through the interaction with a thermostat. The concept of the open system, however, may be extended also to the case without thermostat. In a manybody system, one concentrates only on a few degrees of freedom, trying to obtain closed equations only for these, eliminating higher correlations. This was the case with the classical Boltzmann equation, describing collisions between the particles, where the focus of interest is on the probability to find a particle with a given velocity and coordinate, ignoring higher, correlated probabilities…

Though we have described several derivations of the Master equations, none of them could serve as a proof. They were mere descriptions of the approximations that allowed to obtain - formally though - the desired Markovian equations. A decisive step forward, leading to the understanding of the matter was the paper of van Hove [van Hove (1955)], who tried to give a proof. Although still far from being rigorous from mathematical point of view, he brought in several ideas of fundamental importance…

The following sections are included:

• BEC of the free Bose gas as an equilibrium phase transition

• Markovian real-time description of non-interacting bosons in contact with a thermal bath

• BEC in real time of the free Bose gas

• Spontaneous symmetry breaking in a finite volume. A solvable model of BEC

A new exciting sort of non-equilibrium experiments has emerged in the last decades. With the availability of lasers with carrier frequencies around the band gap of many important semiconductor materials emitting ultrashort pulses in the femtosecond range (1 fms=10⁻¹⁵ sec), we can get a “direct” glimpse on the time evolution of the state of an electron-hole plasma in a semiconductor [Schäfer, Wegener (2002)]…

The following sections are included:

• Equilibrium Green functions

• Non-equilibrium Keldysh-Green Functions

In order to understand the problem of the approach to equilibrium within the various approximation schemes, it is important to know the timereversal property of the non-equilibrium Green functions. Since in interacting many-body systems the second quantization formalism is an essential ingredient, we discuss time-reversal invariance within this formalism. The treatment is borrowed from the quantum field theory [Lee (1981)], by omitting some aspects related to the relativistic formulation…

Within the described diagram technique, the starting point for the calculation of the two-point Green functions is the Dyson equation

Let us consider in this chapter a quantum kinetic approach to the evolution of the electrons and holes created by an ultra-short optical pulse. Here we take into account only the interaction of the carriers with LO-phonons, and neglect the Coulomb interaction. The treatment will be essentially simplified by reducing the true two-time problem of the Green functions to equations for one-time averages through the Kadanoff-Baym approximation. These simplifications are justified for low electron-hole densities and weak polar couplings…

We have seen in the previous Chapter that for a weak coupling to the LO- phonons, the reduction of the true two-time quantum kinetics of Green functions to averages depending on a single time through the Kadanoff- Baym-Ansatz, leads to good phenomenological predictions. However, one could expect that for stronger electron-phonon coupling this approximation does not work. Therefore, in this chapter we shall renounce to this approximation, and formulate a true two-time quantum kinetics. We analyze also the delicate problem of the initial conditions in this non-trivial mathematical problem with two time variables [Gartner, Bányai, Haug (1999)]…

Now, we want to describe the inclusion of the Coulomb interaction beyond the self-consistent Hartree–Fock scheme, within the frame of the Kadanoff- Baym approximation (like in Chapter 26) in order to have a simple onetime quantum kinetic problem. The inclusion of Coulomb screening effects is important for higher carrier densities created by stronger laser pulses…

In this chapter we analyze the problems encountered in the formulation of a full two-times quantum kinetics, in the case of the (screened) Coulomb interaction between the particles. Unfortunately, such an extension - beyond the Kadanoff-Baym Ansatz - is not at all trivial, due to the presence of infrared singularities if the Ward identity is violated. These are not the intrinsic infrared divergences of the Coulomb theory, but spurious ones, introduced by the approximations…

The following sections are included:

• Appendix: H-theorem for bosons

• Bibliography

• Index