# Nonmonotonic confining potential and eigenvalue density transition for
generalized random matrix model

Swapnil Yadav1, Kazi Alam1, K. A. Muttalib1 and Dong Wang2 1Department of
Physics, University of Florida, Gainesville, Florida 32611-8440, USA
2Department of Mathematics, National University of Singapore, Singapore 119076

(Apr 27, 2021)

###### Abstract

We consider several limiting cases of the joint probability distribution for a
random matrix ensemble with an additional interaction term controlled by an
exponent $\gamma$ (called the $\gamma$-ensembles). The effective potential,
which is essentially the single-particle confining potential for an equivalent
ensemble with $\gamma=1$ (called the Muttalib-Borodin ensemble), is a crucial
quantity defined in solution to the Riemann-Hilbert problem associated with
the $\gamma$-ensembles. It enables us to numerically compute the eigenvalue
density of $\gamma$-ensembles for all $\gamma>0$. We show that one important
effect of the two-particle interaction parameter $\gamma$ is to generate or
enhance the non-monotonicity in the effective single-particle potential. For
suitable choices of the initial single-particle potentials, reducing $\gamma$
can lead to a large non-monotonicity in the effective potential, which in turn
leads to significant changes in the density of eigenvalues. For a disordered
conductor, this corresponds to a systematic decrease in the conductance with
increasing disorder. This suggests that appropriate models of
$\gamma$-ensembles can be used as a possible framework to study the effects of
disorder on the distribution of conductances.

##  I Introduction

A generalized random matrix model with additional interactions Alam-Muttalib-
Wang-Yadav20 , called the $\gamma$-ensembles, was introduced recently as a
solvable toy model for three-dimensional (3D) disordered conductors. The joint
probability distribution (jpd) of the $N$ non-negative eigenvalues $x_{i}$ for
these $\gamma$-ensembles has the form

$\displaystyle
p(\\{x_{i}\\};\theta,\gamma)\propto\prod_{i=1}^{N}w(x_{i})\prod_{i<j}|x_{i}-x_{j}||x_{i}^{\theta}-x_{j}^{\theta}|^{\gamma},$
(1) $\displaystyle 0<\gamma,\;\;\;1<\theta<\infty.$

Here we assume the convention $w(x)=e^{-NV(x)}$, so that the empirical
distribution of the particles (a.k.a. the equilibrium measure) converges as
$N\to\infty$. In Alam-Muttalib-Wang-Yadav20 , the parameter $\gamma$ was
restricted to $0<\gamma\leq 1$, but the method developed there allows the
evaluation of the density of eigenvalues of the $\gamma$-ensembles for any
$\gamma>0$, $\theta>1$ and for any well behaved $V(x)$. In particular, it was
shown that the jpd for the $\gamma$-ensembles can be mapped on to the
Muttalib-Borodin (MB) ensembles Muttalib95 ; Borodin99 ; Forrester-Wang15 ;
Zhang15 ; Kuijlaars-Molag19 ; Molag20 ; Wang-Zhang21  (which has the same jpd
as Eq. (1), with $\gamma=1$), by replacing the external potential $V(x)$ with
a $\gamma$-dependent effective potential $V_{\operatorname{eff}}(x;\gamma)$.
This effective potential was calculated explicitly for $\theta=2$ by
numerically solving the Riemann-Hilbert (RH) problem associated with the jpd
of the $\gamma$-ensembles. This allowed the calculation of the corresponding
exact density of the eigenvalues $\sigma(x)$, which can be used to calculate
the average conductance of a disordered conductor.

In terms of the variables in Eq. (1), the average dimensionless conductance
per channel $g_{channel}$ of a disordered conductor (in units of the quantum
conductance $e^{2}/\hbar$) is given by note ; Muttalib-Pichard-Stone87

$g_{channel}=\int_{0}^{\infty}\frac{\sigma(x)}{\cosh^{2}\sqrt{x}}dx.$ (2)

Clearly, a large peak in the density near the origin corresponds to a large
conductance, or a metal, while a density which is small near the origin and
spread out at large values of $x$ will correspond to a small conductance, or
an insulator.

As shown in Alam-Muttalib-Wang-Yadav20 , while the exact solution of the
density for Eq. (1) for $\theta=2$ shows a significant change in the density
as a function of the two-particle interaction parameter $\gamma$, the change
in density is not large enough to affect the conductance $g$ significantly.
Thus the question arises: What is the role of the parameter $\gamma$ in the
transition from metallic to insulating behavior of a disordered quantum
conductor? In this paper we address this question in three steps:

First, we show that if we allow $1<\theta<2$, then the effective potential
near the origin becomes non-monotonic for $\gamma<1$, where the degree of non-
monotonicity increases with decreasing $\gamma$. This is significant because
such non-monotonic effective potential can in principle give rise to a
transition in density from hard-edge to soft-edge, which means a transition
from a diverging to a non-diverging density near the origin, as shown by Clays
and Romano (CR) Claeys-Romano14 . As a bonus, we find that for Laguerre
$\beta$ ensembles, the eigenvalue density for all values of $\beta\geq 1$ can
be obtained by considering the $\theta\to 1$ limit of the $\gamma$-ensembles,
with $\beta=\gamma+1$, as shown in Appendix A.

Second, while the CR model (which belongs to the MB-ensembles) shows a
transition from a diverging to a non-diverging density near the origin by
changing the non-monotonicity parameter $\rho$ of the single-particle
potential $V(x)=x^{2}-\rho x$, we show that for a fixed value of $\rho$, a
similar transition occurs as a function of the two-particle interaction
parameter $\gamma$. This shows that the role of the parameter $\gamma$ in the
$\gamma$-ensembles is qualitatively similar to a non-monotonicity parameter in
the single-particle potential.

Third, we consider a realistic phenomenological single-particle potential for
a disordered conductor of the form $V(x)=\Gamma x-(1/2)\ln\sinh^{2}\sqrt{x}$
where the logarithmic term arises naturally as a Jacobian factor Markos-
Muttalib-Wolfle05  and $\Gamma$ is also a function of $\gamma$. This model
produces a transition in the density from a peak near the origin to a density
with a gap near the origin as $\gamma$ is reduced systematically from 1, the
gap increasing with decreasing $\gamma$. This change in the density is
sufficient to result in a transition from a metallic to an insulating
conductance. While such a toy model is clearly not sufficient to describe
metal-to-insulator transition in actual physical systems, the results suggest
that the $\gamma$-ensembles with appropriate single-particle potentials can be
used as a possible framework to study the distribution of conductances across
the metal-insulator transition.

The paper is organized as follows. In Sec. II we give a brief outline of the
numerical solution to RH problem for $\gamma$ ensembles. The equilibrium
density can be obtained replacing external potential $V(x)$ with $\gamma$
dependent effective potential $V_{\operatorname{eff}}(x;\gamma)$. In Secs. III
, IV and V we follow the three steps mentioned above and systematically
explore the role of the parameter $\gamma$. We summarize our results in Sec.
VI. Results obtained as a bonus for the well-known $\beta$-ensembles as a
$\theta\to 1$ limit of the $\gamma$-ensembles are discussed in Appendix A.
Some mathematical details are given in Appendix B.

##  II The equilibrium problem for $\gamma$ ensemble

Here we give a brief overview of the solution to the RH problem of
$\gamma$-ensembles and the computation of its eigenvalue density. The complete
analysis can be found in Alam-Muttalib-Wang-Yadav20 . Consider the
$\gamma$-ensembles defined by the jpd in Eq. (1). The unique equilibrium
measure $\mu$ that minimizes the energy functional

$\displaystyle\frac{1}{2}\iint\ln\frac{1}{\lvert
x-y\rvert}d\mu(x)d\mu(y)+\frac{\gamma}{2}\iint\ln\frac{1}{\lvert
x^{\theta}-y^{\theta}\rvert}d\mu(x)d\mu(y)$ (3) $\displaystyle+\int
V(x)d\mu(x),$

satisfies the Euler-Lagrange (EL) equation

$\int\ln|x-y|d\mu(y)+\gamma\int\ln|x^{\theta}-y^{\theta}|d\mu(y)-V(x)=\ell$
(4)

if $x$ lies inside the support of density and the equality sign is replaced by
$<$ if $x$ lies outside the support. Here $\ell$ is some constant. In this
section we give formalism for hard-edge support where we assume that the
eigenvalue density lies on support $[0,b]$ for some $b>0$. The similar
formalism for soft-edge for which density lies on support $[a,b]$ with
$b>a>0$, is given in Appendix B. In formulating the RH problem from the above
EL equations, crucial role is played by the Joukowsky transformation (JT) for
hard edge,

$\displaystyle J_{c}(s)={}$ $\displaystyle
c(s+1)(\frac{s+1}{s})^{\frac{1}{\theta}},$ (5)

where $s$ is a complex variable. The points in the complex domain, which are
mapped by the JT on to a real line, form a contour $\nu$ given by,

$r(\phi)=\left.\tan\left(\frac{\phi}{1+\theta}\right)\middle/\left[\sin\phi-\cos\phi\tan\left(\frac{\phi}{1+\theta}\right)\right]\right.,$
(6)

where $0<\phi<2\pi$ is the argument of $s$ in the complex plane. Schematic
Fig. 1 shows mapping of all points on contour $\nu$ to two different regions
in the complex plane by the JT $J_{c}(s)$.

Figure 1:  (Color online) Schematic figure for the mapping of JT for a hard-
edge problem. Here $D$ is the region inside the contour $\nu_{1},\nu_{2}$
($\bar{D}$ is the region outside). $\mathbb{H}_{\theta}$ is the angular region
at the top right between the lines [5],[6]. $\mathbb{C}$ denotes the complex
plane.

By defining complex transforms

$\displaystyle g(z)\equiv{}$ $\displaystyle\int_{0}^{b}\log(z-x)d\mu(x),$
$\displaystyle z\in\mathbb{C}\backslash(-\infty,b],$ (7)
$\displaystyle\tilde{g}(z)\equiv{}$
$\displaystyle\int_{0}^{b}\log(z^{\theta}-x^{\theta})d\mu(x),$ $\displaystyle
z\in\mathbb{H}_{\theta}\backslash(0,b],$

with their derivatives $G(s)\equiv g^{\prime}(s)$,
$\tilde{G}(s)\equiv\tilde{g}^{\prime}(s)$ and the function $M(s)$ as,

$M(s)\equiv\begin{cases}G[J_{c}(s)],&\text{for
}s\in\mathbb{C}\backslash\bar{D},\\\ \tilde{G}[J_{c}(s)],&\text{for }s\in
D\backslash[-1,0],\end{cases}$ (8)

the sum and difference of the EL equations can be written as

$\begin{split}M_{+}(s_{1})+\gamma M_{-}(s_{1})+M_{-}(s_{2})+\gamma
M_{+}(s_{2})={}&2V^{\prime}[J_{c}(s)],\\\
M_{+}(s_{1})-M_{-}(s_{2})+M_{-}(s_{1})-M_{+}(s_{2})={}&0.\end{split}$ (9)

Here $s_{1}\in\nu_{1}$ and $s_{2}\in\nu_{2}$ (see Fig. 1). Equation (9),
together with some of the limits of $M(s)$, form the RH problem for $M(s)$.
The RH problem in terms of $N(s)\equiv M(s)J_{c}(s)$ is then as follows.

#### RH problem for $N$:

  * •

$N$ is analytic in $\mathbb{C}\setminus\nu$.

  * •

$N_{+}(s_{1})+\gamma N_{-}(s_{1})+N_{-}(s_{2})+\gamma N_{+}(s_{2})$
$=2V^{\prime}[J_{c}(s)]J_{c}(s)$

$N_{+}(s_{1})-N_{-}(s_{2})+N_{-}(s_{1})-N_{+}(s_{2})=0.$ (10)

  * •

$N(0)=\theta$ and $N(s)\to 1$ as $s\to\infty$.

We further define a function $f$ such that,

$f[J_{c}(s)]\equiv N_{+}(s)+N_{-}(s).$ (11)

This gives solution to RH problem of $N(s)$ as,

$N(s)=\begin{cases}\frac{-1}{2\pi
i}\oint_{\nu}\frac{f[J_{c}(\xi)]}{\xi-s}\;d\xi+1,&s\in\mathbb{C}\backslash\bar{D},\cr\frac{1}{2\pi
i}\oint_{\nu}\frac{f[J_{c}(\xi)]}{\xi-s}\;d\xi-1,&s\in
D\backslash[-1,0].\end{cases}$ (12)

Also from the RH problem for $N(s)$, the constant $c$ of the JT in Eq. (5)
satisfies the equation

$\frac{1}{2\pi i}{\displaystyle\oint_{\nu}}\frac{f[J_{c}(s)]}{s}ds=1+\theta.$
(13)

Thus the sum equation in the RH problem for $N(s)$ can be rewritten as,

$(1-\gamma)[N_{+}(s_{1})+N_{-}(s_{2})]+2\gamma
f[J_{c}(s)]=2V^{\prime}[J_{c}(s)]J_{c}(s).$ (14)

Defining the inverse mapping of JT as,

$s=J_{c}^{-1}(x)=h(x).$ (15)

with $(s_{1})_{+}=h(y);\ (s_{2})_{-}=\bar{h}(y);\ s_{1}=h(x)\ \text{and}\
s_{2}=\bar{h}(x)$, we substitute for $[N_{+}(s_{1})+N_{-}(s_{2})]$ using Eq.
(12) and the inverse mapping. We finally get the integral equation,

$f(y;\gamma)=\frac{V^{\prime}(y)y}{\gamma}-\frac{1-\gamma}{\gamma}\bigg{[}1+\frac{1}{2\pi}\int_{0}^{b}f(x;\gamma)\phi(x,y)dx\bigg{]},$
(16)

where

$\phi(x,y)=\operatorname{Im}\bigg{[}\left(\frac{1}{h(y)-\overline{h}(x)}+\frac{1}{\overline{h}(y)-\overline{h}(x)}\right)\overline{h}^{\prime}(x)\bigg{]}.$
(17)

We solve Eq. (16) for $f(y;\gamma)$ and Eq. (13) for $c$ numerically, self-
consistently. The new effective potential $V_{\operatorname{eff}}(x;\gamma)$
is related to $f(x;\gamma)$ by

$V^{\prime}_{\operatorname{eff}}(x;\gamma)=\frac{f(x;\gamma)}{x}.$ (18)

The eigenvalue density for this effective potential is given by Alam-Muttalib-
Wang-Yadav20 ,

$\begin{split}\sigma(y;\gamma)={}&\frac{-1}{2{\pi}^{2}\gamma
y}\int_{b}^{0}xV^{\prime}_{\operatorname{eff}}(x;\gamma)\chi(x,y)dx,\\\
\chi(x,y)={}&\operatorname{Re}\bigg{[}\bigg{(}\frac{1}{\overline{h}(y)-h(x)}-\frac{1}{h(y)-h(x)}\bigg{)}h^{\prime}(x)\bigg{]}.\end{split}$
(19)

Figure 2: (Color online) Effective potentials close to the origin and over the
full support, for $\gamma=0.6$ and different values of $\theta$. Near the
origin, the minima of the non-monotonic effective potential first moves away
from the origin and then moves towards the origin as $\theta$ is reduced. Note
that the effective potential is monotonic for $\theta=2$ Alam-Muttalib-Wang-
Yadav20 . Also, consistent with the analytical result for $\theta=1$, the non-
monotonicity of effective potential near the origin reduces as
$\theta\rightarrow 1$.

In summary, starting with a jpd of the $\gamma$-ensemble with some confining
potential $V(x)$, it is possible to map the problem to an MB ensemble
($\gamma=1$), but with an effective potential $V_{eff}(x,\gamma)$ given by Eq.
(18). Then, the density of the eigenvalues for such an MB ensemble can be
obtained using Eq. (19). We will use this prescription in the following
sections to obtain the density of eigenvalues for several different toy
models. We will show that one effect of the parameter $\gamma$ is to add non-
monotonicity to the effective potential.

##  III Nonmonotonic effective potential for $1<\theta<2$

As a first step towards understanding the role of the parameter $\gamma$ in
the $\gamma$-ensembles, we consider a range of the parameter $\theta$, beyond
the value $\theta=2$ considered in detail in Alam-Muttalib-Wang-Yadav20 . The
idea is to show first of all that for certain range of $\theta$, the effective
potential can become non-monotonic near the origin. Within that range, the
goal is then to choose a particular fixed value of $\theta$ that shows a
significant non-monotonicity and systematically study the effective potential
as well as the eigenvalue density as a function of $\gamma$. This would allow
us to focus on the role of $\gamma$ in the $\gamma$-ensembles. We will
restrict ourselves to the case $\gamma<1$, which is expected to be relevant
for disordered quantum conductors.

Figure 2 shows the effective potentials near the origin for $\gamma=0.6$ and a
range of values for $\theta$ between $1$ and $2$. We have shown in Alam-
Muttalib-Wang-Yadav20  that the effective potential for $\theta=2$
monotonically goes to zero at the origin. As $\theta$ is reduced from $2$, the
effective potential develops a non-monotonicity. The minima of the effective
potential gradually becomes deeper and moves away from the origin. Later as
$\theta$ moves closer to $1$, the depth of the minima of the effective
potential decreases and the minima shifts closer to the origin. Thus with
decreasing non-monotonicity, we expect the effective potential to become
linear for $\theta=1$ as predicted by Eq. (27). We have also verified this
expected analytical results for $\gamma>1$ case.

Figure 2 suggests that even for $\theta$ close enough to $\theta=1$, the
effect of $\gamma$ on the non-monotonicity could be observable. We therefore
choose $\theta=1.0001$ and a linear external potential, $V(x)=2x$. Figure 3
shows the effective potential for different values of $\gamma$, where we
include $\gamma>1$ as well to show that the results are qualitatively
different.

Note that the limit $\theta=1$ is identical to the well-known
$\beta$-ensembles with $\beta=\gamma+1$. Analytical results for such Laguerre
$\beta$-ensembles obtained in Appendix A suggest that the non-monotonicity of
the effective potential should disappear at $\theta=1$. The present formalism
allows us to consider the $\theta\to 1$ limit and thereby obtain the effective
potential as well as the density for $\beta$-ensembles for arbitrary $\beta$,
as shown in the Appendix.

Figure 3:  (Color online) Effective potential near the origin for different
$\gamma$, $V(x)=2x$ and $\theta=1.0001$. Figure 4:  (Color online) Effective
potential near the origin for quadratic potential $V(x)=0.2x^{2}$,
$\gamma=0.7$ and $\theta=1.0001$. Figure 5:  (Color online) Effective
potential for $\theta=1.2$, $V(x)=x^{2}-2.35x$ and different $\gamma$.

Figure 6: (Color online) The eigenvalue density, for $\theta=1.2$,
$V(x)=x^{2}-2.35x$, and different values of $\gamma$. Inset shows the
corresponding density near the origin. For $\gamma=0.5$ and $0.4$, the hard-
edge eigenvalue densities become negative near the origin, implying that the
assumption of hard-edge support is wrong and true density has a soft-edge
support. The last two panels (the small kinks in the density are numerical
artifacts and go away with finer grid) show the true eigenvalue density for
$\gamma=0.5$ and $0.4$ with the soft-edge support.

Figure 7: (Color online) The eigenvalue density, for $\theta=1.8$,
$V(x)=\frac{a\gamma}{1+\ln\frac{1-\gamma}{\gamma}}x-\frac{1}{2}\ln(\sinh
2\sqrt{x})$ with $a=0.01$, and different values of $\gamma$. All densities
have soft-edge support.

To explore how the non-monotonicity changes with the single-particle
potential, we consider the $\gamma$-ensemble with a quadratic single-particle
potential $V(x)=\alpha x^{2}$, $\gamma=0.7$ and $\theta\rightarrow 1$. We
choose $\alpha=0.2$ so that the potential is much weaker near the origin
compared to the linear potential. Figure 4 shows that the minima of the
effective potential is shifted significantly away from the origin and is
deeper compared to the effective potentials in Fig. 3.

##  IV Hard-edge to soft-edge transition for eigenvalue density

In the previous section we showed that the effect of decreasing the exponent
$\gamma$ from $1$ in the $\gamma$-ensembles with either a linear or a
quadratic single-particle potential is equivalent to adding a non-monotonicity
in the effective potential for the corresponding MB ensembles. It has been
shown in Claeys-Romano14  that such a minima in confining potential, if deep
enough, can produce a transition from a diverging eigenvalue density at the
hard-edge to a non-diverging density. However, the non-monotonic effective
potentials we have computed in these cases for different $\gamma$ and
different $\theta$ are not sufficient to produce the hard-edge to soft-edge
transition in the eigenvalue density. In this section we show that starting
with a given non-monotonic potential of the form $V(x)=x^{2}-\rho x$, with
fixed $\rho=2.35$ for which the density is still diverging near the origin,
changing $\gamma$ alone is sufficient to produce such a transition. Note that
this is qualitatively different from the CR model Claeys-Romano14 , where a
transition is obtained by changing the non-monotonicity parameter $\rho$ in
the single-particle potential, while we keep $\rho$ fixed, and change the two-
particle interaction parameter $\gamma$ which is expected to be related to the
strength of disorder in a three-dimensional quantum conductor.

We choose the interaction parameter $\theta=1.2$ because the results from Fig.
2 suggest that for a given $\gamma$, the non-monotonicity in the effective
potential is qualitatively the largest for $\theta$ between $1.1$ and $1.5$.
For all $\gamma<1$, we begin with the assumption that the support of density
is hard-edge (i.e. the support starts at the origin)) and we use the hard-edge
formalism to compute the eigenvalue density. If for some $\gamma<1$, our
assumption of hard-edge support for density is wrong and the actual support is
soft-edge (i.e. the support starts away from the origin) then the hard-edge
formalism gives a negative (unphysical) density near origin. In that case, we
switch to the soft-edge formalism described in Appendix B and compute the non-
negative density with soft-edge support. As the $\gamma$ decreases from $1$,
the effective potential increases (becomes more and more non-monotonic) near
origin, as shown in Fig. 5. For some critical value of $\gamma$ between $0.5$
and $0.6$, this added non-monotonicity in the effective potential brings about
the hard-edge to soft-edge transition in the density, see Fig. 6. As $\gamma$
is reduced further, the soft-edge of the support of the density near origin
moves further and further away from origin, increasing the gap in the
spectrum.

##  V Phenomenological model for 3D disordered conductors

In this section we consider a phenomenological model based on results from
Markos-Muttalib-Woelfle-Klauder04 ; Markos-Muttalib-Wolfle05 . We will
restrict ourselves to 3D only; for a brief discussion of how the
dimensionality enters the current formulation, see Appendix C. The jpd for the
ensemble is given by Markos-Muttalib-Woelfle-Klauder04 ; Markos-Muttalib-
Wolfle05 ; Klauder-Muttalib99 ; Gopar-Muttalib02 ; Douglas-Markos-Muttalib14

$\displaystyle
p(\\{x_{i}\\};\gamma)\propto\prod_{i=1}^{N}w(x_{i},\gamma)\prod_{i<j}|x_{i}-x_{j}||s(x_{i})-s(x_{j})|^{\gamma},$
(20) $\displaystyle w(x,\gamma)=e^{-NV(x,\gamma)}.$

where $s(x)=\sinh^{2}\sqrt{x}$.

The Joukowsky transformation for the interaction term,
$|\sinh^{2}\sqrt{x_{i}}-\sinh^{2}\sqrt{x_{j}}|$, is not available and hence
the explicit numerical solution to the RH problem associated with this jpd can
not be obtained. Fortunately, the $x^{\theta}$ interaction term in $\gamma$
ensembles with $\theta=1.8$ and the $\sinh^{2}\sqrt{x}$ interaction term in
Eq. (20) have very similar qualitative behavior over a reasonable range of
support for the eigenvalue density. Thus, we can use the $\gamma$-ensemble
interaction term with $\theta=1.8$ as a solvable toy model. The single-
particle potential $V(x,\gamma)$ has a dominant linear dependence on $x$ in
the strongly disordered regime, whose strength depends on the parameter
$\gamma$. It also includes a logarithmic part arising from a Jacobian of
transformation. In the strong disorder regime, the total single-particle
potential is given by Markos-Muttalib-Wolfle05

$V(x,\gamma)=\Gamma x-\frac{1}{2}\ln(\sinh 2\sqrt{x}),$ (21)

where the coefficient $\Gamma$ depends on disorder, but its functional
relationship with the two-particle interaction parameter $\gamma$ is not known
in general. The relationship has been discussed only in the strongly
disordered insulating regime Markos-Muttalib-Wolfle05  where
$\Gamma\propto\gamma$, with $\gamma\ll 1$. Starting from the strongly
disordered limit, Fig. 7 in Ref. Markos-Muttalib-Wolfle05  suggests a sharp
sigmoidal increase in $\gamma$ as disorder is decreased; this signals a
transition from the strongly disordered insulating regime towards a weakly
disordered metallic regime. Finally in the metallic regime corresponding to
$\gamma\sim 1$, the parameter $\Gamma$ is expected to be very large, although
there is no numerical guideline on its $\gamma$-dependence. A simple one-
parameter model that incorporates the strongly disordered insulating limit as
well as the rapid change at the transition as suggested by the numerical
studies is given by $\Gamma=a\gamma/[1+\ln\frac{1-\gamma}{\gamma}]$, where $a$
is a phenomenological parameter that loosely characterizes the transition
point. In the spirit of a toy model, we do not try to fix $a$. Instead, since
our numerical results converge progressively slowly for $\gamma\leq 0.5$, we
choose $a=0.01$ which generates a transition for $\gamma\sim 0.73$. Starting
from the insulating side and systematically increasing $\gamma$, we stop where
$\Gamma$ diverges [at $\gamma=e/(1+e)$], and therefore reaches the metallic
limit. Note that it is easy to construct a model with more parameters to
include the weakly disordered (metallic) regime within this formulation, but
since our focus is near the transition, which occurs at strong disorder, we
will use the simplest one-parameter model discussed above.

The effect of the logarithm in $V(x,\gamma)$ is two-fold: First, it provides a
starting non-monotonicity when combined with the dominant linear single-
particle potential. Second, it removes any divergence at the origin. Thus
unlike the CR model, a metallic regime in this case will correspond to a peak
in the density near the origin (instead of a diverging density), while an
insulating regime will correspond to zero or exponentially small density (a
gap) over a finite range near the origin. The metal-to-insulator transition in
this case will therefore correspond to the destruction of the peak in the
density of eigenvalues near the origin.

Since there is no divergence at the origin, we use the soft-edge formalism and
compute the eigenvalue densities for different values of $\gamma$. Note that
in this phenomenological model, both the two-particle interaction term and the
single-particle potential change as $\gamma$ is changed. Figure 7 shows the
change in the density as $\gamma$ is increased systematically. At $\gamma=0.7$
the density has a large gap near the origin and is spread out with no peak. As
$\gamma$ increases, the gap becomes smaller and the density starts to develop
a peak near the origin. The peak becomes very large at $\gamma=0.73105$, which
is the largest value our model allows us to consider. Thus there is a clear
“transition” in the density from zero near the origin to a large peak.

Clearly, our simplified solvable toy-models can not provide a quantitative
description of a three-dimensional disordered system. Nevertheless, the toy
model discussed here can provide qualitatively correct behavior for some of
the quantities that are not sensitive to the details of the system parameters.
Here we use Eq. (2) to compute $g_{channel}$, the average conductance per
channel (in units of the conductance quantum $e^{2}/\hbar$). Figure 8 shows
how this quantity changes with $\gamma$. At $\gamma=0.70$ where the density
has a large gap near the origin, the conductance is very small, and it remains
small as long as the gap remains appreciable, up to $\gamma=0.72$. Beyond
$\gamma=0.725$ the gap in the density starts to close and a peak near the
origin starts to grow, and the conductance starts to increase rapidly. It
reaches the value $g_{channel}\sim 1$ for $\gamma=0.73105$ which corresponds
to the metallic regime. Thus a transition in the density from a peak near the
origin to a large gap can be associated with a metal-to-insulator transition
in the conductance.

Figure 8:  (Color online) The average conductance $g_{channel}$, computed from
eigenvalue densities for different $\gamma$ from Figure 7.

##  VI Summary and conclusion

The eigenvalue density of $\gamma$-ensembles has previously been computed by
solving the corresponding Riemann-Hilbert problem. In this paper we use the
same method to explore the role of the parameter $\gamma$ by considering
various solvable toy models. First, we show that for different values of
$\theta$ between $1$ and $2$, the effective potentials for linear as well as
quadratic single-particle potentials can become non-monotonic near the origin
for $\gamma<1$. The minimum of the effective potential shifts further away
from the origin as $\gamma$ is decreased systematically. Second, we show that
in a CR type model with a fixed non-monotonicity, reducing $\gamma$ can give
rise to a transition from a diverging to a non-diverging density. Finally, we
show that a toy model that includes a linear as well as a logarithmic single-
particle potential as suggested for three-dimensional disordered conductors,
$\gamma\sim 1$ gives conductance $g_{channel}\sim 1$, while $\gamma\ll 1$
corresponds to $g_{channel}\ll 1$. For our particular choice of the model, it
also shows a rapid change in the conductance at the transition region between
the two limits. While this by itself cannot describe a true metal to insulator
transition, it provides a framework where in principle one should be able to
study the full distribution of conductances $P(g)$ across a metal-insulator
transition. This is because given a jpd $p(\\{x_{a}\\})$ of the eigenvalues,
the distribution of conductances $P(g)$ can be expressed as Markos-Muttalib-
Woelfle-Klauder04

$P(g)=\int\;\prod_{a}^{N}dx_{a}p(\\{x_{a}\\})\delta\left(g-\sum_{a}\frac{1}{\cosh^{2}\sqrt{x_{a}}}\right).$
(22)

Considering the transition in terms of the full distribution rather than in
terms of the average (or typical) conductance is particularly important. This
is because even in quasi one-dimension, where $\gamma=1$ for all disorder
Beenakker97  and therefore no transition exists Abrahams79 , $P(g)$ has a
highly asymmetric “one-sided log-normal distribution” at the crossover point
Muttalib-Woelfle99 , which is expected to remain qualitatively valid in three
dimensions near the metal-insulator transition that happens at a critical
value $\gamma=\gamma_{c}<1$. It is also known from numerical studies in three
dimensions that at strong disorder, $P(g)$ has a large variance as well as a
finite skewness Douglas-Muttalib09 . The solvable $\gamma$-ensembles with
appropriate single-particle potentials provide a possible framework to
analytically study a broad and highly asymmetric distribution of conductances
across a transition.

As a by-product, we find that the limit $\theta\rightarrow 1$ also corresponds
to the Laguerre $\beta$-ensembles. This allows us to use the model to
numerically compute the eigenvalue density for Laguerre $\beta$-ensembles for
all $\beta>1$. The results agree with various expected analytical expressions
including the ones from the exact analytical solution to the RH problem for
$\theta=1$. This shows the applicability of our method for general
$\gamma$-ensembles with different values of $\theta>1$ and $\gamma>0$.

##  VII Acknowledgment

DW acknowledges support from Ministry of Education, Singapore AcRF Tier 1
Grant No. R-146-000-262-114 and National Natural Science Foundation of China
(NSFC) Grant No. 11871425.

##  Appendix A Laguerre $\beta$-ensembles

The Laguerre $\beta$-ensembles are characterized by the jpd

$\displaystyle
p(\\{x_{i}\\})\propto\prod_{i=1}^{N}w(x_{i})\prod_{i<j}|x_{i}-x_{j}|^{\beta},$
(23) $\displaystyle w(x)=e^{-N\frac{\beta}{2}x},\quad\beta>1.$

The limiting eigenvalue density of Laguerre $\beta$-ensembles for $\beta=1,2,$
and $4$ is known analytically Forrester94 ; Forrester93 , and later it was
shown Baker-Forrester97 ; Johansson98  that the same expression is also valid
for all values of $\beta$. In Eq. (1) if we take limit $\theta\rightarrow 1$
and $V(x)=\frac{\beta}{2}x$, we get the jpd of Laguerre $\beta$ ensembles with
$\beta=1+\gamma$. Thus in the analysis of Sec. III, if we take
$\theta\rightarrow 1$ and $V(x)=\frac{\beta}{2}x$, we can compute the
eigenvalue density for Laguerre $\beta$ ensembles for any $\beta>1$. Note that
Eqs. (16)–(19) are valid only for $\theta>1$. By choosing $\theta=1.0001$ for
the $\theta\rightarrow 1$ limit, we can obtain numerical results valid for the
$\beta$-ensembles. Later in this appendix we analytically solve the RH problem
explicitly for the $\theta=1$ case and show that the results are consistent
with numerical solution for $\theta\rightarrow 1$. As $\theta\rightarrow 1$
the shape of contour $\nu$ approaches a circle.

Once the contour and the mapping (and consequently the inverse mapping) is
known, we solve Eq. (13) and Eq. (16) self-consistently to find $f(x;\beta)$.
Then the effective potential and the eigenvalue density are computed with Eq.
(18) and Eq. (19), respectively, for $\beta=1+\gamma$.

###  A.1 Eigenvalue density for Laguerre $\beta$ ensembles

In Figure 9 we show the effective potentials for different $\beta$ over the
full support of density. The effective potential becomes less and less
converging as $\beta$ increases from $1.4$ to $4$ (or $\gamma$ increases from
$0.4$ to $3$). Figure 11 shows the densities calculated from equation (19) for
different values of $\beta$. These numerical results also agree very well (see
Fig. 10) with the analytical expression

$\sigma(x)=\begin{cases}\frac{2}{\pi}\frac{1}{\beta}(\frac{\beta-x}{x})^{\frac{1}{2}},&\text{for
}0<x<\beta,\\\ 0,&\text{for }x\geq\beta,\end{cases}$ (24)

with the density diverging near the origin as
$\sigma(x)\to\frac{2}{\pi}\beta^{-\frac{1}{2}}x^{-\frac{1}{2}}$ as
$x\rightarrow 0$.

Figure 11 shows that the support of the densities increase as $\beta$
increases. The numerical densities near origin when fitted to curve
$\sigma(x;\beta)=ax^{b}$ show that the exponents b are all $-\frac{1}{2}$ for
different $\beta$. Figure 12 shows the prefactors $a$ as function of $\beta$.

Figure 9:  (Color online) Effective potential for different $\beta$ and
$V(x)=2x$. Figure 10:  (Color online) density for $\beta=4$. The dotted line
shows numerical result (see Eq. (24)) compared to analytical result shown with
bold line. Figure 11:  (Color online) Densities for different $\beta$ and
$V(x)=2x$. Figure 12:  (Color online) Densities near the origin are fitted to
function $\sigma(x;\beta)=ax^{b}$. The points show prefactors $a$ for
different $\beta$. The solid line shows the analytical result from Eq. (24).

###  A.2 RH problem for $\theta=1$

In this section we derive the analytic form of the effective potential for the
Laguerre $\beta$ ensemble by exactly solving the RH problem for $\theta=1$,
$\gamma>0$. The external potential for the Laguerre $\beta$ ensemble is
$V(x)=\frac{\beta}{2}x=\frac{1+\gamma}{2}x$. For $\theta=1$, contour $\nu$ is
a unit circle in the complex plane centered at origin. The regions inside and
outside the contour $\nu$ are both mapped onto the same complex region
$\mathbb{C}\backslash[0,b]$, similar to the contour shown in Fig. 1. Every
point on the contour is mapped onto a point on the real line in $[0,b]$.

When $\theta=1$, Eq. (7) gives $g(z)=\tilde{g}(z)$. $M(s)$ is then defined as

$M(s)\equiv\begin{cases}G[J_{c}(s)],&\text{for
}s\in\mathbb{C}\backslash\bar{D},\\\ G[J_{c}(s)],&\text{for }s\in
D\backslash[-1,0].\end{cases}$ (25)

Now since $\tilde{g}_{+}(x)=g_{+}(x)$, region (1) and region (3) in the
schematic mapping are one and the same. Similarly $\tilde{g}_{-}(x)=g_{-}(x)$
means region (2) and region (4) are the same. In terms of functions $M(s)$
these relations can be written as $M_{+}(s_{1})=M_{+}(s_{2})$ and
$M_{-}(s_{1})=M_{-}(s_{2})$ (see Fig. 1). With $N(s)\equiv M(s)J_{c}(s)$, Eq.
(10) now becomes

$\begin{split}(1+\gamma)[N_{+}(s_{1})+N_{-}(s_{1})]={}&2V^{\prime}[J_{c}(s)]J_{c}(s),\cr\end{split}$
(26)

where $J_{c}(s)=J_{c}(s_{1})=J_{c}(s_{2})=x\in[0,b]$. With $f[J_{c}(s)]$
defined according to Eq. (11) and $V(x)=\frac{\beta}{2}x=\frac{1+\gamma}{2}x$
for Laguerre $\beta$ ensembles, we finally get,

$f(x)=x,\;\;\;V_{\operatorname{eff}}(x)=x.$ (27)

Equation (27) tells us that the non-monotonicity of the effective potentials
previously shown for $\gamma<1$ should disappear when $\theta=1$.

In the RH problem for $\theta=1$, if we choose $V(x)=2x$ instead of
$\frac{\beta}{2}x$, Eq. (26) gives
$f(x)=V_{\operatorname{eff}}(x)=\frac{4}{1+\gamma}x$. The numerical results
obtained for the effective potential of the $\gamma$ ensemble with
$\theta=1.0001$ and $V(x)=2x$ agree very well with this analytic expression
(see Fig. 9).

##  Appendix B soft-edge formalism

For soft-edge support, Joukowsky transfromation is given by,

$\displaystyle J_{c_{1},c_{0}}(s)={}$
$\displaystyle(c_{1}s+c_{0})(\frac{s+1}{s})^{\frac{1}{\theta}}$ (28)

where $s$ is a complex variable. Note that the transformation now contains two
parameters $c_{0}$ and $c_{1}$ to include the two supports for the soft-edges
given by $[a,b]$ where both $a$ and $b$ are real numbers such that $b>a>0$.
The contour $\nu$ (which is a locus of points in the complex plane mapped onto
the real line) corresponding to $J_{c_{1},c_{0}}(s)$ is given by,

$\displaystyle
x(\phi)=\frac{r(\phi)\cos\phi-1}{r^{2}(\phi)-2r(\phi)\cos\phi+1},$ (29)
$\displaystyle
y(\phi)=\frac{-r(\phi)\sin\phi-1}{r^{2}(\phi)-2r(\phi)\cos\phi+1}$ (30)

and $r(\phi)$ solves

$\displaystyle r^{2}(\phi)$ $\displaystyle+$
$\displaystyle\bigg{[}\frac{c_{1}}{c_{0}}\cos\phi-\frac{\sin\phi}{\tan\frac{\phi}{\theta}}-2\cos\phi\bigg{]}r(\phi)$
(31) $\displaystyle+$ $\displaystyle 1-\frac{c_{1}}{c_{0}}=0.$ (32)

where $0<\phi<2\pi$ is the argument of $\frac{s+1}{s}$ in the complex plane.
Schematic Fig. 13 shows contour $\nu$ and mapping by the JT
$J_{c_{1},c_{0}}(s)$.

Figure 13:  (Color online) Schematic figure for the mapping of JT for a soft-
edge problem.

The complex transforms are now defined on the soft-edge support,

$\displaystyle g_{s}(z)\equiv{}$ $\displaystyle\int_{a}^{b}\log(z-x)d\mu(x),$
$\displaystyle z\in\mathbb{C}\backslash(-\infty,b],$ (33)
$\displaystyle\tilde{g_{s}}(z)\equiv{}$
$\displaystyle\int_{a}^{b}\log(z^{\theta}-x^{\theta})d\mu(x),$ $\displaystyle
z\in\mathbb{H}_{\theta}\backslash(a,b],$

with their derivatives $G_{s}(s)\equiv{g_{s}}^{\prime}(s)$,
$\tilde{G_{s}}(s)\equiv\tilde{g_{s}}^{\prime}(s)$ and the function $M_{s}(s)$
as,

$M_{s}(s)\equiv\begin{cases}G_{s}[J_{c_{1},c_{0}}(s)]&\text{for
}s\in\mathbb{C}\backslash\bar{D}\\\
\tilde{G_{s}}[J_{c_{1},c_{0}}(s)]&\text{for }s\in
D\backslash[-1,0];\end{cases}$ (34)

the sum and difference of the EL equations can be written as

$\begin{split}{}&M_{s+}(s_{1})+\gamma M_{s-}(s_{1})+M_{s-}(s_{2})+\gamma
M_{s+}(s_{2})\\\ {}&=2V^{\prime}[J_{c_{1},c_{0}}(s)],\\\
{}&M_{s+}(s_{1})-M_{s-}(s_{2})+M_{s-}(s_{1})-M_{s+}(s_{2})=0.\end{split}$ (35)

Here $s_{1}\in\nu_{1}$ and $s_{2}\in\nu_{2}$ (see Fig. 13). Equation (35),
together with some of the limits of $M_{s}(s)$, form the RH problem for
$M_{s}(s)$. The RH problem in terms of $N_{s}(s)\equiv
M_{s}(s)J_{c_{1},c_{0}}(s)$ is then as follows.

#### RH problem for $N$:

  * •

$N_{s}$ is analytic in $\mathbb{C}\setminus\nu$.

  * •

$N_{s+}(s_{1})+\gamma N_{s-}(s_{1})+N_{s-}(s_{2})+\gamma N_{s+}(s_{2})$
$=2V^{\prime}[J_{c_{1},c_{0}}(s)]J_{c_{1},c_{0}}(s)$

$N_{s+}(s_{1})-N_{s-}(s_{2})+N_{s-}(s_{1})-N_{s+}(s_{2})=0.$ (36)

  * •

$N_{s}(0)=\theta$, $N_{s}(-1)=0$ and $N_{s}(s)\to 1$ as $s\to\infty$.

We further define a function $f_{s}$ such that

$f_{s}[J_{c_{1},c_{0}}(s)]\equiv N_{s+}(s)+N_{s-}(s).$ (37)

This gives the solution to the RH problem of $N_{s}(s)$ as

$N_{s}(s)=\begin{cases}\frac{-1}{2\pi
i}\oint_{\nu}\frac{f_{s}[J_{c_{1},c_{0}}(\xi)]}{\xi-s}\;d\xi+1,&s\in\mathbb{C}\backslash\bar{D}\cr\frac{1}{2\pi
i}\oint_{\nu}\frac{f_{s}[J_{c_{1},c_{0}}(\xi)]}{\xi-s}\;d\xi-1,&s\in
D\backslash[-1,0].\end{cases}$ (38)

Also from the RH problem for $N_{s}(s)$, the constants $c_{1}$ and $c_{0}$ of
the JT in Eq. (28) satisfy the equations

$\begin{split}\frac{1}{2\pi
i}{\displaystyle\oint_{\nu}}\frac{f_{s}[J_{c_{1},c_{0}}(s)]}{s}ds={}&1+\theta,\\\
\frac{1}{2\pi
i}{\displaystyle\oint_{\nu}}\frac{f_{s}[J_{c_{1},c_{0}}(s)]}{s+1}ds={}&1\end{split}$
(39)

Thus the sum equation in the RH problem for $N_{s}(s)$ can be rewritten as,

$\begin{split}{}&(1-\gamma)(N_{s+}(s_{1})+N_{s-}(s_{2}))+2\gamma
f_{s}[J_{c_{1},c_{0}}(s)]\\\
{}&=2V^{\prime}[J_{c_{1},c_{0}}(s)]J_{c_{1},c_{0}}(s).\end{split}$ (40)

Defining the inverse mapping of JT as,

$s=J_{c_{1},c_{0}}^{-1}(x)=h_{s}(x).$ (41)

with $(s_{1})_{+}=h_{s}(y)\ ;\ (s_{2})_{-}=\bar{h_{s}}(y)\ ;\ s_{1}=h_{s}(x)\
\text{and}\ s_{2}=\bar{h_{s}}(x)$, we substitute for
$[N_{s+}(s_{1})+N_{s-}(s_{2})]$ using Eq. (38) and the inverse mapping. We
finally get the integral equation,

$f_{s}(y;\gamma)=\frac{V^{\prime}(y)y}{\gamma}-\frac{1-\gamma}{\gamma}\bigg{[}1+\frac{1}{2\pi}\int_{a}^{b}f_{s}(x;\gamma)\phi_{s}(x,y)dx\bigg{]},$
(42)

where

$\phi_{s}(x,y)=\operatorname{Im}\bigg{[}\left(\frac{1}{h_{s}(y)-\overline{h_{s}}(x)}+\frac{1}{\overline{h_{s}}(y)-\overline{h_{s}}(x)}\right)\overline{h_{s}}^{\prime}(x)\bigg{]}.$
(43)

We solve Eq. (42) for $f_{s}(y;\gamma)$ and Eq. (39) for $c_{1}$ and $c_{0}$
numerically, self-consistently. The new effective potential
$V_{\operatorname{eff}}(x;\gamma)$ is related to $f_{s}(x;\gamma)$ by

$V^{\prime}_{\operatorname{eff}}(x;\gamma)=\frac{f_{s}(x;\gamma)}{x}.$ (44)

The eigenvalue density for this effective potential is given by Alam-Muttalib-
Wang-Yadav20 ,

$\begin{split}\sigma_{s}(y;\gamma)={}&\frac{-1}{2{\pi}^{2}\gamma
y}\int_{b}^{a}xV^{\prime}_{\operatorname{eff}}(x;\gamma)\chi_{s}(x,y)dx,\\\
\chi_{s}(x,y)={}&\operatorname{Re}\bigg{[}\bigg{(}\frac{1}{\overline{h_{s}}(y)-h_{s}(x)}-\frac{1}{h_{s}(y)-h_{s}(x)}\bigg{)}{h_{s}}^{\prime}(x)\bigg{]}.\end{split}$
(45)

##  Appendix C Transport in dimensions other than $d=3$.

In the present work we have focused on transport in three dimensions ($d=3$)
only. In the absence of electron-electron interactions, there exists metal-
insulator transition only for $d>2$ with $d=2$ being a critical dimension
Abrahams79 . How does the dimensionality enter in our formulation? The short
answer is that the parameter $\gamma$ has a highly non-trivial dimensionality
dependence, which results in having, e.g., a true metal-insulator transition
in 3D but only a crossover in quasi one-dimension (Q1D). In this appendix, we
briefly outline how this dimensionality dependence of $\gamma$ is included in
the present formulation.

When a disordered conductor of cross section $L^{d-1}$ and length $L_{z}$ is
connected to two perfect leads, the scattering states at the Fermi energy
defines $N\propto L^{d-1}$ channels. Transport properties are then
characterized by the $2N\times 2N$ transfer matrix $M$ that connects the
outgoing flux to the incoming flux across $L_{z}$. Flux conservation and time-
reversal symmetry allows one to write $M$ in the general form Mello-Pereyra-
Kumar88

$\displaystyle M=\left(\begin{array}[]{cc}u&0\cr
0&u^{*}\end{array}\right)\left(\begin{array}[]{cc}\sqrt{1+\lambda}&\sqrt{\lambda}\cr\sqrt{\lambda}&\sqrt{1+\lambda}\end{array}\right)\left(\begin{array}[]{cc}v&0\cr
0&v^{*}\end{array}\right),$ (52)

where $u$, $v$ are $N\times N$ unitary matrices and $\lambda$ is a diagonal
matrix with non-negative elements. It turns out that in terms of the
parameters of $M$, the $N\times N$ matrix $tt^{{\dagger}}$, where $t$ is the
$N$-channel transmission matrix, can be written as Markos-Muttalib-Wolfle05

$tt^{{\dagger}}=v^{*}(1+\lambda)^{-1}v.$ (53)

Diagonalizing $tt^{{\dagger}}$ gives us $\lambda$ as well as all elements of
the $N\times N$ matrix $v$. The conductance $g$ is then given by $g={\rm
Tr}(tt^{{\dagger}})=\sum_{a=1}^{N}1/(1+\lambda_{a})$, while the matrix $v$
contains information about the dimensionality of the system in the following
way Markos-Muttalib-Wolfle05  :

Consider the $N\times N$ matrix

$\gamma_{ab}=\frac{2K_{ab}}{K_{aa}};\;\;\;K_{ab}=\langle\sum_{\alpha=1}^{N}|v_{a\alpha}|^{2}|v_{b\alpha}|^{2}\rangle,$
(54)

where the angular bracket represents an ensemble average. This matrix appears
in the generalized DMPK (DorokhovMello-Pereyra-Kumar) equation Markos-
Muttalib-Wolfle05 ; Gopar-Muttalib02 ; Douglas-Markos-Muttalib14 ,

$\displaystyle\frac{\partial p(\\{\lambda\\})}{\partial(L_{z}/l)}$
$\displaystyle=$
$\displaystyle\sum_{a=1}^{N}K_{aa}\frac{\partial}{\partial\lambda_{a}}\lambda_{a}(1+\lambda_{a})$
(55) $\displaystyle\times$
$\displaystyle\left[\frac{\partial}{\partial\lambda_{a}}-\sum_{b\neq
a}\frac{\gamma_{ab}}{\lambda_{a}-\lambda_{b}}\right]p(\\{\lambda\\})$ (56)

with $l$ being the mean free path, whose solution gives the evolution of the
jpd $p(\\{\lambda_{a}\\})$ of the eigenvalues $\lambda$ with length, and where
the length and disorder dependence of the matrix $\gamma_{ab}$ contain
information about dimensionality.

For example in Q1D, for a wire of cross-section $L\times L$ and length $L_{z}$
with $L\ll L_{z}$, the localization length $\xi$ is much larger than the
transverse length, $\xi\gg L$, and all channels of transport become equivalent
(this is the definition of a Q1D system). This implies that the eigenvectors
of $v$ are isotropic. In this case Markos-Muttalib-Wolfle05 ,

$K_{ab}^{Q1D}=\frac{1+\delta_{ab}}{N+1},\;\;\;\gamma_{ab}^{Q1D}=1.$ (57)

Note that $\gamma^{Q1D}=1$ is true for all disorder. This is the reason why a
jpd of the form Eq. (20) with $\gamma=1$ implies Q1D. The metal to insulator
crossover in this case is rather unphysical, it occurs with increasing
$L_{z}$; $L_{z}\ll\xi$ being a metal while $L_{z}\gg\xi$ is an insulator
Beenakker97 . (Note that our formulation is based on a large $N$ limit of the
jpd, so it is not valid for strictly $d=1$.)

In higher dimensions, in the absence of isotropy, the matrix $\gamma_{ab}$ is
much more complicated and can not be obtained analytically. However, within a
tight-binding Anderson model with random site energies (with strength $W$) and
nearest neighbor hopping elements, which shows the Anderson metal-insulator
transition at a critical disorder $W_{c}$, it is possible to study the matrix
$K_{ab}$ in the space representation (as opposed to a channel representation)
numerically as a function of both length and disorder. This was done in detail
explicitly for $d=3$ in Ref. [Markos-Muttalib-Wolfle05, ]. The results show
that to a good approximation, one can consider only the most dominant element
$\gamma_{12}=\gamma^{3D}$. Indeed, this one parameter generalization of DMPK
equation was originally conjectured to be applicable beyond the Q1D regime
Klauder-Muttalib99 . With this approximation for $\gamma_{12}$, the solution
to the generalized DMPK equation Eq. (56) gives the jpd for 3D systems as
shown in Eq. (20) where $\gamma=\gamma^{3D}$. A finite-size scaling analysis
shows that for large $L=L_{z}$, as disorder is increased, $\gamma^{3D}\lesssim
1$ for $W<W_{c}$, and $\gamma^{3D}\ll 1$ for $W>W_{c}$, dropping sharply near
$W=W_{c}$ (see, e.g., Figs. 7 and 21 in Ref. [Markos-Muttalib-Wolfle05, ]).
The phenomenological model considered in Sec. V uses these results to relate
the parameter $\Gamma(\gamma)$ to disorder in 3D.

The matrix $K_{ab}$ has not been studied in any other dimensions. However,
given the fact that in 2D in the limit $L=L_{z}\to\infty$ the system is always
an insulator for all finite disorder, one can conjecture that $\gamma^{2D}$
remains much smaller than 1 for all disorder, never reaching the metallic
limit. If the explicit disorder dependence of $\gamma^{2D}$ is known, our
formulation can be adapted to study the conductance distribution $P(g)$ in 2D.
Similarly, given that metal-insulator transitions exist for all higher
dimensions, we expect that the disorder dependence of $\gamma^{dD}$ for $d>3$
will be qualitatively similar to $\gamma^{3D}$, but with different (dimension
dependent) critical disorder $W_{c}$.

In summary, for an arbitrary $d$-dimensional conductor, the current
formulation can be used to obtain the full distribution of conductances for
all disorder if the explicit disorder dependence of the parameter
$\gamma^{dD}$ can be obtained from numerical studies of the matrix $K$ using
the tight-binding Anderson model in $d$-dimensions.

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