1 Jun 1993 - Figure 9: The Feynman diagrams for proper self energy in ladder approxima- tion for particles of spin 1. 2. , which corresponds to Fig. 9 in the ...
June 93
Landau Theory of Fermi-Liquids
Abha Sood Department of Physics, University of Oldenburg D-2900 Oldenburg, Fedral Republic of Germany First Referee: Professor Dr. E. R. Hilf Second Referee: Dr. P. Rujan
Contents 1 Introduction
2
2 Landau’s Theory of Fermi Liquid
3
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Phenomological Derivation . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.3
Microscopic Verification using Green’s Function Theory
. . . . . . . . .
9
2.4
Application: Hard Core Potential . . . . . . . . . . . . . . . . . . . . . .
20
2.4.1
Galitskii’s integral equations . . . . . . . . . . . . . . . . . . . . .
20
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.5
3 Discussion and Conclusion
26
3.1 Suggestions for further study and calculations . . . . . . . . . . . . . . .
26
3.2
28
Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
Introduction
The phase transition of a nucleon gas to nuclear matter still remains to be completely understood. One attempt made in the past is on the basis of an approximation of normal fermi liquids suggested by Landau (1957) which has been further developed by many authors using modern field theoretical methods [AGD63] [Noz64] [GM58]. This approximation is applied to a nucleon gas such that a phase transition to nuclear matter is obtained. The goal of this study was to find a method to extend this technique for higher temperatures using a relativistic treatment. The first step was to estimate and study quantitatively the magnitude of error made as a result of the approximation and the calculation techniques applied. However before the quanitative study may be made the qualitative understanding of the theory is deemed necessary. Thus the first part of this thesis is an exhaustive review study of the Landau approximation for Fermi liquids, presenting the theory in a systematic and concise form. The application to nuclear matter of this approximation is also sketched here. Such a treatment was necessary since the aim of this study is also to understand the error occuring in the determination of physical quantities such as the density and energy of the system due to the approximation undertaken. Since the knowledge of the basic modern many particle theory is absolutely necessary for the understanding of the basic concepts, it is presented briefly in the appendices. It is felt necessary to introduce the field theoretical methods such that a motivated physics graduate student is able to understand this text. For a proper study, however, it is still necessary to refer to the original texts cited there. Most of the calculations leading to the results given are not included since they are either straight forward or may be regarded as good exercises.
2
2 2.1
Landau’s Theory of Fermi Liquid Introduction
Landau’s theory of Fermi liquids has a wide range of applications. The basic concepts and assumptions are therefore presented in this chapter. Its application to matter of short range two body forces, with some modifications, is then considered and the relative advantages of this method with respect to some others are discussed. The phenomenological theory of Landau based on his original ideas [Lan57] is presented in the next section [Noz64]. These ideas are then explored more thoroughly using the methods of quantum field theory [GM58] [Lan59] [AGD63] [Noz64] in the section 2.3. The technical information necessary for the understanding of the text is presented separately in the appendix. The basic assumption of Landau’s theory is that the weakly excited states of a Fermi liquid greatly resemble those of a weakly excited Fermi gas. These states can be described with a set of elementary excitations with spin 12 and momenta close to the Fermi surface. It is then assumed that there is a one to one correspondence between the number of states of a perfect gas to that of a normal Fermi liquid. This may be physically realized by adiabatically switching on the two-particle interaction. The concept of quasi-particles (elementary excitations) is thus introduced. In this manner, however, the important low lying collective states of the liquids are lost which are necessary for the description of (for example) superconductivity. The quasiparticles thus obtained are not the exact stationary states of the system, but a superposition of a large number of exact stationary states of the system [AGD63] with a narrow spread of energy. This leads to the damping of the states. This damping may be explained as the interaction between the quasiparticles, with conserved laws of momentum and energy. This can occur through processes where the excitation decays into several others or where the quasiparticles are scattered by each other. The decay of excitations plays a role only at higher temperatures. On the other hand, for sufficiently high temperature, any system tends to behave like a noninteracting system. If the temperature is sufficiently low, there are only a few low energy quasiparticles which rarely scatter and thus the interaction between the quasiparticles is weak. It is then regarded to be an acceptable approximation for many applications and a perfect gas of quasiparticles only may be considered. In applying the above formalism to the specific problem of nuclear matter, further approximations will become necessary for the calculation of the physically measurable quantities. It is observed that since the nuclear interparticle forces can not be described by a perturbative expansion, the general approach is to calculate the scattering amplitude which remains finite and then using appropriate approximations (such as the ladder approximation). The expansion thus obtained is the ordinary perturbation for the divergent part of the graphs in the small parameter kF a, where a is the range of the interaction and kF corresponds to the Fermi momentum. One of the manners of bypassing the problem created by the assumed singular nuclear potentials has been has been suggested 3
and developed by Galitskii (1958) using the corresponding finite quantities of scattering theory. One alternative method of calculation will be suggested and developed here and will be dealt with in an ensuing study.
2.2
Phenomological Derivation
1
Consider a system of N identical fermions in a volume V assumed to be large, at zero temperature. In case of a simple gas of noninteracting particles, i.e. a perfect gas, the eigenstates are antisymmetric combinations of single particle states, to be taken as plane waves2 . The plane waves are characterized by their wave vector k. To define the eigenstate of the whole system, it is sufficient to indicate which plane waves are occupied with the distribution function n(k). However, the phase information is lost through this process, and the transformation back to the original single particle states before superposition is not unique. Let the ground state of the system correspond to an isotropic distribution n0 (k). The cutoff level in the ground state kF is called the Fermi level and is given by gkF3 /(6π 2) = N/V (Appendix A). If the distribution function is changed by an infinitesimal quantity δn(k) the total energy of the system changes by an amount δE[n] =
X¯ h2 k 2 k
2m
δn(k).
(1)
The functional derivative of the energy with respect to the distribution function is the kinetic energy of a particle with wave vector k, δE[n] X ¯h2 k02 = δkk0 δn(k) 2m k0
where δkk0 :=
δn(k0 ) . δn(k)
(2)
For zero temperature, δn(k) is necessarily positive for k > kF and negative for k < kF since all the energy levels are occupied below kF and are empty above it in the absence of interaction. Since it is assumed that the excitation spectrum of a real fermion system (i.e. Fermi liquid) has the same structure as the excitation spectrum of a perfect Fermi gas. In order to extend these ideas to the Fermi liquid, the interaction is switched on adiabatically, that is without exchange of energy with the surroundings. It is assumed that the states of an ideal gas are transformed to those of a real gas as the interaction gradually increases; the time development of each state can then be studied by means of perturbative treatment3 . 1
The major part of this derivation is based on the notation used in reference [Noz64]. ψ1 is the single particle wave function of the first particle and ψ2 is the wave function of the second particle. Then the corresponding two particle wavefunction obeying the Pauli principle is given by ψ(x1 , x2 ) = ψ1 (x1 )ψ2 (x2 ) − ψ1 (x2 )ψ2 (x1 ). 3 A system with a non-degenerate stationary state cannot make a transition to another state under the action of an infinitely slow perturbation [Lan58]. 2
4
n0 (k)
nk
6
1
6
1
$
0.5 -
kF
&
k
kF
-
k
Figure 1: Landau Distribution Function a) n0k of noninteracting fermions and b) nk of interacting fermions at 0 K temperature It is further assumed that the interaction is only repulsive because the ground state of an attractive system may be radically different from that of the perfect gas. The net repulsive interaction is also assumed to be weak. Since the states in the eigenstate basis vectors of the noninteracting system of the real system are in general unstable and damp out after a certain time period τ , the adiabatic switching on of interaction should require a time much shorter than τ . However, if the interaction is turned on too fast, the final state is no longer an eigenstate of the system. It is therefore necessary that the time period τ should be large which implies that the excited states have a long lifetime. This thus limits the excited states to lie in a low level close to the ground state. When an additional particle of wave vector k with k > kF is added to a perfect gas in the ground state and then the interaction is turned on, an eigenstate of the real gas is obtained. A quasi-particle4 of wave vector k is thus added to the system. The lifetime of the particle defined in this manner is only long near the Fermi surface. Thus the concept of quasi-particle is only valid in the neighbourhood of k = kF . Similarly, a quasi-hole is the removal of a particle of wave vector k with k < kF . Though the same distribution function n(k) now characterizes the real states, it gives the distribution of quasi-particles and not of real particles. It is assumed that the distribution of quasiparticles is spatially homogeneous. The distribution of the quasiparticles in the ground state is thus still given by n0 (k) and the concept of Fermi surface is retained (Fig. 1). The excitation of the system is measured by δn(k) = n(k) − n0 (k). As δn(k) is appreciable only near the Fermi surface the quasi-particles are well defined only in this region and poorly defined elsewhere. In other words, a quasi-particle is an elementary excitation in the neighbourhood of k = kF and gives no information about the ground state of the real particles in the interior of the Fermi surface. The energy of the real system is now a functional of n(k); i.e. E[n(k)]. As shown above, this functional reduces to the sum of energies of the individual particles in case of an ideal gas. If n0 (k) is now altered by an 4
refer to Appendix B.
5
amount δn(k), the variation of energy to first order will be given by δE =
X
k δn(k),
(3)
with k := δE/δn(k) being interpreted as the energy of the quasi-particle. This relation is only valid when the number of quasi-particles added or removed is small as compared to the total number N of particles in the system. The energy of the whole system is, in particular, not the sum of the energies of the quasi-particles. Let kF be the energy required for adding an additional particle to the system at the Fermi surface kF = E0 (N + 1) − E0 (N ) = µ.
(4)
Thus µ := δE/δN is called the chemical potential of the system and the ground state of a system with N + 1 particles is obtained. If the second order effects are not neglected as in the above considerations, the variation of the total energy is given by δE =
X
0k δn(k) +
k
XX k
f (k, k0 ) δn(k) δn(k0 )
(5)
k0
and thus k = 0k +
X
f (k, k0 ) δn(k0 ).
(6)
k0
Then f (k, k 0 ) is the second functional derivative of E and so the variational derivative of k with respect to n(k). Using the classical methods of statistical mechanics, from the condition that the entropy should be conserved when the number of particles and energy are conserved5 , we obtain for the quasi-particle of energy the distribution function n() =
1 . exp ( − µ)β + 1
(9)
5
The following relation of the entropy, derived using purely combinatoric considerations for an assumed perfect gas, may be used to derive the distribution function of the quasi-particles [Hua64] of this model Z S dk = − [n ln n + (1 − n) ln(1 − n)] . (7) V (2π)3 Again this could be achieved by skipping the collective states. Since the energy levels of the quasiparticles are assumed to be similar to that of an ideal fermi gas this equation may also be applied to obtain the quasi-particle distribution (refer to equation 3). Applying the condition for maximum entropy, that the number of particles and the energy are conserved (microcanonical ensemble), the desired distribution is obtained
n() =
1 e(−µ)β
+1
.
(8)
6
This relation is valid because a quasi-particle obtained from fermions is itself a fermion. The probability of finding a particle with energy is n() and µ is the chemical potential adjusted such that the total number of quasi-particles is normalized to N. When T → 0 then n() tends to the step function in Fig. 1 a) and thus the distribution of the quasiparticles is taken to be a Fermi distribution. Including the spin of the particles the above formulation may be restated as E = E0 + δE, where E is the total energy of the system and E0 is the energy of the unperturbed system and on redefining k = (k, σ) as f (k, k0 ) = f (kσ, k0 σ 0 ) = f0 (k, k0 ) + fe (k, k0 ) δσ,σ0 ,
(10)
where fe (k, k 0 ) appears only when the spins are parallel and expresses the exchange interaction between the two quasi-particles. It is clear that f now depends upon the momenta and spin of two quasi-particles. It will be seen later that f is connected with the forward scattering amplitude of two quasi-particles. In an isotropic case, e. g. in the absence of magnetic field the quasi-particle energy does not depend upon spin. The function 0k depends only on k in equation (6) and may be expanded in the series at k ≈ kF as follows: 0k − µ ≈ v(k − kF )
where v ≡
kF ; 0kF ≡ µ. m∗
(11)
The relation between effective mass m∗ and the interaction term f will now be derived. The sum of momenta in a unit volume is equal to a flow of mass. The momentum of a unit volume of the Fermi liquid is the same as the momentum of the quasiparticle in this volume. The current of the particles in the Fermi liquid is equal to the current of quasi-particles. Z
dk kn(k) = m (2π)3
Z
dk vn(k), (2π)3
(12)
where v = ∇k k . Substituting this definition in the above equation and then varying with respect to n where = [n(k)], we obtain Z
dk k δn = (2π)3 m
Z
dk 1 δn∇k k + (2π)3 2
Z
dkdk0 (∇k f (k, k0 ))nδn0 . (2π)6
(13)
which on integration by parts and permution of k to k0 is Z
dk k δn = (2π)3 m
Z
dk 1 δn∇k k − 3 (2π) 2
Z
dkdk0 (∇k0 n0 )f (k, k0 )δn. (2π)6
7
(14)
The average over spin indices is taken since n and do not depend upon spin here. Since δn is arbitrary, it follows that k 1 = ∇k k − m 2
Z
dk0 f (k, k0 )∇k0 n0 . (2π)3
(15)
An estimate for ∇k0 at k 0 ≈ kF is given by the expression, obtained by integration by parts, ∇k0 n0 ≈ −
k0 δ(k0 − kF ). k0
(16)
Since f depends only upon the angle ϕ between k and k0 we obtain the following relation between the effective mass m∗ of the quasiparticles and the mass m of the fermion. The translation invariance property of an isotropic liquid is also used here. 1 1 kF = + ∗ m m 2(2π)3
Z
f (ϕ)cosϕdΩ,
(17)
where f (ϕ) is the value of f (k, k 0 ) at | k |=| k0 |= kF and dΩ is the infinitesimal solid angle. This equation only holds for sufficiently low temperatures. It is additionally assumed that the potential may be described in a self consistent manner (i.e. the interaction is described as a self-consistent field experienced by one quasi-particle due to the presence of all others). In the momentum space, the occupied states form a sphere with radius kF which is termed the Fermi sphere. kF = (
3πN 3 ) . V
(18)
The number of states with kF < k should be equal to the number of particles.
8
2.3
Microscopic Verification using Green’s Function Theory
The modified version of Green’s functions theory applied to the many body problem has been introduced and presented in a compact form in Appendix C. All the results, which have been derived there, will now only be referred to and will be used to derive the Landau theory explicitly. It will be shown that this derivation is equivalent to his original phenomenological derivation presented in the last section 6 . The general equation of motion of a system of N elementary particles which experience two particle interaction is formulated. The poles of the propagator is then calculated and interpreted as quasiparticles. The vertex part of the two particle Greens function is related to the interaction term f (k, k 0 ) in equation (6). The effective mass can then be calculated as shown in the last section. The general form of the N particle Hamiltonian used in the following calculations consists of the free part corresponding to the kinetic energy of the system and the two particle interaction (refer to equation (91) in Appendix C.). It is possible to describe the free part of the Hamiltonian 7 H0 by two field operators with which a single particle Green’s function may be defined whereas the two particle interaction V (x1 , x01 ) may be described by the two particle Green’s function to obtain the equation of motion of a particle added to the N particle ground state. The following equation may be derived using the Heisenberg equation of motion for the field operator ψ(x) and its hermitian conjugate and then applying the definition of the Green’s functions8 . [iδt − H0 (x)]G(x, x0 ) −
Z
d4 x01 V (x − x01 )G(x01 , x; x01 + , x0 ) = δ(x − x0 ).
(19)
Analogous equations for 2- and 3- particle Green’s function etc. may be derived, which are in this approximation not necessary, since we only consider the two particle interaction. The two particle Green’s function may be split into a sum of an uncoupled product of two single particle Green’s functions which also takes the mean Hartree-Fock energy into account and the interacting part which describes the interaction between the two particles (147), (148). The uncoupled part includes not only the zeroth order of the perturbative expansion but all orders to some extent. This statement will be made precise and justified in a later discussion. Substituting (147) in the above equation and separating the free part of the two particle Green’s function we get 0
{iδt − H0 (x)}G(x, x ) − i
Z
d3 r10 V (r − r10 ){G(x, x0 )G(r10 t0 , r10 t0+ )
−G(x, r10 t0+ )G(r10 t0 , x0 ) + δG(x, r10 t0 ; r10 t0+ , x0 )} = δ(x − x0 ) 0
{iδt − H0 (x)}G(x, x ) −
Z
d4 x001 Σ(x, x001 )G(x0 , x001 ) = δ(x − x0 ),
6
(20)
This section follows the presentation of [Noz64] and [Eco83] unless otherwise stated. The free Hamiltonian is defined as H0 = pˆ2 /2m where pˆ is the quantum mechanical momentum operator. The total Hamiltonian is then the sum of the free part and the interacting part H = H0 + V . 8 In this equation, the term adds only an infinitesimal time interval to x01 . In the later equation somewhat different notation is used. 7
9
6 6
ak γk ?
ω-
Figure 2: The Lorentzian peak and its characteristic parameters. where Σ is the self energy or mass operator. It may be also split into an uncoupled part and an interacting part; Σ = ΣF + δΣ with R 3 0 0 0 0 0 ΣF = R−i d Rr1 V (r R− r10 ){G(r t , r t )δ(x − x001 ) − G(x, r0 t0+ )δ(t − t0 )δ(r10 − r100 )} and R 4 1 0 1 0+ 0 4 0 4 0 4 0 δΣ = d r1 d x1 d x2 d x3 G(r1 , x1 )G(x, x02 )Γ(x01 , x02 ; x03 , x001 )G(x03 , r01 ) The more complicated second term may now be neglected. This is justified with the observation that the Fourier transform of δΣ with respect to the space variables for a large interval in time becomes incoherent and the result goes to zero. However, it must be noted that in case the interaction is strong and long ranged, this term will dominate over the effect of the first two terms. The observation of the effect of the uncoupled terms in this case can not be made. The Fourier transform of the resulting equation is (0k − ω − ΣL (k, ω))G(k, ω) = 1.
(21)
ΣL is the effective self energy or the effective mass operator which has been calculated in equation (155) and (156) to the lowest order in the perturbation series. A better approximation of the self energy is possible using self consistent equations to obtain a solution of the Green’s function (19) for a chosen singular potential (refer to the following section). ΣL (k, ω) is thus merely the average Hartree-Fock energy felt by the particle with wave vector k as mentioned before. Thus the only effect of the interaction in this approximation, is a correction to the kinetic energy of the particles. Since ΣL (k, t − t0 ) is localized in time the resultant Fourier transform is a smooth function of ω. The poles of the equation (21) may now be determined and it is assumed that there is only a single solution for each value of k. This is the basic Landau assumption. The Lorentz curve (refer to Fig. 2) is the natural extension of the line in the line spectrum to the normed curve in the band spectrum for exponential damping. It is assumed that the states of the interacting particles experience an exponential damping when expanded with respect to the orthonormal basis of the states of the noninteracting particles9 . These 9
refer to the Lehmann representation of Green’s function presented in Appendix C.
10
states of the interacting particles are not the eigenstates and thus not stable but tend to wander away from the original states in a finite time interval. They thus have only a finite lifetime. If the states are damped by a factor eγk t then the inverse of γk is the measure of their lifetime. The characteristic features of the Lorentz curve are sketched in Fig. 2. In order to define the position and the characteristics of the poles the following procedure is adopted, analogous to the development in [Eco83] and [Wyl86]. The function ˜ ω) is defined as G(k, ˜ ω) := GR (k, ω) − GA (k, ω), G(k,
(22)
where GA,R are defined in the equation (136) which are the advanced and the retared ˜ ω) is defined as Greens functions respectively and the term A(k, ˜ ω) := iG(k, ˜ ω). A(k,
(23)
The purpose of this definition is obtain a function A˜ such that its analytic continuation ˜ ω) is real in the complete complex may be obtained. It may be observed now that A(k, if ω is real, and for fermions it is always nonnegative. It can be shown that Z
dω ˜ A(k, ω) = 1. 2π
(24)
Γ(k, ω) is defined by the equation
Γ(k, ω) =
Z
˜ ω0 ) dω 0 A(k, . 2π ω − ω 0
(25)
One can obtain a relation between GR,A and the Γ using the complex analysis methods of integrating over the poles as follows: GR (k, ω) = lim+ Γ(k, ω + is). s→0
GA (k, ω) = lim+ Γ(k, ω − is).
(26)
s→0
Γ(k, ω) is an analytic function in the complex plane except for the region on the real ˜ ω) 6= 0. It follows directly from the equation (25) that axis where A(k, ˜ ω) Γ(k, ω + is) − Γ(k, ω − is) = −iA(k,
when s −→ 0+ .
(27)
This quantifies the jump over the real axis. It is useful to define the above quantities in order to obtain a suitable function Γ(k, ω) which may be analytically continued over the ˜ ω) branch cut onto the next Riemann sheet using Cauchy integral formula. Since A(k, 11
may be regarded as the evaluation of an analytic function on the real axis10 , it is possible to obtain the required analytic continuation onto the next Riemann sheet rather simply as11 a.c.{Γ(k, ω)} = Γ(k, ω) + a.c.i{A˜b (k, ω)} + iA˜0 (k, ω)} f or Im ω > 0, a.c.{Γ(k, ω)} = Γ(k, ω) − a.c.i{A˜b (k, ω)} − iA˜0 (k, ω)} f or Im ω < 0,
(28)
˜ ω) is assumed to have the form of a Lorentzian peak over a where the function A(k, smooth background (Fig. 2) given by ˜ ω) = A˜b (k, ω) + A˜0 (k, ω) A(k, 2 | γk | ak 2 | γk | ak where A˜0 (k, ω) = = 2 2 (ω − k ) + γk (ω − k + iγk )(ω − k − iγk )
(29)
and A˜b (k, ω) is the smooth background which is a contribution of the incoherent part of the mass operator and will be neglected in the calculations. Using equation (26) and the above results, the following analytic continuation of the the retarded and the advanced Greens functions may be obtained. ( R
a.c.{G (k, ω)} = ( A
a.c.{G (k, ω)} =
Γ(k, ω) for Im {ω} > 0 ˜ Γ(k, ω) − a.c.i{A(k, ω)} for Im {ω} < 0
(30)
Γ(k, ω) for Im {ω} < 0 ˜ Γ(k, ω) + a.c.i{A(k, ω)} for Im {ω} > 0
(31)
Thus we see that the poles lie only in the lower half plane for GR (k, ω) and in the upper half plane for GA (k, ω). A further simplification may be considered at zero temperature for fermions, since the poles (if any exist) of the analytic continuation of G(k, ω) lie only in one quadrant of the complex plane. Thus it is seen that (
a.c.{G(k, ω)} = (
a.c.{G(k, ω)} =
Γ(k, ω) for Im {ω} > 0, Re {ω} > µ a.c.{Γ(k, ω)} for Im {ω} > 0, Re {ω} < µ Γ(k, ω) for Im {ω} < 0, Re {ω} < µ a.c.{Γ(k, ω)} for Im {ω} < 0, Re {ω} > µ
(32)
Thus, analogous to the reasoning presented above and using the equations (29) and (32), we see that the poles lie only in the lower half plane for Re {ω} > µ and in the upper half plane for Re {ω} < µ. It may thus be inferred that there exists a branch cut for the analytic function G(k, ω) at the line µ = Re{ω} and the poles cross the real axis with the Re{ω} increasing from 0 to ∞ at µ = Re{ω}. This surface defines the Fermi surface of the interacting particles. The peaks of the poles on the positive/negative complex plane of the analytic continuation of the Green’s function which appear, are the quasiparticles. ˜ ω) where A(k, ω) and B(k, ω) The quantity 2πi(A(k, ω) + B(k, ω)) corresponds to A(k, are the spectral norm density in the equation (136). Thus it can be seen, that the peak 10
A prerequisite for analytic continuation of a function over a branch cut in a straightforward manner. In general, an analytic continuation over a branch cut for an arbitrary function is not a trivial problem. 11 a.c. is the abbreviation for analytic continuation.
12
6
B(k, ω) A(k, ω)
for | k |< kF | k |> kF
A˜b (k, ω)
ω -
Figure 3: A qualitative presentation of the shape and the position of the poles of the analytic continuation of the Γ(k, ω). ˜ ω) appears as a peak in A(k, ω) if k > µ which is interpreted as a quasiparticle of A(k, whereas for k < µ it appears as peak in B(k, ω), which is then interpreted as a quasihole. Inside the Fermi surface are the quasiholes, whereas outside it are the quasiparticles for a given value of Re{ω} (Fig. 3). Thus it is assumed that the pole of the Green’s function is not a singular Dirac delta function but contain a sharp Lorentzian peak12 . γk is the width of the curve (at half maximum) which defines the lifetime of the state, τk = 1/γk . The weight of the peak can be shown to correspond to the residue of the pole ak . It is the measure13 of the extent to which the concept of independent particle, i.e. the bare part of the quasiparticle, is retained; and (1 − ak ) is the proportion of the particle participating in the interaction, i.e. the dressed part of the quasiparticle. As the interaction tends to zero the quasiparticle tends to the corresponding bare particle, as required, and the Lorentian peak converges to Dirac delta peak. The position of the peak curve given by k corresponds to the average energy of the state [Eco83]. The behaviour of the poles is diagramatically presented in Fig. 3. ˜ ω) described in this manner are well deIt is crucial to observe that the peaks of A(k, fined only close to the real axis, if their lifetime is well defined (refer to equation (142) in Appendix C. and its discussion). Due to the short lifetime the peak of the quasiparticle will dissipate before they can develop. This restricts the position of the poles only close to the Fermi surface. Thus, for the physically interesting properties of interacting fermions which are described through the quasiparticle, it is reasonable to study only the peaks rather than the complete Green’s function. A qualitative picture of the poles is illustrated in Fig. 4. 12 13
in the sense of mass distribution not in the mathematical sense
13
Im {ω} 6
n(k) 6 1 zk
µ
ak
Re{ω} -
kkF Figure 5: The distribution function n(k) of the interacting fermions - quasiparticles in the independent particle eigenfunctions.
Figure 4: The poles of the analytic continuation of Γ(k, ω).
It is further assumed, that the interaction γ(k, t) is continuous at the surface of discontinuity of Γ and thus the self-energy Σ too is continuous. This then assures the “continuity” of the pole of G(k, ω). It can be shown14 that the energy of the quasi-particle and quasihole at the Fermi surface is equal to the chemical potential µ. The residue ak of G can be written as ak = −
1 . 1 + ∂Σ/∂ω |ω=k
(33)
The figure 4 concisely presents the results. The slope of the curve on which the poles lie, at the point of intersection with the real axis (ωk = µ), is zero. This allows the physical interpretation of the residue of the pole as the magnitude of jump experienced at the Fermi surface. This quantifies the extent to which the concept is retained. The discontinuity at the Fermi surface of the distribution of noninteracting fermions is exactly one, whereas that of the interacting particles can be shown to be equal to the residue ak . As the interaction increases the discontinuity tends to zero and the concept of quasi-particles is no longer valid [Noz64]. The distribution of the interacting fermions can be understood as if the next unoccupied free state above the Fermi surface (the lowest excited state) is now occupied through interaction by a quasiparticle and at the same time a quasihole appears below the Fermi surface (Fig. 5). The distribution of the particles15 is calculated as follows †
N( 1 ) (k) = N(− 1 ) (k) = hak, 1 , ak, 1 i = −2i lim 2
14 15
2
2
Z
∞
t→0+ −∞
2
by the Van Hove Theorem [Noz64]. The index 12 refers to the spin of the particles.
14
G(k, ω)eiωt
dω . 2π
(34)
q
k1
k1 + k
U
U
^
7 s k2 k1 + k2 − q
q+k
k1
k2 − k
k2 − k
k1 + k
q
q
k1
w
I
k1 + k
/
-
M
k2 − k k1 − k2 + k + q k2
k2
Figure 6: The vertex pert of a two particle Greens function in the lowest order of perturbation theory. The difference16 between N 1 (k) for | k |< kF and | k |> kF is exactly due to the presence 2 of the pole and is equal to the residue a. N( 1 ) (kF + 0) − N( 1 ) (kF − 0) = a. 2
(35)
2
Thus the jump in the momentum distribution of the interacting particles takes place at the same place as for the non-interacting particles. Since 0 ≤ N( 1 ) (k) ≤ 1 it follows that 2 the size of the jump also lies in this range. The consequences of the Landau assumption will now be precisely formulated17 . With the help of the quasiparticle description of the interacting particles, the single particle concept and the Fermi surface concept are still partially valid, though their lifetimes and the number are restricted. The two particle interaction is described through the vertex part of the two particle Greens function (refer also to equations (147) and (148)). The vertex part has fermion permutation symmetry18 since the particles under consideration are fermions. Intermediate states can occur in the formulation of the vertex part, which corresponds to different number of particles in the system19 in accordance with the appropriate time ordered product of the Ψ20 operators in the Greens functions. The vertex function of a pair of nearly equal wave vectors k1 , k3 and k2 , k4 with a small momentum tranfer21 k can be written as k3 = k1 + k and k4 = k2 − k and the vertex function22 is written as Γαβ,γδ (k1 , k2 ; k1 + k, k2 − k). In the lowest order of the perturbation theory the diagrams in Fig. 6 are obtained. The internal part of the vertex function correspond to the following pairs of propagators. • G(q)G(k1 + k2 − q) 16
Since the rest of the integral may be made small for a proper choice of t The derivation contained in this section is based on the presentation found in the references [AGD63] and [Lan59]. 18 Γαβ,γδ (k1 , k2 ; k3 , k4 ) = −Γβα,γδ (k2 , k1 ; k3 , k4 ) 19 i.e. N or N + 2 or N − 2 20 These Ψ are defined as the field operators in Appendix B (refer to equation (84). 21 k := (| k |, ω) with | k | kF and ω µ 22 In short we can write Γαβ,γδ (k1 , k2 ; k1 + k, k2 − k) = Γαβ,γδ (k1 , k2 ; k). 17
15
• G(q)G(k + q) • G(q)G(k1 − k2 + k − q)
It is observed that the second diagram is divergent for small k. This has a singular contribution to the vertex function whereas the other two contribute a nonsingular term. The vertex part can now be calculated with the ladder summation of all the diagrams which consist partly of singular and partly of nonsingular interaction. This summation can be implicitly done for the second term such as to include ladder diagrams to all orders. Γαβ,γδ (k1 , k2 ; k) =
(1) Γαβ,γδ (k1 , k2 ) − i
Z
d4 q (1) Γ (k1 , q)G(q)G(q + k)Γηβ,ξδ (q, k2 ; k), (36) (2π)4 αξ,γη
where in Γ(1) k has been safely set to 0. The estimate for the propagator has already been calculated as G(p) ≈
a , /¯h − v(| p | −kF ) + iδ sign(| k | −kF )
(37)
where | p ≈ kF and is nearly 0. The term v(| p | −kF ) is obtained from equation (11). The integral in equation (36) may be split to give a contribution of the region close to the point | p |= kF and the region far away from it. The first region contains the singularities and thus the corresponding integral determines the contribution of these singularities. For small k only a small region close this point comes into consideration. Since the arguments of the two Greens function lie close together, the other quanitities in the integrand may be assumed to vary only slowly with respect to q. There is thus a contribution from the poles only if they lie on the opposite sides of the real axis. This implies either • if | q |< kF then | q + k |> kF or • if | q |> kF then | q + k |< kF
Since k is assumed to be small, the product G(q)G(q + k) can be replaced by Aδ(q )δ(| q | −kF ) for the part of the integral with respect to q obtained by going around the poles. The coefficient A can be determined by integrating the estimate of the product G(q)G(q + k) (36) which is given by Z
a q /¯h − v(| q | −kF ) + iδ sign(| q | −kF ) a × k /¯h + ω − v(| k + q | −kF ) + iδ sign(| k + q | −kF ) 2πia2 vk = , v ω − vk
A =
d4 q
16
(38)
where v is in the direction along q with | v |= v. The contribution of the rest of the region is regular and is simply represented by φ(q). Thus the total product G(q)G(q + k) is G(q)G(q + k) =
2πia2 vk δ(q )δ(| q | −kF ) + φ(q). v ω − vk
(39)
Substituting the above expression in the equation (36) and in the singular part carrying out the integration with respect to dqd where d4 q = q 2 dqddo and do is the solid angle in the direction of q we get Z
d4 q (1) Γ (k1 , q)φ(q)Γηβ,ξδ (q, k1 ; k) (2π)4 αξ,γη Z d4 q (1) 2πia2 vk −i Γ (k , q) δ(q )δ(| q | −kF )Γηβ,ξδ (q, k1 ; k) 1 (2π)4 αξ,γη v ω − vk Z d4 q (1) (1) = Γαβ,γδ (k1 , k2 ; k) − i Γ (k1 , q)φ(q)Γηβ,ξδ (q, k1 ; k) (2π)4 αξ,γη Z do (1) vk 2πia2 kF Γ (k1 , q) Γηβ,ξδ (q, k1 ; k). + v (2π)4 αξ,γη ω − vk (1)
Γαβ,γδ (k1 , k2 ; k) = Γαβ,γδ (k1 , k2 ; k) − i
(40)
(41)
The limit k → 0 of the vertex part is not unique. There are two different ways of obtaining the limit as are shown below Γωαβ,γδ (k1 , k2 ) = Γkαβ,γδ (k1 , k2 ) =
lim
Γαβ,γδ (k1 , k2 ; k)
lim
Γαβ,γδ (k1 , k2 ; k).
k→0;k/ω→0
k→0;ω/k→0
(42)
It is possible to relate the two quantities to each other. Taking the first limit of equation (36), we obtain (1)
Γωαβ,γδ (k1 , k2 ) = Γαβ,γδ (k1 , k2 ) − i
Z
d4 q (1) Γ (k1 , k2 )φ(q)Γωξβ,ηδ (k1 , k2 ). (2π)4 αξ,γη
(43)
It is simple to eliminate Γ(1) from equation (36) using the above equation to obtain
Γαβ,γδ (k1 , k2 ; k) =
Γωαβ,γδ (k1 , k2 ) +
a2 kF2 (2π)3 v
Z
Γωαξ,γη (k1 , q)Φ(q)Γξβ,ηδ (q, k2 )
vk do, (44) ω − vk
where iΦ denotes the second term of the product G(q)G(q + k) in equation (39). Now taking the second limit of the above equation we obtain the desired relation between the two quantities. Γkαβ,γδ (k1 , k2 ) = Γωαβ,γδ (k1 , k2 ) +
a2 kF2 (2π)3 v
Z
Γωαξ,γη (k1 , q)Φ(q)Γkξβ,ηδ (q, k2 )do.
17
(45)
Now the vertex part will be shown to be equal to the scattering amplitude f (k, k0 ) which is given in the orignal phenomological paper of Landau [Lan57]. Since for small k and ω, i.e. close to the poles of Γ, it can be shown that Γ Γω and thus Γω may reasonably be neglected in the equation (44). Γ may be represented by the product χαγ (k1 ; k)χ0βδ (k2 ; k) close to its pole. Substituting this expression in the equation (44), and using the definition ναγ (n) = ω−nk vnk χα,γ (n); n = unit vector along q, we get k2 a2 (ω − vnk)ναγ (n) = nk F 2 (2π)
Z
Γωαξ,γη (n, l)νηξ (l)do.
(46)
It can be made plausible that the scattering amplitude of quasi-particles for small momentum transfer k is a2 Γαβ,γδ (k1 , k2 ; k) and the scattering amplitude for forward scattering is a2 Γ(k) [AGD63]. The quantity a2 Γω itself does not have any physical significance but it is related to a2 Γk (45) and | k1 |=| k2 |= kF . This is explained as the collision of the two particles involved when the momentum transfer tend to zero at the Fermi surface, the energy exchange must also be strictly 0 (i.e. ω = 0). The other limit Γk allows a small energy tranfer even when the momentum transfer is strictly equal to zero. This implies that the momentum remains on the energy shell. Thus the process of the first limit is non-physical at the Fermi surface and no scattering processes take place. The k and ω refer to the energy and momentum difference respectively. The effective mass can now be determined which has been defined at the end of the last section. Some relations of the behavior of the system near the pole necessary for the derivation of this result have been calculated and listed below [AGD63]. If the system is in an infinitely weak field δU, which is homogenous in space and slowly varying in time, the first order expansion of the Green’s function of the system in the power series of δU can be used to derive the relation ∂G−1 (p) 1 i = =1− ∂ a 2
Z
Γωαβ,αβ (p, q){G2 (q)}ω
d4 q . (2π)4
(47)
If the particles are assumed to have an infinitesimally small charge δe and the system is assumed to be in a weakly spacially homogeneous magnetic field which is constant in time, in the limit where δe → 0 and k → 0 the following relation may be obtained near the pole. −1
∇p G
v p p i =− =− ∗ =− + a m a m 2
Z
Γkαβ,αβ (p, q)
d4 q q . {G2 (q)}k m (2π)4
(48)
The change in the Greens function when the system moves as a whole with a small and slowly varying velocity δu in the limit where ω → 0 and δu can be calculated and the following relation be derived near the pole, p
∂G−1 p i = =p− ∂ a 2
Z
Γωαβ,αβ (p, q)q{G2 (q)}ω
18
d4 q . (2π)4
(49)
The last important relation can be derived observing the change of the Greens function in an infinitesimally small field δU which is constant in time and weakly homogeneous in space and using the equilibrium condition µ + δU =const. in the limit where k → 0 and δU → 0. ∂G−1 i =1− ∂µ 2
Z
Γkαβ,αβ (p, q){G2 (q)}k
d4 q . (2π)4
(50)
Sustituting the relation between Γk and Γω (45) into the equation (48) and then using formula (39) we obtain the desired result 1 kF 1 = ∗+ m m 2(2π)3
Z
a2 Γωαβ,αβ (ϕ)cos(ϕ)do.
(51)
With some what more effort and using the results derived above it is possible to obtain [AGD63] the relation N 8π kF3 , = V 3 (2π)3
(52)
which is however well known from the noninteracting limit.
19
2.4
Application: Hard Core Potential
The distant goal of the Landau theory is to describe the nuclear matter. Nuclear matter is assumed to consist of 4 (approximately) degenerate nucleons and there exists a short range two-body interaction in the first approximation which contains a repulsive as well as an attractive component (for example, a hard core repulsion modelling the influence of a strongly interacting meson cloud over a very short range and a Yukawa attraction which rapidly diminishes with increasing distance). The first step towards this goal is to solve the hard core problem. This problem is not trivial to solve since the first order of the perturbation theory of an infinite hard core potential is divergent in momentum space. Thus an alternative approach has been developed relating the appropriate terms developed with this method to their corresponding terms in the scattering theory [Gal58]. This formalism is briefly presented here and further developed to calculate the physical quantities of interest. The nuclear matter is only one of the many possible applications of this method. Thus the treatment here will be kept quite general such that the results may be directly applied elsewhere.
2.4.1
Galitskii’s integral equations
This is an attempt to calculate the energy spectrum of a non-ideal Fermi gas in a radially symmetric, short-ranged (i.e. na 1 where n is the density of the particles and a is the range), positive (i.e. repulsive) but strong hard-core potential V (r) [Gal58], [FW71]. This approximation may be justified as follows [Mig77]. The case where na 1 corresponds to the “gas approximation”. The gas approximation refers to the case where the density of particles is such that the interaction between more than two particles at one time may be safely neglected. This approximation may not be directly applied to nuclear matter. In the case where the temperature is low, the number of excitations is small and instead of a gas of interacting particles, a gas of elementary excitations or quasi- particles may be considered and the gas approximation can be applied. For nuclear mater kF a ≈ 1/3 [Eco83]. The interaction between the particles is assumed to be not retarded23 and localized in time: V (x − x0 ) = U(r − r0 )δ(t − t0 ). The excitation of the system is s = (k1 ) − (k2 ) where k1 > kF > k2 , kF being the momentum at the Fermi surface and k1 and k2 are momenta of a particle being excited from k2 to k1 or vice versa. In comparision with the last section, it is noted that this interaction is included in the vertex part Γ (127). The effective self energy of the quasiparticle (quasihole) will now be determined. In the first order of approximation of the perturbation series for one praticle Greens function, it is seen that the self energy Σ(k) corresponds to the two diagrams of the equations (155) and (156). For a singular potential these diagrams diverge and a simple perturba23 i.e., the particles and holes are not allowed to interact. This corresponds to the non-relativistic approximation where c = ∞
20
tion expansion does not lead to a reasonable answer. Some of the higher terms can be obtained by either increasing the lines of interaction always only between the particles the ladder approximation; or by adding lines of interaction between particles and holes; and/or by increasing the number of closed particle-hole loops (156). The ladder approximation is the simplest approximation for two particle Greens function. This corresponds to the higher order Born approximations for forward scattering. For the strong repulsive interaction, it is necessary to include all orders of this series. This is possible by solving an implicit equation for the self energy. Since the coupling constant of the interaction is not small, it is not necessary that the perturbation series converges. However, the parameter f0 kF is assumed to be small, where f0 is the real part of the scattering amplitude24 for small momenta which is related to the range of the potential25 a and kF corresponds to the Fermi wavenumber (refer to Appendix D and [FW71]). Since the two body scattering amplitude in presence of a medium remains finite even though the potential in the momentum representation is infinite to all orders of the perturbation series, it is justified to make an expansion in terms of a parameter which can be made small. The first three terms of the expansion are explicitly given below which do not correspond to the orders of a normal perturbative expansion but may be fully accounted by using only ladder diagrams. In the limit kF → 0 we get f (k, k0 ) → −a and the following expansion is used (refer to Appendix D). E k2 ¯h2 = F [A + BkF a + CkF2 a2 ] N 2m
(53)
where the constants A,B and C are to be determined. The summation of all the ladder diagrams is necessary since the higher order diagrams account for a change in the wave function in case of singular potential. The ladder diagrams are kept and summed over since this corresponds to the picture that the repeated interaction between particles produces a change in the wave function. The interpretation of the equation (53) can now be given. The first term evidently is the energy of the noninteracting system; the second term which is linear in scattering length, is the forward scattering (both direct and exchange) from the particles of the medium; and the third term takes into account the Pauli priciple which reduce the number of free intermediate states available to the fermion and only appears if the fermion is first excited and then deexcited. [FW71] It is assumed that the higher order collisions arising from the particlehole interaction may be safely neglected in the approximation used so far. It is convenient to represent the summation of the self-energy part of the ladder diagrams as a block which is the effective interaction of the vertex part Fig. 7. The diagram representation of the equation (20) is shown in Fig. 8. It is necessary now to obtain a good estimate for the self energy. The self energy can be written as follows [Gal58] Σ(k) = −2i
Z
4 0
0
0
0
Z
d k G0 (k )Γ(k, k ; k, k ) + i
d4 k0 G0 (k0 )Γ(k, k0 ; k0 , k)
(54)
f0 := f (k, k0 ) where | k |=| k0 | a may be also regarded as the s-wave scattering length. For | k |→ 0 ⇒ f0 → −a (refer to the Appendix D.) 24 25
21
p
p
p
Figure 8: The diagrammatic representation of the equation (148) which takes the two-particle interaction into account.
Figure 7: The Feynman diagrams for effective interaction (vertex part) in ladder approximation
Figure 9: The Feynman diagrams for proper self energy in ladder approximation for particles of spin 12 , which corresponds to Fig. 9 in the diagram representation. The effective interaction (vertex part) Γ which is required in the above equation is first calculated and then expressed in terms of the scattering function f (k, k0 ). Since the calculations are rather tricky they are not presented here (refer to the orignal literature by Galitskii [Gal58] or [FW71]). The final result obtained is 4π (4π)2 Z d3 q 0 Γ(k1 , k2 ; k3 , k4 ) ≡ Γ(k, k ; g) = − f (k, k ) + f (k, q)f ∗ (q, k0 ) m m (2π)3 N(q) 1 ×{ + 2 } (55) 2 mE − q + iδN(q) q − k 02 + iδ 0
This equation is written in the center of mass coordinate system where k = k1 − k2 ; k 0 = k3 − k4 and g = (k1 + k2 )/2 = (k3 + k4 )/2. The quanitity N(q) := 1 − n0g+q − n0g−q where n0k := θ(kF − k) implies that N(q) is 1(−1) if both the state g ± q are outside (inside) the Fermi sphere and zero otherwise. In the first approximation, the effective interaction Γ is equal to the scattering amplitude f in the above equation. Since one integration must still be done, the value of f (k, k0 ) for all k, k0 must be known (i.e. values of f off the energy shell26 must also be calculated.) The Dirac equation may be used in order to calculate if | k |=| k0 | and the following result 26
This point still remains unclear and must be studied in more detail.
22
may be obtained by using the generalized optical theorem for the scattering amplitude. 4π (4π)2 Z d3 q 1 ∗ 0 f (k, k0 ) + f (k, q)f (q, k )P m m (2π)3 q 2 − k 02 2 Z 3 (4π) dq N(q) 1 ∗ 0 + f (k, q)f (q, k ){ + } m (2π)3 mE − mq + iδN(q) q 2 − k 02 + iδ Γ(k, k0 ; g) = Re
(56)
The proper self energy for relative coordinates may be written as follows where k˜ := (k − k 0 )/2: Σ(k) = −2i
Z
4 0
0
˜ k; ˜ g) + i d k G0 (k )Γ(k,
Z
˜ k; ˜ g) d4 qG0 (q)Γ(k,
(57)
which may be calculated by substituting the equation (56) into (57) to the second order. The partial wave expansion is used using the long wavelength expansion δ0 ≈ −ka (refer to Appendix D) and we get f (k, k0 ) ≈ −4πa + 4πika2 + O(k2 a3 )
(58)
for | k |=| k0 | → 0. Thus the “exact” Green’s function in this approximation may be calculated and is given by the following equation G−1 (k, ) ≈ 0 − 0k − Σ(2) (k, 0 )
(59)
where 0 := − Σ(1) . Σ(1) and Σ(2) can be calculated now. The first term is trival and only gives a shift in the ground state energy. The second term is however very complicated and may only be calculated in an approximation close to the Fermi surface. The energy level which are the poles of the Greens function (59) have been calculated by Galitskii [FW71]. These results are used in the Chapter 3. This brief treatment is only attempted such that all the concepts used are so documented.
23
2.5
Discussion
The Landau theory is an ingenious simplification of an extremely complex and almost incalculable many body system in an elegant manner such that a relatively good approximation to all orders of perturbation may be obtained. It still remains to be discussed how accurate this method actually is and therefore all the approximations used in theory are listed and discussed here. The Hartree-Fock method fails to properly include the correlations between the particles. Thus, the 2-particle Greens functions are included only partially without the vertex part. It is the vertex part which accounts for the correlations. The ladder approximation is a simple but self consistent manner to sum a subclass of Feynman diagrams to all orders of the perturbation theory. In the ladder diagrams repeated interactions are allowed before the particles leave the region where the interaction can take place. These diagrams also correspond to the Born series in the order of the interaction V (r − r0 ). This method is useful for obtaining a rough estimate of the binding energy for high density, strongly interacting fermion systems. Thus this method is an inprovement to the Hartree-Fock method. The two particle interaction is used here. It is well known that the cluster expansion does not converge in general for strong interaction. Thus considering only this simple case here may be problematic. But in the low density approximation, this truncation is justified. The potential is assumed to be retarded and thus instantaneous. It is thus denoted by horizontal lines in the Feynman diagrams. The higher temperatures where the energy of the particles is relativistic, this approximation no longer holds, since all calculations here are in the nonrelativistic limit. This limit is taken instead of the relativistic one as appropriate for low densities and low temperature. A proper extension to this theory to the description of relativistic phenomena requires the implementation of relativistic propagators and an analogue calculation. However, it must be made compatible to the approximations used here and the restrictions imposed on the density and excitation energy of the system. Some useful preliminary references on the subject are [Eli62], [AP60] and [BC76]. In the microscopic description of the many body systems, Greens functions are normally used instead of the wavefunction. This does not give us a complete description of the system but allows us to concentrate on most essential features of the system for finding the states and marcoscopic properties. The weakly excited states of a system of interacting particles can be described as an aggregrate of elementary excitations - quasi-particles. The excited state of the system may so be described by fewer parameters than is necessary for an exact description. The elementary state is not a stationary state but a packet of stationary states with a narrow energy spread. If the energy of the spread is small compared to the excitation energy then the description of the states by means of elementary excitations is possible. These excitations are analogous to phonons. In contrast to phonons, which are bosons, 24
these quasi-particles arising from fermions also obey the Pauli principle and hence are fermions. The simple quasi-particles are one-particle excitations which may be considered to originate due to a transition of a particle from a state under the Fermi momentum surface to a state above the Fermi surface. This may be interpreted as a particle-hole creation in a background Fermi sea. These quasi-particles have a mass different from the free particle mass. The one particle excitation in a real fermi system is equivalent to the excitation of an ideal gas composed of quasi-particles with a Fermi distribution with respect to energy. Another unsolved problem as yet is that of including the degeneracy of the system explicitly, which hasn’t been attempted due to lack of time but should be relatively straight forward. In principle, since we have now a weakly interacting gas of quasi-particles it should be possible to include the degeneracy in the same manner as is done for the Fermi gas. Thus the quasi-particle method is very interesting due to its conceptual simplicity and as it is applicable to a wide range of phenomena such as electrons in a conductor (jellium model), superfluidity and clusters. Thus it is still an interesting challenge not only to use this method to obtain physical results for specific systems but also to improve this technique further.
25
3
Discussion and Conclusion
The Landau approximation with one application has been presented so far which has been studied using the cited literature. It was the aim of this study to quantitatively determine the errors of this method. Therefore it is necessary to compare it with other methods which are based on nonperturbative calculations to obtain an estimate of deviation from the expected results. However, this further goal has not been accomplished due to the lack of time.
3.1
Suggestions for further study and calculations
Consider a model of a many body system consisting of N nucleons which interact with each other [Men92]. The nucleon interaction is assumed to consist of a repulsive hard core potential27 and an attractive scalar Yukawa potential with a σ boson obtainable in a similar fashion to the Nambu-Jona-Lasino model. This model has been solved in the first order perturbation theory and then using the path-integral formalism nonpertubatively in the first quantization [Men93]. The results are consistent and may be compared to the lattice gauge calculations using first principles, also made by the author. The degeneracy of the nucleons is set to be 40 in this model in order to compare it with the results of the lattice calculations28 , and degeneracy = 4 is used to make a comparison with the extrapolated values to the ones obtained in the nature. This model may perhaps also be calculated using the Landau approximation revised in this thesis. The first step will be to apply the results of the hard core problem obtained by Galitiskii as roughly sketched in Section 2.4. The states lying close to the Fermi surface have been calculated by Galitskii [Gal58]) and the following result is obtained upto the second order of the expansion parameter where k :=| k | with k ≈ kF . (k) =
h2 kF2 h2 kF2 ¯ 2 k2 4 h ¯h2 kF2 4 ¯ 16 ¯ + (kF a)( )+(kF a)2 [ (11−2 ln 2)− (7 ln 2−1)(¯hk−¯ hkF )](60) 2m 3π 2m 15π 2m 15π 2 2m
At k = kF we obtain the result (kF ) =
h2 kF2 ¯ 2 kF2 h ¯h2 kF2 4 ¯ 4 + (kF a)( ) + (kF a)2 [ (11 − 2 ln 2)] 2m 3π 2m 15π 2m
(61)
which is equal to the chemical potential µ29 . Since kF may be written as a function of density, ρ, we obtain a relationship between the energy and ρ. The plot of this graph may be directly compared with a similar graph of the model calculated in the same manner 27
which is included through excluded volume This degeneracy is an effect of using staggerd fermions and is only an artifact of the upto date lattice calculations. 29 This statement is only exact for T = 0 28
26
as in [Men93] above setting the attractive part of the potential equal to 0. The Taylor expansion may be made close to the Fermi surface k = kF +
∂k |k (k − kF ) ∂k F
(62)
and so it follows comparing the equation (60) and (61) that the effective mass m∗ is given by m∗ 8 =1+[ (11 − 2 ln 2)(akF )2 ] m 15π
(63)
The next step would be to calculate the propagator for the complete problem which is described above, i. e. now the attractive Yukawa potential would be included in the Hamiltonian, and a self consistent Dyson equation can be obtained in the momentum space (refer to Appendix C. equation (158)). The equation (59) is the “exact” Green’s function G00 for the hard core potential. Now the exact Green’s function G0 for the complete problem may be solved using equation (159) where Σ∗ (k) may be calculated using equation (160) to the first order. The poles of the Greens function again give the new dispersion relation with the help of which the energy levels of the complete system can be obtained. This is however only a rough sketch of the proposed method. The details such as how to obtain the energy levels far from the Fermi surface (Galitskii equations) the extention to finite temperature and the legitimacy of using an attractive interaction while using the Landau approximation have to be dealt with before attempting the calculations. The problem solved so far, is applicable only for non-degenerate case. It is proposed to include the degeneracy in a consistent manner and then to the compare the results with those presented above. In this manner, some quantative control over magnitude of the error due to the approximation is sought. Even for the non-degenerate case it may be checked whether the two methods converge to the same result. One more control over the results can be achieved by comparing the above results with those determined by independent pair approximation (Brueckner’s Theory) [FW71]. The potential is modelled here by a repulsive hard core and an attractive square well. Instead of Galitski’s integral equations, the Bethe-Salpeter equations are used to obtain the solution of the hard core potential which give the same ground state energies at low density. The effect of the square well is assumed to be small, thus it does not produce a change in the wave function. For the hard core solution the Born approximation is used in the region of the square well. This last approximation, however, is not strictly necessary and the coupled problem could be solved exactly [MS19]. The more realistic physical problem may be solved using the derivation for the nucleonnucleon interaction constructed as follows. Let the potential is constituted of a short range repulsive meson exchange, an intermediate attractive scalar meson exchange and a 27
long range light π-meson exchange. These contituents will not be discussed here [BBN85] but the effective potential for two interacting nucleons may be used in order to obtain the quasi-particle energy spectrum and thus the statistics and relevant quantities.
3.2
Results and Conclusion
A preliminary study was undertaken using the Landau approximation and the Thomas Fermi model for nuclear matter [KWH74]. A simple extension to relativitic energies and densities was undertaken without closer scrutiny. The results obtain were not comparable to those of the model presented in the last section and would therefore not be discussed here. This result was however the motivation for a closer examination if the Landau theory. The main aim of the study was to gather a better understanding of the nuclear manybody problem and the phase transition of a nucleon to nuclear matter at low density and temperature. Eventually, the further development of the methods learnt here to the more complicated problem of finite temperature, higher density and degeneracy was also planned which too has been partially solved. The latter has been omitted here and left as suggestion to further study. The complexity of the problem comples one to use complex and difficult field theorectical methods a few of which are described here. A direct application of these specific methods has also been developed but due to the lack of time, could not be calculated. An ensuing study would deal with these calculation. It was however observed that these methods are very general and may be applied with success to other fields, one of the best examples of which is cluster physics. The discussion 2.5 makes it clear that much better results in this field are to be expected than with other methods since all the problem concerning the strong interactions are absent there. Thus this treatement is a useful introduction for students to the application of this method to the related fields.
28
Appendix A.
Perfect Gas: a statistical mechanical view point.
30
Statistical mechanics relates the thermodynamical functions to the Hamiltonian of the many particle system. In the grand canonical ensemble, where the total energy and the total number of particles of the entire system, the average energy and the average number of particles of a large number of identical subsystems are conserved, the partition function is given by ZG :=
XX
e−β(E(j,N )−µN ) ,
(64)
j
N
that is, the summation overt all states j and for each number of particles N present in the subsystem (from 0 to infinite). Here j is the j-th state for a fixed number of particles N of the subsystem and β is the inverse of temperature T times Boltzmann constant kB . The partition function ZG is related to quantum mechanics with the relation ZG =
XX
ˆ
ˆ
ˆ
ˆ
hjN | e−β(H−µN ) | jN i = T r e−β(H−µN ) .
(65)
j
N
Furthermore, the thermodynamical potential Ω is defined as Ω(T, V, N ) = −kB T ln ZG .
(66)
Any thermodynamical quantities may now be calculated with the thermodynamical potential. For a non-interacting Fermion gas in occupation number representation, the ˆ =H ˆ 0 may be written as partition function for the ground state H ZG = T r e−β(H0 −µN ) = ˆ
ˆ
X
hn1 · · · | e−β(H0 −µN ) | n1 · · ·i. ˆ
ˆ
(67)
n1 ,···
Since these states are the eigenstates of the Hamiltonian and the occupation number operator, the operators may be replaced by their eigenvalues. The exponent is now a c-number with respect to the state vectors. The sum of the expectation values would factorize to the product of the trace of single particle expectation value. This is, however, only possible because there is no interaction between the particles and the total energy of the subsystem is the sum of the energy of individual particles. ZG =
X
X
n1
ni
hn1 | e−β(E1 −µ)n1 | n1 i · · · = Πi
hni | exp−β(Ei −µ)ni | ni i.
30
(68)
The notation used in this section by and large corresponds to the one found in Fetter and Walecka, 1971 [FW71]
29
Since the occupation number of fermions is either 0 or 1 this reduces to ZG = Πi [1 + e−β(Ei −µ) ].
(69)
Now the thermodynamical potential and the average number of particles may be calculated using equations (66) and (69). Ω = −kB T
X
ln (1 + exp−β(Ei −µ) )
(70)
i
and ˆi = hN
X
n0i =
i
X i
1 1+
e−β(Ei −µ)
.
(71)
It is assumed that the ensemble is contained in a large volume with periodic boundary conditions. The sum over single fermion levels can be replaced by an integral over wave number Z
X in continuum limit where L is large gV
7→
(2π)3
i
d3 k,
(72)
where g is the degeneracy in the single particle momentum state. The maximum wave number kF (Fermi surface) for T = 0 is determined by computing the expectation value of the number operator in the ground state G, where σ is the spin index ˆ | Gi = Nave = hG | N in continuum limit
=
X
X
k,σ
k,σ
hG | nk,σ | Gi =
gV X Z gV 3 dk θ(k − k) = · k3 F (2π)3 σ 6π 2 F Nave (6π 2 ) ⇒ kF3 = . gV
θ(kF − k)
(73)
The density of states, that is the number of states of energy between k+dk, may now be written as
D() =
gV gV 2m 4πk2 dk = 2 (2π)3 4π ¯h2
3/2
1/2 d.
(74)
D() may more precisely be called the density of orbitals because it refers to the solutions of single particle problem and not to the states of the N particle system. D() · n() is then the density of occupied orbitals [KK80].
30
Appendix B.
Quasi-Particle Method: Second Quantization
This theory may be restated in the language of second quantization where the observables are expressed in the occupation number representation of quantum theory. The algebra of the wave functions is now transformed to the observables which now obey the quantization rules. This is realized in the following formalism. A system consists of N particles, where the states of the k-th particle are described by the space Lk of square integrable functions on set Mk , and the states of the system are described by square integrable functions of N variables x1 , x2 , · · · , xN where xk ∈ Mk , form a Hilbert space, denoted by HN , where the scalar product is given by Z
(φ1 , φ2 ) :=
φ1 (x1 , x2 , · · · , xN ) · φ2 (x1 , x2 , · · · , xN ) dN x,
(75)
where dN x := dx1 . · · · .dxN is a differential measure on (Mk )N . [Ber66] HN is the direct (tensor) product of N single particle Hilbert spaces H1 , 1 HN = H11 ⊗ H21 ⊗ · · · ⊗ HN ,
(76)
and the states31 | α1 · · · αN ) := | α1 > · · · | αN >, that is, the product of N single-particle states, which form a complete orthonormal basis [BR86]. (α1 · · · αN | α01 · · · α0N ) = < α1 | α01 > · · · < αN | α0N > = δα1 ,α01 · · · δαN ,α0N .
(77)
If the system consists of N identical particles, the sets Mk coincide with each other, and it is unnecessary to consider the whole space HN [Ber66]. N Let the subspace of the system HB ⊂ HN consist of all totally symmetric states and the subspace HFN ⊂ HN consist of all antisymmetric states in HN . Then the particles N lying in HB are called bosons and those lying in HFN are called fermions [Ber66]. The state of the system consisting of an indefinite number of particles are described by vectors of the space32 H which is the direct sum of all HN and the one-dimensional space H0 which corresponds to the vacuum state33 . The subspace HF ⊂ H is now for example defined as HF :=
∞ M
HFN
where HF0 = H0
0
| αi >:= ψαi (·) and αi := (ni , li , mi ) , i.e., the set of quantum numbers describing the i-th particle called Fock space 33 the state in which no particles are present 31
32
31
These completely symmetric or antisymmetric states may be written in terms of creation aα † and annihilation aα operators acting on the vacuum state [BR86]. | α > = a†α | 0 > < α | = < 0 | aα .
(78)
The algebra of these operators is derived from the physical restriction of permutation symmetry imposed upon the wave function. The algebra of fermions thus obtained is given below [BR86] {aα , aα0 } := aα aα0 + aα0 aα {aα , aα0 † } = δα,α0 = {aα0 † , aα } {aα , aα0 } = 0 = {aα † , aα0 † }.
(79) (80)
The state of an isolated fermion in an external field, created by other particles, is deterˆ mined by the Hamiltonian34 H(x). If α and φα are the eigenvalue and the eigenfunction ˆ 1 (x) in the state α, then the corresponrespectively of the single particle Hamiltonian H ding N particle Hamiltonian in the non-interacting case [Dav65] may be calculated as follows. We have ˆ N (x1 , · · · , xN ) = H
N X
ˆ 1 (xi ). H
(81)
i=1
The N particle wave function in the same representation is φ(x1 , · · · , xN ). The operator of the number of particles in the state α is of the form n ˆ α = a†α aα
(82)
which is the operater for the wave function in the occupation number representation. ˆ 1 (x) − α )φα = 0 is the equation for determining the Thus the eigenvalue equation (H single particle states | α >, the total Hamiltonian of the system of independent fermions can be written in the form Z
ˆ := H
ˆ † (x)H ˆ ˆ 1 (x)Ψ(x) Ψ dx.
(83)
ˆ in the occupation number representation, are expressed in terms The field operators Ψ, of the operators aα ˆ Ψ(x, t) =
X
aα φα (x)e−iα t .
(84)
α 34
x := (x, σ), where x is the spatial component and σ is the spin component.
32
The algebra for field operators is derived from the corresponding algebra of the creation and annihilation operators and by using the conditions of orthonormality and completeness of φ’s ˆ ˆ † (x0 )} = δ(x − x0 )δσ,σ0 = {Ψ ˆ † (x), Ψ(x ˆ 0 )} {Ψ(x), Ψ ˆ † (x), Ψ ˆ † (x)}. ˆ ˆ 0 )} = 0 = {Ψ {Ψ(x), Ψ(x
(85)
The free Hamiltonian for the wave equation in the occupation number representation may now be written with equation ( 84) ( 85) as X
ˆ0 = H
α a†α aα
(86)
α
and the number operator as X
ˆ= N
a†α aα .
(87)
α
In general, any observable in the occupation number representation is obtained from the coordinate representation with ˆ = F
Z
X
ˆ † (x)F(x) ˆ ˆ Ψ(x) Ψ dx =
ˆ|β> a†α aβ < α | F
(88)
α,β
where
Z
ˆ|β> = (118) ( Ψn (r)Ψ†n(r0 ) if t > t0 † 0 where T (Ψn (r) · Ψn (r )) := −Ψ†n (r0 )Ψn (r) if t < t0 and | φ > is the actual normalized ground state of the system. It is clear that if t > t0 then the function G may be interpreted as the probability amplitude of the system such that when a particle is added to it in the ground state (r0 , t0 ) it is found later at (r, t) without change of spin σ, again in the ground state. Thus the analogous formulation for G if t < t0 is then that of the probability amplitude of the system where a particle is annihilated at (r, t) in the ground state and the system is found at (r0 , t0 ) without change in the state. Thus G is the propagator of the additional particle or hole. It isolates the information relative to the ground state without being concerned about the exact structure of the ground state [Noz64]. The method of obtaining the solutions to the Schr¨ odinger equation here is by first deriving the equations using appropriate Green’s functions and then using the correct boundary conditions.
38
It is possible to extend the definition of the time-ordering operator such that it operates on an arbitrary number of field operators as follows [AGD63] T (O1 O2 · · · ON ) := (−1)P OP1 (tP1 )OP2 (tP2 ) · · · OPN (tPN ) such that tP1 > · · · > tPN . Before this formalism can be applied to a many particle system the notation of the interaction picture has to be introduced in the second quantization notation. The interaction picture is defined by isolating from the Hamiltonian41 Hint , and then transforming the Schr¨odinger wavefunction into the interaction picture as follows [AGD63] Φ(I) = eiH0 t Φ(S) .
(119)
We obtain on differentiating with respect to time ∂Φ(I) ˆ int Φ(I) , i = H ∂t ˆ int := eiH0 t Hint e−iH0 t . H (120) ˆ in this representation is obtained from the corresponding Schr¨ Every observable F odinger operator using 120., and hence it satisfies the following equation ˆ ∂F ˆ] = i [ H0 , F ∂t
(121)
As the Hamiltonians Hint (t) do not commute for different times, the solution is not simply Z
t
Φ(I) (t) = const. exp{−i
Hint (t0 )dt0 }Φ(I) (t = 0),
but the time-ordered product of Hamiltonians42 and the solution is now written as Z
Φ(I) (t) = S(t, t0 )Φ(I) (t0 )
where S(t, t0 ) := T exp (−i
t
t0
Hint (t0 )dt0 ).
(123)
41
H = H0 + Hint where H0 is the Hamiltonian of the free particle and Hint is the Hamiltonian for the interaction between particles. Rt 42 ˆ int (t0 )Φ(I) (t0 ) and the The integral form of the equation 120 is Φ(I) (t) = Φ(I) (t0 ) + i dt0 H (0) (1) (2) (0) solution for this may be written in the form Φ(I) (t) = Φ(I) + Φ(I) + Φ(I) + · · · where Φ(I) = R t (1) ˆ int (t1 )Φ(I) (t0 ) is the first approximatiΦ(I) (t0 ) is the zeroth approximation, Φ(I) = −i t0 dt1 H R R t (2) ˆ int (t1 ) t1 dt2 H ˆ int (t2 )Φ(I) (t0 ) is the second approximation and Φ(n) = on, Φ(I) = (−i)2 t0 dt1 H (I) t0 R R t ˆ int (t1 ) · · · tn−1 dtn H ˆ int (tn )Φ(I) (t0 ) is the n-th approximation where t > t1 > t2 > · · · > (−i)n t0 dt1 H t0 tn . Now the solution can be written in the compact form [AGD63] using the relation Z t Z tn−1 Z Z t (−i)n t (n) n ˆ ˆ ˆ ˆ int (tn )Φ(I) (t0 ) Φ(I) = (−i) dt1 Hint (t1 ) · · · dtn Hint (tn )Φ(I) (t0 ) = T dt1 Hint (t1 ) · · · dtn H n! t0 t0 t0 t0 Z t
as Φ(I) (t) = const. T exp{−i
t0
ˆ int (t0 )dt0 }Φ(I) (t0 ) H
39
(122)
In the Heisenberg picture the wave functions ΦH are time independent and we get Φ(S) (r, t) = e−iHt Φ(H) (r).
(124)
Thus Φ(I) (t) = Q(t)Φ(H) where Q(t) is an unitary operator. Then Φ(t) = S(t, tα ) P ΦH where P := eiH0 α e−iHα .
(125)
If the interaction is now turned on adiabatically 44 , (i.e. H(t) → H0 for t → ±∞), we obtain 45 Φ(I) (t) = S(t, −∞)ΦH . 43
The representation of observables in the Heisenberg picture F˜ with respect to the observables in the interaction picture F is then F˜ (t) = S −1 (t)F (t)S(t).
(126)
The Heisenberg operators averaged over the ground state may be calculated as follows: ˜1 (t1 ) · · · O ˜N (tn )) | Φ0 > < Φ0H | T (O H 0 −1 =< ΦH | S (t1 )O1 (t1 )S(t1 ) · · · S −1 (tN )ON (tN )S(tN ) | Φ0H > =< Φ0H | S −1 (∞)S −1 (∞, t1 )O1 (t1 )S(t1 , t2 ) · · · ON (tN )S(tN , ∞)S(∞) | Φ0H > =< Φ0H | S −1 (∞)T (O1 · · · ON S(∞)) | Φ0H > (127) with t1 > t2 > · · · > tN , where this last condition is only required for the calculation steps. The equation (127) holds in general, but the permutation symmetry must normally be taken into account. S(∞) Φ0H is the function Φ(I) (∞) obtained from the ground state function Φ(I) (−∞) by adiabatically turning on the interaction between the particles. Since the ground state of the system is assumed to be non-degenerate, and a stationary non-degenerate state cannot make a transition to another state under the action of an infinitely slow perturbation. It may be concluded that only a contribution to the phase factor L can be made. This may be generalized to all non-degenerate states. Thus Φ(I) (∞) = S(∞)Φ0H = eiL Φ0H < Φ0H | Thus hΦ(I) (∞) |=< Φ0H | S −1 (∞) = < Φ0H | S(∞) | Φ0H > 0 0 (127) ˜1 · · · O ˜ N ) | Φ0H >= < ΦH | T (O1 · · · ON S(∞)) | ΦH > =⇒< Φ0H | T (O < Φ0H | S(∞) | Φ0H >
(128)
whereΦ(I) = eiH0 t Φ = S(t, α)P eiHt Φ ⇒ eiH0 t = S(t, α)P eiHt where there is no interaction at t = −∞, which then implies that for α → −∞ we get P → 1 45 using the relation S(t2 , t1 )S(t1 , t0 ) = S(t2 , t0 ) where t2 > t1 > t0 ⇒ S(t2 , t1 ) = S(t2 , −∞)S −1 (t1 , −∞) =: S(t2 )S −1 (t1 ) and it can be shown that S(t2 , t1 ) = S −1 (t1 , t2 ) and define S(t) := S(t, −∞) ⇒ S −1 (t) := S(∞, t) 43
44
40
This relation (128) is only valid for the ground state of the system, since any other energy level if multiple degenerate can in general make a transition to another state as a result of collision between particles. The definition (118) is also the starting point of the physical interpretation of the Green’s function by means of the Lehmann representation. The purpose of using this representation is to obtain the elementary spectrum of system consisting of an arbitrary number of particles. Since some of the features of the single particle Greens functions follow directly from the quantum mechanical principles thus remaining independent of the specific form of the interaction, it is useful to exploit them to obtain the desired spectrum [FW71]. A complete set of Heisenberg states46 is inserted between the (Heisenberg) field operators. These states are eigenfunctions of the Hamiltonian plus 3-momentum (or 4-momentum). X
[< φ | ΨαH (x) | φn >< φn | Ψ†βH (x0 ) | φ > θ(t − t0 )
Gαβ (x, x0 ) = i −< φn | ΨαH (x) | φ > θ(t0 − t)]
En −E )(t−t0 ) h ¯
< φ | ΨαS (r) | φn >< φn | Ψ†βS (r0 ) | φ > θ(t − t0 )
(129)
n
−ei(
En −E )(t−t0 ) h ¯
< φ | Ψ†βS (r0 ) | φn >< φn | ΨαS (r) | φ > θ(t0 − t)]
Since the momentum of the system is taken to be a constant of motion this inserted set of eigenstates may be regarded as a set of eigenfunctions with respect to the momentum operator. Thus from the relations p ˆ
p ˆ
Ψσ (r) = e−i h¯ ·ˆr Ψσ (0)ei h¯ ·ˆr and pˆφ = 0 we obtain the result Gαβ (x, x0 ) = i
X
[e−i(
En −E )(t−t0 ) h ¯
p ˆn ·(r−r 0 )
e−i h¯
< φ | Ψα (0) | φn >< φn | Ψ†β (0) | φ > θ(t − t0 )
n −E n ·(r−r 0 ) i( Enh )(t−t0 ) i pˆh ¯ ¯
−e
e
< φ | Ψ†β (0) | φn >< φn | Ψα (0) | φ > θ(t0 − t)]
(130)
and thus the Green’s function depends only upon the differences r − r0 and t − t0 . This result may be used to calculate the Fourier transform Gα,β (k, ω) of the Green’s function Gαβ (x − x0 ). Z
Gαβ (k, ω) =
d3 (r − r 0 )
Z
0
0
d(t − t0 )e−ik(r−r ) eiω(t−t ) Gαβ (x − x0 )
(131)
It follows with substitution of equation (130) in equation (131) and including the term ±iη in order to ensure the convergence of the integral. Gαβ (k, ω) = V 46
P
n[
ω−(En −E)/¯ h+iη ˆ
+
] ω+(En −E)/¯ h−iη
(132)
In Heisenberg picture the states eiHt/¯h φS (t) = φH are time independent whereas the operators are ˆ ˆ time dependent according to the relation OH (t) = eiHt/¯h OS e−iHt/¯h
41
where V is the volume of the system obtained on integration. The denominators in the above expression in the first sum written in detail is ω−
=ω−
En (N +1)−E(N ) h ¯
En (N +1)−E(N +1) h ¯
−
E(N +1)−E(N ) h ¯
=ω−
n (N +1) h ¯
−
µ h ¯
where n is the excitation energy of the N + 1 particle system and µ is the chemical potential of the system and an analogous term is obtained for the second denominator for an N − 1 particle system. ω+
En (N −1)−E(N ) h ¯
=ω+
n (N −1) h ¯
−
µ h ¯
On substituting these term in the expression for G(k, ω) we obtain the desired Lehmann representation. Gα,β (k, ω) = h ¯V
P
n[
hω−µ−nk (N +1)+iη ¯
+
] hω−µ+n,−k (N −1)−iη ¯
(133)
At this stage a simplification of the matrix structure of Gαβ may be undertaken in case of elementary particles with spin 12 using the Pauli matrices σ and the identity matrix as a basis. Since G must be scalar under spatial rotations, it necessarily takes the form G(k, ω) = aI + bk · σ. and a and b may only be functions of k 2 and ω. If in addition, the Hamiltonian is invariant under reflections then G must have this property too, and b vanishes because k · σ is a pseudoscalar and G has the following simple structure proportional to the unit matrix. Gαβ (k, ω) = δαβ G(| k |, ω).
(134)
It is observed that the function G(k, ω) is a meromorphic function of h ¯ ω , with simple poles at the exact excitation energies of the interacting system corresponding to momentum h ¯ k. Thus the singularities of the Green’s function immediately yield those excited states for which the numerator does not vanish. In an interacting system, the field operator connects the ground state with the numerous excited states of a system of N+1 particles [FW71]. In the infinite volume limit, i.e. the thermodynamic limit, the analytic structure of G(k, ω) is completely altered because the discrete poles merge to form a branch line. This limit is taken by going from the discrete case (133) to the continuous case and through change of variable from dn to dω. The resulting expression is Z
"
#
A(k, ω 0 ) B(k, ω 0 ) G(k, ω) = dω 0 + ω − µ/¯h − ω 0 + iη ω − µ/¯h + ω 0 − iη 0 dn where A(k, ω) = const. V |< n, k | Ψ˜† (0) | Ψ0 >|2 dω dn ˜ and B(k, ω) = const. V |< n, −k | Ψ(0) | Ψ0 >|2 . dω# " Z ∞ A(k, ω 0) B(k, ω 0) R,A 0 and G (k, ω) = dω + ω − µ/¯h − ω 0 ± iη ω − µ/¯h + ω 0 ± iη 0 ∞
42
(135)
(136) (137)
The real and the imaginary part of the Green’s function in this representation would then be given using Dirac’s equation as Re{G(k, ω)} = P (
Im{G(k, ω)} =
Z
∞
"
dω 0
0
A(k, ω 0 ) B(k, ω 0) + ω − µ/¯h − ω 0 ω − µ/¯h + ω 0
#
−πA(k, ω − µ/¯h) for ω > h ¯ −1 µ πB(k, ω − µ/¯h) for ω < h ¯ −1 µ
(138)
If the spacing between adjacent energy level is of the order of ∆, the discrete levels can only be resolved in time scale τ which is long compared to ¯h/∆. Inversely the condition ∆ ≈ h ¯ /τ has to be fulfilled for discrete resolution which is not possible for a macroscopic sample and only a level density may be determined47 [FW71]. The time evolution of the elementary excitation spectrum may now be studied using the fourier transformation of G(k, ω) with respect to t. G(k, t) =
Z
dω G(k, ω)e−iωt 2π
(139)
Since G(k, ω) has a complex analytic structure the integral is split into two parts G(k, t) =
Z ¯h −∞
dω G(k, ω)e−iωt + 2π
Z
∞
µ/¯ h
dω G(k, ω)e−iωt 2π
(140)
These integrals can be evaluated using complex analysis techniques in the case where a simple pole48 exists at ω = k − iγk with k ≈ µ and k > µ and the following expression is obtained G(k, t) =
Z ¯h−i∞ µ/¯ h
dω −iωt A [G (k, ω) − GR (k, ω)] − iae−ik t/¯h e−γk t e 2π
(141)
Since the integral in the above equation can be made arbitarily small with an appropriate < choice of t (| t | h ¯ /(k − µ) and | t | γk ∼1), the remaining expression G(k, t) ≈ −iae−ik t/¯h e−γk t
(142)
can be interpreted as the propagation of an additional particle in an approximate eigenstate (quasiparticle) with the frequency k /¯h, attenuation γk and the residue a of the pole corresponds to the amplitude of the wave packet [FW71]. An estimate of the integral in equation (141) is now made in order to obtain an estimate Since it is assumed that ∆ ¯h/τ , where τ = t − t0 with t0 = 0 If in case there are more than one simple poles, this analysis has to be made for each case separately. When k < µ an analogous analysis may be made with different signs and the case of the propagation of a quasihole is obtained. 47 48
43
of the maximum lifetime of such a quasiparticle. For k close to the Fermi surface GR (k) has the form GR (k) ≈
a . k /¯h − µ/¯h − v(k − kF ) + iγk
(143)
Thus this expression has a pole at k − µ = h ¯ v(k − kF ). The estimate of the integral leads then to the following condition where the above approximation is valid. −γk ¯h2 ae−iµt/¯h −iae−ik t/¯h e−γk t . πt[k − µ]2
(144)
It is also possible to obtain the particle density and the ground state energy of the system directly from the definition of the Green’s function N = hΨ†α (x)Ψα (x)i = −i V
Gαα (x − x0 ) lim 0 r→r t0 → t + 0
(145)
and by solving the above expression for µ(N) and using the formula µ=
∂E0 ∂N
! V
respectively. The definition (118) can now easily be extended to the N particle case as follows: GN (x1 , · · · , xN ; x01 , · · · , x0N ) := −i < T (Ψ(x1 ) · · · Ψ(xN )Ψ† (x0N ) · · · Ψ† (x01 ) > .
(146)
The thermodynamical properties including the density of states in the energy of the system can be related to single particle Green’s function. However, 2-particle interaction requires the use of 2-particle Green’s function. This is clear observing equations (92) and (146). In general, the two particle Green’s function contains three parts: G(x1 , x2 ; x01 , x02 ) = G(x1 , x01 )G(x2 , x02 ) − G(x1 , x02 )G(x2 , x02 ) + δG(x1 , x2 ; x01 , x02 )
(147)
where the first two terms represent the free propagation of two particle and two holes which is thus called the “free” part of G. The last term is called the “bound” part of G as it describes the interaction between the particles. This last term can explicitly be written as follows δG(x1 , x2 ; x01 , x02 ) =
Z Z Z Z
d˜ x1 d˜ x2 d˜ x01 d˜ x02 G(x1 , x ˜1 )G(x2 , x ˜2 )Γ(˜ x1 , x ˜2 ; x ˜01 , x˜02 )G(x01 , x ˜01 )G(x02 , x ˜02 )(148)
44
where the interaction is localized, that is, restricted to a small region in space and time. The free part in comparison has an infinite range. The function Γ describes the effective interaction of two elementary excitations which is called the vertex part or proper vertex and the rest of the integrand is the propagation from r1 , r2 to r10 , r20 of two particles or two holes or a particle-hole pair [Noz64]. This result can also be made clear on considering all feasible time ordering in the twoparticle Green’s function. In the first case where the given time is either t1 , t2 > t01 , t02 or t1 , t2 < t01 , t02 , we obtain the two cases of propagation of particle and hole pairs. If, however, the observed time is t1 , t02 > t2 , t01 or vice versa then the propagation of a particle-hole pair is observed49 .
In general, the time order of an arbitrary number of operators can be calculated using Wick’s theorem (149). It states that the time ordering of an arbitrary number of operators in interaction picture is equal to the sum over the normal order of all possible contraction of the operators. The normal order of the operators is obtained when all the creators are ordered to the left and all the annihilators are ordered to the right. This result can be obtained after appropriate number of permutations of the operators in accordance with the anticommutator rules. N(O1 · · · ON ) := (−1)p OPA1 · · · OPAi OPCi+1 · · · OPCN . The contraction of 2 arbitrary operators is defined as z }| {
O1 O2 := T (O1 O2 ) − N(O1 O2 ), and it follows T (O1 O2 · · · ON ) =
X
N(O1 O2 · · · ON ),
all possible contractions
= N(O1 O2 · · · ON ) +
X
z
}|
{
N(O1 O2 · · · Oi · · · Oj · · · ON ) + · · · .
(149)
i (150) < S(∞) > Z ∞ ∞ X −i (i)n Z ∞ = ··· dt1 · · · dtn < T (Ψ(x)Ψ† (x0 )Hint (t1 ) · · · Hint (tn )) > < S(∞) > n=0 n! −∞ −∞
G(x, x0 ) =
Let Hint be a spin-independent interaction for acting between a pair of identical fermions. Then Hint has the form Hint 49
1Z Z = dxdx0 Ψ† (x)Ψ† (x0 )V (x − x0 )Ψ(x0 )Ψ(x); V (x − x0 ) = U(r − r0 )δ(t − t0 ) 2
refer to the defintion in equation (146) of the two particle Green’s function
45
Wick’s theorem can be applied to calculate the above expression. It is observed that the average of a T product of any number of field operators can be expressed as the sum of the products of free Green’s functions ( here without the spin indices) as follows G(x, x0 ) = G0 (x, x0 ) + δG1 (x, x0 ) + δG2 (x, x0 ) + · · · with
Z
−1 0 d4 x1 d4 x2 V (x1 − x2 )[iG0 (x, x1 )G0 (x2 , x2 )G0 (x1 , x(151) ) 2 < S(∞) > −iG0 (x, x1 )G0 (x1 , x2 )G0 (x2 , x0 ) + iG0 (x, x2 )G0 (x1 , x1 )G0 (x2 , x0 ) −iG0 (x, x2 )G0 (x2 , x1 )G0 (x1 , x0 ) − iG0 (x, x0 )G0 (x1 , x1 )G0 (x2 , x2 ) +iG0 (x, x0 )G0 (x2 , x1 )G0 (x1 , x2 )]
δG1 (x, x0 ) =
This result can be represented systematically using the Feynman diagram technique. This is an attempt to facilitate calculations and to obtain a graphical representation of the complicated integral equations [BR86]. This formalism would be demostrated here on the example of a two body propagator which would later be used in the Landau theory. The first step is to draw all topologically distinct diagrams for n vertices, where n= 0, 1, 2, .... One-to-one correspondence between a diagram and a system of contractions is referred to as the faithful representation. The diagrams of the faithful representation are referred to as the labeled Feynman diagrams. However, since the integrals corresponding to the labeled diagrams are all integrated over the same time interval, they all contribute the same amount to the summation. Thus for each n-th order expansion, there are 2n (internal) vertices to which correspond 2n n! different diagrams, some of which with the same contribution. The corresponding representative diagrams, where diagrams contributing the same amount are omitted is termed the unlabeled Feynman diagram. It is, however, necessary to take into account those diagrams which are topologically invariant under a change of time label, which defines the symmetry factor S. Thus, the total number of topologically distinct diagrams for 2n vertices are n!/S. (This treatement is only meant to present a brief simple introduction to the topic which is much too complex and rich to be comlpetely presented here.) The second step is to determine the contribution of vertices. For a two-body interaction in the second quantization notation of the form in equation 92 X
V (t) =
α,α0 ,β,β 0
ˆ | α0 β 0 > = < αβ | V
Z Z
ˆ | α0 β 0 > a†α a†β aα0 aβ 0 < αβ | V
ˆ φˆ∗α (x)φˆ∗β (x0 )V(x, x0 )φˆα0 (x)φˆβ 0 (x0 ) dx dx0 ,
each vertex has the contribution
α
β
A ,,
K A
J ]V(t) J
ˆ | α0 β 0 > = < αβ | V
β0 α0 Contraction of a pair of operators are represented by oriented lines called propagators. A normal propagator g contributes 46
α, t 0
0
< α, t | g | α , t > =
z
}|
, ,
{
aα (t)a†α0 (t0 )
= , α0 , t0 , and the abnormal propagators gˆ and gˇ contribute α, t z
}|
z
}|
, ,
{
< 0 | gˆ | α, t; α0, t0 >= a† (t)α a†α0 (t0 ) =
,
α0 , t0 ,
α, t , ,
{
< α, t; α0 , t0 | gˇ | 0 >= aα (t)aα0 (t0 ) =
,
α, t ,
to the diagram. The single particle state labels are then summed and integration over time interval, from tinitial to tf inal is undertaken. As the next step one-to-one correspondence between a diagram and a system of contractions is achieved by multiplying the contribution of the diagram by (−1)nL where nL is the number of closed fermion loops. In general, the diagrams may consist of many disconnected parts, where each of the disconnected parts are a connected fermion loop. The expectation value in (151) is equal to the sum of the connected diagrams only and the disconnected diagrams are factorized and cancelled with the denominator. The calculation of the many particle Green’s function is completely analogous to the above formulation. The external lines are replaced by thick lines, which denote the complete Green’s function. They implicitly include the vacuum perturbation and the diagrams where a two particle interaction between the four external legs of the propagator occurs can be represented by a square box. The disconnected parts of the diagrams are already excluded. The first and second order terms of a two particle Green’s function constructed with the help of the prescription thus described are diagrammatically represented below. )@ , ( @, ) ,@ (, @
) ( ) (
) n
(
,
)@
) ) ( ( ) ) ( (
n @,
,@
(,
@
) )@ , ( ( @, ) ) ,@ ( (, @
In the particle-hole case, where the particle may not be regarded as identical to the holes, the following diagrams for the interaction upto second order are obtained @
@
@ @
, ,
, ,
) ( ) (
,
)@
) ) ( ( ) ) ( (
n @,
,@
(,
@
) )@ , ( ( @, ) ) ,@ ( (, @
It is, however, necessary to simplify the calculation of the propagator by first calculating the Fourier transforms with respect to the time coordinate. The advantage of this transformation is that the different cases of time ordering are automatically considered and G(x, x0 ; ω) has a simple form. In the case of uniform and isotropic systems, that is, where the condition Gαβ (x, y) = δαβ G(x − y) holds, the spatial and temporal inva47
riance allows a full Fourier transformation of the Green’s function and the two particle interaction as follows Gαβ (x, y) = (2π)−4
Z
d4 ke−ik·(x−y) Gαβ (k)
(152)
Uλλ0 ,µµ0 (k, k 0 ) = Vλλ0 ,µµ0 (k − k0 )δ(t − t0 ) −4
Z
= (2π)
0
dk 3 e−ix·(k−k ) Vλλ0 ,µµ0 (x − x0 )δ(t − t0 )
The Feynman rules for momentum space may be now derived from the rules defined in the coordinate space. The basic differences are that the energy and momenta at each vertex has to be conserved q 00 Z K ,, A 0 00 q0 A d4 x ei(q−q +q )·x ∼ (2π)4 δ 4 (q − q 0 + q 00 ) =
the free Green’s function corresponds to the factor
(0)
Gαβ (k) = δαβ G(k, ω) = δαβ
θ(| k | −kF ) θ(kF − | k |) + ω − ωk + iη ω − ωk − iη
q
,
(153)
and the two particle interaction corresponds to Uλλ0 ,µµ0 (q) = Vλλ0 ,µµ0 (q).
(154)
The interaction in momentum space would be frequency independent since it is the Fourier transform of a time variable which occurs in the form δ(t − t0 ). The closed loops contribute a factor (−1)F where F is the number of closed loops contained in the diagram. Each closed loop is associated with eiωη G(k, ω) where η → 0+ at the end of the calculation. The topological structure of the diagrams remains identical to the structure in the coordinate space, but the interpretation and the labeling is naturally quite different. Thus the first order of the perturbative expansion gives the terms
(1a)
i G0 (k) ¯h · (2π)4
(1b)
i G0 (k) ¯h · (2π)4
_^
Gαβ (k) =
Gαβ (k) =
Z
Z
d4 k1 − U (0)αβ,µµ G0 (k1 )eiω1 η G0 (k)
(155)
d4 k1 U (k − k1 )αµ,µβ G(0) (k1 )eiω1 η G0 (k)
(156)
Since the interaction in this case is assumed to be spin independent we get Uαα0 ,ββ 0 (q) = U(q)δαα0 δββ 0 . It implies in spin space Uαβ,µµ (q) = 2U(q)δαβ and Uαµ,βµ (q) = U(q)δαβ . The Dyson equation in momentum space is now to be calculated, where Σαβ (k) is the self energy of the system (or the mass operator). The self energy Σ may be split into several parts each of which are irreducible, i.e., the self energy cannot be separated into
48
parts by cutting only a single free Green’s function. The total roper self energy is then the sum of all orders of the proper self energies. Σ∗ = Σ∗0 + Σ∗1 + Σ∗2 + · · ·
(157)
The corresponding Dyson equation for the proper self energy operator would be Gαβ (k) = Gαβ (k) + Gαβ Σ∗ (k)αβ Gαβ (k) (0)
(0)
(158)
This self consistent equation which may be explicitly solved for exact Green’s function as follows Gαβ (k) =
1 [G(0) (k)]−1 − Σ∗ (k)
(159)
where [G(0) (k)]−1 = ω − h ¯ −1 0k The perturbative expansion of the proper self energy corresponding to the the first order terms is Σ
∗ (1)
i (k) = (2π)4
Z
d4 k1 [−2V (0) + V (k − k1 )]G0 (k1 )eiω1 η
(160)
It is observed that the first term of the first order is only a shift in energy whereas the second term of the first order corresponds to the Born approximation for forward scattering (refer to Appendix D). In order to investigate the thermodynamical properties of the system at finite temperatures we require the grand canonical partition function ZG and the thermodynamical potential Ω of the system50 . ZG = e−βΩ = T re−β(H(λ)−µN ) Ω(λ) = −(β)−1 ln T re−β(H(λ)−µN ) ,
(161)
ˆ := Hˆ0 + µN ˆ may be defined to obtain the modified Heisenberg where a new operator K and interaction pictures. H S OK (r, τ ) = eKτ /¯h OK (r, τ )e−Kτ /¯h
(162)
As long as τ is real. Thus an analogous finite temperature Green’s functions theory may be formulated and 50
H(λ) = H0 + λU
49
all the results obtained above for zero temperature can now be extended to finite temperatures for the new Hamiltonian (which is now no longer time dependent). The first derivative of Ω is the expectation value of the potential energy U of the system. ∂Ω(λ) U e−βH(λ) = Tr = hU i ∂λ Z
(163)
50
Appendix D. Scattering Theory A short range two particle interaction may be described as a scattering process between two particles. This theory is especially appropriate to describe the hard core interaction. The main features of this theory for this particular application would be briefly presented here. The Hamiltonian of a particle may be written as H = H0 + V , where H0 stands for the kinetic energy operator and V for the potential energy operator. If | φi is eigenstate of the particle in case the potential tends to zero then the equation H0 | φi = E | φi may be obtained, where E is the eigenvalue of H0 . Similarly, if | Ψi the is the eigenstate of the complete Hamiltonian H, then the equation (H0 + V ) | Ψi = E | Ψi is obtained. The solution of the above equation under the boundary condition the Ψ → φ in case V → 0 would be | Ψi =
1 | Ψi+ | φi E − H0
(164)
1 since the operator E−H is singular, a further appropriate condition is supplied by making 0 the denominator slightly complex.
| Ψ(±) i =
1 V | Ψ(±) i+ | φi E − H0 ± i
(165)
This result is known as the Lippmann-Schwinger equation. The physical interpretation is that the positive or negative solutions are the sum of the incident unperturbed plane wave and an outgonig or incoming spherical wave obtained through scattering. The system which is experimentally easy to prepare is the case of the positive solution where an incident plane wave is scattered and an outgoing spherical wave plus the incident plane wave is obtained. The Lippmann-Schwinger equation is independent of the basis. The coordinate and the momentum representations are easily obtained as follows: hx | Ψ
(±)
i =
Z
d3 x0 hx |
1 | x0 ihx | V | Ψ(±) i + hx | φi E − H0 ± i0
2m Z 3 0 (±) 0 d x G (x , x)hx | V | Ψ(±) i + hx | φi h ¯2 1 hp | Ψ(±) i = hp | V | Ψ(±) i + hp | φi E − (p2 /2m) ± i
⇐⇒ hx | Ψ(±) i =
51
(166) (167)
The Born approximation consists of substituting the plane wave solution for outgoing or incoming spherical waves. A transition operator T may be defined as V | Ψ(+) i = T | φi
(168)
for the purpose of obtaining the higher order Born approximations (also using the Lippmann-Schwinger equations) which become necessary in the presence of more than one scatterer. The implicit equation in the case of denumerably infinite number of scatteres is T | φi = V | φi + V
1 T | φi. E − H0 + i
(169)
It can be shown in the coordinate representation, in the limit where | x | | x0 | and where it is assumed that the potential is local, that hx | Ψ(±) i ' eik·x +
eik·|x| f (k0 , k) |x|
(170)
where the first term corresponds to the incident plane wave and the second term corresponds to the outgoing spherical wave observed asymptotically. The scattering amplitude f (k 0 , k) is the equivalent to f (k0 , k) ≡ −
1 2m 0 (+) hk | V | ψk i. 2 4π ¯h
(171)
dσ The physically measureable quantity, the differential crosssection dΩ , is the ratio of the number of particles scattered into an infinitesimal area dσ of the substended solid angle dΩ per unit time to the number of incident particles crossing the unit area per unit time integrated over the whole solid angle. It is related to the scattering amplitude as follows
dσ =| f (k0 , k) |2 . dΩ
(172)
The optical theorem describes a relation between scattering amplitude and the total crosssection. The physical interpretation is that the shadow cast in the forward scattering is the attenuation of the the intensity of the incident beam so that the scattered particles are removed from in it in an amount proportional to the total crosssection σtot . This theorem is obtained as a direct consequence of the conservation of the probability flux.
σtot =
4π Imf (θ = 0) |k|
(173)
52
The partial wave expansion, i. e. the expansion of a spherical wave in an infinite superposition of plane waves, is the most effective when only a small number of the partial waves contribute which is the case at low energies. Thus if a partial wave expansion of plane waves with respect to spherical wave is made, the scattering process can be understood as the change in coefficients of the expansion of the outgoing spherical wave. In case of a singular potential this method is useful since the wave function of the non-interacting system are no longer a good approximation of the wave function after the scattering process. Through a change in coeffiecients of the expansion, the new wave function is obtained. The scattering amplitude51 remains finite even though the scattering potential is singular. ˜ |) = f (θ) = f (| k
∞ 1 X ˜|) ˜ |)eiδl (i|k (2l + 1)al (| k Pl (cosθ) ˜ | k | l=0
(174)
˜ | the absolute value of the wave propagation vector, where l is the angular momentum, | k ˜ ˜ |) := eiδl (|k|) sinl (|k˜|) and Pl the Legendre polynome under the condition that m = 0, al (| k ˜| |k
˜ |) → 0 for | k ˜ |→ 0. δ0 (| k In the low energy limit only the s wave makes a contribution to the total cross section since only the first term of the expansion is relavent where l = 0. Then the scattering ˜ is related to the scattering length52 a [Joa79]. amplitude f0 (:= f (k)), Thus using the above equations we get f0 → −a
51
dσ → a2 dΩ
˜ |:=| k, k0 | |k
where a := − lim ˜ |k|→0 hard core potential. 52
˜|) tan δ0 (|k ˜| |k
σtot → 4πa2
if
˜ |→ 0 |k
(175)
, which corresponds to the radius of the hard sphere in case of the
53
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55
Acknowledgement I take this oppertunity to express my gratitude to Prof. E. Hilf for giving me this topic and encouraging me at each phase of this work. I also thank Dr. E. Mendel for inspiring and encouraging discussions and Dipl. Phys. G. Nolte for very helpful and constructive suggestions. I would also like to acknowledge Dr. B. Kleihaus, Dr. L. Polley and Dr. P. Rujan for reading and critizing this thesis. I especially thank M. Thiele for encouraging me to study physics and giving me moral and financial support throughout this period of study.
56
Hiermit versichere ich, dass ich diese Arbeit selbstst¨ andig verfasst and keine anderen als die angegebenen Quellen benutzt habe.
Unterschrift
57
