Green's Functions and Condensed Matter

INDEX

Chapter 1

Introduction or Why Green’s Functions?

§1.1 Classical Green’s functions

Anyone who has used the Coulomb potential due to a point charge has used a Green’s function. If the charge is e, situated at the point with position vector r′, the electric potential at a second point r is

.

From this potential it is possible to write down the potential due to a number of charges or, in the limit of a continuous distribution, that due to a charge distribution p(r). The resulting potential is

and satisfies Poisson’s equation

Consequently, the Green’s function,

enables the solution of Poisson’s equation for an arbitrary charge density to be written as a quadrature. This illustrates the great power and usefulness of the Green’s function. In general, Green’s functions provide explicit solutions of differential equations.

To exemplify some of the methods that are used in this book, let us prove that (1.1) is a solution of equation (1.2). First, we note that the Green’s function is the Coulomb potential due to a unit charge situated at r′. Now the charge density p(r) of a unit point charge at r′ is zero if r ≠ r′ and has an integral over all space which is the total charge present, namely unity. Hence,

.

The function which satisfies these equations is Dirac’s three-dimensional delta function, δ(r — r′). Hence

.

The Green’s function (1.3) must therefore satisfy Poisson’s equation,

We now show that the Green’s function (1.3) is indeed a solution of equation (1.4). We usually have to satisfy the boundary condition that the potential tends to zero at infinity. We can, therefore, introduce the Fourier transforms,

and

Equation (1.4) is transformed to

,

whence

We confirm that G(R) satisfies Poisson’s equation for a point charge.

Now the potential given by equation (1.1) can be written

Hence

and we can use equation (1.4) to obtain

This confirms that ϕ(r) satisfies Poisson’s equation.

The solution is unique because in using the Fourier transform we have assumed that ϕ(r) tends to zero sufficiently rapidly as |r| → ∞. This ensures that ϕ(r) automatically satisfies the spatial boundary conditions, an advantage of the use of a Green’s function.

A second familiar example of the use of Green’s functions comes from the solution of Maxwell’s equations. As these involve time as well as space we consider them in some detail too. We look for the solution of the equations for the potentials A, ϕ,

Both equations can be solved using the Green’s function G (R, s) which satisfies

We again use the spatial Fourier transform G (k, s) which now satisfies

Here k is a parameter. This equation has the same form as that for a simple harmonic oscillator with a unit impulsive force at time s = 0. G plays the role of displacement, c–2 is the inertia and k² the spring constant. The solution depends on the displacement G and velocity ∂G/∂s, at the initial time s = 0. It is a general feature of problems involving time that we need to define boundary conditions which, in turn, are determined by the physical nature of the problem. Thus a given set of differential equations is satisfied by many different Green’s functions, and the physics of the problem is required to determine the appropriate one to use. This means that one deals with the boundary conditions at the stage of determining the Green’s function (independently of ρ and j in this case). Once we have found G we can find A and ϕ for a large number of different problems. This is another advantage of the use of Green’s functions.

For the moment let us now solve equation (1.10) with G = 0 = ∂G/∂s at s = 0. Since the impulse is of unit strength, the momentum of the oscillator is unity immediately after s = 0; since its mass is c–2, the velocity is ∂G/∂s = c². Formally this result can be obtained by the following argument. Since ∂G/∂s exists G is continuous at s = 0. If equation (1.11) is integrated from s = — (< 0) to s = , one finds

.

tend to zero. Since G is continuous the integral tends to zero. Hence

as we argued before.

The solution of equation (1.10) for s > 0 and G = 0, ∂G/∂s = c² at s = 0 is then

.

We can also include the fact that G (k, s) is zero for s < 0 by writing

,

where the theta-function is defined by

The spatial Fourier transform of G (k, s), and therefore a solution of equation (1.10), is

where we have used the Fourier transform of the δ-function in the last step. Since R and s are positive, the last term is zero, θ(s) is redundant in the first, and

.

Hence a pulse at one point produces an effect at another point a distance R away, at a later time R/c. The solution G, therefore, describes the retarded interaction. The corresponding solution of equation (1.8) is

the usual form of the retarded potential. We can prove that ϕ(r, t) does indeed satisfy (1.8) by acting on it with the D’Alembertian. Thus

From our examples we have learnt a number of things. The response of our systems to linear perturbations can be expressed in terms of Green’s functions which are independent of the perturbation. The choice of Green’s function to be used depends on the boundary conditions for the problem. A complete set of boundary conditions determines the Green’s function uniquely and so this form of solution already includes the boundary conditions. The form of the solution as an integral also expresses the superposition principle for a linear system, namely that the effect arising from the sum of two causes is the sum of the effects arising from the causes separately. The use of Green’s functions reduces the solution of partial differential equations to a number of quadratures. The power of their use is well known from examples in electromagnetism, where they were originally introduced by Green (1828).

In all of the problems we have to discuss, the spatial boundary condition is that the Green’s function tends to zero at infinity. This is usually ensured if one finds the spatial Fourier transform of G. The temporal boundary condition usually has to be dealt with more carefully.

§ 1.2 Linear response of quantum systems

We now show that the linear response of a quantum system can analogously be expressed in terms of functions which we call Green’s functions. The response of a quantum mechanical system to an external field or force is to be obtained by solving the appropriate time-dependent Schrödinger equation or, equivalently, for macroscopic systems to be described statistically, by solving the equation for the density matrix:

Here H is the Hamiltonian of the unperturbed system, which we assume to be independent of time, and H′ is linear in the external field. Usually, the system is in equilibrium before the external field acts. Hence, the initial value of ρ is, for a grand canonical ensemble, (see, for example, Kubo 1971)

where

T being the absolute temperature, and μ the chemical potential. Provided that H conserves particle number and so commutes with N, ρo is a solution of equation (1.12) with H′ zero.

To find the linear response of the system, we need to solve equation (1.12) to first order in H′. We begin by making a canonical transformation to remove the term linear in H from the equation. The appropriate transformation is

where S is unitary and satisfies

This takes us to the Heisenberg representation for the Hamiltonian H. If H does not depend explicitly on t, then

In general

From equations (1.12) and (1.18) it follows that

where

and is still linear in H′(t). At the initial time t = 0,

Hence the solution of equation (1.19) to first order in H′ is, by iteration,

If H is independent of time, S and ρo commute and

Usually one is interested in comparing the value of some macroscopic variable (for example, charge density, magnetic moment) with observation. The variable is represented by a macroscopic operator M, say, and the quantity to be compared with experiment is the quantum and thermal average of M at time t. This is

The first term is the equilibrium value of the quantity. The second is the change δ〈M〉 induced by the field. If one uses the cyclic property of the trace of a product of operators, namely,

,

one finds that

where, for any operator α, we write

The result (1.24) is often known as Kubo’s formula after its discoverer (Kubo 1957).

It is useful to show the dependence of Ĥ′(t) on the external field explicitly. If at time t this is a classical field A (t), we can write

where B is an operator belonging to the system being perturbed. (A could be the magnetic field at a point and B the magnetic moment density operator.) In general, H′(t) would be a sum of terms of the type (1.26) but since we are considering only the linear response the effects of these terms can be considered separately and the total response obtained by adding the individual contributions. Thus the use of (1.26) does not limit the applicability of the result. If (1.26) is used in equation (1.24) one finds

where

This result, which yields the dependence of the effect on the cause, is analogous to the classical result (1.11), and therefore G(t, t′) is called a Green’s function. The θ-function ensures that the effect at time t depends on the cause only at preceding times. Hence, the Green’s function in (1.28) is called a retarded Green’s function. It is possible to define an advanced Green’s function,

the use of which is more formal.

The part of G (t, t′) which multiplies the θ-function, that is

,

is often called the response function or after-effect function.

Any definition of a Green’s function to within a constant factor is possible and many different definitions are to be found in the literature. Care is always required in checking the constant of proportionality when comparing equations involving Green’s functions.

It follows from this discussion that we know the linear properties of a quantum mechanical system once we know the appropriate Green’s functions (1.29). These are sufficient to describe most electromagnetic emission, absorption and transmission experiments, accoustic attenuation, resonance experiments such as electron spin and nuclear spin, and many others. As we shall see in §2.8, they also determine the thermodynamic potential and hence the thermo-dynamic properties. Hence the Green’s functions contain all the information we need about the system and, if we can find the Green’s functions without first determining all the eigenstates and energies of the unperturbed system, we may save a considerable amount of work and obtain useful results economically. In any case, even if it were possible to compute the eigenvalues and eigenstates of a macroscopic system it would not be possible to hold the information in a useful store. However, we can hope to hold the information contained in the Green’s functions.

We have stated that absorption can be related to Green’s functions using the linear response of the system to an appropriate external field. While this is true, it is common to discuss absorption in terms of Fermi’s golden rule. We therefore show explicitly how this relates absorption to Green’s functions.

Suppose that a classical field A (ω) frequency ω interacts with a quantum system, through a term in the Hamiltonian

.

Then, according to Fermi’s rule the rate at which the quantum system changes its state from |m〉 to |n〉 is

.

If we sum over all possible final states |n〉 and thermally average over initial states |m〉, the rate at which energy ω is absorbed from the field is

.

If we introduce the Fourier transform of the δ-function this can be written

where H is the Hamiltonian of the unperturbed system so that

.

The sums over m and n can be performed to yield

.

The integrand has a similar structure to a Green’s function although it is not exactly of the same form. However, the methods of §3.2 can be used to relate the two explicitly. An example using this expression is given in §7.2 where the elastic scattering of neutrons by phonons is discussed.

§ 1.3 The simple harmonic oscillator

As a simple example of the use of a Green’s function in quantum mechanics we consider the problem of a one-dimensional simple harmonic oscillator of unit mass forced to oscillate by a time dependent force f(t). This simple problem is a useful guide to the physical interpretation of the formal properties of Green’s functions. If x is the displacement of the oscillator, the interaction term in the Hamiltonian due to the external force is the potential

.

According to the general theory of the previous section the displacement at time t is (for an ensemble of oscillators)

and the relevant Green’s function is

.

Heisenberg’s equations of motion for a quantum oscillator are the same as the classical ones, with the same solutions. Hence

.

Using the commutation relations of x and p this leads to

As is often the case in quantum mechanics, the simple harmonic oscillator provides an exactly soluble model. The Green’s function oscillates with the angular frequency ω of the oscillator. This is not surprising as it is the response to a δ-function force. This is partly the result of having no dissipation in the system. If there were dissipation the system would eventually return to equilibrium after an external force were removed. One would then have the condition,

This is the case for all macroscopic systems. In this sense the problem of the simple harmonic oscillator is artificial and atypical. Nevertheless, the simple harmonic oscillator provides a useful model of G for some purposes. In particular one would expect that for a system which can oscillate at several frequencies, the Green’s functions will contain components which oscillate at these frequencies. In quantum mechanics, these frequencies become the excitation energies divided by h. Hence, we expect that an analysis of the Green’s functions will yield the excitation energies of the system.

The parallel between the quantum and classical systems can usefully be pushed further. Suppose we look at the classical response of a simple harmonic oscillator to a force f(t) and include a dissipative term. The equation of motion is

.

The classical’s Green’s function is found by solving

with

.

The solution can be found in the same way as that of equation (1.10). For the case cor > 1 (little dissipation, ordinary damping)

This agrees with the quantum mechanical result in the limit of no dissipation (τ → ∞). Also, as expected, it satisfies condition (1.33). It is, therefore, a reasonable model Green’s function for a system in which energy is dissipated. Unfortunately, just because energy is not conserved, it is not possible to write down a simple Hamiltonian from which to derive equation (1.34) and so we cannot obtain the analogue of (1.35) as easily from quantum mechanics.

As we shall later be led to consider the time Fourier transform of Green’s functions, we look now at the Fourier transform of (1.35). We have,

Considered as a function of the complex variable s, G (s) is analytic in the upper half-plane and has poles at

.

For τ large, these are close to the real axis at the natural frequencies of the system. The displacement from the real axis depends on τ the time for the exponential decay of the Green’s function.

There are many important physical systems which behave as simple harmonic oscillators and for which the discussion of this section is directly relevant. In particular, the basic excitations of the lattice of a solid are phonons which behave as simple harmonic oscillators. We shall discuss these excitations in some detail in Chapter 7.

§1.4 Single-particle Green’s functions

Whenever we come to evaluate explicitly the Green’s functions obtained in §1.2 in a many-body problem, we usually find that we are quickly led to consider similar functions with a simpler structure, the single-particle Green’s functions. To illustrate this, consider the following Green’s function

where ρ(r, t) is the Heisenberg operator for the particle density at r at time t and |0〉 is the ground state. Let us evaluate G for a system of dynamically independent particles whose single-particle normalized eigenstates are ϕm (rm. The Green’s function can be evaluated using either many-body wave functions or second quantization. As we have to use the latter method in most of this book we shall use that method here. For convenience the main properties of the second quantized operators and states are listed in Appendix A. (For a derivation of the properties readers are referred to an advanced text on quantum theory, for example Ziman (1969)).

In these terms

and

where cn destroys the state n and

.

Hence, for t >t′,

be taken in independent particle states, the average will be non-zero only when the product of operators has the net effect of not creating or destroying any particles. In other words, the product c1(t)cn(tcreates. This requires that the pair of labels l, n is the same (apart from order) as the pair k, m. Hence, for a non-zero average

.

Therefore

The first commutator is zero. The second term is zero if n = l .

If n ≠ l, the degrees of freedom n and l are independent and one has

and this leads us naturally to consider objects such as

.

+(r, t) is the operator which creates a particle at the position r at time t (see Appendix A). If we denote

then |r, t〉 describes a state which comprises the ground state with an extra particle at r at time t. Similarly, |r′ t′〉 has an extra particle at r′ at time t′. Therefore

is the probability amplitude for finding a particle at r at time t, given that a particle is placed at r′ at the time t′. It therefore describes in quantum mechanical terms the way a particle travels from r′t′ to r, t. +(r, t(r′, t′)|0〉 describes the travel of a hole from r′, t′ to r, t.

The new functions are not themselves Green’s functions because that term is reserved for functions which are singular or which have singular derivatives when the arguments become equal. It is easy, however, to construct Green’s functions, known as single-particle Green’s functions from them. For example, we can write down the Green’s functions

and

There are two things to be learnt from this example. The first is that the basic building blocks for Green’s functions in many-body problems are likely to be the single-particle Green’s functions. The second is that with the Green’s functions we have so far constructed, there is not a simple relation between the Green’s function G(r, t; r′t′) and gA or gR. Two important steps which have advanced the theory and made Green’s functions a useful tool are:

(i) the discovery of Green’s functions in which the relationship between those involving many operators can, at least for independent particles, be simply related to those involving fewer operators; and

(ii) the discovery of relations between these (formal) Green’s functions and the observable ones.

The procedure in any calculation is to establish an approximation for calculating the former Green’s functions and then to use (ii) to calculate the observable Green’s functions from them. Section 2.2 is concerned with (ii). Most of the remainder of the book is concerned with the approximations.

Although we have come to single-particle Green’s functions as basic building blocks, there are occasions when they are the observable functions. We give two examples.

(a) As we have seen, the single-particle Green’s function describes the behaviour of an extra particle added to the system. An experiment where such behaviour is directly observed is one where particles can tunnel quantum mechanically through a barrier from one material to another. The results of such experiments can be given in terms of single-particle Green’s functions (cf. §8.7).

(b) The thermodynamic properties including the density of states in energy of systems can be related to single-particle Green’s functions (see §2.8).

§1.5 Correlation functions

The operator

+(r(r)|〉 is just the mean density at r. In a macroscopic system it will be the actual density at r. If the state, | 〉, contains no particles at r the mean value is zero, otherwise it is finite. The integral (1/N) f ρ(r)dv integrated over a small volume Δv then has a mean value P(r) Δv which is the probability of finding a particle in the small volume Δv. P(r) is the probability density and

where the integral is over the whole volume. Now consider the operator

.

Since ρ(r) and ρ(r′) commute the product is Hermitian. Its expectation value 〈|ρ(r)ρ(r′)|〉 is zero in any state, | 〉, for which there is no particle at either r and r′ and is finite otherwise. The integral product

,

where the integrals are over small volumes Δv, Δv′, then has a mean value f P(r, r′) ΔvΔv′ and, for a macroscopic system,

is the probability density that there is one particle at r, and another at r′.

If there were no correlation between the probability of finding a particle at r, and the probability of finding a particle at r′, we should have

,

that is,

This is true for a system of dynamically and statistically independent particles. For the usual systems of interest, interactions between particles are important and (1.40) is not satisfied. The difference between the two sides

therefore represents the correlation between the positions of the particles. For this reason

is called the density–density correlation function. This is a slight misnomer because (1.41) represents the correlation, but it is very commonly used.

Particles in a quantum system always obey either Fermi or Bose statistics and are, therefore, never statistically independent. The density-density correlation function for such a system is therefore never zero.

The definition of a correlation function is very easily generalized to the case of other commuting observables. If A and B commute,

is the correlation function between them. If A and B do not commute, it is not possible to measure them separately and so it is not strictly possible to talk about correlations between them. Nevertheless, for macroscopic systems the measurement of one may well not perturb the other by much and the correlation between them is in practice, measurable. Even when this is not the case it is possible formally to define correlation functions. Because of the non-commutability of the observables, the definitions are not unique and there is no universal agreement. Some authors call any expectation value of the form (1.43) a correlation function. Others (see Kubo 1957) generalize the function so that it is real and reduces to (1.43) for commuting observables. In this case,

When we allow observables which do not commute, there is no need for them to depend on the same time. We can therefore consider correlation functions

and

The Green’s function which appears in the linear response can be written in terms of correlation functions of the type in equation (1.45). This Green’s function is

Notice a basic difference between Green’s functions and correlation functions. The former (or their time derivatives) are discontinuous at t = t′ whereas the latter are not. The response of the system can be expressed equally well in terms of either but, in practice, it is usually easier to derive the Green’s functions. The results of