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Methods of Quantum Field Theory in Statistical Physics
By A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinki, Richard A. Silverman
Dover Publications, Inc.Copyright © 1963 Richard A. Silverman
All rights reserved.
GENERAL PROPERTIES OF MANY-PARTICLE SYSTEMS AT LOW TEMPERATURES
I. Elementary Excitations. The Energy Spectrum and Properties of Liquid He4 at Low Temperatures
1.1. Introduction. Quasi-particles. Statistical physics studies the behavior of systems consisting of a very large number of particles. In the last analysis, the macroscopic properties of liquids, gases and solids are due to microscopic interactions between the particles making up the system. Obviously, a complete solution of the problem, involving determination of the behavior of each individual particle, is out of the question. Fortunately, however, the overall macroscopic characteristics are determined only by certain average properties of the system.
To be explicit, we now consider some thermodynamic properties. The macroscopic state of a system is specified by giving three independent thermodynamic variables, e.g., the pressure P, the temperature T, and the average number N of particles in the system. From a quantum-mechanical point of view, a closed system of N particles is characterized by its energy levels En. Suppose that from the system we single out a volume (subsystem) which can still be regarded as macroscopic. The number of particles in such a subsystem is still very large, whereas the interaction forces between particles act at distances whose order of magnitude is that of atomic dimensions. Therefore, apart from boundary effects, we can regard the subsystem itself as closed, and characterized by certain energy levels (for a given number of particles). Since the subsystem actually interacts with other parts of the closed system, it does not have a fixed energy and a fixed number of particles, and, in fact, it has a nonzero probability of occupying any energy state.
As is familiar from statistical physics (see e.g., L8), the microscopic derivation of thermodynamic formulas is based on the Gibbs distribution, which gives the following probability of finding the subsystem in the energy state with a number of particles equal to N:
(1.1) [MATHEMATICAL EXPRESSION OMITTED]
In this formula, T denotes the absolute temperature, μ the chemical potential, and Z a normalization factor which is determined from the condition
(1.2) [MATHEMATICAL EXPRESSION OMITTED]
According to (1.1), we have
(1.3) [MATHEMATICAL EXPRESSION OMITTED]
The quantity Z is called the grand partition function. If the energy levels EnN are known, the partition function can be calculated. This immediately determines the thermodynamic functions as well, since the formula
(1.4) Ω = -T ln Z
relates the quantity Z to the thermodynamic potential Ω (involving the variables V, T and μ).
Obviously, the simplest use that can be made of these formulas is to calculate the thermodynamic functions of ideal gases, since in this case the energy is just the sum of the energies of the separate particles. However, in general, it is impossible to determine the energy levels of a system consisting of a large number of interacting particles. Therefore, so far, interactions between particles in quantum statistics have been successfully taken into account only when the interactions are sufficiently weak, and perturbation-theory calculations of thermodynamic quantities have been carried out only to the first or second approximations. In the majority of physical problems, where the interaction is far from small, an approach based on the direct use of formulas (1.1)–(1.4) is unrealistic.
The case of very low temperatures is somewhat exceptional. As T -> 0, the important energy levels in the partition function are the weakly excited states, whose energies differ only very little from the energy of the ground state. The character of the energy spectrum of the system in this region of energies can be ascertained in some detail, by using very general considerations which are valid regardless of the magnitude and specific features of the interaction between the particles.
As an example illustrating the subsequent discussion, consider the excitation of lattice vibrations in a crystal. As long as the vibrations are small, we can regard the lattice as a set of coupled harmonic oscillators. Introducing normal coordinates, we obtain a system of 3N linear oscillators with characteristic frequencies ωi(N is the number of atoms). According to quantum mechanics, the energy spectrum of such a system is given by the formula
[MATHEMATICAL EXPRESSION OMITTED]
where the ni are arbitrary nonnegative integers (including zero), and various sets of numbers ni correspond to various energy levels of the system. The lattice vibrations can be described as a superposition of monochromatic plane waves, propagating in the crystal. Each wave is characterized by a wave vector and a frequency, and also by an index s specifying the type of wave involved. Because of the possibility that various types of waves can propagate in the crystal, the frequency to, regarded as a function of the wave vector k, is not a single-valued function, but rather consists of several branches ωs(k), where the total number of branches equals 3r (r is the number of atoms belonging to one unit cell of the crystal). For small momenta, three of these branches (the acoustic branches) are characterized by the fact that the frequency depends linearly on the wave vector:
ωs(k) = us(θ, φ)|k|.
For other momenta, the curve ωs(k) begins with some finite value for k = 0, and depends weakly on k in the region of small wave numbers.
From a knowledge of the frequency spectrum, the energy levels and the matrix elements of the displacements of the atoms of the lattice (the coordinates of the oscillators), we can calculate completely, at least in principle, both the thermodynamic and the kinetic characteristics of the vibrating lattice. However, instead of the model of coupled oscillators, it turns out in practice to be very convenient to use another, equivalent model. This model can be obtained by applying the quantum-mechanical correspondence principle, which states that every plane wave corresponds to a set of moving "particles," with momentum determined by the wave vector k and energy determined by the frequency ωs(k). Thus, an excited state of the lattice can be thought of as an aggregate of such "particles" (called phonons), moving freely in the volume occupied by the crystal. This leads to an expression for the energy levels of the system which is analogous to that for an ideal gas. In fact, ni can be interpreted as the number of phonons in the state i, where i = (k,s), and the numbers ni range over all nonnegative integers. It follows that phonons obey Bose statistics, even when the atoms making up the system have half-integral spins.
At very low temperatures, the most important role is played by phonons with small energies. According to what was just said about the branches of the frequency spectrum, the phonons with the smallest energies correspond to the acoustic branches, in the region of small momenta. In this case, the function ω(k) is linear, and this fact alone permits us to draw a number of qualitative conclusions, e.g., to deduce that the heat capacity of the lattice is proportional to T3.
For quantitative calculations, the isotropic Debye model is often used instead of the spectrum of the actual lattice. In this model, it is assumed that the low-frequency part of the spectrum, instead of having three acoustic branches, is the same as that of an isotropic body, i.e., consists of longitudinal phonons with energy ωi(k) = uik and transverse phonons with two possible polarizations and the same dependence ωt(k) = utk of the energy on the momentum. Furthermore, it is assumed that the momenta of the phonons do not exceed a certain upper bound kD determined by a normalization involving the appropriate number of degrees of freedom. Then it is clear that
kD ~ 1/a,
where a is the interatomic distance. This model leads to Debye's well-known interpolation formula for the heat capacity of solids. Later on, we shall use this model to study the interaction of electrons and phonons in a metal.
If we take into account the small anharmonic terms in the potential energy of the vibrating lattice, the expression for the energy (given above) is no longer exact, and transitions between states with different sets of numbers ni are now possible. This fact can also be interpreted in phonon language, in terms of various interaction processes between phonons, leading to scattering of phonons by phonons and the creation of new phonons. In other words, in a rigorous analysis it is only an approximation to regard the phonons as freely moving particles. The role of the anharmonic terms becomes greater as the amplitude of the vibrations increases, i.e., as the temperature is raised. In the phonon model, the number of phonons increases as the temperature is raised, and this increases the importance of interactions between phonons. Therefore, the very concept of phonons as freely moving particles is applicable only in the region of temperatures that are not too high (considerably lower than the melting point).
We now consider the general case. By analogy with the example just considered, to construct a model of the energy spectrum for the weakly excited states of a system, we make the basic assumption that to a first approximation the structure of the energy levels obeys the same principle as that of the energy levels of an ideal gas. In other words, it is assumed that any energy level can be obtained as the sum of the energies of a certain number of "quasi-particles" or "elementary excitations", with momentum p and energy ε(p), moving in the volume occupied by the system. [Generally speaking, the dispersion law ε(p) for these excitations will be different from the expression ε0(p) = p2/2m for the energy of free particles.] It should be emphasized that the elementary excitations are the result of collective interactions of the particles of the system, and therefore pertain to the system as a whole and not to its separate particles. In particular, the number of elementary excitations is certainly not the same as the total number of particles in the system.
All energy spectra can be divided into two broad categories, spectra of the Bose type and spectra of the Fermi type. In the first case, the excitations have integral-valued intrinsic moments (spins) and obey Bose statistics. In the second case, the excitations have half-integral spins and obey Fermi statistics. According to quantum mechanics, the spin of any system can only change by an integer. It follows that Bose excitations can appear or disappear one at a time, whereas Fermi excitations always appear and disappear in pairs.
As already mentioned in the example given above (involving lattice vibrations), the statistics of the elementary excitations do not have to be the same as the statistics of the particles making up the system. It is obvious only that a Bose system cannot have excitations with half-integral spins.
The elementary excitations do not correspond to exact statistical states of the system, but instead represent wave packets, i.e., superpositions of large numbers of exact stationary states with a narrow spread in energy. As a result, transitions from one state to another have nonzero probability. This leads to spreading of the packet, i.e., attenuation (or damping) of the excitations. Therefore, a description of the system by using elementary excitations is possible only as long as the (energy) width of the packet determining its attenuation is small compared to the energy of the packet.
Spread of the packet and the concomitant attenuation of the elementary excitations can be regarded as the result of interactions between the "quasi-particles," during which the laws of energy and momentum conservation are satisfied. Clearly, all such processes can be divided into processes in which one excitation "decays" into several others, and processes in which excitations are "scattered" by each other. As we shall see below, decay of excitations can take place only at sufficiently high energies. Moreover, scattering processes are important only when the number of excitations is sufficiently large. Thus, at low temperatures, where low-energy excitations are important and there are few of them, both types of processes leading to the attenuation of excitations are unimportant. The weakness of the interactions between excitations at low temperatures allows us to regard them as an ideal gas of "quasi-particles."
At present, because of both experimental data and direct theoretical calculations, the ideas just presented, concerning the structure of energy spectra, are well-established facts. Of course, the energy spectra of different physical objects (e.g., liquids consisting of the isotopes He3 and He4, metals, dielectrics, etc.) are completely different. For example, the spectrum of liquid He is of the Bose type, while the spectrum of liquid He and the electronic spectra of metals are of the Fermi type.
1.2. The spectrum of a Bose liquid. An example of a system with a spectrum of the Bose type is the Bose liquid, i.e., a liquid consisting of atoms with integral-valued spins. In nature, there exists only one such liquid which does not solidify at absolute zero, namely, liquid helium (more precisely, the isotope He4). Since the atoms of He4 have spin zero, for all practical purposes we can coniine ourselves to this case.
The dependence of the excitation energy of a Bose liquid on the momentum in the limit of small values of the momentum can be determined by very general considerations. The region of small values of p corresponds to long-wavelength oscillations of the liquid. But such oscillations are just ordinary sound waves. From this we conclude at once that for small p, the elementary excitations are identical with acoustic quanta (phonons), for which the relation between energy and momentum is well known. In fact, noting that the acoustic frequency ω is related to the wave vector by the formula ω = uk, where u is the velocity of sound, we immediately find that the desired relation between ε and p is
(1.5) ε = up.
Thus, for small momenta, the energy of an excitation in the Bose liquid depends linearly on its momentum, and the coefficient of proportionality is just the velocity of sound.
As the momentum increases, the function ε(p) ceases to be linear, and the subsequent behavior of the curve ε(p) cannot be determined quite so easily. Then we have recourse to the following argument, which allows us to draw a variety of conclusions concerning the behavior of ε(p) for arbitrary momenta: The energy of the liquid is a functional of its density ρ(r) and its hydrodynamic velocity v(r), i.e.,
(1.6) [MATHEMATICAL EXPRESSION OMITTED]
where E(1) is the part of the energy which is independent of the velocity. If we consider small oscillations, then
ρ(r) = [bar.ρ] + δρ(r),
where ρ is the equilibrium density, which is independent of the coordinates, and δρ(r), v(r) are small quantities describing the oscillations. It should be noted that by definition
[MATHEMATICAL EXPRESSION OMITTED]
where V is the volume occupied by the liquid.
Excerpted from Methods of Quantum Field Theory in Statistical Physics by A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinki, Richard A. Silverman. Copyright © 1963 Richard A. Silverman. Excerpted by permission of Dover Publications, Inc..
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