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\title{Chapter 6: The Fermi Liquid}
\author{L.D.\ Landau}
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\maketitle
\tableofcontents
\pagebreak
\Large
\section{introduction: The Electronic Fermi Liquid}
As we have seen, the electronic and lattice degrees of freedom
decouple, to a good approximation, in solids. This is due
to the different time scales involved in these systems.
\beq
\tau_{ion}\sim 1/\om_D\;\gg\;\tau_{electron}\sim\frac{\hbar}{E_F}
\eeq
where $E_F$ is the electronic Fermi energy. The electrons may be thought of
as instantly reacting to the (slow) motion of the lattice, while remaining
essentially in the electronic ground state. Thus, to a good approximation
the electronic and lattice degrees of freedom separate, and the small
electron-lattice (phonon) interaction (responsible for resistivity,
superconductivity etc) may be treated as a perturbation (with $\om_D/E_F$
as an expansion parameter); that is if we are
capable of solving the problem of the remaining purely electronic system.
At first glance the remaining electronic problem would also appear to be
hopeless since the (non-perturbative) electron-electron interactions are
as large as the combined electronic kinetic energy and the potential energy
due to interactions with the static ions (the latter energy, or rather
the corresponding part of the Hamiltonian, composes the solvable portion
of the problem). However, the Pauli principle keeps low-lying orbitals
from being multiply occupied, so
is often justified to ignore the electron-electron interactions, or treat
them as a renormalization of the non-interacting problem (effective mass) etc.
This will be the initial assumption of this chapter, in which we will cover
\begin{itemize}
\item the non-interacting Fermi liquid, and
\item the renormalized Landau Fermi liquid (Pines \& Nozieres).
\end{itemize}
These relatively simple theories resolve some of the most
important puzzles involving metals at the turn of the century. Perhaps
the most intriguing of these is the metallic specific heat. Except in
certain ``heavy fermion'' metals, the electronic contribution to
the specific heat is always orders of magnitude smaller than the
phonon contribution. However, from the classical theorem of equipartition,
if each lattice site contributes just one electron to the conduction
band, one would expect the contributions from these sources to be
similar ($C_{electron}\approx C_{phonon}\approx 3Nrk_B$). This puzzle
is resolved, at the simplest level: that of the non-interacting
Fermi gas.
\section{The Non-Interacting Fermi Gas}
\subsection{Infinite-Square-Well Potential }
We will proceed to treat the electronic degrees of freedom,
ignoring the electron-electron interaction, and even the electron-lattice
interaction. In general, the electronic degrees of freedom are split into
electrons which are bound to their atomic cores with wavefunctions which are
essentially atomic, unaffected by the lattice, and those valence (or near
valence) electrons which react and adapt to their environment. For the
most part, we are only interested in the valence electrons. Their
environment described by the potential due to the ions and the core
electrons-- the core potential. Thus, ignoring the electron-electron
interactions, the electronic Hamiltonian is
\beq
H=\frac{P^2}{2m} + V(\r)\,.
\eeq
As shown in Fig.~\ref{fig:corepot}, the core potential $V(\r)$, like the lattice,
is periodic
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{1d.pdf}}
\caption[]{\em{Schematic core potential (solid line) for a one-dimensional
lattice with lattice constant $a$.}}
\label{fig:corepot}
\end{figure}
For the moment, ignore the core potential, then the electronic wave
functions are plane waves $\psi\sim e^{i\k\cdot\r}$. Now consider the
core potential as a perturbation. The electrons will be strongly effected
by the periodicity of the potential when $\lambda=2\pi/k\sim a$
\footnote{Interestingly, when $\lambda\sim a$, the Bragg condition
$2d\sth\approx a\approx \lambda$ may easily be satisfied, so the electrons,
which may be though of as DeBroglie waves, scatter off of the lattice.
Consequently states for which $\lambda=2\pi/k\sim a$ are often
forbidden. This is the source of gaps in the band structure, to be
discussed in the next chapter.}. However, when $\k$ is small so that
$\lambda\gg a$ (or when $\k$ is large, so $\lambda\ll a$) the structure of
the potential may be neglected, or we can assume $V(\r)=V_0 $ anywhere
within the material. The potential still acts to confine the electrons (and
so maintain charge neutrality), so $V(\r)=\infty $ anywhere outside the
material.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{squarewell.pdf}}
\caption[]{\em{Infinite square-well potential. $V(\r)=V_0$ within
the well, and $V(\r)=\infty$ outside to confine the electrons and
maintain charge neutrality.}}
\label{fig:squarewell}
\end{figure}
Thus we will approximate the potential of a cubic solid with linear
dimension $L$ as an infinite square-well potential.
\beq
V(\r)=\left\{
\begin{array}{ll}
V_0 & 00$ and may be excluded.
Solutions with $n_i=0$ cannot be normalized and are excluded (they correspond
to no electron in the state).
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{1stquad.pdf}}
\caption[]{\em{Allowed k-states for an electron confined by a infinite-square
potential. Each state has a volume of $\lep\pi/L\rip^3$ in k-space.}}
\label{fig:1stquad}
\end{figure}
The eigenenergies of the wavefunctions
are
\beq
-\frac{\hbar^2\lap}{2m}\psi =\frac{\hbar^2}{2m}\sum_ik_i^2
=\frac{\hbar^2\pi^2}{2mL^2} \lep n_x^2+n_y^2+n_z^2\rip
\eeq
and as a result of these restrictions, states in k-space are confined to the
first quadrant (c.f.\ Fig.~\ref{fig:1stquad}).
Each state has a volume $\lep\pi/L\rip^3$ of k-space. Thus as $L\to\infty$,
the number of states with energies $E(\k)\frac12\AA$!
\subsection{The Mott Transition}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\textwidth,clip=true]{screenpot.pdf}}
\caption[]{\em{Screened defect potentials. As the screening length increases,
states that were free, become bound.}}
\label{fig:screenpot}
\end{figure}
Now consider an electron bound to an ion in Cu or some other metal. As shown
in Fig.~\ref{fig:screenpot} the screening length decreases, and bound states
rise up in energy.
In a weak metal (i.e., something like YBCO), in which
the valence state is barely free, a reduction in the number of carriers
(electrons) will increase the screening length, since
\beq
r_{TF}\sim n^{-1/6}\,.
\eeq
This will extend the range of the potential, causing it to trap or bind
more states--making the one free valance state bound.
Now imagine that instead of a single defect, we have a concentrated system
of such ions, and suppose that we decrease the density of carriers
(i.e., in Si-based semiconductors, this is done by doping certain
compensating dopants, or even by modulating the pressure). This will in
turn, increase the screening length, causing some states that were free
to become bound, causing an abrupt transition from a metal to an insulator,
and is believed to explain the MI transition in some transition-metal oxides,
glasses, amorphous semiconductors, etc.This metal-insulator transition was
first proposed by N.\ Mott, and is called the Mott transition, more significantly
Mott proposed a criterion based on the relevant electronic density for
when this transition should occur.
In Mott's criterion, a metal-insulator transition occurs when this potential,
in this case coming from the addition of an ionic impurity, can just bind an
electronic state. If the state is bound, the impurity band is localized.
If the state is not bound, then the impurity band is extended. The critical
value of $\lambda=\lambda_c$ may be determined numerically\cite{Yukawa} with
$\lambda_c/a_0 = 1.19$, which yields the Mott criterion of
\beq
2.8 a_0 \approx n_c^{-1/3} \, ,
\eeq
where $a_0$ is the Bohr radius. Despite the fact
that electronic interactions are only incorporated in the extremely weak coupling
limit, Thomas-Fermi Screening, Mott's criterion even works for moderately and
strongly interacting systems\cite{mottworkswell}.
\pagebreak
\subsection{Fermi liquids}
The purpose of these next several lectures is to introduce you to
the theory of the Fermi liquid, which is, in its simplest form, a
collection of Fermions in a box plus interactions.
In reality , the only physical analog is a gas of $^3$He, which due
its nuclear spin (the nucleus has two protons, one neutron), obeys
Fermi statistics for sufficiently low energies or temperatures. In addition,
simple metals, from the first or second column of the periodic table,
for which we may approximate the ionic potential
\beq
V(\R)=V_0
\eeq
are a close approximant to Fermi liquids.
Moreover, Fermi Liquid theory only describes the "gaseous" phase of these
quantum fermion systems. For example, $^3$He also has a superfluid
(triplet), and at least in $^4$He-$^3$He mixtures,
a solid phase exists which is not described by Fermi Liquid Theory. One
should note; however, that the Fermi liquid theory state does serve as the starting
point for the theories of superconductivity and super fluidity.
One may construct Fermi liquid theory either starting from a many-body
diagrammatic or phenomenological viewpoint. We, as Landau, will
choose the latter. Fermi liquid theory has 3 basic tenants:
\begin{enumerate}
\item momentum and spin remain good quantum numbers to describe
the (quasi) particles.
\item the interacting system may be obtained by adiabatically turning
on a particle-particle interaction over some time $t$.
\item the resulting excitations may be described as quasi-particles
with lifetimes $\gg t$.
\end{enumerate}
\subsection{Quasi-particles}
The last assumption involves a new concept, that of the
quasiparticles which requires some explanation.
\subsubsection{Particles and Holes}
Particles and Holes are excitations of the non-interacting system at
zero temperature. Consider a system of $N$ free Fermions each of mass $m$ in
a volume $V$. The eigenstates are the anti-symmetrized combinations
(Slater determinants) of $N$ different single particle states.
\beq
\psi_\p(\r)=\frac{1}{\sqrt{V}}e^{i\p\cdot\r/\hbar}
\eeq
The occupation of each of these states is given by $n_\p=\theta(p-p_F)$
where $p_F$ is the radius of the Fermi sphere. The energy of the system
is
\beq
E=\sum_\p n_\p \frac{p^2}{2m}
\eeq
and $p_F$ is given by
\beq
\frac{N}{V}=\frac1{3\pi^2}\lep\frac{p_F}{\hbar}\rip^3
\eeq
Now lets add a particle to the lowest available state $p=p_F$ then,
for $T=0$,
\beq
\mu=E_0(N+1)-E_0(N)=\pde{E_0}{N}=\frac{p_F^2}{2m}\,.
\eeq
If we now excite the system, we will promote a certain number of
particles across the Fermi surface $S_F$ yielding particles above and an
equal number of vacancies or holes below the Fermi surface. These are our
elementary excitations, and they are quantified by $\delta n_p=n_p-n_p^0$
\beq
\delta n_p=
\left\{
\begin{array}{ll}
\delta_{p,p'} \mbox{ for a particle $p'>p_F$}\\
-\delta_{p,p'} \mbox{ for a hole $p'< p_F$}
\end{array}
\right.\,.
\eeq
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{ph_excitation.pdf}}
\caption[]{\em{Particle and hole excitations of the Fermi gas.}}
\label{fig:ph_excitation}
\end{figure}
If we consider excitations created by thermal fluctuations, then
$\delta n_p\sim 1$ only for excitations of energy within $k_BT$ of
$E_F$. The energy of the non-interacting system is
completely characterized as a functional of the occupation
\beq
E-E_0=\sum_\p\frac{p^2}{2m} (n_p-n_p^0)=\sum_\p\frac{p^2}{2m}\delta n_p\,.
\eeq
\begin{figure}[htb]
\centerline{\includegraphics[width=0.49\textwidth,clip=true]{ph_fe.pdf}}
\caption[]{\em{Since $\mu=p_F^2/2m$, the free energy of a particle or a hole
is $\delta F = \left| p^2/2m-\mu\right| >0$, so the system is stable to
these excitations.}}
\label{fig:ph_fe}
\end{figure}
Now lets take our system and place it in contact with a particle
bath. Then the appropriate potential is the free energy, which
for $T=0$, is $F=E- \mu N$, and
\beq
F-F_0=\sum_p\lep\frac{p^2}{2m}-\mu\rip\delta n_p\,.
\eeq
The free energy of a particle, with momentum $\p$ and
$\delta n_{\p'}=\delta_{\p,\p'}$ is $\frac{p^2}{2m}-\mu$ and it corresponds
to an excitation outside $S_F$. The free energy of a hole
$\delta n_{\p'}=-\delta_{\p,\p'}$ is $\mu-\frac{p^2}{2m}$, which corresponds
to an excitation within $S_F$. However, since $\mu=p_F^2/2m$, the free energy
of either at $p=p_F$ is zero, hence the free energy of an excitation is
\beq
\left| p^2/2m-\mu\right|\,,
\eeq
which is always positive; ie., the system is stable to excitations.
\subsubsection{Quasiparticles and Quasiholes at $T=0$}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.49\textwidth,clip=true]{fl_sp.pdf}}
\caption[]{\em{Model for a fermi liquid: a set of interacting
particles an average distance $a$ apart bound within an infinite
square-well potential.}}
\label{fig:fl_sp}
\end{figure}
Now let's consider a system with interacting particles an average
distance $a$ apart, so that the characteristic energy of interaction is
$\frac{e^2}{a}e^{-a/r_{TF}}$. We will imagine that this system evolves slowly
from an ideal or noninteracting system in time $t$ (i.e., the interaction
$U\approx \frac{e^2}{a} e^{-a/r_{TF}}$ is turned on slowly, so that the
non-interacting system evolves while remaining in the ground state into
an interacting system in time $t$).
If the eigenstate of the ideal system is characterized
by $n_p^0$, then the interacting system eigenstate will evolve
quasistatistically from $n_p^0$ to $n_p$. In fact if the system is
isotropic and remains in its ground state, then $n_p^0 = n_p$. However,
clearly in some situations (superconductivity, magnetism) we
will neglect some eigenstates of the interacting system in this way.
Now let's add a particle of momentum $\p$ to the non-interacting ideal
system, and slowly turn on the interaction.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\textwidth,clip=true]{qpevolve.pdf}}
\caption[]{\em{We add a particle with momentum $\p$ to our noninteracting
($U=0$) Fermi liquid at time $t=0$, and slowly increase the interaction
to its full value $U$ at time $t$. As the particle and system evolve, the
particle becomes dressed by interactions with the system (shown as a shaded
ellipse) which changes the effective mass but not the momentum of this
single-particle excitation (now called a quasi-particle).}}
\label{fig:qpevolve}
\end{figure}
As $U$ is switched on, we slowly begin to perturb the particles
close to the additional particle, so the particle becomes
dressed by these interactions. However since momentum is conserved, we
have created an excitation (particle and its cloud) of momentum $\p$.
We call this particle and cloud a quasiparticle. In the
same way, if we had introduced a hole of momentum $\p$ below the
Fermi surface, and slowly turned on the interaction, we would have produced
a quasihole.
Note that this adiabatic switching on procedure will have
difficulties if the lifetime of the quasi-particle $\tau p_F$) interacts with one of the particles
below the Fermi surface with momentum $\p_2$. As a result, two new
particles appear above the Fermi surface (all other states are full)
with momenta $\p_3$ and $\p_4$..}}
\label{fig:billiards}
\end{figure}
To estimate this lifetime consider the following argument
from AGD: A particle with momentum $\p_1$ above the Fermi
surface ($p_1 > p_F$) interacts with one of the particles
below the Fermi surface with momentum $\p_2$. As a result, two new
particles appear above the Fermi surface (all other states are full)
with momenta $\p_3$ and $\p_4$. This may also be interpreted as
a particle of momentum $\p_1$ decaying into particles with momenta
$\p_3$ and $\p_4$ and a hole with momentum $\p_2$. By Fermi's
golden rule, the total probability of such a process if proportional to
\beq
\frac{1}{\tau}\propto \int\delta\lep\vep_1+\vep_2-\vep_3-\vep_4\rip d^3p_2d^3p_3
\eeq
where $\vep_1=\frac{p_1^2}{2m}-E_F$, and the integral is subject to the
constraints of energy and momentum conservation and that
\beq
p_2 p_F\,,\;\;\;p_4=\left|\p_1+\p_2-\p_3\right|>p_F
\eeq
It must be that $\vep_1+\vep_2=\vep_3+\vep_4>0$ since both particles
3 and 4 must be above the Fermi surface. However, since
$\vep_2<0$, if $\vep_1$ is small, then $|\vep_2|\alt\vep_1$ is also small,
so only of order $\vep_1/E_F$ states may scatter with the state
$\k_1$, conserve energy, and obey the Pauli principle. Thus, restricting
$\vep_2$ to a narrow shell of width $\vep_1/E_F$ near the Fermi surface,
and reducing the scattering probability $1/\tau$ by the same factor.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\textwidth,clip=true]{quasilife.pdf}}
\caption[]{\em{A quasiparticle of momentum $\p_1$ decays via a particle-hole
excitation into a quasiparticle of momentum $\p_4$. This may also be
interpreted as a particle of momentum $\p_1$ decaying into particles with
momenta $\p_3$ and $\p_4$ and a hole with momentum $\p_2$. Energy
conservation requires $|\vep_2|\alt\vep_1$. Thus, restricting $\vep_2$ to a
narrow shell of width $\vep_1/E_F$ near the Fermi surface. Momentum
conservation $\k_1-\k_3=\k_4-\k_2$ further restricts the available
states by a factor of about $\vep_1/E_F$. Thus the lifetime of a
quasiparticle is proportional to $\lep\frac{\vep_1}{E_F}\rip^{-2}$.}}
\label{fig:quasilife}
\end{figure}
Now consider the constraints placed on states $\k_3$ and
$\k_4$ by momentum conservation
\beq
\k_1-\k_3=\k_4-\k_2\,.
\eeq
Since $\vep_1$ and $\vep_2$ are confined to a narrow shell around the
Fermi surface, so too are $\vep_3$ and $\vep_4$. This can be seen
in Fig.~\ref{fig:quasilife}, where the requirement that $\k_1-\k_3=\k_4-\k_2$
limits the allowed states for particles 3 and 4. If we take $\k_1$
fixed, then the allowed states for 2 and 3 are obtained by rotating
the vectors $\k_1-\k_3=\k_4-\k_2$; however, this rotation is severely limited
by the fact that particle 3 must remain above, and particle 2 below, the
Fermi surface. This restriction on the final states further reduces the
scattering probability by a factor of $\vep_1/E_F$.
Thus, the scattering rate $1/\tau$ is proportional to
$\lep\frac{\vep_1}{E_F}\rip^2$ so that excitations of sufficiently small
energy will always be sufficiently long lived to satisfy the constraints of
reversibility. Finally, the fact that the quasiparticle only interacts with
a small number of other particles due to Thomas-Fermi screening (i.e., those
within a distance $\approx r_{TF}$), also significantly reduces the scattering
rate.
\subsection{Energy of Quasiparticles.}
As in the non-interacting system, excitations will be
quantified by the deviation of the occupation from the ground
state occupation $n_\p^0$
\beq
\delta n_\p=n_\p-n_\p^0\,.
\eeq
At low temperatures $\delta n_\p\sim 1$ only for $p\approx p_F$ where the
particles are sufficiently long lived that $\tau \gg t$.
It is important to emphasize that only $\delta n_\p$ not $n_\p^0$
or $n_\p$, will be physically relevant. This is important since it does
not make much sense to talk about quasiparticle states, described by $n_\p$,
far from the Fermi surface since they are not stable and cannot be excited
by the perturbations we are considering (i.e., thermal, or in a transport
experiment).
For the ideal system
\beq
E-E_0=\sum_\p\frac{p^2}{2m} \delta n_\p\,.
\eeq
For the interacting system $E[n_\p]$ becomes much more complicated. If however
$\delta n_\p$ is small (so that the system is close to its ground state)
then we may expand:
\beq
E[n_\p]=E_o +\sum_\p \ep_\p \delta n_\p +{\cal{O}}(\delta n_\p^2)\,,
\eeq
where $\ep_\p=\delta E/\delta n_\p$. Note that $\ep_\p$ is intensive
(ie.\ it is independent of the system volume). If
$\delta n_\p=\delta_{\p,\p'}$, then $E\approx E_0+\ep_{\p'}$; i.e., the
energy of the quasiparticle of momentum $\p'$ is $\ep_{\p'}$.
In practice we will only need $\ep_p$ near the Fermi surface
where $\delta n_\p$ is finite. So we may approximate
\beq
\ep_\p\approx \mu+(\p-\p_F)\cdot\grad_p\left.\ep_\p\right|_{p_F}
\eeq
where $\grad_p\ep_\p=v_\p$, the group velocity of the quasiparticle.
The ground state of the $N+1$ particle system is obtained by adding
a particle with $\ep_\p=\ep_F=\mu=\pde{E_0}{N}$ (at zero temperature);
which defines the chemical potential $\mu$. We make learn more
about $\ep_\p$ by employing the symmetries of our system.
If we explicitly display the spin-dependence,
\beqa
\ep_{\p,\sigma}&=&\ep_{-\p,-\sigma}\;\;\mbox{under time-reversal}\\
\ep_{\p,\sigma}&=&\ep_{-\p,\sigma}\;\;\mbox{under BZ reflection}
\eeqa
So $\ep_{\p,\sigma}=\ep_{-\p,\sigma}=\ep_{\p,-\sigma}$; i.e., in the
absence of an external magnetic field, $\ep_{\p,\sigma}$ does not depend
upon $\sigma$ if. Furthermore, for an isotropic system $\ep_\p$
depends only upon the magnitude of $\p$, $|\p|$, so $\p$ and
$ v_\p=\grad \ep_\p(|\p|)=\frac{\p}{|p|} \der{\ep_\p(|\p|)}{|\p|}$
are parallel. Let us define $m^*$ as the constant of proportionality
at the fermi surface
\beq
v_{p_F}=p_F/m^*
\eeq
Using $m^*$ it is useful to define the density of states at the
fermi surface. Recall, that in the non-interacting system,
\beq
D(E_F)=\frac1{2\pi^2}\lep\frac{2m}{\hbar^2}\rip^{3/2} E_F^{1/2}
=
\frac{mp_F}{\pi\hbar^3}
\eeq
where $p=\hbar k$, and $E=p^2/2m$.
Thus, for the interacting system at the Fermi surface
\beq
D_{interacting}(E_F)=\frac{m^*p_F}{\pi\hbar^3}\,,
\eeq
where the $m^*$ (generally $>m$, but not always) accounts for the
fact that the quasiparticle may be viewed as a dressed particle, and must
``drag'' this dressing along with it. I.e., the effective mass
to some extent accounts for the interaction between the particles.
\section{Interactions between Particles: Landau Fermi Liquid}
\subsection{The free energy, and interparticle interactions}
The thermodynamics of the system depends upon the free energy $F$,
which at zero temperature is
\beq
F-F_0=E-E_0-\mu(N-N_0)\,.
\eeq
Since our quasiparticles are formed by
adiabatically switching on the interaction in the $N+1$ particle
ideal system, adding one quasiparticle to the system adds one
real particle. Thus,
\beq
N-N_0=\sum_\p\delta n_\p\,,
\eeq
and since
\beq
E-E_0\approx \sum_\p \ep_\p \delta n_\p\,,
\eeq
we get
\beq
F-F_0\approx\sum_\p \lep\ep_\p-\mu\rip\delta n_\p\,.
\eeq
As shown in Fig.~\ref{fig:fs_distort}, we will be interested in excitations of the
system which distort the Fermi surface by an amount proportional to $\delta$.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.45\textwidth,clip=true]{fs_distort.pdf}}
\caption[]{\em{We consider small distortions of the fermi surface,
proportional to $\delta$, so that $\frac1{N}\sum_\p|\delta n_\p |\ll 1$.}}
\label{fig:fs_distort}
\end{figure}
For our theory/expansion to remain valid, we must have
\beq
\frac1{N}\sum_\p|\delta n_\p |\ll 1\,.
\eeq
Where $\delta n_\p\neq 0$, $\ep_\p-\mu$ will also be of order $\delta$.
Thus,
\beq
\sum_\p \lep\ep_\p-\mu\rip\delta n_\p\sim \CO(\delta^2)\,,
\eeq
so, to be consistent we must add the next term in the Taylor series
expansion of the energy to the expression for the free energy.
\beq
F-F_0=\sum_\p \lep\ep_\p-\mu\rip\delta n_\p
+\frac12\sum_{\p,\p'} f_{\p,\p'} \delta n_\p\delta n_{\p'}
+\CO(\delta^3)
\label{eq:deltafree}
\eeq
where
\beq
f_{\p,\p'} =\frac{\delta E}{\delta n_\p\delta n_{\p'}}
\eeq
The term, proportional to $f_{\p,\p'}$, was added (to the Sommerfeld theory)
by L.D.\ Landau. Since each sum over $\p$ is proportional to the
volume $V$, as is $F$, it must be that $f_{\p,\p'}\sim 1/V$.
However, it is also clear that $f_{\p,\p'}$ is an interaction between
quasiparticles, each of which is spread out over the whole volume $V$, so
the probability that they will interact is $\sim r_{TF}^3/V$, thus
\beq
f_{\p,\p'}\sim r_{TF}^3/V^2
\eeq
In general, since $\delta n_\p$ is only of order one near the
Fermi surface, we will only care about $f_{\p,\p'}$ on the Fermi surface
(assuming that it is continuous and changes slowly as we cross the Fermi
surface.
\beq
\mbox{Interested only in $\left. f_{\p,\p'}\right|_{\ep_\p=\ep_{\p'}=\mu}$ !}
\eeq
Thus, $f_{\p,\p'}$ only depends upon the angle between $\p$ and $\p'$.
We can also reduce the spin dependence of $f_{\p,\p'}$ to a
symmetric and anti symmetric part. First in the absence of an external
field, the system should be invariant under time-reversal, so
\beq
f_{\p\sigma,\p'\sigma'}=f_{-\p-\sigma,-\p'-\sigma'}\,,
\eeq
and, in a system with reflection symmetry
\beq
f_{\p\sigma,\p'\sigma'}=f_{-\p\sigma,-\p'\sigma'}\,.
\eeq
Then
\beq
f_{\p\sigma,\p'\sigma'}=f_{\p-\sigma,\p'-\sigma'}\,.
\eeq
It must be then that $f$ depends only upon the relative orientations of the
spins $\sigma$ and $\sigma'$, so there are only two independent components
$f_{\p\uparrow,\p'\uparrow}$ and $f_{\p\uparrow,\p'\downarrow}$. We can
split these into symmetric and antisymmetric parts.
\beq
f_{\p,\p'}^a=\frac12 \lep f_{\p\uparrow,\p'\uparrow}- f_{\p\uparrow,\p'\downarrow}\rip
\;\;\;\;\;\;
f_{\p,\p'}^s=\frac12 \lep f_{\p\uparrow,\p'\uparrow}+ f_{\p\uparrow,\p'\downarrow}\rip \,.
\eeq
$f_{\p,\p'}^a$ may be interpreted as an exchange interaction, or
\beq
f_{\p\sigma,\p'\sigma'}=
f_{\p,\p'}^s + \bsi\cdot\bsi' f_{\p,\p'}^a
\eeq
where $\bsi$ and $\bsi'$ are the Pauli matrices for the spins.
Our ideal system is isotropic in momentum. Thus, $f_{\p,\p'}^a$ and
$f_{\p,\p'}^s$ will only depend upon the angle $\theta$ between $\p$
and $\p'$, and so we may expand either $f_{\p,\p'}^a$ and $f_{\p,\p'}^s$
\beq
f_{\p,\p'}^\alpha =\sum_{l=0}^\infty f_l^\alpha P_l(\cth)\,.
\eeq
Conventionally these f parameters are expressed in terms of reduced units.
\beq
D(E_F) f_l^\alpha=\frac{Vm^*p_F}{\pi^2\hbar^3} f_l^\alpha =F_l^\alpha\,.
\eeq
\subsection{Local Energy of a Quasiparticle}
\label{sec:local_energy}
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{qp_inhom.pdf}}
\caption[]{\em{The addition of another particle to a homogeneous system
will yeilds in forces on the quasiparticle which tend to restore equilibrium.}}
\label{fig:qp_inhom}
\end{figure}
Now consider an interacting system with a certain distribution of excited
quasiparticles $\delta n_{\p'}$. To this, add another quasiparticle of
momentum $\p$ ($\delta n_\p'\to \delta n_\p'+ \delta_{\p,\p'}$). From
Eq.~\ref{eq:deltafree}
the free energy of the additional quasiparticle is
\beq
\epti_\p-\mu=\ep_\p-\mu +\sum_{\p'}f_{\p',\p}\delta n_{\p'}\,,
\label{eq:addfree}
\eeq
(recall that $f_{\p,\p'}=f_{\p',\p}$). Both terms here are $\CO(\delta)$.
The second term describes the free energy of a quasiparticle due to the other
quasiparticles in the system (some sort of Hartree-like term).
The term $\epti_\p$ plays the part of the {\em{local energy of a
quasiparticle}}. For example, the gradient of $\epti_\p$ is the force the system exerts on the additional quasiparticle. When the quasiparticle is
added to the system, the system is inhomogeneous so
that $\delta n_{\p'}=\delta n_{\p'}(\r)$. The system will react to this
inhomogeneity by minimizing its free energy so that $\grad_rF=0$.
However, only the additional free energy due the added particle
(Eq.~\ref{eq:addfree}) is inhomogeneous, and has a non-zero gradient. Thus,
the system will exert a force
\beq
-\grad_r\epti=-\grad_r\sum_{\p'} f_{\p',\p} \delta n_{\p'}(\r)
\eeq
on the added quasiparticle resulting from interactions with other
quasiparticles.
\subsubsection{Equilibrium Distribution of Quasiparticles at Finite $T$}
$\epti_\p$ also plays an important role in the finite-temperature
properties of the system. If we write
\beq
E-E_0=\sum_\p\ep_\p \delta n_\p +
\frac12\sum_{\p,\p'}f_{\p',\p} \delta n_{\p'}\delta n_{\p}
\eeq
Now suppose that $\sum_\p\left|\langle \delta n_{\p}\rangle\right| \ll N$,
as needed for the expansion above to be valid, so that
\beq
\delta n_{\p}= \langle \delta n_{\p}\rangle
+\lep \delta n_{\p}-\langle \delta n_{\p}\rangle \rip
\eeq
where the first term is $\CO(\delta)$, and the second $\CO(\delta^2)$.
Thus,
\beq
\delta n_{\p}\delta n_{\p'}\approx
-\langle \delta n_{\p}\rangle\langle \delta n_{\p'}\rangle
+\langle \delta n_{\p}\rangle\delta n_{\p'}
+ \langle \delta n_{\p'}\rangle\delta n_{\p}
\eeq
We may use this to rewrite the energy of our interacting system
\beqa
E-E_0&\approx&
\sum_\p\ep_\p\delta n_\p
-\frac12\sum_{\p,\p'}f_{\p',\p}
\langle \delta n_{\p}\rangle\langle \delta n_{\p'}\rangle
+ \sum_{\p,\p'}f_{\p',\p}\langle \delta n_{\p}\rangle\delta n_{\p'}\nonumber\\
&\approx&
\sum_\p \lep \ep_\p +\sum_{\p'}f_{\p',\p}\langle \delta n_{\p'}\rangle
\rip\delta n_\p
-\frac12\sum_{\p,\p'}f_{\p',\p}
\langle \delta n_{\p}\rangle\langle \delta n_{\p'}\rangle\nonumber\\
&\approx&
\sum_\p \langle\epti_\p\rangle \delta n_\p
-\frac12\sum_{\p,\p'}f_{\p',\p}
\langle \delta n_{\p}\rangle\langle \delta n_{\p'}\rangle+\CO(\delta^4)
\eeqa
At this point, we may repeat the arguments made earlier to
determine the fermion occupation probability for non-interacting
Fermions (the constant factor on the right hand-side has no effect).
We will obtain
\beq
n_\p(T,\mu)=\frac1{1+\exp{\beta(\langle\epti_\p\rangle-\mu)}}\,,
\eeq
or
\beq
\delta n_\p(T,\mu)=\frac1{1+\exp{\beta(\langle\epti_\p\rangle-\mu)}}
-\theta(p_f-p)\,.
\eeq
However, at least for an isotropic system, this expression bears
closer investigation. Here, the molecular field (evaluated
within $k_BT$ of the Fermi surface)
\beq
\langle\epti_\p-\ep_\p\rangle=\sum_{\p'}f_{\p',\p}\langle \delta n_{\p'}\rangle
\eeq
must be independent of the location of $\p$ on the Fermi surface (and of
course, spin), and is thus constant. To see this, reconsider the
Legendre polynomial expansion discussed earlier
\beqa
\langle\epti_\p-\ep_\p\rangle
&=&\sum_{\p'}f_{\p',\p}\langle \delta n_{\p'}\rangle\nonumber\\
&\propto& \sum_l\int d^3p f_lP_l(\cth)\langle \delta n_{\p'}\rangle\nonumber\\
&\propto& f_0\int d^3p \langle \delta n_{\p'}\rangle=0\nonumber\\
\eeqa
In going from the second to the third line above, we made use of the
isotropy of the system, so that $\langle \delta n_{\p'}\rangle$ is
independent of the angle $\th$. The evaluation in the third line,
follows from particle number conservation. Thus, to lowest order in
$\delta$
\beq
n_\p(T,\mu)=\frac1{1+\exp{\beta(\ep_\p-\mu)}} + \CO(\delta^4)
\eeq
\subsubsection{Local Equilibrium Distribution}
Now suppose we introduce a local weak perturbation such as the
type discussed in Sec.~\ref{sec:local_energy} to an isotropic system at
zero temperature. Such a perturbation could be caused by, eg. a sound
wave or a weak magnetic field (which you will explore in your homework,
to calculate the sound velocity and susceptibility of a Landau Fermi
liquid). This pertubation will cause a small deviation of the
equilibrium distribution function, leading to a new "local equilibrium"
distribution
\beq
\bar n_\p= n_\p(\epti_\p-\mu)
\eeq
where the argument of the RHS indicates that this is the distribution
corresponding to the local energy discussed above. The gradient of
the local energy yields a force which tries to restore the equilibrium
distribution $n_\p(\ep_\p-\mu)$ derived above). The deviation from
true equilibrium is
\beqa
\delta n_\p &=& n_\p -\bar n_\p \\
&=& \delta \bar n_\p + \frac{\partial n_\p (\ep_\p-\mu)}
{\partial \ep_\p}
\lep \epti_\p - \ep_\p \rip
\label{eq:deltanp}
\eeqa
Using Eq.~\ref{eq:addfree}, we find
\beq
\delta n_\p = \delta \bar n_\p - \frac{\partial n_\p }{\partial \ep_\p}
\sum_{\p'} f_{\p,\p'} \delta n_{\p'}
\eeq
At zero temperature, the factor $\frac{\partial n_\p }{\partial \ep_\p}=
-\delta (\ep_\p-\mu)$, so both $\delta n_\p$ and $ \delta \bar n_\p$
are restricted to the fermi surface. Since the pertubation of interest
is small, we may expand both $\delta n_\p$ and $ \delta \bar n_\p$
in a series of Legendre polynomials, and we will also split
them into symmetric and antisymmetric parts (as we did with $f_{\p,\p'}$
previously). For example,
\beq
\delta n_\p^s = \sum_l \delta (\ep_\p-\mu) \delta n_l^s P_l
\eeq
If we make a similar expansion for the antisymmetric and symmetric
parts of $\delta \bar n_\p$, and substitute this back into
Eq.~\ref{eq:deltanp}, then we find
\beqa
\delta \bar n_l^a &=& \lep 1+ \frac{F_l^a}{2l + 1}\rip \delta n_l^a\\
\delta \bar n_l^s &=& \lep 1+ \frac{F_l^s}{2l + 1}\rip \delta n_l^s
\eeqa
\subsection{Effective Mass $m^*$ of Quasiparticles}
This argument most closely follows that of AGD, and we will follow their
notation as closely as possible (without introducing any new symbols).
In particular, since an integration by parts is necessary, we will
use a momentum integral (as opposed to a momentum sum) notation
\beq
\sum_\p\to V\int \frac{d^3p}{(2\pi\hbar)^3}\,.
\eeq
The net momentum of the volume $V$ of quasiparticles is
\beq
\P_{qp}=2V \int \frac{d^3p}{(2\pi\hbar)^3} \p n_\p\mbox{ net quasiparticle momentum}
\eeq
which is also the momentum of the Fermi liquid. On the other
hand since the number of particles equals the number of quasiparticles,
the quasiparticle and particle currents must also be equal
\beq
\J_{qp}=\J_{p}=2V \int \frac{d^3p}{(2\pi\hbar)^3} \v_\p n_\p\mbox{ net
quasiparticle and particle current}
\eeq
or, since the momentum is just the particle mass times this current
\beq
\P_{p}=2Vm \int \frac{d^3p}{(2\pi\hbar)^3} \v_\p n_\p\mbox{ net quasiparticle
and particle current}
\eeq
where $\v_\p=\grad_p\epti_\p$, is the velocity of the quasiparticle. So
\beq
\int \frac{d^3p}{(2\pi\hbar)^3} \p n_\p =
m \int \frac{d^3p}{(2\pi\hbar)^3} \grad_p\epti_\p n_\p
\label{eq:massequal}
\eeq
Now make an arbitrary change of $n_\p$ and recall that $\epti_\p$ depends
upon $n_\p$, so that
\beq
\delta \epti_\p=V \sum_{\sigma'} \int \frac{d^3p}{(2\pi\hbar)^3} f_{\p,\p'} \delta n_{\p'}\,.
\eeq
For Eq.~\ref{eq:massequal}, this means that
\beqa
\int \frac{d^3p}{(2\pi\hbar)^3} \p \delta n_\p&=&
m \int \frac{d^3p}{(2\pi\hbar)^3} \grad_p\epti_\p \delta n_\p \\
&& +
m V \int \frac{d^3p}{(2\pi\hbar)^3} \sum_{\sigma'}
\int \frac{d^3p'}{(2\pi\hbar)^3}
\grad_p\lep f_{\p,\p'} \delta n_{\p'}\rip n_\p\,,\nonumber
\eeqa
or integrating by parts (and renaming $\p\to\p'$ in the last part), we get
\beqa
\int \frac{d^3p}{(2\pi\hbar)^3} \frac{\p}{m} \delta n_\p &=&
\int \frac{d^3p}{(2\pi\hbar)^3} \grad_p\epti_\p \delta n_\p \\
&&-
V \sum_{\sigma'}\int \frac{d^3p'}{(2\pi\hbar)^3} \int \frac{d^3p}{(2\pi\hbar)^3}
\delta n_\p f_{\p,\p'} \grad_{p'} n_{\p'} \,,\nonumber
\eeqa
Then, since $\delta n_\p$ is arbitrary, it must be that the
integrands themselves are equal
\beq
\frac{\p}{m}=\grad_p\epti_\p
-\sum_{\sigma'}V \int \frac{d^3p'}{(2\pi\hbar)^3}
f_{\p,\p'} \grad_{p'} n_{\p'}
\eeq
The factor $\grad_{p'} n_{\p'}=-\frac{\p'}{p'}\delta(p'-\p_F)$. The
integral may be evaluated by taking advantage of the system isotropy,
and setting $\p$ parallel to the z-axis, since we mostly interested
in the properties of the system on the Fermi surface we take $p=p_F$,
let $\th$ be the angle between $\p$ (or the z-axis) and $\p'$, and finally
note that on the Fermi surface
$\left|\left.\grad_p\epti_p\right|_{p=p_F}\right|=v_F=p_F/m^*$. Thus,
\beq
\frac{p_F}{m}=\frac{p_F}{m^*}
+
\sum_{\sigma'}\int \frac{p'^2 dpd\Omega}{(2\pi\hbar)^3}
f_{\p\sigma,\p'\sigma'} \frac{\p'}{p'} \delta(p'-p_F)
\eeq
However, since both $p$ and $p'$ are restricted to the Fermi surface
$\frac{\p'}{p'}=\cth$, and evaluating the integral over $p$, we get
\beq
\frac{1}{m}=\frac{1}{m^*}+
\frac{Vp_F}{2}\sum_{\sigma,\sigma'}
\int \frac{d\Omega}{(2\pi\hbar)^3}
f_{\p\sigma,\p'\sigma'} \cth \,,
\eeq
where the additional factor of $\frac12$ compensates for the additional
spin sum. If we now sum over both spins, $\sigma$ and $\sigma'$,
only the symmetric part of $f$ survives (the sum yields $4f^s$), so
\beq
\frac{1}{m}=\frac{1}{m^*}+
\frac{4\pi Vp_F}{(2\pi\hbar)^3}
\int d\lep\cth\rip f^s(\th) \cth \,,
\eeq
We now expand $f$ in a Legendre polynomial series
\beq
f^\alpha(\th)=\sum_l f_l^\alpha P_l(\cth)\,,
\eeq
and recall that $P_0(x)=1$, $P_1(x)=x$, .... that
\beq
\int_{-1}^1 dx P_n(x) P_m(x) dx = \frac{2}{2n+1}\delta_{nm}
\eeq
and finally that
\beq
D(0) f_l^\alpha=\frac{Vm^*p_F}{\pi^2\hbar^3} f_l^\alpha =F_l^\alpha\,,
\eeq
we find that
\beq
\frac{1}{m}=\frac{1}{m^*} + \frac{F_1^s}{3m^*}\,,
\eeq
or $m^*/m= 1+F_1^s/3$.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|l|}\hline
Quantity & Fermi Liquid & Fermi Liquid/Fermi Gas\\\hline
Specific Heat & $C_v=\frac{m^*p_F}{3\hbar^3}k_B^2 T$ &$\frac{C_V}{C_{V0}}=\frac{m^*}{m}=1+F_1^s/3$\\
Compressibility & &$\frac{\kappa}{\kappa_0}=\frac{1+F_0^s}{1+F_0^s/3}$\\
Sound Velocity & $c^2=\frac{p_F^2}{3mm^*} \lep 1+F_0^s\rip$ &$\lep\frac{c}{c_0}\rip^2=\frac{1+F_0^s}{1+F_1^s/3}$\\
Spin Susceptibility & $\chi=\frac{m^*p_F}{\pi^2\hbar^3}\frac{\beta^2}{1+F_0^a}$ &$\frac{\chi}{\chi_0}=\frac{1+F_1^s/3}{1+F_0^a}$\\
\hline
\end{tabular}
\caption[]{\em{Fermi Liquid relations between the Landau parameters $F^\alpha_n$
and some experimentally measurable quantities. For the latter, a zero
subscript indicates the value for the non-interacting Fermi gas.}}
\label{fig:Lanparam}
\end{center}
\end{table}
The effective mass cannot be experimentally measured directly;
however, it appears in many physically relevant measurable quantities,
including the specific heat
\beq
C_V=\lep\pde{E/V}{T}\rip_{VN}=\frac1{V}\pde{}{T}\sum_\p \epti_\p n_\p.
\eeq
To lowest order in $\delta$, we may neglect $f_{\p,\p'}$ in both $\epti_\p$
and $n_\p$, so
\beq
C_V=\frac1{V}\sum_\p \ep_\p \pde{n_\p}{T}\,.
\eeq
Recall that the density of states $D(E)=\sum_\p\delta(E-\ep_p)$, and
making the same assumption that we made for the non-interacting system,
that $\pde{\mu}{T}$ is negligible, we get,
\beq
C_V=\frac{1}{V} \int d\ep D(\ep)\ep\pde{}{T}\frac{1}{\exp{\beta(\ep-\mu)}+1}\,.
\eeq
This integral is identical to the one we had to evaluate for the
non-interacting system, and yields the result
\beqa
C_V&=& \frac{\pi^2}{3V} k_B^2TD(E_F)\nonumber\\
&=& \frac{k_B^2Tm^*p_F}{3\hbar^3}\,.
\eeqa
Thus, measuring the electronic contribution to the specific heat $C_V$
yields information about the effective mass $m^*$, and hence $F_1^s$.
Other measurements are related to some of the remaining Landau parameters,
as summarized in table~\ref{fig:Lanparam}.
\begin{thebibliography}{00}
%% \bibitem must have the following form:
%% \bibitem{key}...
%%
\bibitem{Yukawa} https://arxiv.org/pdf/hep-ph/0407258v2.pdf
\bibitem{mottworkswell} https://arxiv.org/pdf/1411.4372.pdf
\end{thebibliography}
\edo