# Scattering and transport properties of the three classical Wigner-Dyson
ensembles at the Anderson transition

A. M. Martínez-Argüello  0000-0002-6422-0673 Instituto de Física, Benemérita
Universidad Autónoma de Puebla, Apartado Postal J-48, 72570 Puebla, Pue.,
Mexico  M. Carrera-Núñez  0000-0002-1526-4864<EMAIL_ADDRESS>Departamento de Ciencias Naturales y Exactas, Universidad de Guadalajara,
Carretera Guadalajara–Ameca Km. 45.5 C.P. 46600. Ameca, Jalisco, Mexico  J. A.
Méndez-Bermúdez  0000-0002-1748-9901 Instituto de Física, Benemérita
Universidad Autónoma de Puebla, Apartado Postal J-48, 72570 Puebla, Pue.,
Mexico

###### Abstract

An extensive numerical analysis of the scattering and transport properties of
the power-law banded random matrix model (PBRM) at criticality in the presence
of orthogonal, unitary, and symplectic symmetries is presented. Our results
show a good agreement with existing analytical expressions in the metallic
regime and with heuristic relations widely used in studies of the PBRM model
in the presence of orthogonal and unitary symmetries. Moreover, our results
confirm that the multifractal behavior of disordered systems at criticality
can be probed by measuring scattering and transport properties, which is of
paramount importance from the experimental point of view. Thus, a full picture
of the scattering and transport properties of the PBRM model at criticality
corresponding to the three classical Wigner-Dyson ensembles is provided.

###### pacs:

72.20.Dp, 03.65.Nk, 71.30.+h, 73.23.-b

##  I Introduction

Scattering and transport properties of disordered mesoscopic conductors have
been of interest for a long time (see Refs. Lee1985 ; Beenakker1997 ;
MelloBook  and the references therein). Among the diverse phenomena observed
in these systems, the disorder-induced localization-delocalization transition
of electronic states, known as Anderson or metal-insulator transition (MIT),
has received special attention Lee1985 ; Janssen1994 ; Imada1998 ; Evers2008 ;
Rodriguez2010 ; EdwardsBook . At the MIT, the energy spectra show anomalous
behavior while the electronic states present multifractal characteristics and
strong amplitude fluctuations. The latter are usually described by an infinite
set of critical exponents and represent one of the most important
characteristics of the MIT Janssen1994 ; Evers2008 ; Janssen1998 ;
Huckestein1995 ; Mirlin2000a ; Fyodorov1994 ; Fyodorov1995 ; Falko1995 ;
Wegner1980 ; Schreiber1991 ; Mildenberger2002 ; Backer2019 ; Carnio2019 . At
criticality, i.e. at the MIT, both the dimensionality and the symmetries
present in the system play an important role.

The statistical properties of ordinary disordered samples are well described
by the random matrix theory (RMT) Guhr1998  for which three universal symmetry
classes are known: the orthogonal [the symplectic] class describing systems in
the presence of time reversal and presence [absence] of spin-rotation
symmetry, and the unitary class for systems with broken time-reversal
symmetry. In the Dyson scheme these symmetries are labeled by the indices
$\beta=1,2,$ and 4, for the orthogonal, the unitary, and the symplectic
classes; respectively MehtaBook ; Dyson1962a ; Dyson1962b .

Until now, many important features of disordered systems at the MIT have been
analyzed using numerical techniques. This is due to the complexity in
obtaining analytical expressions at criticality, some of which are available
only perturbatively. In particular, the so-called power-law banded random
matrix (PBRM) model has widely been used since it captures all the key
features of the Anderson critical point and is also convenient for its low
computational cost Evers2008 ; Mirlin1996 ; Kravtsov1999 ; Varga2000 . For the
closed system on the one hand, since the appearance of the PBRM model
originally proposed by Mirlin et al. Mirlin1996 , a plethora of studies
regarding different aspects of the model with $\beta=1$ symmetry have been
performed Mirlin1996 ; Evers2008 ; Mirlin2000a ; Varga2000 ; Mendez2012 ;
Cuevas2001 ; Kravtsov1997 ; Kravtsov2000 ; Varga2002 ; Mendez2014 ; Mirlin2000
; Evers2000 ; Rao2022  while less studies regarding $\beta=2$ symmetry are
reported Mendez2014 ; Kravtsov2011 ; Mirlin2000a ; Mirlin2000 . Moreover, the
energy spectra and multifractal behavior of the PBRM model in the presence of
the symplectic symmetry ($\beta=4$) have recently been analyzed Carrera2021 .
For the open system, on the other hand, several scattering and transport
properties of the model when the system is in the presence of time-reversal
invariance ($\beta=1$) Antonio2009 ; Mendez2005 ; Mendez2006 ; Mendez2010 ;
Kottos2002 ; Mendez2005 ; Mendez2014 , which are in agreement with the ones
obtained by using the three-dimensional Anderson model at MIT Mendez2010 ;
Cuevas2001 , have been investigated. However, to our knowledge, the analysis
of the scattering and transport properties of the PBRM model for the unitary
case are scarce Alcazar2009  while the symplectic case has been left
unexplored.

It is the purpose of the present paper to deepening the understanding of the
scattering and transport properties of critical systems belonging to the
symplectic class; that is, we study the open symplectic PBRM ensemble at
criticality. The regime of a small number of single-mode leads attached to the
PBRM model is studied in detail. Nevertheless, the multichannel or multiple
single-mode leads setup is also studied for some scattering and transport
quantities of interest. In order to provide a full picture of the PBRM model
for the three classical Wigner-Dyson ensembles, the scattering and transport
properties of the PBRM model for the $\beta=1$ and $\beta=2$ cases, previously
considered in the literature, are also reviewed and extended when appropriate.
Our results are also compared with RMT predictions in the appropriate limits.

The organization of the paper is as follows. In the next section, the
generalized PBRM model in the presence of the three symmetry classes and the
scattering setup are described. The perfect coupling regime, the Wigner delay
time, and the resonance widths when the PBRM supports one open channel are
analyzed in detail in Sec. III. This is of particular interest, since it shows
that the multifractal properties of the isolated PBRM model can be probed by
measuring transport properties. The analysis of the scattering and transport
properties of the PBRM model in the two and four open-channel setups is the
subject of Sec. IV. In the same section, the scattering and transport
properties in the multichannel setup are also presented. Finally, the
conclusions are given in Sec. V.

##  II Model and scattering setup

The PBRM model describes one-dimensional (1D) tight-binding wires of length
$N$ with random long-range hoppings Evers2008 ; Mirlin1996 . In the presence
of the three symmetry classes, it is represented by $N\times N$ real symmetric
($\beta=1$), complex Hermitian ($\beta=2$), or $2N\times 2N$ quaternion-real
Hermitian ($\beta=4$) matrices whose elements are statistically independent
random variables drawn from a normal distribution with zero mean and variance
given by Carrera2021

$\displaystyle\langle|H_{ii}|^{2}\rangle$
$\displaystyle=\beta^{-1}\quad\mathrm{and}$
$\displaystyle\langle|H_{ij}|^{2}\rangle$
$\displaystyle=\frac{1}{2(1+\delta_{\beta,4})}\frac{1}{1+\left[\sin\left(\frac{\pi|i-j|}{N}\right)\Big{/}\left(\frac{\pi
b}{N}\right)\right]^{2\mu}}.$ (1)

The matrix sizes are $L=N$ for $\beta=1$ and 2, and $L=2N$ for $\beta=4$. The
PBRM model is a random matrix ensemble with off-diagonal matrix elements
decaying away from the diagonal in a power-law fashion. Also, in (1) the PBRM
model is in its periodic version, i.e., the 1D wire is in a ring geometry,
where $\mu$ and $b$ are the parameters of the model and $\delta_{\beta,4}$ is
the Kronecker delta. In particular, for the symplectic case ($\beta=4$) the
PBRM model preserves the quaternion structure of the Hamiltonian where each
eigenvalue is two-fold degenerate due to Kramers degeneracy (for more details
see Ref. Carrera2021 ). The power-law decay $\mu=1$ sets the PBRM model at the
MIT critical point Evers2008 ; Mirlin2000a ; Mirlin1996 ; Varga2000 ;
Cuevas2001 ; Carrera2021 ; Kravtsov1997 ; Kravtsov2000 ; Varga2002 .
Furthermore, regardless of the value of $\mu$, insulating- to metallic-like
behavior may be induced by varying the effective bandwidth $b$ from small $b$
to large $b$ values, respectively. Here, the scattering and transport
properties of the PBRM model of Eq. (1) at criticality, $\mu=1$, is the focus
of this work.

The isolated wire is opened by attaching to it $M$ semi-infinite single-
channel leads, each one described by a 1D tight-binding Hamiltonian

$H_{\text{lead}}=\sum_{n=1}^{-\infty}(|n\rangle\langle n+1|+|n+1\rangle\langle
n|),$ (2)

thus $M$ establishes the number of open channels or propagating modes. The
$M\times M$ scattering matrix, $S(E)$, can be written as Mahaux1969 ;
Verbaarschot1985 ; Rotter1991 ; Kottos2002

$S(E)=\left(\begin{array}[]{ccc}r&t^{\prime}\\\
t&r^{\prime}\end{array}\right)\\\ =\openone_{M}-2i\pi
W^{T}(E\openone_{L}-H_{\text{eff}})^{-1}W,$ (3)

where $r$ [$r^{\prime}$] and $t$ [$t^{\prime}$] are the reflection and
transmission amplitudes when the incidence is from the left [right],
$\openone_{n}$ stands for the unit matrix of dimension $n$, $E$ is the energy,
and the superscript $T$ indicates the matrix transposition operation. In Eq.
(3), $H_{\text{eff}}$ is the non-Hermitian effective Hamiltonian given by

$H_{\text{eff}}=H-i\pi WW^{T},$ (4)

where $H$ is the $L\times L$ Hamiltonian matrix of the PBRM model that
describes the isolated wire with $L$ resonant states and $W$ is an $L\times M$
energy independent matrix that couples those resonant states to the $M$
propagating modes in the leads. The elements of $W$ are
$W_{ij}=w_{0}\delta_{ij_{0}}$, where $w_{0}$ is the coupling strength between
the wire and the leads and $j_{0}=1,\ldots,M$ are the sites at which the leads
are attached. According to the symmetry present in the Hamiltonian, the $S$
matrix is unitary symmetric, unitary, and unitary self-dual matrix for
$\beta=1,2,$ and 4, respectively Beenakker1997 . Additionally, due to the ring
geometry of the isolated wire under consideration, the scattering and
transport properties do not depend on which site the leads are attached to.
Then, for simplicity and without loss of generality, in this work the leads
are attached at consecutive sites of the wire.

##  III PBRM model with one open channel

In this section, the statistical properties of scattering phases, Wigner delay
time, and resonance widths, when the scattering system supports one open
channel and is in the presence of the $\beta=1,2,$ and 4 symmetries are
analyzed. The relation between Wigner delay times and the properties of the
spectra and eigenstates of the corresponding isolated wire are also discussed.

###  III.1 Perfect coupling regime

Figure 1: Absolute value of the optical $S$ matrix for the PBRM model at
criticality as a function of the coupling strength, $w_{0}$, for values of $b$
indicated in each panel and several wire lengths $N$ as indicated in panel
(a). The symmetries are $\beta=1$ (first column), 2 (second column), and 4
(third column). The insets show numerically obtained histograms (red lines)
compared with the phase distribution (5) (black continuous lines) for
$|\langle S\rangle|=0.5$. The error bars are smaller than the symbol size, so
they are not displayed.

In the one-channel setup, the scattering matrix of Eq. (3) reduces to a phase
given by $S(E)=\mathrm{e}^{\mathrm{i}\Phi(E)}$. This case corresponds to a
single-channel lead attached to the wire (1). For the PBRM model in the limit
$b\gg 1$ (metallic-like regime), that phase is distributed according to the
following expression Ossipov2005

$\mathcal{P}(\Phi)=\frac{1}{2\pi}\frac{1}{\gamma+\sqrt{\gamma^{2}-1}\cos\Phi},$
(5)

where $\gamma=(1+|\left\langle S\right\rangle|^{2})/(1-|\left\langle
S\right\rangle|^{2})$. In the so-called _perfect coupling regime_ , the
averaged scattering matrix $\left\langle S\right\rangle$, also known as
optical matrix, vanishes and the phase is uniformly distributed over the unit
circle.

Before proceeding with the analysis of the phase distribution (5), some
general statements about the numerical simulations are given. For the
statistical analysis shown throughout this work, most calculations are
performed in the perfect coupling regime since in this limit a number of
analytical predictions from RMT are known. Also, the calculations are
performed around $E\sim 0$ considering wire lengths of $N=50,100,200,400$, and
800 with $10^{6},10^{6},10^{5},10^{5}$, and $10^{5}$ random realizations,
respectively. All fittings are performed through the nonlinear least-squares
Marquardt-Levenberg algorithm as implemented by gnuplot. The error bars are
computed by the jackknife method, unless specified otherwise. Histograms for
the probability distributions and for the cumulative probabilities do not
contain error bars since the statistics is done with a large amount of data
such that the error is not significant.

Figure 2: Phase distribution in the perfect coupling regime, $|\langle
S\rangle|~{}\sim 0$, for the PBRM model at criticality for several wire
lengths $N$ as indicated in panel (g) and several values of $b$, as indicated
in each panel. The symmetries are $\beta=1$ (first column), 2 (second column),
and 4 (third column). Symbols correspond to numerical results while black
lines correspond to the RMT prediction (5).

In Fig. 1, the modulus of the optical matrix as a function of the coupling
strength $w_{0}$ for the PBRM model at criticality with $\beta=1$ (first
column), $\beta=2$ (second column), and $\beta=4$ (third column) is shown.
Different values of the bandwidth $b$ and wire lengths $N$ are considered. It
is observed that the perfect coupling, $|\langle S\rangle|\approx 0$, does not
depend on the wire length, but shows a strong dependence on $b$. The perfect
coupling is attained for values of $w_{0}<0.5$, $0.45<w_{0}<1$, and
$w_{0}\approx 1$, when the system is in the insulator-like (top panels), in
between the insulator-like and the metallic-like (middle panels), and close to
the metallic-like regime (bottom panels), that is for $b=0.1$, 1, and 10,
respectively. In the insets, the phase distribution $\mathcal{P}(\Phi)$ is
shown for a coupling strength $|\langle S(w_{0})\rangle|\approx 0.5$. The
histograms in red lines correspond to numerical results while the continuous
black lines correspond to the analytical distribution of Eq. (5). For $b=0.1$
and 1, the histograms show two peaks around $\Phi=\pi$ which vanish for
$b=10$, i.e., when the system displays a metallic behavior. In the latter, a
good agreement with the RMT prediction is observed.

The phase distribution in the perfect coupling regime for the PBRM model at
criticality with $\beta=1$ (first column), 2 (second column), and 4 (third
column) symmetries is shown in Fig. 2. The wire lengths are indicated in Fig.
2(g) while the values of parameter $b$ are indicated in each panel. The
symbols are obtained by numerical simulations while the solid lines are the
RMT prediction of Eq. (5). Again, two big peaks around $\Phi=\pi$ show up when
the system is close to the insulator-like regime $b=0.1$ (top panels) which
tend to disappear as $b$ increases (middle and bottom panels). When the system
attains a metallic behavior, $b=10$ (bottom panels), the phase is uniformly
distributed around $1/2\pi$, in accordance with the RMT prediction (5). It is
also clear that $\mathcal{P}(\Phi)$ does not depend on the wire length nor the
symmetry (orthogonal, unitary, or symplectic), as expected in the one open-
channel setup MelloBook .

Figure 3: Distribution of the Wigner delay time for the PBRM model at
criticality for several wire lengths $N$, as indicated in panel (a). The
symmetries are $\beta=1,2$, and 4, in the first, second, and third columns,
respectively. The different values of $b$ are indicated in each panel. The
histograms are obtained through Eq. (6), with the normalization constant
$w_{0}^{2}\pi/(2L)$, in the perfect coupling regime. The dashed black lines
(bottom panels) correspond to the RMT prediction of Eq. (7).

###  III.2 Wigner delay time and resonance widths

The delay experienced by a quantum particle due to its interactions with a
scattering region is described by the so-called Wigner delay time
$\tau_{\text{w}}$. Near the center of the spectrum ($E=0$), it is given by
Steinbach2000 ; Kottos2002

$\tau_{\text{w}}(E=0)=\left.\frac{d\Phi(E)}{dE}\right|_{E=0}=-2\;\text{Im}\;\text{Tr}\left.\left(E-H_{\text{eff}}\right)^{-1}\right|_{E=0}.$
(6)

In the metallic regime and for the one open-channel setup, the distribution of
the Wigner delay time is known for all symmetry classes $\beta=1,2,$ and 4
Gopar1996 ; Fyodorov1997a ; Fyodorov1997b ; Angel2017 , namely

$\mathcal{P}(\tau_{\text{w}})=\frac{2/\beta}{(\beta/2)!}\left(\frac{\beta}{2\tau_{\text{w}}}\right)^{2+\beta/2}\mathrm{e}^{-\beta/2\tau_{\text{w}}}.$
(7)

On the other hand, the poles of the scattering matrix show up as resonances
which in turn are the complex eigenvalues
$\mathcal{E}_{n}=E_{n}-i\Gamma_{n}/2$ of the effective non-Hermitian
Hamiltonian $H_{\text{eff}}$ [see Eq. (3)], with $E_{n}$ and $\Gamma_{n}$
respectively the position and width of the $n$th resonance. Furthermore, the
resonance width $\Gamma_{n}$ is related to the life time of the $n$th
resonance as $\tau_{n}=1/\Gamma_{n}$, and hence a relationship between
$\Gamma_{n}$ and the Wigner delay time is expected. These quantities, delay
times and resonance widths, have been of pivotal importance in the realm of
complex scattering both theoretically and experimentally Mendez2005 ;
Kottos2002 ; Steinbach2000 ; Gopar1996 ; Fyodorov1997a ; Fyodorov1997b ;
Angel2017 ; Kukulin1989 ; Doron1990 ; Kottos2005 ; Fyodorov2012 ; Kuhl2008 ;
Fyodorov2015 ; Novaes2022 , where recent progress has been made to extend
their study to wave-chaotic scattering systems in the presence of absorption
Chen2021a ; Chen2021b . For the PBRM model at the critical point, to our
knowledge, exact theoretical results for the resonance statistics are scarce,
nonetheless recent progress has been reported for resonance statistics in
standard banded matrices in both weak and strong localization regimes
Fyodorov2022 .

The distribution of the Wigner delay time for the PBRM model at criticality is
reported in Fig. 3 for the three symmetry classes $\beta=1,2$, and 4, in the
first, second, and third column, respectively. The values of the bandwidth $b$
are shown in each panel. For $b=0.1$ (top panels),
$\mathcal{P}(\tau_{\text{w}})$ has its maximum at $\tau_{\text{w}}\sim 0$;
that is, the system is in the localized regime and conduction is suppressed.
For $b=1$ (middle panels), relatively small time delays dominate, meaning that
the system is neither an insulator nor a conductor. For $b=10$ (bottom
panels), the distribution of $\tau_{\text{w}}$ is well described by its RMT
prediction (7) which sets the system in a metallic-like regime.

The logarithm of the distribution of the resonance widths, normalized to its
typical value
$\Gamma^{\text{typ}}\equiv\exp\left\langle\ln\Gamma\right\rangle$, for the
PBRM model at criticality with $\beta=1$ (first column), 2 (second column),
and 4 (third column) are shown in Fig. 4. The wire lengths under consideration
are indicated in Fig. 4(g), and the different values of $b$ in each panel. For
the histograms, only 25% of the eigenvalues around the center of the spectrum,
$E=0$, are used. The typical value $\Gamma^{\text{typ}}$ follows a power-law
with respect to the wire length $N$, $\Gamma^{\text{typ}}\propto
N^{-\lambda}$, as observed in the insets of the same figure. There, the dashed
lines are the best fittings to the numerical data. The resulting exponents
$\lambda$ for each case are reported in Table 1.

###  III.3 Wigner delay time vs spectral and eigenstate properties of the
isolated PBRM model

It is well known that the spatial fluctuations of the eigenstates of
disordered systems at criticality show multifractal behavior Janssen1994 ;
Janssen1998 ; Evers2008 ; Hashimoto2008 ; Faez2009 ; Richardella2010 . This
behavior is characterized by a set of _generalized dimensions_ $D_{q}$ or
_multifractal dimensions_ , where $q$ is a real number. Furthermore, the
multifractal properties of the eigenstates can also be studied through the
Wigner delay time, as shown below.

For disordered systems at criticality, a relationship between the inverse
moments of the Wigner delay time and the multifractal dimensions of the
eigenstates of the corresponding isolated system $D_{q}$ is given by
Ossipov2005 ; Mendez2005 ; Mirlin2006 ,

$\left\langle\tau_{\text{w}}^{-q}\right\rangle\propto
N^{-\sigma_{q}},\quad\text{where}\quad\sigma_{q}\equiv q\;D_{q+1}$ (8)

and for the PBRM model at criticality the following functional form for
$\sigma_{q}$ as a function of the bandwidth $b$ has been proposed Mendez2014

$\sigma_{q}\approx\dfrac{q}{1+(\alpha_{q+1}b)^{-1}}$ (9)

with $\alpha_{q}$ being fitting constants. In addition, for PBRM models at
criticality, the typical value of the Wigner delay time, defined as
$\tau_{\text{w}}^{\text{typ}}\equiv\exp\left\langle\ln\tau_{\text{w}}\right\rangle$,
obeys the scaling law given by Mendez2006

$\tau_{\text{w}}^{\text{typ}}\propto
N^{\sigma_{\tau}},\quad\text{where}\quad\sigma_{\tau}\equiv D_{1}.$ (10)

Figure 4: Distribution of logarithm of resonance widths, normalized to its
typical value, for the PBRM model at criticality with $\beta=1,2,$ and 4, in
the first, second, and third columns, respectively. The system sizes $N$ are
indicated in panel (g). The values of the bandwidth are indicated in each
panel. The insets show the scaling law $\Gamma^{\text{typ}}\propto
N^{-\lambda}$. The black dashed lines are the best fitting of the scaling law
to the numerical data. The error bars are smaller than the symbols. Figure 5:
Distribution of the first inverse moment of the Wigner delay time for the PBRM
model at criticality in the presence of the $\beta=1$ (first column), 2
(second column), and 4 (third column) symmetries. In top panels $b=0.1$, in
middle panels $b=1$, and in bottom panels $b=10$. The wire lengths $N$ are
indicated in panel (g). The insets show the best fitting (black dashed lines)
using the scaling law (8) to the numerical data (symbols). The error bars are
smaller than the symbols size. Figure 6: Distribution of the Wigner delay time
normalized to its typical value for the PBRM model at criticality. The
symmetry classes are $\beta=1$ (first column), 2 (second column), and 4 (third
column). The bandwidth $b$ is indicated in the panels and the wire lengths $N$
are indicated in panel (g). The insets show fittings of the scaling law (10)
(dashed lines) to the numerical data (dots). The error bars are smaller than
the symbol size.

For completeness, the _level compressibility_ $\chi$, a quantity often used to
characterize the fluctuations of spectra of disordered systems at criticality,
is also analyzed. In the metallic [insulator] regime $\chi=0$ [$\chi=1$] while
at intermediate regimes (neither metallic nor insulator) $0<\chi<1$
Chalker1996 ; Klesse1997 . For PBRM models in the presence of $\beta=1$ and 2
symmetries, the level compressibility as a function of $b$ is given by
Evers2008 ; Kravtsov2011

$\chi=\left\\{\begin{array}[]{lr}1-4b,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\;~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\beta=1,&b\ll
1,\\\ 1-\pi\sqrt{2}b+\frac{4}{3}(2-\sqrt{3})\pi^{2}b^{2},\,\,\,\beta=2,&b\ll
1,\\\ 1/2\beta\pi b,&b\gg 1.\end{array}\right.$ (11)

In addition a heuristic relation between $\chi$ and $\sigma_{q}$ is also
known, namely Mendez2014

$\chi\approx\frac{q-\sigma_{q}}{q(\sigma_{q}+1)}.$ (12)

In Fig. 5, the distribution of the logarithm of the first inverse moment of
the Wigner delay time for the PBRM model at criticality is reported. The
values of the bandwidth $b$ and wire lengths $N$ are indicated in the panels.
The insets show $\ln\langle\tau_{\text{w}}^{-1}\rangle$ vs $\ln N$ as dots
while the dashed lines correspond to fittings to the numerical data with Eq.
(8). The resulting exponents $\sigma_{q}$ from the fittings are reported in
Table 1. A good agreement between the numerical data and the scaling law of
Eq. (8) is observed.

The behavior of the logarithm of the distribution of
$\ln\big{(}\tau_{\text{W}}/\tau_{\text{W}}^{\text{typ}}\big{)}$ for the PBRM
model at criticality with the three symmetry classes $\beta=1,2,$ and 4, for
different wire lengths $N$ [see Fig. 5(g)], and different bandwidths $b$, is
plotted in Fig. 6. The insets show the logarithm of
$\tau_{\text{w}}^{\text{typ}}$ as a function of $\ln N$. The dots correspond
to numerical results while the dashed lines correspond to fittings with the
scaling law of Eq. (10) to the numerical data. The exponents $\sigma_{\tau}$
resulting from the fittings are reported in Table 1. A good agreement between
the numerical data and the scaling law (10) is observed.

Table 1:  Values of the parameters obtained from the power-law fittings to the data of the insets in Figs. 4, 5, and 6. | FIG. 4 | FIG. 5 | FIG. 6
---|---|---|---
|  |  | $\Gamma^{\text{typ}}\propto N^{-\lambda}$ | $\langle\tau_{\text{W}}^{-1}\rangle\propto N^{-\sigma_{q}}$ | $\tau^{\text{typ}}\propto N^{\sigma_{\tau}}$
| $b$ | Inset panel | $\lambda$ | $\sigma_{q}$ | $\sigma_{\tau}$
$\beta=1$ | 0.1 | (a) | $1.7647\pm 0.0072$ | $0.1404\pm 0.0037$ | $0.2507\pm 0.0048$
1 | (d) | $1.2265\pm 0.0080$ | $0.7133\pm 0.0068$ | $0.8465\pm 0.0055$
10 | (g) | $1.0276\pm 0.0213$ | $0.9439\pm 0.0180$ | $0.9586\pm 0.0163$
$\beta=2$ | 0.1 | (b) | $1.6424\pm 0.0036$ | $0.2282\pm 0.0034$ | $0.3744\pm 0.0035$
1 | (e) | $1.1490\pm 0.0063$ | $0.8469\pm 0.0023$ | $0.9217\pm 0.0027$
10 | (h) | $1.0187\pm 0.0244$ | $0.9609\pm 0.0192$ | $0.9685\pm 0.0204$
$\beta=4$ | 0.1 | (c) | $1.7223\pm 0.0245$ | $0.1699\pm 0.0052$ | $0.2990\pm 0.0023$
1 | (f) | $1.1596\pm 0.0068$ | $0.8499\pm 0.0060$ | $0.9257\pm 0.0040$
10 | (i) | $1.0218\pm 0.0456$ | $0.9590\pm 0.0175$ | $0.9680\pm 0.0196$

In Fig. 7, the multifractal dimension $D_{q}$ as a function of the bandwidth
$b$ for several values of $q$ is plotted. Figures 7(a), 7(b), and 7(c)
correspond to the system in the presence of the $\beta=1,2,$ and 4 symmetry,
respectively. The empty symbols are computed by direct diagonalization of the
isolated PBRM mode at criticality (see also Ref. Carrera2021  for more
details). The filled symbols are obtained from the scaling law (8) and the
dashed lines correspond to Eq. (9) with the $\alpha_{q+1}$ taken from Ref.
Carrera2021 . These values of $\alpha_{q+1}$ are: 3.33 (5.70, 4.82), 2.55
(4.45, 3.82), 2.11 (3.73, 3.22), 1.52 (2.74, 2.41), and 1.21 (2.22, 1.98) for
$\beta=1\;(2,4)$ and $q=0.2,0.6,1,2$, and 3, respectively Carrera2021 . For
the three symmetry classes $\beta=1$ (a), 2 (b), and 4 (c) a good
correspondence between the direct calculation of $D_{q}$ (empty symbols) and
the analytics (dashed lines) is found. These results show that for the one
channel setup, the multifractal properties of the isolated PBRM model at
criticality can be directly extracted from Wigner delay time, a transport
property. This is convenient from an experimental point of view.

Figure 7: Multifractal dimensions $D_{q}$ for the PBRM model at criticality as
a function of the bandwidth $b$. The empty symbols are obtained from direct
diagonalization of the closed system (1). The filled symbols correspond to
results obtained from the scaling law (8). The dashed lines correspond to the
heuristic relation (9). The error bars are the rms of the residuals. For the
sake of clarity, the symbols (lines) are displaced vertically upward.

In Fig. 8 the spectral compressibility of the PBRM model at criticality is
reported. The symbols are obtained from Eq. (12) with the $\sigma_{q}$
extracted from the scaling law (8). The dashed lines correspond to the
theoretical prediction of Eq. (11). For the PBRM model in the presence of the
symmetry classes, $\beta=1$ and 2, Figs. 8(a) and 8(b), respectively, a good
agreement with the analytics is observed. For the $\beta=4$ case, there is no
theoretical prediction available to compare with. However, the following
recursive relation Mendez2014 :

$\sigma_{q}(q-\sigma_{q})\approx\alpha_{q+1}b$ (13)

is known. Then, the PBRM model in the $\beta=4$ case can also be contrasted
with this last relation. For this purpose, in the insets of Fig. 8, the
recursive relation (13) (symbols) is shown. The dashed lines are $\sim b$. A
good agreement with the numerical data (symbols) is observed.

Figure 8: Level compressibility of the PBRM model at criticality as a function
of the bandwidth $b$. The symmetries are $\beta=1,2,$ and 4, in panels (a),
(b), and (c), respectively. The symbols are obtained from Eq. (12), the dashed
red lines correspond to expression (11). Inset: Recursive relation (13), the
dashed red lines ($\sim b$) are plotted to guide the eye. The error bars are
the rms of the residuals.

To this point, the analysis of the PBRM model at criticality with one open
channel has been performed. In the next sections, the setup in which the
scattering system supports $M$ open channels is studied and contrasted with
available RMT predictions.

##  IV PBRM model with $M$ open channels

In this section, the analysis of the scattering and transport properties of
the PBRM model at criticality with $M$ open channels is presented. As in the
previous section, this is performed following the scattering matrix approach
of Eq. (3). As shown in Sec. III for the one open-channel setup, the
scattering and transport properties of the PBRM model at criticality do not
change significantly with the wire length $N$. A similar behavior has also
been found for the $M$ open-channels setup in the $\beta=1$ case Mendez2005 ;
Mendez2006 ; Mendez2010 . Here, this has also been verified for the $\beta=2$
and 4 cases (not shown). Therefore, in what follows the wire length is set to
$N=50$. For the statistical analysis, $10^{6}$ realizations of the scattering
matrix $S$ in the perfect coupling regime are considered. In the same line as
in Sec. III, known analytical RMT predictions are presented first and later
they are compared with numerical simulations in the appropriate limits.

###  IV.1 Scattering properties

Within the RMT approach of quantum transport, the average of the magnitude of
the elements of the scattering matrix $S$ has been obtained Beenakker1997 .
That is

$\left\langle|S_{nm}|^{2}\right\rangle_{\text{RMT}}=\frac{1-(1-2/\beta)\delta_{n,m}}{M-1+2/\beta},$
(14)

where $\beta$ is the symmetry class present in the system, $M$ is the number
of open channels, and $\delta_{nm}$ is the usual Kronecker delta.

From Eq. (14) and based on numerical simulations for the $M=2$ open-channel
case, it has been conjectured that the average of the $S$-matrix elements,
$\langle|S_{12}|^{2}\rangle$ and $\langle|S_{11}|^{2}\rangle$, can be
expressed as a function of the bandwidth $b$ Alcazar2009 ; Mendez2010 . These
are given by

$\displaystyle\left\langle|S_{12}|^{2}\right\rangle(b)$
$\displaystyle=\dfrac{\langle|S_{12}|^{2}\rangle_{\text{RMT}}}{1+(\varepsilon
b)^{-2}}\quad\mathrm{and}$ (15)
$\displaystyle\left\langle|S_{11}|^{2}\right\rangle(b)$
$\displaystyle=1-\left\langle|S_{12}|^{2}\right\rangle,$ (16)

where $\varepsilon$ is a free parameter to be determined by the best fitting
to the numerical data. Furthermore, by using a phenomenological expression
that relates the bandwith $b$ with the correlation dimension $D_{2}$, which is
a broadly accepted measure of the spatial extension of the eigenfunctions of
disordered systems at the critical point, Eqs. (15) and (16) can also be
written in terms of $D_{2}$ Alcazar2009 ; Mendez2010 . For the $\beta=1$ case
one gets

$\displaystyle\left\langle|S_{12}|^{2}\right\rangle(D_{2})$
$\displaystyle=\dfrac{\langle|S_{12}|^{2}\rangle_{\text{RMT}}}{1+(\kappa/\varepsilon)^{2}\;(D_{2}^{-1}-1)^{2}}\quad\mathrm{and}$
(17) $\displaystyle\left\langle|S_{11}|^{2}\right\rangle(D_{2})$
$\displaystyle=1-\left\langle|S_{12}|^{2}\right\rangle(D_{2}),$ (18)

where $\kappa$ is a fitting parameter and $\varepsilon$ is obtained from Eq.
(15). Meanwhile, for the $\beta=2$ and 4 cases one gets

$\displaystyle\left\langle|S_{12}|^{2}\right\rangle(D_{2})$
$\displaystyle=\left\\{\begin{array}[]{lr}\dfrac{\left\langle|S_{12}|^{2}\right\rangle_{\text{RMT}}}{1+\left(\kappa
D_{2}/\pi\right)^{-2}},&b\ll 1,\\\\[14.22636pt]
\dfrac{\left\langle|S_{12}|^{2}\right\rangle_{\text{RMT}}}{1+\left[2\pi(1-D_{2})/\rho\right]^{2}},&b\gg
1,\end{array}\right.$ (21)
$\displaystyle\left\langle|S_{11}|^{2}\right\rangle(D_{2})$
$\displaystyle=1-\left\langle|S_{12}|^{2}\right\rangle(D_{2}),$ (22)

where $\kappa$ and $\rho$ are fitting parameters, different for each symmetry
class $\beta$. Notice that the expressions (15), (17), and (21) were obtained
heuristically and its analytical derivation remains to be proven, while the
relations (16), (18), and (22) are a consequence of the flux conservation
condition of the $S$ matrix. However, as shown below, these equations describe
well the numerical data.

In Fig. 9 the average of the $S$-matrix elements, when the PBRM model at
criticality supports $M=2$ open channels, is shown for the $\beta=1,2,$ and 4
symmetries, in the first, second, and third columns, respectively. The symbols
correspond to numerical simulations and the dashed blue lines are the RMT
prediction (14) for which
$\left\langle|S_{12}|^{2}\right\rangle_{\text{RMT}}=1/3,1/2,$ and $2/3$ for
$\beta=1,2,$ and 4, respectively. In the same figure,
$\left\langle|S_{11}|^{2}\right\rangle_{\text{RMT}}=1-\left\langle|S_{12}|^{2}\right\rangle_{\text{RMT}}$
is also shown. The dashed red curves in Figs. 9(a), 9(b), and 9(c) correspond
to the generalizations of Eqs. (15) and (16) for the $\beta=1,2,$ and 4
symmetry, respectively. Figures 9(d), 9(e), and 9(f), show
$\left\langle|S_{12}|^{2}\right\rangle(D_{2})$ with the values of $D_{2}$
extracted from Fig. 7. The dashed red lines are Eqs. (17)-(22). The obtained
fitting parameters $\varepsilon$, $\kappa$, and $\rho$, for each symmetry
class are reported in Table 2. In all the cases, it is observed that for large
$b$ and $D_{2}$ the model is well described by the RMT predictions while the
corresponding generalizations work well for any $b$ and $D_{2}$.

Figure 9: Average of the $S$-matrix elements for the PBRM model at criticality with $M=2$ open channels. The symmetries are $\beta=1$ (first column), 2 (second column), and 4 (third column). The symbols correspond to numerical simulations, the dashed blue lines are the corresponding RMT prediction (14). Top panels: Average of the $S$-matrix elements as a function of the bandwidth $b$. The dashed red lines are Eq. (15) and (16). Bottom panels: same as top panels but as a function of the correlation dimension $D_{2}$ compared with Eqs. (17)-(22). Error bars are smaller than the symbol size.  Table 2:  Values of the parameters obtained by fittings to the data of figure 9. $\beta$ | $\varepsilon$ | $\kappa$ | $\rho$
---|---|---|---
1 | $2.5129\pm 0.0017$ | $2.1043\pm 0.0242$ |
2 | $3.0618\pm 0.0022$ | $4.7867\pm 0.0017$ | $2.7857\pm 0.0065$
4 | $1.9359\pm 0.0030$ | $3.9909\pm 0.0078$ | $1.8036\pm 0.0038$

###  IV.2 Transmission and shot noise power

In this section, known analytical results from RMT for the scattering and
transport properties of complex media are first revised. Those results are
expected to describe the PBRM model at criticality in the case of $b\gg 1$,
i.e., in the metallic-like regime.

It is well established that given a scattering problem, the transmission
coefficient can be obtained from the elements of the scattering matrix as

$T=\text{Tr}(t\;t^{\dagger}),$ (23)

where Tr is the trace operation, $t$ is the transmission amplitude [see Eq.
(3)], and the symbol $\dagger$ represents the adjoint of $t$.

For the $M=2$ open-channel setup, or two single-channel leads attached to a
complex scattering media, the transmission distribution is given by
Beenakker1997

$\mathcal{P}(T)=\frac{1}{2}\beta T^{-1+\beta/2},\quad 0<T<1,$ (24)

for the symmetry class labeled by $\beta$.

Also, for the $M=4$ open-channel setup, or four single-channel leads attached
to a complex scattering media, the distribution of $T$ for the $\beta=1$ case
is MelloBook

$\displaystyle\mathcal{P}(T)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rl}\frac{3}{2}T,\qquad 0<T<1,\\\
\frac{3}{2}(T-2\sqrt{T-1}),\qquad 1<T<2,\end{array}\right.$ (27)

while for the $\beta=2$ case it is MelloBook

$\displaystyle\mathcal{P}(T)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rl}2T^{3},\qquad 0<T<1,\\\
2(2-T)^{3},\qquad 1<T<2.\end{array}\right.$ (30)

Now, for the $\beta=4$ symmetry, following Ref. MelloBook  it is
straightforward to arrive at

$\mathcal{P}(T)=\left\\{\begin{array}[]{rl}\frac{12}{7}T^{7},\quad
0<T<1,\\\\[5.69046pt] \frac{12}{7}(2-T)^{5}(T^{2}+10T-10),\quad
1<T<2.\end{array}\right.$ (31)

In general, the tail of the transmission distribution for all $\beta$ and any
number of open channels $M$ decays as MelloBook

$\mathcal{P}(T)\propto T^{-1+\beta M^{2}/2}.$ (32)

Furthermore, for a complex scattering media attached to two leads supporting
respectively $N_{1}$ and $N_{2}$ open channels, the average transmission and
its variance are given by Baranger1994 ; Beenakker1997

$\left\langle T\right\rangle=\frac{N_{1}N_{2}}{N_{1}+N_{2}-1+2/\beta}$ (33)

and

$\mathrm{var}\left(T\right)=\frac{2N_{1}N_{2}(N_{1}-1+\frac{2}{\beta})(N_{2}-1+\frac{2}{\beta})}{\beta(N_{1}+N_{2}-2+\frac{2}{\beta})(N_{1}+N_{2}-1+\frac{4}{\beta})(N_{1}+N_{2}-1+\frac{2}{\beta})^{2}},$
(34)

for all symmetry classes labeled by $\beta$.

Another transport quantity of interest is the so-called shot noise power $P$,
defined as
$P=\left\langle\text{tr}\left(tt^{\dagger}-tt^{\dagger}\;tt^{\dagger}\right)\right\rangle$,
whose probability distribution is given by Savin2006

$P=\frac{N_{1}\left(N_{1}-1+\frac{2}{\beta}\right)N_{2}\left(N_{2}-1+\frac{2}{\beta}\right)}{\left(K-2+\frac{2}{\beta}\right)\left(K-1+\frac{2}{\beta}\right)\left(K-1+\frac{4}{\beta}\right)},$
(35)

where $K=N_{1}+N_{2}$ is the total number of open channels. Note that although
the distribution (35) was derived for the $\beta=1$ and 2 symmetries, it also
encompasses the symplectic case, $\beta=4$, as will be verified below for the
PBRM model with $b\rightarrow\infty$.

Figure 10: Transmission distribution of the PBRM model at criticality with
$M=2$ open channels and several values of $b$. The symmetries are $\beta=1,2,$
and 4, in the first, second, and third columns, respectively. The red lines
correspond to histograms obtained from numerical simulations while the dashed
blue lines are the RMT prediction (24) for the respective $\beta$. A smooth
transition from localized-like to metallic-like regime is observed for the
three symmetry classes.

In Eqs. (33)-(35) the number of open channels, $N_{1}$ and $N_{2}$, may be
different. However, in order to compare these quantities with numerical
simulations of the PBRM model, we set $N_{1}=N_{2}$ with $M=N_{1}+N_{2}$ the
total number of open channels.

For the PBRM model at criticality with two single-channel leads attached to it
($M=2$ open channels), the transmission distribution for several values of $b$
and $\beta=1,2,$ and 4 (first, second, and third column, respectively), is
shown in Fig. 10. The red lines correspond to histograms obtained from
numerical simulations while the dashed blue lines are Eq. (24) for the
respective $\beta$. For the three symmetry classes under consideration, a
smooth transition from a localized-like to a metallic-like regime is observed
as the bandwidth $b$ increases. The metallic-like regime is reached when $b=4$
for which a good agreement with the RMT prediction (24) is obtained [see Figs.
10(j), 10(k), and 10(l)].

Figure 11: Distribution of the logarithm of transmission for the PBRM model at
criticality with $M=2$ and several values of the bandwidth $b$, as indicated
in panel (g). The symmetries are $\beta=1$ (first column), 2 (second column),
and 4 (third column). The error bars in panels (d)-(f) are smaller than the
symbol size. See text for the discussion.

It is instructive to look at the behavior of the transmission distribution
$\mathcal{P}(T)$ in the insulator-like regime $b\ll 1$ where RMT predictions
are not available. In this regime where $\mathcal{P}(T)\approx 0$ it is more
convenient to analyze the distribution of $\ln T$. For the PBRM model at
criticality with $M=2$, the distribution of the logarithm of $T$ for $\beta=1$
(first column), 2 (second column) and 4 (third column) and several values of
the $b<1$, as indicated in Fig. 11(g), is shown in Fig. 11. In Figs. 11(a),
11(b), and 11(c), it is observed that the shape and width of
$\ln\mathcal{P}(\ln T)$ do not change despite the fact that $b$ varies. This
means that $\ln\mathcal{P}(\ln T)$ should be scale invariant, a property that
was confirmed before for $\beta=1$ with the typical transmission
$T^{\text{typ}}=\exp\left\langle\ln T\right\rangle$ as scaling parameter. Also
note in Figs. 11(d)–(f) that $\left\langle\ln T\right\rangle\sim\ln b$ for
$b<1$. Indeed, all distributions $\ln\mathcal{P}(\ln T)$ fall on top of each
other when plotted as a function of $\ln(T/T^{\text{typ}})$, as shown in Figs.
11(g), 11(h), and 11(i). We stress that this behavior has previously been
reported for the PBRM model in the presence of the $\beta=1$ symmetry but
neither for the $\beta=2$ (periodic) nor for the $\beta=4$ symmetries.

Since the tails of the distribution $\mathcal{P}(T)$ from the PBRM model in
the metallic-like regime ($b\rightarrow\infty$) are expected to correspond to
Eq. (32), then it is reasonable to assume that those tails may be described by

$\mathcal{P}(T)\propto T^{\nu}$ (36)

for all $b$, with $\nu$ a fitting parameter. Furthermore, for the PBRM model
with $\beta=1$ symmetry, in Ref. Antonio2009  the following relationship
between the exponent $\nu$ and the correlation dimension $D_{2}$:

$\nu(D_{2})\propto(1-D_{2})^{2},\quad b\gg 1,$ (37)

has been proposed.

Figure 12: Exponents $\nu$ extracted from the best fitting of Eq. (36) to the
numerical transmission distribution of the PBRM model at criticality for
$M=2$. The symmetries are $\beta=1,2,$ and 4, in the first, second, and third
columns respectively. Top panels: $\nu$ as a function of the bandwidth $b$.
The red inverted triangles correspond to numerical results for $T>0.5$, the
dashed blue lines are the RMT prediction (32) for the corresponding symmetry
classes and the dashed black lines are the best fitting of $b^{2}$ to the
data. Bottom panels: Same as top panels but as a function of the correlation
dimension $D_{2}$ whose relationship with $b\gg 1$ is given through
$D_{2}(b)=1-(2\pi b)^{-1}$ Evers2008 ; Antonio2009 . The fittings are
performed by using $(1-D_{2})^{2}$. The error bars are the rms of the residual
and are smaller than the symbol size.

The asymptotic behavior of the transmission distribution for the PBRM model at
criticality as a function of the bandwidth $b$ and of the correlation
dimension $D_{2}$ is analyzed in Fig. 12 for the $\beta=1,2,$ and 4 symmetries
in the first, second, and third columns, respectively. In Figs. 12(a), 12(b),
and 12(c), for each value of $b$ the corresponding exponent $\nu$ (red
inverted triangles) is obtained from the best fitting of Eq. (36) to the
numerical data for $T>0.5$. The exponent $\nu$ as a function of $D_{2}$ is
shown in Figs. 12(d), 12(e), and 12(f). The relationship between $D_{2}$ and
the bandwidth $b\gg 1$ is given by $D_{2}(b)=1-(2\pi b)^{-1}$ Evers2008 ;
Antonio2009 . In all panels the dashed blue lines are the RMT prediction (32)
according to the symmetry class. The dashed black lines are the best fittings
to the numerical data, which are proportional to $~{}b^{-2}$ (top panels) and
to $(1-D_{2})^{2}$ (bottom panels). The latter shows that the relationship
(37) is valid for $D_{2}>0.75$.

Figure 13: Transmission distribution of the PBRM model at criticality for
$M=4$ and several values of the effective bandwidth $b$, as indicated in the
panels. The symmetries are $\beta=1,2,$ and 4 in the left, middle, and right
columns, respectively. The red lines are histograms obtained by numerical
simulations and the dashed blue lines are the corresponding RMT predictions
(27)-(31).

The transmission distribution for the PBRM model at criticality with four
single-channel leads attached it ($M=4$) is reviewed in Fig. 13. The
symmetries are $\beta=1,2,$ and 4, in the left, middle, and right panels,
respectively, and the considered values of $b$ are indicated in the panels.
The red lines are histograms obtained from numerical simulations, and the
dashed blue lines are the RMT predictions (27)–(31) according to the symmetry
class. For each case, a smooth transition from an insulator-like to a
metallic-like regime is observed. In Figs. 13(j) and 13(k), the metallic-like
regimen is reached for values of $b=4$ when the system is in the presence of
the orthogonal and the unitary symmetry, respectively. However, when the
system is in the presence of the symplectic symmetry, a larger value of $b$ (=
10) is required to reach the metallic-like phase, as shown in Fig. 13(l).
Close to the insulator-like regime ($b<0.5$), the transmission distribution
shows a pretty similar behavior to that obtained for the $M=2$ case, as
observed in Fig. 14 (see also Fig. 11). There, $\ln\mathcal{P}(\ln T)$ also
shows a scaling property, i.e., all the curves fall on top of each other when
normalized to its typical value $T^{\text{typ}}=\exp\left\langle\ln
T\right\rangle\sim b^{2}$, as observed in Figs. 13(g), 13(h), and 13(i). Also,
$\left\langle\ln T\right\rangle$ is a linear function of $\ln b$ for $b<1$,
see Figs. 13(d), 13(e), and 13(f).

Figure 14: Distribution of the logarithm of transmission of the PBRM model at
criticality with $M=4$ and several values of the bandwidth $b$, as indicated
in panel (g). The symmetries are $\beta=1$ (first column), 2 (middle column),
and 4 (third column). The error bars in panels (d)-(f) are smaller than the
symbol size.

For the PBRM model at criticality with $\beta=1$ symmetry, it has been
conjectured that the averages $\left\langle
T\right\rangle,\;\text{Var}(\left\langle T\right\rangle)$, and $P$, as a
function of the bandwidth $b$, obey the following expression

$X(b)=X_{\text{RMT}}\left[\frac{1}{1+(\delta\;b)^{-2}}\right],$ (38)

where $X$ may be $\left\langle T\right\rangle,\;\text{Var}(\left\langle
T\right\rangle)$, or $P$; $X_{\text{RMT}}$ is the corresponding RMT prediction
and $\delta$ is a fitting parameter. In what follows, this conjecture is
verified for the PBRM model at criticality for the $\beta=2$ and 4 symmetries.
For completeness, the $\beta=1$ case is also reviewed. For the numerical
analysis, the wire length is set to $N=200$ and $10^{5}$ realizations of the
$S$ matrix of Eq. (3) are generated.

Figure 15: Average transmission for the PBRM model at criticality with
symmetry $\beta=1,2,$ and 4, in the first, second, and third columns,
respectively. Top panels: $\left\langle T\right\rangle$ as a function of $M$
for several values of the bandwidth $b$. The symbols correspond to numerical
simulations, the dashed blue lines are the RMT predictions (33), and the red
line at $\left\langle T\right\rangle=0$ is plotted to guide the eye. Bottom
panels: Conjecture (38) for $\left\langle T\right\rangle$, the symbols
correspond to numerical simulations. Insets: Fitting parameter from (38) for
each $M$, the error bars are the rms of the residuals and are smaller than the
symbol size.

In Figs. 15(a), 15(b), and 15(c), for the PBRM model at criticality
$\left\langle T\right\rangle$ as a function of the open channels $M$ is shown
for $\beta=1,2,$ and $4$, in the first, second, and third columns,
respectively, and for several values of $b$. The symbols correspond to
numerical results while the dashed blue lines are the RMT predictions (33)
according to the symmetry class. The red horizontal line at $\left\langle
T\right\rangle=0$ is shown to guide the eye. In those panels, it is observed
that for the three symmetry clases at small $b<0.2$, $\left\langle
T\right\rangle\approx 0$ since the system is in the insulator-like regime. As
$b$ increases ($0.2\leq b<4$) the average transmission also increases until it
reaches the RMT prediction ($b=10$) where the system is in the metallic-like
regime. For the three symmetry classes labeled by $\beta$, Eq. (33) gives an
accurate description even though deviations appear for the $\beta=4$ case when
$M\geq 4$. In the bottom panels the conjecture (38) for $\left\langle
T\right\rangle$, normalized to its RMT prediction (33), as a function of
$\delta b$ for $\beta=1$ [Fig. 15(d)], 2 [Fig. 15(e)], and 4 [Fig. 15(f)], is
shown. The symbols are obtained from numerical simulations while the dashed
blue and dashed red lines, at $\left\langle T\right\rangle=1$ and 0
respectively, are shown to guide the eye. In this case, the conjecture
correctly describes the numerical data as can be seen by the shape described
by the symbols and more clearly in the small error bars shown in the insets,
which are not even visible since they are smaller than the symbols.

Figure 16: Variance of transmission for the PBRM model at criticality in the
presence of the symmetry $\beta=1$ (first column), 2 (second column), and 4
(third column). Top panels: $\text{var}(T)$ as a function of $M$ for several
values of the bandwidth $b$, the symbols are the numerical results, dashed
blue lines are the RMT predictions (34), and the red line at $\text{var}(T)=0$
is shown to guide the eye. Bottom panels: Conjecture (38) for $\text{var}(T)$,
normalized to its corresponding RMT prediction. The symbols correspond to
numerical simulations while the dashed blue line is the expression (38).
Insets: Fitting parameter from (38) for each $M$. The error bars are the rms
of the residuals.

In Fig. 16, the results for the variance of $T$ as a function of $M$ for the
PBRM model at criticality are reported for several values of $b$ and for the
symmetries $\beta=1,2$, and 4, in the first, second, and third columns,
respectively. In the top panels, the symbols are the numerical data, the
dashed blue lines are the corresponding RMT prediction (34), and the red line
at $\text{var}(T)=0$ is shown to guide the eye. It can be observed that for
$M\geq 2$ and certain interval of $b$, the fluctuations are greater than the
RMT predictions. That behavior is more notorious for the $\beta=2$ and 4 cases
in the interval $0.4\leq b\leq 4$. However, in the metallic regime ($b=10$), a
good agreement with the RMT predictions is obtained.

The $\text{var}(T)$ normalized to its corresponding RMT prediction as a
function of $\delta b$ is plotted in Fig. 16(d), 16(e), and 16(f), for
$\beta=1,2,$ and 4, respectively. For $2<\delta b<40$ and $M\geq 2$ large
deviations between the conjecture (38) (dashed blue line) and the numerical
data (symbols) are observed, even though these deviations are smaller for the
$\beta=1$ case. This can also be appreciated in the insets where the error
bars are relatively large.

Finally, the shot noise power $P$ as a function of $M$ and for several values
of $b$ is reported in the top panels of Fig. 17. A transition from an
insulator-like ($b=0.01$) where $P=0$ (red horizontal line) to a metallic-like
($b=10$) regime (symbols) is clearly observed. In the metallic-like regime,
the numerical results (symbols) are in agreement with the RMT predictions (35)
(dashed blue lines). In the bottom panels of the same figure, the conjecture
(38) for $P$ as a function of $\delta b$ and several values of $M$ is shown. A
good agreement between the numerical data and expression (38) is obtained for
the three symmetry classes $\beta$, as revealed by the shape described by the
symbols and the small error bars obtained (see insets).

Figure 17: Shot noise power $P$ for the PBRM model at criticality with
symmetry $\beta=1$ (first column), 2 (second column), and 4 (third column).
Top panels: $P$ as a function of $M$ for several values of the bandwidth $b$,
the symbols correspond to numerical results, the dashed blue lines are the RMT
predictions (35), and the red line at $P=0$ is shown to guide the eye. Bottom
panels: Conjecture (38) for $P$, the symbols are obtained by numerical
simulation. Insets: Fitting parameter from (38) for each $M$, the error bars
are the rms of the residuals.

##  V Conclusions

In this paper, an extensive numerical study of the scattering and transport
properties of the PBRM model at criticality in the presence of the three
symmetry classes, orthogonal, unitary, and symplectic, has been presented. For
the sake of completeness, some known results for the orthogonal case
previously studied in the literature have also been reviewed, while new ones
for the unitary (in the periodic model) and the symplectic cases, which had
remained unexplored in the context of the PBRM models, are reported.
Surprisingly, the results presented confirm that the scattering and transport
properties of the symplectic PBRM model can be well described by existing
analytical and heuristic relations widely used in studies of the PBRM model in
the presence of the $\beta=1$ and 2 symmetries. Importantly, for the three
symmetry classes the multifractal properties of the isolated model were
obtained from scattering and transport properties, which is very convenient
from the experimental point of view since direct access to the eigenfunctions
is not required.

Additionally, an analytical result for the transmission distribution in the
presence of the symplectic symmetry with $M=4$ open channels (31), which
applies to the symplectic PBRM model at criticality in the metallic regime,
was provided. Also, in this study, the results for $\beta=1$ are in accordance
with those reported in, for example, Ref. Mendez2010 . Moreover, to our
knowledge, our results for $\beta=2$ have not been reported before; only the
nonperiodic version of the PBRM model with broken time-reversal symmetry
($\beta=2$) has been studied in Ref. Alcazar2009 . And the symplectic case,
$\beta=4$, has not been reported in neither the periodic nor in the
nonperiodic version of the PBRM model so far. Thus, with the present study a
more clear panorama about the scattering and transport properties of the PBRM
model at criticality in the presence of the three classical Wigner-Dyson
symmetries is given.

###### Acknowledgements.

We thank I. Varga for his useful comments on the manuscript. A. M. M-A
acknowledges financial support from CONACyT. M.C.-N. thanks financial support
from CONACyT (Grant “Ciencia de Frontera 2019”, No. 10872) and the facilities
provided by the “Centro de Análisis de Datos y Supercómputo (CADS)” from the
University of Guadalajara through of the Leo-Atrox Supercomputer. J.A.M.-B.
thanks support from CONACyT (Grant No. 286633), CONACyT-Fronteras (Grant No.
425854), VIEP-BUAP (Grant No. 100405811-VIEP2022), and Laboratorio Nacional de
Supercómputo del Sureste de México (Grant No. 202201007C), Mexico.

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