Possible Quantum Critical Region tuned by \\the Disorder and Pauli-Blocking Effects
A. Kwang-Hua Chu
%\Large Based on the acoustic analog, we investigate both of the effects : disorder and (Pauli-blocking) interaction to the possible localization in electron gases by using the quantum discrete kinetic model. Effects of the disorder (or free-orientation : $\theta$ which is related to the relative direction of scattering of particles w.r.t. to the normal of the propagating plane-wave front) which is introduced into the \"{U}hling-Uhlenbeck equations together with those of the Pauli-blocking are presented. We obtain the possible phase diagram (related to the strength of disorders and the mean free path) which resembles qualitatively that proposed by Abrahams$^{1}$. \vspace{2mm} \noindent PACS : \hspace*{2mm}71.10.Hf; 71.30.+h; 73.43.Nq; 73.20.Jc; 73.20.Fz \newline
Chu-AKH.tex
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\title{Possible Quantum Critical Region tuned by \\the Disorder and Pauli-Blocking Effects}
\author{A. Kwang-Hua Chu} %\cite{CHU:2004}}  %\date  affiliation \address %affiliation
\date{Department of Physics, Xinjiang University, \\
Urumqi 830046, PR China}
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\maketitle
\begin{abstract}           %\Large
Based on the acoustic analog, we investigate both of the effects :
disorder and (Pauli-blocking) interaction to the possible
localization in electron gases by using the quantum discrete
kinetic model. Effects of the disorder (or free-orientation :
$\theta$ which is related to the relative direction of scattering
of particles w.r.t. to the normal of the propagating plane-wave
front) which is introduced into the \"{U}hling-Uhlenbeck equations
together with those of the Pauli-blocking  are presented. We
obtain the possible phase diagram (related to the strength of
disorders and the mean free path) which resembles qualitatively
that proposed by Abrahams$^{1}$.

\vspace{2mm} \noindent PACS : \hspace*{2mm}71.10.Hf; 71.30.+h;
73.43.Nq; 73.20.Jc; 73.20.Fz
\newline
\end{abstract}
%\pacs{ }
%71.10.-w; 05.30.-d; 43.35.Gk; 67.40.Mj; 43.35.Ae; 42.50.Ar; 42.50.Gy; 42.25.Bs; 34.50.-s}
% 73.43.Nq : quantum phase transitions;  73.20.Qt : electron solids;   72.15.Rn : localization effects
% Produces the title.
%02.10.Jf; 34.10.+x; 43.35.Ae; 43.35.Lq; 67.40.Kh; 67.40.Mj}  %
\doublerulesep=6mm    %\parskip=12 pt
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\bibliographystyle{plain}
%\section{Introduction}
An intriguing aspect of quantum mechanics is that even at absolute
zero temperature quantum fluctuations prevail in a system, whereas
all thermal fluctuations are frozen out. These quantum
fluctuations are able to induce a macroscopic phase transition in
the ground state of a many-body system, when the ratio of two
competing terms in the underlying Hamiltonian is varied across a
critical value [1-4]. The instability of the quantum critical
behavior with respect to (w.r.t.) the disorder can be interpreted
as a signal for phenomena of localization. Recently some of the
new developments in the localization problem \cite{QCP:2000} have
become a major theme in the condense matter research. One example
is the strongly-interacting electron (non-Fermi) liquid with
different strengths of disorder \cite{RMP:2001A}. Interesting
issues are the quantum phase transition (QPT) and quantum critical
point (QCP).  Both effects of disorder and interaction are closely
relevant to the weak and strong localization
\cite{Lat:Disord,MIT:JPC}. They are then related to the
metal-insulator transition in two-dimensions (2D). Although most
of theories proposed before are based on the Fermi liquid
behavior, new insights could be obtained considering the possible
analogy with superconducting transition which might be related to
the bosonic system [2-3]. \newline %The possibility of a
%metal-insulator transition (MIT) in two dimensions (2D) has been
%the subject of intensive research in recent years but the physics
%behind this phenomenon is still not understood. It is well
%established that the MIT occurs in a regime where both Coulomb
%(electron-electron) interactions and disorder are strong.
In the last two decades a variety of metals have been discovered
which display thermodynamic and transport properties at low
temperatures which are fundamentally different from those of the
usual metallic systems which are well described by the Landau
Fermi-liquid theory. The resistivity in a variety of high mobility
2D electron/hole systems is seen experimentally to exhibit a
number of interesting anomalies that don't as yet have an adequate
theoretical understanding.
%The high quality of the samples allows measurements to be made at very low carrier densities
%corresponding to $r_s $ being larger than around 10, where $r_s
%=E_{e-e}/E_F$ is the ratio of the Coulomb energy to Fermi energy.
Note that, at sufficiently low electron densities, an ideal
two-dimensional electron systems becomes strongly correlated,
because the kinetic energy is overpowered by energy of
electron-electron interactions (exchange and correlation energy).
The interaction strength is normally described by the Wigner-Seitz
radius, $r_s =1/(\pi n_s)^{1/2} a_B$ (where $n_s$ is the electron
density and $a_B$ is the effective Bohr radius in semiconductor).
%Theoretically it is well known that, in the strongly localized
%limit, the competition between electron-electron interactions and
%disorder leads to glassy dynamics (electron or Coulomb glass). %
%In a previous paper we have developed a theoretical framework for studying interaction corrections
%to conductivity of the two-dimensional electron gas (2DEG) due to electron-electron interactions
%and disorder for the arbitrary relation between temperature and the elastic mean free path.
%
% (The metallic behavior is defined
%by a decrease of the resistivity (say, ${\bar \rho}$) for
%decreasing temperature $T$ for certain range of density larger
%than a critical density) However,
Till now, for rather large $r_s$ ($\gg 1$) and together with
disorder, the nature of this metal-insulator transition (MIT)
remains the subject of ongoing debate [6,8-9]. %\newline
%In 1957 Landau proposed the Fermi liquid model of the metallic state, emphasizing
%a potentially wide validity of this model when the Pauli principle
%is combined with the effects of screening of the bare Coulomb
%interaction.
\newline Motivated by the analogy
between electrons in periodic or disordered metals and waves in
classical acoustical systems [10-12] an investigation for
observing possible QCP in  MIT or relevant localization
\cite{Lat:Disord} using the quantum discrete kinetic model [13-14]
was performed and will be presented here. In present approach the
\"{U}hling-Uhlenbeck collision term [13] which could describe the
collision of a gas of dilute hard-sphere Fermi- or Bose-particles
by tuning a parameter $\gamma$ : a Pauli-blocking factor (or
$\gamma f$ with $f$ being a normalized (continuous) distribution
function giving the number of particles per cell) is adopted
%in this paper, we plan to investigate the possible static \cite{Local:1996} and/or dynamic localization
%which relates to dispersion relations of plane waves propagating in electron gases by introducing
together with a disorder or free-orientation ($\theta$ which is
related to the relative direction of scattering of particles
w.r.t. to the normal of the propagating plane-wave front) into the
quantum discrete kinetic model which can be used to obtain
dispersion relations of (plane) sound waves propagating in quantum
gases. We then study the quantum critical behavior based on the
acoustical analog [15-17] which has been verified before. The
possible phase diagram for MIT  and/or resistance(or
disorder)-scattering amplitude curves (as the temperature is
decreased ) we obtained resemble qualitatively those proposed in
[5] (cf Fig. 4 therein). The non-Fermi liquid behavior was also
clearly illustrated here.
%This presentation will give us more clues for the
%studies of the quantum wave dynamics in 2D electron gases and the
%possible appearance of the localization which is directly linked
%to the particles (number) density and their energy states.
\newline We firstly introduce the concept of acoustical
analog [15] in brief.
%Note that, studies of classical wave mechanical systems
%have some important advantages over quantum mechanical wave
%systems even there are similarities in-between.
In a mesoscopic system, where the sample size is smaller than the
mean free path for an elastic scattering, it is satisfactory for a
one-electron model to solve the time-independent Schr\"{o}dinger
equation :
%\begin{displaymath}
 $-({\hbar^2}/{2m}) \nabla^2 \psi + V' (\vec{r}) \psi = E \psi$
%\end{displaymath}
or (after dividing by $-\hbar^2/2m$)
%\begin{equation}
 $\nabla^2 \psi + [q^2 - V (\vec{r})] \psi = 0$,
%\end{equation}
where $q$ is an (energy) eigenvalue parameter, which for the
quantum-mechanic system is $\sqrt{2mE/\hbar^2}$. Meanwhile, the
equation for classical (scalar) waves is
%\begin{displaymath}
 $\nabla^2 \psi - ({\partial^2 \psi}/{c^2 \,\partial t^2})
 =0$
%\end{displaymath}
or (after applying a Fourier transform in time and contriving a
system where $c$ (the wave speed) varies with position $\vec{r}$)
%\begin{equation}
 $\nabla^2 \psi + [q^2 - V (\vec{r})] \psi = 0$,
%\end{equation}
here, the eigenvalue parameter $q$ is $\omega/c_0$, where $\omega$
is a natural frequency and $c_0$ is a reference wave speed.
Comparing the time dependencies one gets the quantum and classical
relation $E= \hbar \omega$ [15-17].\newline %It seems the control and
%observability of the classical experimental analogs may be matched
%by analytical works or numerical simulations. However, classical
%systems could be used to study time-dependent potential fields and
%nonlinear effects, which are very difficult and time-consuming to
%treat numerically or analytically.
%\newline
%The plane (sound) waves propagation in dilute monatomic
%(hard-sphere) gases had been successfully investigated by
%continuous and/or discrete kinetic models since 1960s
%\cite{Kneser:1961,Chu:PhD} (please see the detailed cited
%references therein). Relevant initial and/or boundary value
%problems, i.e., the former being central to the analytical or
%numerical approach because of the propagation of the forced sound
%from certain origin while the latter being almost related to the
%experimental environment due to the sensors and transducers
%located somewhere downstream, must be well defined and then solved
%to obtain the complex spectra or dispersion relations (real part :
%sound dispersion, imaginary part : sound attenuation or
%absorption).
%In comparison with experiments, results of the
%continuous velocity approach gave better fit than the discrete
%velocity one. The integral form of the former \cite{Kneser:1961},
%however, may smooth out some peculiar phenomena or only give {\it
%bulk} physical behavior considering the continuous distribution of
%particles' velocities. The discrete form of the latter, i.e.,
%particles' velocities (and the associated number density) being a
%finite set while keeping the space and time continuous, provides
%us possibilities to adjust the discrete velocity, e.g., the free
%orientation of it in the 2D plane (which could be thought as a
%kind of disorders for co-planar velocity models) and solve
%relevant problems in order to gain more physical insights for
%specific interests. For instance, molecular beam interacting with
%surfaces (solids or liquids) will normally depend on some specific
%incident or reflecting angles. Waves propagation in random or
%disordered media might be another case
%\cite{Loc:CMP,John:1991,Figotin:1993}. Both theories and
%measurements are in rapid progress, for the latter, acoustical
%analogs considering continuum-mechanic and quantum-mechanic
%approaches included \cite{RMP:Loc2001}. \newline
%In the following presentation, based on the acoustical analogs
%(considering the continuum-mechanic and quantum-mechanic
%approaches in-between \cite{RMP:Loc2001,Chu:2001}), we shall
%demonstrate the possible phase diagram related to the route to the
%quantum critical point (QCP) which was proposed by Abrahams
%\cite{QCP:2000}. Note that, as shown in our previous results
%\cite{Chu:2001,ChuA:2001,ChuA:2002A} which, as $\theta$ being a
%disorder parameter, indicate that for $\theta=0$ (larger disorder)
%and $\theta=\pi/4$ (smaller disorder),  there exist gaps of
%spectra and possible (dynamical) localization which are similar to
%those reported in Refs. 9-12. %\cite{John:1991,Figotin:1993,RMP:Loc2001,Local:1996}.
%Our (quantum kinetic) approach, as it includes the non-uniform
%variation of those transport coefficients, like viscosity and
%thermal conductivity which are directly linked to the mean free
%path of the gas \cite{Chu:PhD}, thus will give researchers more
%insights for similar problems especially for those current
%interests in interacting 2D electron gases.
%\section{Formulations}
We assume that the gas is composed of identical hard-sphere
particles of the same mass \cite{U:U,Chu:PhD,Platkowski:1988}. The
velocities of these particles are restricted to, e.g., : ${\bf
u}_1, {\bf u}_2, \cdots, {\bf u}_p$, $p$ is a finite positive
integer. The discrete number densities of particles are denoted by
$N_i ({\bf x},t)$ associated with the velocities ${\bf u}_i$ at
point ${\bf x}$ and time $t$. If only nonlinear binary collisions
and the evolution of $N_i$ are considered, we have
\begin{equation}
 \frac{\partial N_i}{\partial t}+ {\bf u}_i \cdot \nabla N_i
 = F_i \equiv \frac{1}{2}\sum_{j,k,l} (A^{ij}_{kl} N_k N_l - A_{ij}^{kl}
 N_i N_j),
%\sum^p_{j=1} \sum_{(k,l)} (A^{ij}_{kl} N_k N_l - A^{kl}_{ij}
% N_i N_j),
\hspace*{3mm} i \in\Lambda =\{1,\cdots,p\},
% i=1,\cdots, p,
\end{equation}
where $(i,j)$ and $(k,l)$ ($i\not=j$ or $k\not=l$) are admissible
sets of collisions [13-14,16-18]
%\cite{Chu:PhD,U:U,Platkowski:1988,Chu:2001,ChuA:2001,ChuA:2002A}.
%We may also define the right-hand-side of above equation as
%\begin{equation}
%\end{equation}
%with $i \in$ $\Lambda$ =$\{1,\cdots,p\}$.
Here, the summation is taken over all $j,k,l \in \Lambda$, where
$A_{kl}^{ij}$ are nonnegative constants satisfying [13-14,18]
%\cite{Chu:PhD,U:U,Platkowski:1988}
%\begin{displaymath}
 $ A_{kl}^{ji}=A_{kl}^{ij}=A_{lk}^{ij}$, % \hspace*{25mm}
%  \mbox{indistinguishability of the particles in collision},
%\end{displaymath}
%\begin{displaymath}
 $ A_{kl}^{ij} ({\bf u}_i +{\bf u}_j -{\bf u}_k -{\bf u}_l )=0$,
% \hspace*{10mm} \mbox{conservation of  momentum in the collision},
%\end{displaymath}
%\begin{displaymath}
 and $A_{kl}^{ij}=A_{ij}^{kl}$. %,  \hspace*{37mm} \mbox{microreversibility condition}  .
%\end{displaymath}
The conditions defined for the discrete velocities above require
that there are elastic, binary collisions, such that momentum and
energy are preserved, i.e.,
%\begin{displaymath}
 ${\bf u}_i +{\bf u}_j = {\bf u}_k +{\bf u}_l$,
 $|{\bf u}_i|^2 +|{\bf u}_j|^2 = |{\bf u}_k|^2 +|{\bf u}_l|^2$,
%\end{displaymath}
are possible for $1\le i,j,k,l\le p$.  %The collision operator is
%now simply obtained by joining $A_{ij}^{kl}$ to the corresponding
%transition probability densities $a_{ij}^{kl}$ through $
%A_{ij}^{kl}$ =$ S|{\bf u}_i-{\bf u}_{j}|$ $a_{ij}^{kl}$, where,
%\begin{displaymath}
% $a_{ij}^{kl} \ge 0$, %\hspace*{12mm}
% $\sum^p_{k,l=1}  a_{ij}^{kl}=1$, $\hspace*{3mm} \forall i,j=1,\cdots,p$;
%\end{displaymath}
%with $S$ being the effective collisional cross-section
%\cite{Chu:PhD,U:U,Platkowski:1988,Chu:2001,ChuA:2001,ChuA:2002A}.
%If all $n$ ($p=2n$) outputs are assumed to be equally probable,
%then $a_{ij}^{kl}$=$1/n$ for all $k$ and $l$, otherwise
%$a_{ij}^{kl}$= 0.%\newline
%The term $ S|{\bf u}_i-{\bf u}_{j}| dt$ is the volume spanned by
%the particle with ${\bf u}_i$ in the relative motion w.r.t. the
%molecule with ${\bf u}_j$ in the time interval $dt$. Therefore, $
%S|{\bf u}_i$ $-{\bf u}_{j}| N_j$ is the number of $j$-particles
%involved by the collision in unit time. Collisions satisfying the
%conservation and reversibility conditions which have been stated
%above are defined as {\it admissible collisions}
%\cite{Chu:PhD,Chu:2001,ChuA:2001,ChuA:2002A}.
%\newline The simplified discrete kinetic model, i.e., the $2\times
%n$-velocity model
%\cite{Chu:PhD,Platkowski:1988,Chu:2001,ChuA:2001,ChuA:2002A}, is
%to consider a one-component discrete velocity gas such that the
%particles can attain 2$n$ velocities in the 2D-plane. In
%particular, the velocity discretization is characterized by (i)
%$|{\bf u}_i |= c$, (ii) ${\bf u}_i +{\bf u}_{i+n}= 0$, (iii) ${\bf
%u}_i \cdot {\bf u}_{i+1}$= $c^2 \cos (\pi/n)$, $i=1,\cdots, 2n$;
%where the index is to be intended modulo 2$n$, i.e. $i\equiv
%i+2n$. Such a model is called the planar 2$n$-velocity model. If
%only elastic collisions are taken into account, then the
%non-trivial admissible ones (where this term is used to denote
%those collisions which produce non-vanishing terms in the
%collision operator) are
%\begin{displaymath}
% \mbox{ head-on encounters :} \hspace*{8mm} ({\bf u}_i,{\bf u}_{i+n})
%  \longleftrightarrow ({\bf u}_j,{\bf u}_{j+n}) \hspace*{6mm} \forall j \not= i,
%  i=1,\cdots, 2n.
%\end{displaymath}
%Meanwhile, the momentum and energy are presumably preserved
%\begin{displaymath}
% {\bf u}_i+{\bf u}_{i+n}={\bf u}_j+{\bf u}_{j+n} ,  \hspace*{12mm}
%%\end{displaymath}
%%\begin{displaymath}
% {\mid {\bf u}_i \mid}^2+{\mid {\bf u}_{i+n} \mid}^2={\mid {\bf u}_j \mid}^2
%  +{\mid {\bf u}_{j+n} \mid}^2 .
%\end{displaymath}
%Moreover, all the velocity directions after collisions are assumed
%to be equally probable. For example, there are admissible
%collisions $({\bf u}_1,{\bf u}_{3}) \longleftrightarrow ({\bf
%u}_2,{\bf v}_{u})$ as $n=2$
%\cite{Chu:PhD,Platkowski:1988,Chu:2001,ChuA:2001,ChuA:2002A}.
We note that, the summation of $N_i$ ($\sum_i N_i$) : the total
discrete number density here is related to the macroscopic density
: $\rho \,(= m_p \sum_i N_i)$, where $m_p$ is the mass of the
particle \cite{Platkowski:1988}.
\newline
%\setlength{\unitlength}{1mm}
%\begin{picture}(40,40)(-20,0)
%\put(20,20){\vector(1,0){10}} \put(32,20){\makebox(0,0)[bl]{1}}
%\put(20,20){\vector(-1,0){10}} \put(6,20){\makebox(0,0)[bl]{3}}
%\put(20,20){\vector(0,-1){10}} \put(22,8){\makebox(0,0)[bl]{4}}
%\put(20,20){\vector(0,1){10}} \put(21,32){\makebox(0,0)[bl]{2}}
%\put(-12,0){\makebox(0,0)[bl]{\small Fig. (i)\hspace*{1mm}
%Schematic plot of a admissible collision $(v_1, v_3)
%  \leftrightarrow (v_2, v_4)$ for hard-sphere particles. }}
%\end{picture}
%\vspace{5mm} \noindent
Together with the introducing of the \"{U}hling-Uhlenbeck
collision term \cite{U:U} :
%\begin{equation}
 $F_i$ $=\sum_{j,k,l} A^{ij}_{kl} \,[ N_k N_l$ $(1+\gamma N_i)(1+\gamma N_j)$ $-
 N_i N_j (1+\gamma N_k)(1+\gamma N_l)]$,
%\end{equation}
into equation (1), for $\gamma <0$ (normally, $\gamma=-1$), we can
then obtain a quantum discrete kinetic equation for a gas of
Fermi-particles; while for $\gamma
> 0$ (normally, $\gamma=1$) we obtain one for a gas of Bose-particles,
and for $\gamma =0$ we recover the equation (1).  \newline
Considering binary  collisions only, from equation above, the
model of quantum discrete kinetic equation for Fermi or Bose gases
proposed in [13-14] is then a system of $2n(=p)$ semilinear
partial differential equations of the hyperbolic type :
\begin{displaymath}
 \frac{\partial}{\partial t}N_i +{\bf v}_i \cdot\frac{\partial}{\partial
 {\bf x}} N_i =\frac{c S}{n} \sum_{j=1}^{2n} N_j N_{j+n}(1+\gamma N_{j+1})
 (1+\gamma N_{j+n+1})-
\end{displaymath}
\begin{equation}
 \hspace*{18mm} 2 c S N_i  N_{i+n} (1+\gamma N_{i+1})(1+\gamma
 N_{i+n+1}),\hspace*{24mm} i=1,\cdots, 2 n,
\end{equation}
where $N_i=N_{i+2n}$ are unknown functions, and ${\bf v}_i$ =$ c
(\cos[\theta+(i-1) \pi/n], \sin[\theta+(i-1)\pi/n])$; $c$ is a
reference velocity modulus and the same order of magnitude as that
($c$, the sound speed in the absence of scatters) used in Ref. 9,
$\theta$ is the orientation starting from the positive $x-$axis to
the $u_1$ direction and could be thought of as a parameter for
introducing a {\it disorder}
\cite{RMP:Loc2001,Local:1996,Chu:2001,ChuA:2001,ChuA:2002A}, $S$
is an effective collision cross-section for the collision system.
\newline Since passage of the sound wave will cause a small departure
from an equilibrium state and result in energy loss owing to
internal friction and heat conduction, we linearize above
equations around a uniform equilibrium state (particles' number
density : $N_0$) by setting $N_i (t,x)$ =$N_0$ $(1+P_i (t,x))$,
where $P_i$ is a small perturbation. %The equilibrium state here is
%presumed to be the same as in Refs. 7, 13, and 16. Firstly, we
%have, (say, i=m)
%\begin{displaymath}
% \frac{\partial }{\partial t} P_m +{\bf v}_m\, \cdot
%%% \frac{\partial }{\partial t} P_m +c_m\, \cos\frac{(m-1)\pi}{n}
% \frac{\partial }{\partial {\bf x}} P_m +2 c S N_0 [(P_m +P_{m+n} )+
% \gamma N_0 (P_m +P_{m+n}+P_{\small \sum})+\cdots]=
%\end{displaymath}
%\begin{equation}
% \frac{c S N_0}{n} \sum_{k=1}^{2n} [(P_k +P_{k+n}+
% \gamma N_0 (P_k +P_{k+n}+P_{\small sum})+\cdots],
%\end{equation}
%here, $m=1,\cdots,2n$, $P_{\small sum}=0$ for n=2 because of the
%restriction for the total perturbations in an equilibrium state
%and the remaining terms in both sides are higher order terms
%related to $(\gamma N_0)^2$. The linearized version of above
%equation  is
%\begin{equation}
% \frac{\partial }{\partial t} P_m +{\bf v}_m\, \cdot
% \frac{\partial }{\partial {\bf x}} P_m +2 c S N_0 (P_m +P_{m+n} )(1+\gamma N_0)=
% \frac{2 c S N_0}{n} \sum_{k=1}^{2n} P_k (1+\gamma N_0).
%\end{equation}
%In these equations after replacing the index $m$ with $m+n$ and
%using the identities $P_{m+2n}$ = $P_m$, then we have
%\begin{equation}
% \frac{\partial }{\partial t} P_{m+n} -{\bf v}_m\, \cdot
%%% \frac{\partial }{\partial t} P_m +c_m\, \cos\frac{(m-1)\pi}{n}
% \frac{\partial }{\partial {\bf x}} P_{m+n} +2 c S N_0 (P_m +P_{m+n} )(1+\gamma N_0)=
% \frac{2 c S N_0}{n} \sum_{k=1}^{2n} P_k (1+\gamma N_0) .
%\end{equation}
%Combining above two equations, firstly adding then subtracting,
%with $A_m =(P_m + P_{m+n})/2$ and $B_m=(P_m -P_{m+n})/2$, we can
%have
%\begin{equation}
% \frac{\partial }{\partial t} A_m -c\,
% \cos[\theta+\frac{(m-1)\pi}{n}]
% \frac{\partial }{\partial x} B_m +4 c S N_0 A_m (1+\gamma N_0)=
% \frac{4 c S N_0}{n} \sum_{k=1}^{2n} A_k (1+\gamma N_0) , %\hspace*{3mm} m=1,\cdots,2n,
%\end{equation}
%\begin{equation}
% \frac{\partial }{\partial t} B_m +c\,
% \cos[\theta+\frac{(m-1)\pi}{n}]
% \frac{\partial }{\partial x} A_m =0 , \hspace*{3mm} m=1,\cdots,2n.
%\end{equation}
%Note that $\partial P_m /\partial y =0$ as $P_m$ only varies along
%the wave propagating direction : x-axis direction. From $P_{m+2
%n}=P_m$, and with $A_m =(P_m +P_{m+n})/2$ and $B_m=(P_m
%-P_{m+n})/2$, we can have $A_{m+n}= A_m$, $B_{m+n}$ =$-B_m$.
%\newline
After some similar manipulations as mentioned in [14-17], with
$B=\gamma N_0 <0$ \cite{U:U,Chu:PhD}, which gives or defines the
(proportional) contribution from the Fermi gases (if $\gamma < 0$,
e.g., $\gamma=-1$), we then have
\begin{equation}
 [\frac{\partial^2 }{\partial t^2} +c^2
 \cos^2[\theta+\frac{(m-1)\pi}{n}]
 \frac{\partial^2 }{\partial x^2} +4 c S N_0 (1+B) \frac{\partial
 }{\partial t}] D_m= \frac{4 c S N_0 (1+B)}{n} \sum_{k=1}^{n} \frac{\partial
 }{\partial t} D_k  ,
\end{equation}
where $D_m =(P_m +P_{m+n})/2$, $m=1,\cdots,n$, since $D_1 =D_m$
for $1=m$ (mod $2 n)$. \newline %\section{Dependence of Wavenumber on the Frequency}
We are ready to look for the solutions in the form of plane wave
$D_m$= $a_m$ exp $i (k x- \omega t)$, $(m=1,\cdots,n)$, with
$\omega$=$\omega(k)$. This is related to the dispersion relations
of 1D (forced) plane wave propagation in Fermi gases. So we have
\begin{equation}
 (1+i h (1+B)-2 \lambda^2 cos^2 [\theta+\frac{(m-1)\pi}{n}]) a_m -\frac{i h (1+B)}{n}
 \sum_{k=1}^n a_k =0  , \hspace*{6mm} m=1,\cdots,n,
\end{equation}
where
\begin{displaymath}
\lambda=k c/(\sqrt{2}\omega),  \hspace*{18mm} h=4 c S N_0 /\omega
\hspace*{6mm} \propto \hspace*{2mm} 1/K_n,
\end{displaymath}
where $h$ is the rarefaction parameter of the gas; $K_n$ is the
Knudsen number which is defined as the ratio of the mean free path
of gases to the wave length of the plane (sound) wave.
\newline
%Let $a_m$ = ${\cal{C}}/(1+i h_b-2 \lambda^2 \cos^2
%[\theta+(m-1)\pi/n])$, where ${\cal{C}}$ is an arbitrary, unknown
%constant, since we here only have interest in the eigenvalues of
%above relation. The eigenvalue problems for different $2\times
%n$-velocity model reduces to
%\begin{equation}
%  1-\frac{i h_b}{n} \sum^n_{m=1} \frac{1}{1+i h_b -2 \lambda^2
%    \cos^2\,[\theta+\frac{(m-1)\pi}{n}]} =0.
%\end{equation}
%\section{Results and Discussions}
We can obtain the complex spectra ($\lambda=\lambda_r +$ i
$\lambda_i$; $\lambda_r = k_r c/(\sqrt{2}\omega)$: sound
dispersion, a relative measure of the sound or phase speed;
$\lambda_i = k_i c/(\sqrt{2}\omega)$ : sound attenuation or
absorption) from the complex polynomial equation above. Here, $B$
could be related to the occupation number of different-statistic
particles of gases
%as we noticed the similarity in-between from Ref. 7.
To examine the critical region possibly tuned by the
Pauli-blocking measure $B=\gamma N_0$ and the disorder $\theta$,
as evidenced from our preliminary results : $\lambda_i =0$ for
cases of $B=-1$ or $\theta=\pi/4$ [17], we firstly check those
spectra near $\theta=0$, say, $\theta=0.005$ and $\theta=\pi/4
\approx 0.7854$, say, $\theta=0.78535$ for a $B$-sweep ($B$
decreases from 1 to -1). We plot them into figures 1, 2,
respectively. Note that, as the disorder or free-orientation
$\theta$ is not zero, there will be two kinds of propagation of
the disturbance wave : sound and diffusion modes [16-17,19-20].
The latter (anomalous) mode has been reported in Boltzmann gases
[19,21] and is related to the propagation of entropy wave which is
not used in the acoustical analog here. The absence of (further)
diffusion (or maximum absorption) for the sound mode at certain
state ($h$, corresponding to the inverse of energy $E$; cf. Refs.
11 or 15) is classified as a localized state [11,15,17]. The state
of decreasing $h$ corresponds to that of $T$ (absolute
temperature) decreasing as the mean free path is increasing
(density or pressure decreasing).
\newline We can observe the max. $\lambda_i$ (absorption of sound
mode, relevant to the localization length according to the
acoustical analog [15,17]) drop to around four orders of magnitude
from $\theta=0.005$ to $0.78535$! This is a clear demonstration of
the effect of disorder. Meanwhile, once the Pauli-blocking measure
($B$) increases or decreases from zero (Boltzmann gases), the
latter (Fermi gases : $B <0$) shows opposite trend compared to
that of the former (Bose gases : $B>0$) considering the shift of
the max. $\lambda_i$ state ($\delta h$). $\delta h >0$ is  for
Fermi gases ($|B|$ increasing), and  the reverse ($\delta h <0$)
is for Bose gases ($B$ increasing)! This illustrates partly the
electron-electron interaction effect (through the Pauli exclusion
principle). These results will be crucial for further obtaining
the phase diagram (flow to metallic or insulating state as the
density or temperature is decreased) tuned by both disorder and
the (electron-electron) interaction below. Here, $B=-1$ or
$\theta=0, \pi/4$ might be fixed points commented in [22].
\newline
%To further present the results linked to the localization, we must
%introduce the concept of acoustical analog [13] in brief.
%%Note that, studies of classical wave mechanical systems
%%have some important advantages over quantum mechanical wave
%%systems even there are similarities in-between.
%In a mesoscopic system, where the sample size is smaller than the
%mean free path for an elastic scattering, it is satisfactory for a
%one-electron model to solve the time-independent Schr\"{o}dinger
%equation :
%\begin{displaymath}
% $-({\hbar^2}/{2m}) \nabla^2 \psi + V' (\vec{r}) \psi = E \psi$
%\end{displaymath}
%or (after dividing by $-\hbar^2/2m$)
%\begin{equation}
% $\nabla^2 \psi + [q^2 - V (\vec{r})] \psi = 0$,
%\end{equation}
%where $q$ is an (energy) eigenvalue parameter, which for the
%quantum-mechanic system is $({2mE/\hbar^2})^{1/2}$. Meanwhile, the
%equation for classical (scalar) waves is
%\begin{displaymath}
% $\nabla^2 \psi - ({\partial^2 \psi}/{c^2 \,\partial t^2})
% =0$
%\end{displaymath}
%or (after applying a Fourier transform in time and contriving a
%system where $c$ (the wave speed) varies with position $\vec{r}$)
%\begin{equation}
%  $\nabla^2 \psi + [q^2 - V (\vec{r})] \psi = 0$,
%\end{equation}
%here, the eigenvalue parameter $q$ is $\omega/c_0$, where $\omega$
%is a natural frequency and $c_0$ is a reference wave speed.
%Comparing the time dependencies one gets the quantum and classical
%relation $E= \hbar \omega$ [12-13]. \newline
To check what happens
when the temperature is decreased (or the density or $h$ is
decreased) to near $T=0$, we collect all the data based on the
acoustical analog from the dispersion relations (especially the
absorption of sound mode) we calculated for ranges in different
degrees of disorder (here, $\theta$ is up to $\pi/4$ considering
single-particle scattering and binary collisions; in fact, effects
of $\theta$ are symmetric w.r.t. $\theta=\pi/4$ for
$0\le\theta\le\pi/2$ [17]) and Pauli-blocking measure. After that,
we plot the possible phase diagram for the (dimensionless)
conductivity vs. the absolute (dimensionless) temperature into
Fig. 3 (for different $B$s : $B=-0.9, -0.7, -0.5, -0.3, 0.1$).
Here, MFP is
%($S_b $ $\propto B K_n$ or the scattering amplitude, $K_n \propto r_s$ or
the mean free path and the temperature vs. MFP relations could be
traced  from [23] (cf. Fig. 3 therein). Note that, the resistivity
or resistance is proportional to the strength of disorder in 2D
(in the sense that for weak disorder it is given by $1/(k_F l)$,
in units of $\hbar/e^2$; $k_F$ is the Fermi wave number, $l$ is
the mean free path associated with the usual Drude conductivity)
[5]. This figure shows that as the temperature decreases to a
rather low value, the resistivity (or strength of disorder) will
decrease sharply (at least for Bose or Fermi gases). There is no
doubt that this result resembles qualitatively that proposed
before [5,24-25].
\newline
To know the detailed effects of electron-electron interactions
(tuned by $B$s here), we plot the corresponding figure as shown in
Fig. 4. Each contour line (flow path relevant to a specific $B$
tuned by the disorder) represents the behavior when $T$ is
decreased, and different (flow) lines represent different values
of scattering amplitude ($S_b$ $\propto K_n$ or $r_s$).
Interesting results are (i) there seems to be a quantum phase
transition boundary (interface or regime) for bosonic-like
particles ($B>0$) which resembles that of high-temperature
superconducting phase transition due to doping if we treat the
strength of disorder to be equivalent to the doping amount! (ii)
as evidenced in the top part of this plot for the Bose gases (near
QCPs for smaller resistivity together with rather large arrows),
it confirms Larkin's comment (cf page 793 in [2] by Larkin) : both
Bose and Fermi approached are important for the (QP) transition;
Bose approach is more useful in a small region close to the
transition!
\newline To make sure we already recover previous proposed results
(possible renormalization group (RG) flows for disorder plus
interactions or scattering amplitude-resistance curves or
suggested phase diagram tuned by $r_s$ and disorder, cf. [5] or
[24-25]), we summarize our results by illustrating them into a 3D
plot as demonstrated in Fig. 5. Crossover lines separate those
flows which begin at $T>0$ at small resistance (disorder  or the
inverse of conductance : $1/G$) and flow initially, as the
temperature is lowered, toward larger resistance but then are
repelled by the fixed point and flow to large $S_b$ (or $r_s$) and
large G (metallic behavior) from those which begin at larger
disorder (or smaller G) and flow toward very-large disorder
(insulating behavior). Again, this result resembles that proposed
before (cf. [5] or [24-25] therein). Note that, this flow is also
similar to those in [24-25] (e.g., cf. Fig. 41 by Aoki [24]).
Meanwhile, as expected before, at $T=0$, a 2D system would become
a Wigner crystal phase at $r_s \ge 37$ [26]. This lies possibly
near or above the rather large $S_b$ position with zero strength
of disorder in our illustrations. Possible QCPs happen around $S_b
\sim 80$ for $B=0.85$ but $S_b \sim 10$ for $B=0.1$.
% Note that, $K_n$ plays the role of $r_s$
%($\propto$ the ratio of Coulomb energy to Fermi energy as defined in Ref. 1)
%since the long-range dominated cases ($r_s$ is large)
%correspond to our cases of $h$ being small or $K_n$ being large
%(collisionless regime or rather-low density regime where long
%range interactions can still exist \cite{QCP:2000,RMP:2001A}).
\newline To conclude in brief, our illustrations here, although
are based on the acoustical analog  of our quantum discrete
kinetic calculations, can indeed show the non-Fermi liquid and
quantum critical behavior for the metal-insulator transition in 2D
[27-31] as the temperature is decreased to rather low values.
%i.e., the possible localized behavior of the spectra near the
%range of disorders : $\theta=0$ and $\theta=\pi/4$ for different
%Pauli blocking parameters seem to be the same as the acoustical
%analog (cf Ref. 11) of the localization found elsewhere
%\cite{Loc:CMP,John:1991,RMP:Loc2001,Local:1996}even though the
%physical length-scale parameter used here is the mean free path of
%the electron gases subjected to continuous collisions.
%We found, for sufficiently strong interactions (which would occur
%at low density, say, $h \ll 1$ in our approach), a
%non-Fermi-liquid state of interacting electrons ($\gamma \not =
%-1$ in our formulation) is stable in the presence of disorder.
%As the temperature decreases, the conductivity increases sharply. We
%shall investigate more complicated problems in the future [22-23].

%
%{\small Acknowledgements. The author is partially supported by the
%China Post-Dr. Science Foundation (Grant No. : 1999-17) and
%National Natural Science Foundation of China (NSFC) under Grant
%No. 10274061.}
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\newpage

\psfig{file=chufg1.eps,bbllx=0.0cm,bblly=18cm,bburx=10cm,bbury=28cm,rheight=8cm,rwidth=8cm,clip=}

\begin{figure}[h]
%\vspace{22mm}

\hspace*{12mm} Fig. 1 \hspace*{3mm}  {\small Variations of the
smaller (Sound mode) and larger (Diffusion mode [19,21])
\newline \hspace*{12mm} $\lambda_i$
w.r.t. $h$ for the same disorder : $\theta=0.005$ in different
Pauli-blocking measures \newline \hspace*{12mm} ($B>0$ corresponds
to Bose gases). The maximum absorption states for sound modes
\newline \hspace*{12mm} correspond to possible localized states  as the diffusion stops.
$B$ decreases from $1$
\newline \hspace*{10mm}  (the highest)  to
$-1$ with the decrement being $-0.2$. $\lambda_i =0$ as $B=-1$. $h
=4 c S N_0/\omega$. \newline \hspace*{10mm} $N_0$ is the number density
and $S$ is the effective collision cross-section.}
\end{figure}

\psfig{file=chufg2.eps,bbllx=0.0cm,bblly=18cm,bburx=10cm,bbury=28cm,rheight=8cm,rwidth=8cm,clip=}

\begin{figure}[h]
\hspace*{10mm} Fig. 2 \hspace*{1mm} {\small Pauli-blocking ($B$)
effects on the absorption $\lambda_i$. (Note that the \newline
\hspace*{10mm} localization length $\propto 1/\lambda_i$, cf.
[11,16-17]. The energy $E$
 (relevant to the \newline \hspace*{10mm} illustration of localized states
in [11]) corresponds to $\hbar \omega$ [15-16]. $E \propto 1/h$
\newline \hspace*{10mm}  once $c\,S\,N_0$ is fixed.) $B$ decreases
from $1$ (the highest)  to $-1$ with the
\newline \hspace*{10mm} decrement  being $-0.2$. Bose and Fermi
gases are in opposite trend for
\newline \hspace*{10mm}   the same
increment $|B|=0.2$. $\theta=0.78535$ and $\lambda_i=0$ as
$\theta=\pi/4 \approx 0.7854$.}
\end{figure}

\newpage

\psfig{file=chu-fg3.ps,bbllx=0.0cm,bblly=12.8cm,bburx=14cm,bbury=24.2cm,rheight=9.6cm,rwidth=9.6cm,clip=}

\begin{figure}[h]
\hspace*{10mm} Fig. 3 \hspace*{1mm} Possible phase diagram for
non-Fermi gases w.r.t. the disorder and \newline \hspace*{10mm}
the scattering amplitude $S_b$. $S_b \propto  B K_n$ and $K_n
\propto$ the mean free path. Note that  \newline \hspace*{10mm}
the resistivity is proportional to  the strength of disorder in 2D
[5-6]. Here, once
\newline \hspace*{10mm}     the
temperature is  lowered down the resistivity firstly increases and
 then decreases.
%($\theta$) effects  on the attenuation ($\lambda_i$). \newline \hspace*{10mm}
\end{figure}

\psfig{file=chu-fg4.ps,bbllx=0.0cm,bblly=13cm,bburx=14cm,bbury=24.2cm,rheight=9.5cm,rwidth=9.5cm,clip=}

\begin{figure}[h]
\hspace*{10mm} Fig. 4 \hspace*{1mm} Possible flow pattern for
$S_b$ w.r.t. the disorder and Pauli-blocking \newline
\hspace*{10mm}
 measure ($B$). $S_b \propto  B K_n$ and $K_n
\propto$ the mean free path. Note that  the \newline
\hspace*{10mm} resistivity is proportional to  the disorder [5-6]
in 2D. Each continuous flow
\newline \hspace*{10mm} represents the behavior when $T$ is
decreased, and different paths represent \newline \hspace*{10mm}
different values of Pauli-blocking measure. (cf Fig. 4 in [5])
\end{figure}

\newpage

\psfig{file=chu-fig-5.ps,bbllx=0.0cm,bblly=12.8cm,bburx=14cm,bbury=24.2cm,rheight=10cm,rwidth=10cm,clip=}

\begin{figure}[h]
\hspace*{10mm} Fig. 5 \hspace*{1mm} Possible scattering amplitude
($S_b$) vs. disorder and Pauli-blocking \newline \hspace*{10mm}
measure curves. $S_b \propto B K_n$, and $K_n \propto$ the mean
free path. Note that \newline \hspace*{10mm} the resistivity  is
proportional to the strength of disorder in 2D [6].  Here,
\newline \hspace*{10mm} the
flow (arrow) direction means that the temperature is decreasing.
\newline \hspace*{10mm}
These results resemble qualitatively those proposed by Abrahams in
[5-6].
\end{figure}
\end{document}             % End of document.
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%-----
%
%\subsubsection*{Acknowledgements}
%{\small  This reseach is partially supported by the China
%Post-Dr. Science Foundation.}
%------------------------------------------------------------------------
%
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