††institutetext: Yukawa Institute for Theoretical Physics, Kyoto University,
Sakyo-ku, Kyoto 606-8502, Japan

# Phase diagram near the quantum critical point
in Schwinger model at $\theta=\pi$:
analogy with quantum Ising chain

Hiroki Ohata<EMAIL_ADDRESS>
###### Abstract

The Schwinger model, one-dimensional quantum electrodynamics, has CP symmetry
at $\theta=\pi$ due to the topological nature of the $\theta$ term. At zero
temperature, it is known that as increasing the fermion mass, the system
undergoes a second-order phase transition to the CP broken phase, which
belongs to the same universality class as the quantum Ising chain. In this
paper, we explore the phase diagram near the quantum critical point (QCP) in
the temperature and fermion mass plane using first-principle Monte Carlo
simulations, while avoiding the sign problem by using the lattice formulation
of the bosonized Schwinger model. Specifically, we perform a detailed
investigation of the correlation function of the electric field near the QCP
and find that its asymptotic behavior can be described by the universal
scaling function of the quantum Ising chain. This finding indicates the
existence of three regions near the QCP, each characterized by a specific
asymptotic form of the correlation length, and demonstrates that the CP
symmetry is restored at any nonzero temperature, entirely analogous to the
quantum Ising chain. The range of the scaling behavior is also examined and
found to be particularly wide.

††preprint: YITP-23-140

##  1 Introduction

Phase transitions are a fundamental concept in physics, observed in numerous
systems, including water, magnets, superconductors, and more. These
transitions occur on a variation of an external control parameter, such as
temperature, magnetic field, pressure, and coupling constant. Identifying and
characterizing the phases in the parameter plane, i.e., establishing the phase
diagram, is a crucial step toward a comprehensive understanding of the model.

The critical point is particularly intriguing in the phase diagram.
Remarkably, in the vicinity of the critical point, various microscopic models
sharing the same symmetry pattern and dimensionality exhibit qualitatively the
same behaviors, characterized by the same critical exponents, due to the
emergent scale invariance at the critical point. Thus, universality provides a
unified understanding of various macroscopic models near the critical point
and also enables us to analyze the critical behavior of a complex physical
system in terms of a much simpler, and sometimes exactly solvable, model.

In the context of high-energy physics, several model studies suggest the
existence of a critical point at finite temperature and baryon chemical
potential region in the phase diagram of strongly interacting matter governed
by quantum chromodynamics (QCD) Asakawa:1989bq ; Barducci:1989wi ;
Barducci:1994 ; Halasz:1998qr ; Berges:1998rc ; harada:1998zq ;
Kitazawa:2002jop , and universality has been applied to the study of critical
behaviors Pisarski:1983ms ; Hatta:2002sj . However, although the Monte Carlo
simulation based on the lattice formulation of QCD Wilson:1974sk ;
Creutz:1980zw  is at present the only first-principle method to explore the
phase structure of QCD, the conventional lattice QCD simulation suffers from
the sign problem at finite baryon chemical potential. Consequently, the
location or even the existence of the QCD critical point has not been
established yet. The detection of the QCD critical point has been one of the
main purposes of relativistic heavy-ion collision experiments Pandav:2022xxx .

In this paper, we explore a phase diagram in a prototype gauge theory that
possesses a critical point, the Schwinger model Schwinger:1962tp . The
Schwinger model is one-dimensional quantum electrodynamics and shares many
low-energy phenomena with QCD, such as confinement, chiral symmetry breaking,
and nontrivial topological $\theta$ vacuum. Due to the topological nature of
the $\theta$ term, the Scwhinger model has CP symmetry at $\theta=\pi$. At
zero temperature, it is already established that as increasing the fermion
mass, the system undergoes a second-order phase transition to the CP broken
phase, which belongs to the same universality class as the quantum Ising chain
Hamer:1982mx ; Byrnes:2002nv ; Shimizu:2014fsa ; Azcoiti:2017mxl ;
Thompson:2021eze . However, compared to the comprehensive studies on the
second-order phase transition itself, the phase structure near the critical
point at finite temperature remains much less known.

Our present analysis is based on universality with the quantum Ising chain and
first-principle Monte Carlo simulations. Although the Monte Carlo simulation
of the Schwinger model at $\theta=\pi$ is hindered by the severe sign problem
when using the conventional lattice formulations, the author recently pointed
out Ohata:2023sqc  that this problem can be avoided by using the lattice
formulation of the bosonized Schwinger model Bender:1984qg . In this work, we
establish the phase diagram of the Schwinger model near the critical point by
verifying the scaling behavior of the correlation function, which is predicted
from universality with the quantum Ising chain, through Monte Carlo
simulations. Our study provides a comprehensive understanding of the phase
structure near the critical point in the Schwinger model at $\theta=\pi$ and
demonstrates the power of universality in a gauge theory.

This paper is organized as follows. In section 2, we review the spontaneous CP
symmetry breaking at zero temperature from both analytical and numerical
sides, along with introducing the Schwinger model and the $\theta$ term. In
section 3, we first review the quantum Ising chain and then conjecture the
phase diagram of the Schwinger model at $\theta=\pi$ based on universality
with the quantum Ising chain. Our strategy to establish the conjectured phase
diagram using the Monte Carlo method is also explained in this section. In
section 4, we perform a detailed numerical investigation of the correlation
function and establish the conjectured phase diagram. Section 5 is devoted to
summary and conclusion.

##  2 Spontaneous CP symmetry breaking at zero temperature

In this section, we briefly review the spontaneous CP symmetry breaking at
zero temperature in the Schwinger model at $\theta=\pi$.

Let us begin by introducing the Schwinge model and the $\theta$ term. The
Schwinger model is a one-dimensional $\mathrm{U}(1)$ gauge theory coupled with
a Dirac fermion Schwinger:1962tp . The Euclidean action of the Schwinger model
is given by

$S_{E}=\int
d^{2}x\,\overline{\psi}\quantity(\not{\partial}+g\not{A}+m)\psi+\frac{1}{4}F_{\mu\nu}F_{\mu\nu}.$
(1)

Here $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength of the $\mathrm{U}(1)$ gauge field $A_{\mu}$, $\psi$ the Dirac
fermion, $g$ the dimensionful gauge coupling, and $m$ the fermion mass. The
Euclidean action (1) is symmetric under the CP transformation. It is possible
to introduce the so-called $\theta$ term without breaking the $\mathrm{U}(1)$
gauge symmetry or spoiling the renormalizability:

$i\theta Q,\quad Q=\int d^{2}x\,\frac{g}{4\pi}\epsilon_{\mu\nu}F_{\mu\nu},$
(2)

where $\epsilon_{\mu\nu}$ is the anti-symmetric tensor. Introducing the
$\theta$ term into the action typically results in the explicit breaking of
the CP symmetry, since the term is anti-symmetric under the CP transformation.
However, there is one nontrivial exception: the CP symmetry is not explicitly
broken at $\theta=\pi$ because the $\theta$ term appears as $\exp(i\theta Q)$
in the path-integral, and $Q$ is known to take an integer value for any gauge
configuration. The fate of this CP symmetry is one of the key issues in this
paper.

The spontaneous CP symmetry breaking at $\theta=\pi$ was first predicted by
Coleman with an intuitive analytical picture using the bosonized form of the
Schwinger model Coleman:1976uz . After bosonization and integrating out the
gauge fields, the Hamiltonian of the bosonized Schwinger model reads
Coleman:1974bu ; Mandelstam:1975hb ; Coleman:1975pw

$H=\int
dx\,\frac{1}{2}\pi^{2}+\frac{1}{2}\quantity(\partial_{x}\phi)^{2}+\frac{g^{2}}{2\pi}\phi^{2}-\frac{e^{\gamma}}{2\pi^{3/2}}mg\mathcal{N}_{g/\sqrt{\pi}}\cos(2\sqrt{\pi}\phi-\theta).$
(3)

Here $\pi$ is the conjugate momentum, $\gamma$ Euler’s constant, and
$\mathcal{N}_{g/\sqrt{\pi}}$ denotes the normal ordering with respect to the
boson mass $g/\sqrt{\pi}$. In this formulation, the scalar field $\phi$ is
related to the electric field $E=F_{01}$ as

$\frac{\phi}{\sqrt{\pi}}=\frac{E}{g},$ (4)

through the Gauss law constraint. Thus, $\phi$ serves as an order parameter of
the CP symmetry. The effective potential of $\phi$ reads Coleman:1976uz

$V(\phi)=\frac{g^{2}}{2\pi}\phi^{2}-\frac{e^{\gamma}}{2\pi^{3/2}}mg\cos(2\sqrt{\pi}\phi-\theta).$
(5)

At $\theta=0$, the cosine term forms a pocket at $\phi=0$, indicating a unique
CP-symmetric vacuum at any fermion mass. This changes drastically at
$\theta=\pi$. Figure 1 shows the effective potential at $\theta=\pi$ at
various fermion masses.

Figure 1:  The effective potential of the bosonized Schwinger model (5) at
$\theta=\pi$.

For small fermion masses, the bosonic mass term dominates the potential, and
we expect a unique vacuum. On the other hand, for large fermion masses, the
cosine term forms a potential wall at $\phi=0$, and we expect degenerate
vacua, corresponding to the spontaneous CP symmetry breaking. In an
intermediate fermion mass region, where the CP symmetry is either maintained
or at least well-preserved, by expanding the cosine to the second order, the
effective potential becomes Byrnes:2002nv

$V(\phi)\simeq\frac{1}{2}\quantity{\frac{g}{\sqrt{\pi}}\quantity(1-\sqrt{\pi}e^{\gamma}\frac{m}{g})}^{2}\phi^{2}.$
(6)

As $m/g$ increases, the effective mass decreases and reaches zero at

$m_{c}/g=1/\quantity(\sqrt{\pi}e^{\gamma})\simeq 0.317,$ (7)

where we expect a second-order phase transition. It is important to note that
this second-order phase transition is driven purely by quantum fluctuations.
Hence the critical point at zero temperature $(m_{c},T=0)$ is referred to as
the quantum critical point (QCP) in the following.

First-principle numerical simulations are needed for a more quantitative and
conclusive understanding. Unfortunately, the conventional Monte Carlo method
is not applicable due to the severe sign problem at $\theta=\pi$, when using
the Euclidean lattice fermion formulations. Consequently, numerical
investigations of the QCP have been mainly performed using the finite-
dimensional spin Hamiltonian, which is derived by applying the Jodan–Wigner
transformation to the Kogut–Susskind fermion formulation of the Schwinger
model Hamer:1982mx ; Ranft:1982bi ; Byrnes:2002nv ; Buyens:2017crb .

The first numerical evidence of the above analytical picture was provided by
Hamer et al. Hamer:1982mx , in which the spin Hamiltonian was combined with
the finite-size scaling method Hamer:1980  to identify the QCP. They estimated
the critical mass to be $m_{c}/g=0.325(20)$ and a critical exponent
$\nu=0.9(1)$. A more precise investigation was done by Byrnes et al.
Byrnes:2002nv  using the density matrix renormalization group method
White:1992zz ; White:1993zza . They obtained at present the most precise
estimate of the critical mass

$\frac{m_{c}}{g}=0.3335(2),$ (8)

and two critical exponents

$\nu=1.01(1),\quad\beta/\nu=0.125(5).$ (9)

These two critical exponents are precisely consistent with those of the
quantum Ising chain ($\nu=1,\beta=1/8$), indicating that these two models are
in the same universality class. As a quite different approach, Shimizu et al.
Shimizu:2014fsa  applied the Grassmann tensor renormalization group Gu:2010gr
to the conventional Wilson fermion formulation in Euclidean space-time. Using
the Lee–Yang and Fisher zero analyses, they provided further evidence that the
QCP belongs to the Ising universality class. Other approaches include, for
example, a conversion method from the imaginary $\theta$ term Azcoiti:2017mxl
, and one using quantum computing Thompson:2021eze .

##  3 Phase diagram near the quantum critical point

Compared to the comprehensive studies on the second-order phase transition at
zero temperature, the phase structure near the QCP remains much less known. To
the best of the author’s knowledge, the only study addressing this issue was
conducted by Buyens et al. Buyens:2016ecr , in which they investigated the
fate of the spontaneously broken CP symmetry at finite temperature using the
matrix product operators Verstraete:2004ve ; Zwolak:2004zw  based on the spin
Hamiltonian formulation. Through a detailed investigation of the electric
field at finite temperature near $\theta=\pi$, they obtained an indication
that the CP symmetry is restored at any nonzero temperature.

In this paper, we explore broad characteristics near the QCP from the
perspective of universality with the quantum Ising chain, using a completely
different lattice formulation from previous studies. In this section, we first
review the quantum Ising chain, in particular its phase structure at finite
temperature, and then explain our strategy to explore the phase diagram of the
Schwinger model at $\theta=\pi$.

###  3.1 Quantum Ising chain

We here review the phase diagram of the quantum Ising chain following a review
Sachdev:2011 . The quantum Ising chain is the quantum analog of the two-
dimensional classical Ising model with no external magnetic field. The
Hamiltonian of the quantum Ising chain is given by

$H_{I}=-J\sum_{i_{x}}\quantity(\sigma_{i_{x}}^{z}\sigma_{i_{x}+1}^{z}+g\sigma_{i_{x}}^{x}),$
(10)

where $J>0$ is an overall energy scale, $g>0$ is the dimensionless coupling
constant, and $\sigma_{i_{x}}^{z},\sigma_{i_{x}}^{x}$ are the Pauli matrixes

$\sigma^{z}=\matrixquantity(1&0\\\
0&-1),\quad\sigma^{x}=\matrixquantity(0&1\\\ 1&0)$ (11)

which reside on a spatial site $i_{x}$. The Hamiltonian is symmetric under the
$\mathbb{Z}_{2}$ transformation, generated by the unitary operator
$\prod_{i_{x}}\sigma_{i_{x}}^{x}$.

Unlike the massive Schwinger model, the quantum Ising chain is exactly
solvable in the sense that all eigenstates are obtained analytically by
applying the Jordan–Wigner transformation and diagonalizing the Hamiltonian by
the Bogoliubov transformation Lieb:1961fr ; Pfeuty:1970pf . At zero
temperature, as decreasing the coupling constant, the system undergoes a
second-order phase transition to the ferromagnetic ($\mathbb{Z}_{2}$ broken)
phase at $g_{c}=1$, which is analogous to the Schwinger model at $\theta=\pi$.

The phase diagram of the quantum Ising chain in the temperature and coupling
constant plane can be deduced from the correlation function of the order
parameter at finite temperature 111Note that we do not subtract the connected
part $\expectationvalue{\sigma^{z}}^{2}$ from the correlation function.

$C(x)=\expectationvalue{\sigma_{0}^{z}\sigma_{i_{x}}^{z}}=\left.\tr(\sigma_{0}^{z}\sigma_{i_{x}}^{z}e^{-H_{I}/T})\middle/\tr(e^{-H_{I}/T})\right..$
(12)

A crucial feature that we will exploit throughout this paper is the explicit
asymptotic form of the correlation function, derived by Sachdev Sachdev:1995bf
:

$\lim_{x\to\infty}C(x)=ZT^{1/4}G_{I}(\Delta/T)\exp(-\frac{Tx}{c}F_{I}(\Delta/T)),\quad\Delta=r(g_{c}-g),$
(13)

where $Z,c,r$ are the nonuniversal constants, and $F_{I}$ and $G_{I}$ are the
universal scaling functions of the quantum Ising chain. Their explicit forms
are Sachdev:1995bf

$\displaystyle F_{I}(s)$
$\displaystyle=\absolutevalue{s}\theta(-s)+\frac{1}{\pi}\int_{0}^{\infty}dy\,\ln\coth\frac{\quantity(y^{2}+s^{2})^{1/2}}{2},$
(14) $\displaystyle\ln G_{I}(s)$
$\displaystyle=\int_{s}^{1}\frac{dy}{y}\quantity[\quantity(\frac{dF_{I}(y)}{dy})^{2}-\frac{1}{4}]+\int_{1}^{\infty}\frac{dy}{y}\quantity(\frac{dF_{I}(y)}{dy})^{2}.$
(15)

Figure 2 plots the universal scaling functions $F_{I},G_{I}$, illustrating
their smoothness throughout the entire region.

Figure 2:  The universal scaling functions of the quantum Ising chain (14,
15).

The correlation length is defined from the exponential decay of the
correlation function (13) as

$\xi^{-1}=\frac{T}{c}F_{I}\quantity(\frac{\Delta}{T}).$ (16)

At any nonzero temperature, the correlation length is finite, indicating that
the correlation function vanishes in the long-distance limit. Simultaneously,
the correlation function should converge to its connected part as

$\lim_{x\to\infty}C(x)=\expectationvalue{\sigma^{z}}^{2}.$ (17)

Thus, we can deduce that the system is in the paramagnetic ($\mathbb{Z}_{2}$
symmetric) phase at any nonzero temperature.

In the vicinity of the QCP, even a slight shift in the coupling constant
results in a substantial change in $\Delta/T$. It is meaningful to categorize
the phase into three depending on the value of $\Delta/T$, each characterized
by a specific asymptotic form of the correlation length Sachdev:1995bf :

$\xi=\begin{cases}c\sqrt{\frac{\pi}{2\Delta T}}e^{\Delta/T},&\Delta/T\gg 1,\\\
\frac{4c}{\pi T},&\absolutevalue{\Delta}/T\ll 1,\\\
\frac{c}{\absolutevalue{\Delta}},&\Delta/T\ll-1.\end{cases}$ (18)

In the low-temperature region with $\Delta>0$, the correlation length diverges
exponentially toward the zero temperature limit, corresponding to the
ferromagnetic phase at zero temperature. Conversely, the long-range order at
zero temperature is thermally destroyed. This region is commonly referred to
as the thermally disordered region in the context of condensed matter physics
Vojta:2003 . In the other low-temperature region with $\Delta<0$, the
correlation length saturates at a finite value in the zero temperature limit,
resulting in a disordered phase even at zero temperature. This region is
called the quantum disordered phase Vojta:2003 . The final region with
$\absolutevalue{\Delta}/T\ll 1$ is the so-called quantum critical region,
where the physics is controlled by the thermal excitations of the quantum
critical ground state Vojta:2003 . The resulting phase diagram of the quantum
Ising chain is shown in the left panel of Fig. 3. The universality with the
quantum Ising chain implies the phase diagram of the Schwinger model at
$\theta=\pi$ near the QCP, as shown in the right panel of Fig. 3.

Figure 3:  (Left) Phase diagram of the quantum Ising chain. (Right)
Conjectured phase diagram of the Schiwnger model at $\theta=\pi$.

###  3.2 Schwinger model at $\theta=\pi$

We here explain our strategy to establish the conjectured phase diagram of the
Schwinger model at $\theta=\pi$. In the previous subsection, we observed that
the phase diagram of the quantum Ising chain can be deduced from the
asymptotic form of the correlation function (13). Therefore, our approach is
rather straightforward: we calculate the correlation function of the electric
field at $\theta=\pi$ at finite temperatures near the QCP and examine whether
the correlation function shares the same asymptotic form as the quantum Ising
chain (13). If this confirmation is obtained at a certain temperature, the
scaling behavior should also hold at lower temperatures, allowing us to
establish the phase diagram of the Schwinger model near the QCP. Our method
does not aim to approach the QCP itself, but rather inspect the scaling
behavior near the QCP. Consequently, we need not employ the finite-size
scaling method because the correlation length would be always finite in our
analysis. The most significant difficulty is that we must circumvent the sign
problem at $\theta=\pi$ in some way.

Recently, the author pointed out Ohata:2023sqc  that by utilizing the lattice
formulation of the bosonized Schwinger model Bender:1984qg , Monte Carlo
simulations can be performed without encountering the sign problem. The
lattice bosonized Schwinger model is obtained after removing the normal
ordering appearing in the cosine term in the bosonized Hamiltonian (3). In the
continuum, using the Wick theorem, the normal ordering can be removed as
Coleman:1974bu

$\mathcal{N}_{g/\sqrt{\pi}}\exp(i2\sqrt{\pi}\phi)=\exp{2\pi\Delta\quantity(x=0;\frac{g}{\sqrt{\pi}})}\exp(i2\sqrt{\pi}\phi),$
(19)

where $\Delta(x,g/\sqrt{\pi})$ is the Feynman propagator for the scalar field
of mass $g/\sqrt{\pi}$. The lattice counterpart of this relation is obtained
by just replacing the Feynman propagator with that on the lattice
Bender:1984qg

$\Delta\quantity(x;\frac{g}{\sqrt{\pi}};\frac{1}{a})=\int^{\pi}_{-\pi}\frac{d^{2}k}{\quantity(2\pi)^{2}}e^{ikx}\quantity(4\sum_{\mu}\sin^{2}\quantity(\frac{k_{\mu}}{2})+\quantity(\frac{ag}{\sqrt{\pi}})^{2})^{-1},$
(20)

where $a$ is the lattice spacing. Consequently, the lattice counterpart of the
continuum Hamiltonian (3) can be expressed without using the normal ordering

$Ha=\sum_{x=0}^{L_{x}-1}\frac{1}{2}\quantity(a\pi_{x})^{2}+\frac{1}{2}\quantity(\partial_{x}\phi_{x})^{2}+\frac{\quantity(ag)^{2}}{2\pi}\phi_{x}^{2}-\frac{e^{\gamma}}{2\pi^{3/2}}\frac{m}{g}\quantity(ag)^{2}\mathcal{O}(1/ag)\cos(2\sqrt{\pi}\phi_{x}-\theta).$
(21)

Here, $\partial_{x}$ denotes the forward derivative
$\partial_{x}f_{x}\coloneqq f_{x+1}-f_{x}$, and the factor $\mathcal{O}(1/ag)$
is defined as

$\mathcal{O}(1/ag)\coloneqq\exp{2\pi\Delta\quantity(0;\frac{g}{\sqrt{\pi}};\frac{1}{a})},\\\
$ (22)

and behaves $\mathcal{O}(1/ag)\simeq 10/ag$ for $ag\ll 1$ Ohata:2023sqc . The
validity of this lattice formulation was explicitly confirmed in Ref.
Ohata:2023sqc .

The correlation function of the electric field at temperature
$T=1/(aL_{\tau})$ can be expressed using the path-integral

$\displaystyle C(x)$
$\displaystyle=\frac{1}{g^{2}}\expectationvalue{E_{x}E_{0}}$ (23a)
$\displaystyle=\frac{1}{\pi}\expectationvalue{\phi_{x}\phi_{0}}$ (23b)
$\displaystyle=\frac{1}{\pi}\left.\tr(\phi_{x}\phi_{0}e^{-H/T})\middle/\tr
e^{-H/T}\right.$ (23c) $\displaystyle=\frac{1}{\pi}\left.\int
D\phi\,\phi_{x,0}\phi_{0,0}e^{-S_{E}}\middle/\int D\phi\,e^{-S_{E}},\right.$
(23d)

where $S_{E}$ is the lattice Euclidean action of the bosonized Schwinger model

$S_{E}=\sum_{\tau=0}^{L_{\tau}-1}\sum_{x=0}^{L_{x}-1}\frac{1}{2}\quantity(\partial_{\tau}\phi_{x,\tau})^{2}+\frac{1}{2}\quantity(\partial_{x}\phi_{x,\tau})^{2}+\frac{\quantity(ag)^{2}}{2\pi}\phi_{x,\tau}^{2}\\\
-\frac{e^{\gamma}}{2\pi^{3/2}}\frac{m}{g}\quantity(ag)^{2}\mathcal{O}(1/ag)\cos(2\sqrt{\pi}\phi_{x,\tau}-\theta).$
(24)

The periodic boundary condition is imposed for both directions in this paper.
The lattice Euclidean action (24) is obviously real and bounded below even at
$\theta\neq 0$, meaning no sign problem. The Monte Carlo configurations can be
easily generated by combined use of the heat-bath algorithm and the rejection
sampling, just like the case of $\mathrm{SU}(2)$ Yang–Mills theory
Creutz:1980zw .

##  4 Numerical results

In this work, we use a sufficiently fine and large lattice of
$ag=0.2,L_{x}=1792$. We later verify that the finite lattice spacing and
finite spatial length effects are indeed nearly negligible in our analysis. As
for the temperature, we mainly consider two low temperatures $L_{\tau}=112,56$
to inspect the scaling behavior near the QCP.

###  4.1 Autocorrelation and lattice artifacts

In this subsection, we investigate the autocorrelation and lattice artifacts
in preparation for the large-scale numerical simulations in the next
subsection.

We first investigate the autocorrelation among Monte Carlo configurations near
the QCP for reliable error estimates. The autocorrelation can be characterized
by the autocorrelation function Sokal:1997

$A(t)=\frac{1}{N-1}\sum_{i=1}^{N-t}\phi_{i}\phi_{i+t}-\expectationvalue{\phi}^{2},$
(25)

where $\phi_{i}$ is the mean of $\phi_{x,\tau}$:

$\phi_{i}=\frac{1}{L_{x}L_{\tau}}\sum_{x,\tau}\phi_{x,\tau}$ (26)

at the $i$-th Monte Carlo configuration, and $\expectationvalue{\phi}$ is
$\expectationvalue{\phi}=\frac{1}{N}\sum_{i}\phi_{i}$. From the normalized
autocorrelation function

$C(t)=A(t)/A(0),$ (27)

the integrated autocorrelation time with finite lattice data is defined as

$\tau_{\mathrm{int}}(t)=\frac{1}{2}+\sum_{i=1}^{t}C(i).$ (28)

Twice the integrated autocorrelation time

$\tau_{\mathrm{int}}=\lim_{t\to\infty}\tau_{\mathrm{int}}(t)$ (29)

gives an estimate of the number of iterations required to generate mostly
independent configurations. Figure 4 shows the integrated autocorrelation time
(28) at $m/g=0.2,0.33,0.4$. From such behaviors, we estimate the integrated
autocorrelation time (29).

Figure 4:  The integrated autocorrelation time with finite lattice data (28)
at $m/g=0.2,0.33,0.4$.

The estimated integrated autocorrelation times at the two lattices and various
fermion masses are summarized in Table 1.

$ag$ | $L_{x}\times L_{t}$ | $m/g$ | $\tau_{\mathrm{int}}$
---|---|---|---
0.2 | $1792\times 112$ | 0.2 | $\sim 8\times 10^{2}$
0.2 | $1792\times 112$ | 0.33 | $\sim 6\times 10^{4}$
0.2 | $1792\times 112$ | 0.4 | $\sim 2\times 10^{6}$
0.2 | $1792\times 56$ | 0.1 | $\sim 2\times 10^{2}$
0.2 | $1792\times 56$ | 0.2 | $\sim 8\times 10^{2}$
0.2 | $1792\times 56$ | 0.33 | $\sim 1.4\times 10^{4}$
0.2 | $1792\times 56$ | 0.4 | $\sim 1.2\times 10^{5}$
0.2 | $1792\times 56$ | 0.5 | $\sim 5\times 10^{5}$
Table 1: Estimated integrated autocorrelation times at the two lattices and
various fermion masses.

We find that the integrated autocorrelation time is longer at larger fermion
masses and lower temperatures, which seems to be related to the expected
correlation length near the QCP, as shown in the right panel of Fig. 3. In
this work, we set the number of iterations to approximately
$\tau_{\mathrm{int}}/2$ and use the binning of ten to eliminate the remaining
autocorrelation.

We also investigate the finite lattice spacing and finite spatial length
effects. For this purpose, we compare correlation functions at $m/g=0.32$
calculated using three different lattices:

  * •

$ag=0.2,L_{x}\times L_{\tau}=1792\times 112$,

  * •

$ag=0.2,L_{x}\times L_{\tau}=896\times 112$,

  * •

$ag=0.1,L_{x}\times L_{\tau}=3584\times 224$.

By comparing the first and the second lattices, we can assess the finite
spatial length effect. The finite lattice spacing effect can be examined by
comparing the first and the third lattices. Figure 5 shows the correlation
functions at these three lattices.

Figure 5:  Correlation functions and fit results using the single exponential
function (30) at three lattices of $ag=0.2,L_{x}\times L_{\tau}=1792\times
112$, $ag=0.2,L_{x}\times L_{\tau}=896\times 112$, and $ag=0.1,L_{x}\times
L_{\tau}=3584\times 224$, For visibility, only one-fifth and one-tenth of the
data points are plotted for $ag=0.2$ and $ag=0.1$, respectively.

For each correlation function, we perform a correlated fit using the single
exponential function

$C(x)=A\exp(-x/\xi),$ (30)

with the correlation length $\xi$ and the amplitude $A$ being the fit
parameters. The fit range is set as $xg\in\quantity[25,30]$. The fit results
are also shown in Fig 5. We find that the resulting correlation lengths and
amplitudes are consistent with each other, suggesting that both finite lattice
spacing and finite spatial length effects are nearly negligible in our
analysis.

###  4.2 Correlation function near the quantum critical point

We now perform a detailed investigation of the correlation function near the
QCP. To carefully examine the behavior near the QCP, we analyze correlation
functions over a broad range of fermion masses. The simulation parameters
relevant in this subsection are summarized in Table 2.

$ag$ | $L_{x}\times L_{t}$ | $m/g$ | num. of iterations | num. of configurations
---|---|---|---|---
0.2 | $1792\times 112$ | 0.16, 0.18, 0.2, 0.22, 0.24, 0.26 | $4\times 10^{2}$ | $10^{5}$
0.2 | $1792\times 112$ | 0.28, 0.3, 0.32, 0.34, 0.36, 0.38 | $3\times 10^{4}$ | $2\times 10^{3}$
0.2 | $1792\times 112$ | 0.4, 0.42, 0.44 | $10^{6}$ | $10^{3}$
0.2 | $1792\times 56$ | 0.08, 0.10, 0.12, 0.14 | $10^{2}$ | $10^{5}$
0.2 | $1792\times 56$ | 0.16, 0.18, 0.2, 0.22, 0.24, 0.26 | $4\times 10^{2}$ | $10^{5}$
0.2 | $1792\times 56$ | 0.28, 0.3, 0.32, 0.34, 0.36, 0.38 | $7\times 10^{3}$ | $10^{4}$
0.2 | $1792\times 56$ | 0.4, 0.42, 0.44 | $6\times 10^{4}$ | $10^{3}$
0.2 | $1792\times 56$ | 0.48, 0.5, 0.52 | $1.3\times 10^{5}$ | $10^{3}$
Table 2: Summary of the simulation parameters.

For each correlation function, we perform a correlated fit using the single
exponential function (30) and extract the correlation length $\xi$ and
amplitude $A$. For demonstration, we show the correlation functions and their
fit results at $m/g=0.2,0.26,0.34,0.44$ at the lower temperature in Fig 6. The
fit ranges are set as
$xg\in\quantity[20,25],\quantity[20,25],\quantity[25,30],\quantity[30,35]$,
for $m/g=0.2,0.26,0.34,0.44$, respectively.

Figure 6:  Correlation functions and fit results by the single exponential
function (30) at $m/g=0.2,0.26,0.34,0.44$. The lattice is $ag=0.2,L_{x}\times
L_{\tau}=1792\times 112$. For visibility, only one-fifth of the data points
are plotted.

We next perform fits to the correlation length and amplitude at the lower
temperature, independently. Universality with the quantum Ising chain suggests
the asymptotic form of the correlation function in the Schwinger model at
$\theta=\pi$ as

$\lim_{x\to\infty}C(x)=Z\quantity(T/g)^{1/4}G_{I}(\Delta/T)\exp(-\frac{Tx}{c}F_{I}(\Delta/T)),\quad\Delta=r(m-m_{c}),$
(31)

where $Z,c,r$ are nonuniversal dimensionless constants, and the functional
forms of $F_{I}$ and $G_{I}$ are given in Eqs. (14, 15). Figure 7 shows the
correlation length and amplitude as functions of the fermion mass, fitted
using

$\frac{T}{cg}F_{I}(r(m-m_{c})/T),\quad Z(T/g)^{1/4}G_{I}(r(m-m_{c})/T),$ (32)

respectively. In these fits, we set the critical mass to $m_{c}/g=0.3335$,
using the wisdom obtained by Byrnes et al. (8). From the fits, the
nonuniversal constants are found to be

$Z=0.2435(16),\quad c=0.978(11),\quad r=1.593(46).$ (33)

Note that the value of $r$ is also obtained from the fit to the correlation
length, as shown in the left panel of Fig. 7. In the following analysis,
however, we use the value obtained from the amplitude since the fit seems to
be more reliable, although they are consistent with each other.

Figure 7:  Fits to the correlation length (left) and to the amplitude (right)
using the universal scaling functions of the quantum Ising chain (14, 15). The
lattice is $ag=0.2,L_{x}\times L_{\tau}=1792\times 112$.

We rescale the correlation lengths and amplitudes at the two temperatures
using the nonuniversal constants (33) and compare them with the universal
scaling functions of the quantum Ising chain (14, 15). Figure 8 demonstrates
that the rescaled data align beautifully with the expected analytical curves.
In particular, we observe that data at the higher temperature also match the
analytical curves, providing strong evidence that the correlation function of
the Schwinger model at $\theta=\pi$ shares the same asymptotic form with the
quantum Ising chain. Based on the argument in Section 3, we conclude that the
phase diagram of the Schwinger model at $\theta=\pi$ is entirely analogous to
the quantum Ising chain near the QCP, as schematically shown in Fig. 3.

Figure 8:  Rescaled correlation length (left) and amplitude (right) using the
nonuniversal constants (33), compared with the universal scaling functions of
the quantum Ising chain (14, 15).

###  4.3 Range of scaling behavior

With the established asymptotic form of the correlation function (31) and the
explicit values of the nonuniversal constants (33), we can quantitatively
predict the behavior of certain observables near the QCP. In this subsection,
we roughly estimate the range of the scaling behavior by comparing the
predictions to direct numerical results.

We first examine how well the scaling behavior holds at zero temperature by
comparing our prediction with Byrnes et al.’s results Byrnes:2002nv . At zero
temperature in the CP symmetric phase ($m<m_{c}$), the inverse of the
correlation length, i.e., an energy gap, is given by

$\quantity(\xi
g)^{-1}=\frac{1}{c}\frac{\absolutevalue{\Delta}}{g}=\frac{r}{c}\quantity(\frac{m_{c}}{g}-\frac{m}{g})=0.543(17)-1.629(50)\frac{m}{g}.$
(34)

This relation is quite similar to their estimate of the energy gap
Byrnes:2002nv

$\frac{\Delta E}{g}\simeq 0.569-1.72\frac{m}{g}.$ (35)

Note that their relation was obtained from a linear fit through the data in
the range $m\in\quantity[0,m_{c}]$, so exact agreement is not expected.
Nevertheless, the similarity between Eqs. (34, 35) indicates that the scaling
behavior holds surprisingly well in the region $m\in\quantity[0,m_{c}]$ at
zero temperature.

We can also predict the electric field in the CP broken phase ($m>m_{c}$) at
zero temperature from the asymptotic form of the correlation function (31). In
this region, using the zero temperature limit of $G_{I}$ Sachdev:1995bf

$G_{I}(\Delta/T)\to(\Delta/T)^{1/4},\quad\Delta/T\gg 1,$ (36)

the correlation function behaves as

$C(x)\to
Z(T/g)^{1/4}\quantity(\Delta/T)^{1/4}=Z\quantity(\Delta/g)^{1/4}=Zr^{1/4}\quantity(\frac{m}{g}-\frac{m_{c}}{g})^{1/4}.$
(37)

Therefore, the electric field in the CP broken phase is given by

$\absolutevalue{\expectationvalue{E}}=Z^{1/2}r^{1/8}\quantity(\frac{m}{g}-\frac{m_{c}}{g})^{1/8}.$
(38)

We plot Eq. (38) using the nonuniversal constants (33) in Fig. 9.

Figure 9:  Fermion mass dependence of the electric field at zero temperature.

We observe a rather sharp but continuous increase in the electric field near
the QCP, corresponding to the small critical index $\beta=1/8$. Byrnes et al.
calculated the electric fields at various fermion masses in the CP broken
phase, as shown in Fig. 15 in Ref. Byrnes:2002nv . From the figure, we can
obtain rough estimate of their results, for example,
$(m/g,\absolutevalue{\expectationvalue{E}}/g)\simeq(0.35,0.31),(0.62,0.44),(1.0,0.48)$,
which approximately agree with the analytical curve in Fig. 9, indicating that
the scaling behavior holds well in the region $m/g\in\quantity[m_{c}/g,1]$ at
zero temperature.

The similarities with Byrnes et al.’s results at zero temperature reinforce
the validity of our estimate for the nonuniversal constants (33) and indicate
that the scaling behavior holds well in the region $m/g\in[0,1]$ at zero
temperature.

We also examine the range of the scaling behavior at the critical fermion
mass. In this region, the inverse of the correlation length is

$\quantity(\xi g)^{-1}=\frac{\pi T}{4cg},$ (39)

and the amplitude is given by

$A=0.8587Z\quantity(\frac{T}{g})^{1/4},$ (40)

from the behavior of $G_{I}$ Sachdev:1995bf :

$G_{I}(\Delta/T)=0.8587...,\quad\absolutevalue{\Delta}/T\ll 1.$ (41)

In Fig. 10, we compare Eqs. (39, 40) with our direct numerical results, which
are obtained in the same way as in the previous subsection. The simulation
parameters are summarized in Table. 3.

$ag$ | $L_{x}\times L_{t}$ | $m/g$ | $\tau_{\mathrm{int}}$ | num. of iterations | num. of configurations
---|---|---|---|---|---
0.2 | $1792\times 28$ | 0.3335 | $\sim 2\times 10^{3}$ | $10^{3}$ | $10^{5}$
0.2 | $1792\times 16$ | 0.3335 | $\sim 10^{3}$ | $5\times 10^{2}$ | $10^{5}$
0.2 | $1792\times 12$ | 0.3335 | $\sim 5\times 10^{2}$ | $2.5\times 10^{2}$ | $10^{5}$
0.2 | $1792\times 10$ | 0.3335 | $\sim 3\times 10^{2}$ | $1.5\times 10^{2}$ | $10^{6}$
Table 3: Simulation parameters and estimated autocorrelation times for
investigating temperature dependence of the correlation length and the
amplitude at the critical fermion mass.

Figure 10:  Temperature dependence of the correlation length (left) and the
amplitude (right) at the critical fermion mass.

We find that the direct numerical results agree with the analytical curves
with an accuracy of at least ninety percent up to $T/g\simeq 0.4$. This
ensures that the scaling behavior holds well in the region $T/g\in[0,0.4]$ at
the critical fermion mass.

In Fig. 11, we finally present the inverse of the correlation length in the
temperature and fermion mass plane obtained from the asymptotic form (31) and
the nonuniversal constants (33). Based on the analysis in this subsection, the
values are reliable at zero temperature or at the critical fermion mass within
the range plotted in Fig. 11. At $m=0$, the Schwinger model is equivalent to
the free boson theory (3), resulting in an energy gap of $g/\sqrt{\pi}\simeq
0.564g$ at any temperature. In Fig. 11, we indeed observe approximate
temperature independence up to $T/g\simeq 0.3$, suggesting that Fig. 11 is to
some extent reliable except for the two edges on the finite-temperature side.

Figure 11:  Inverse of the correlation length in the temperature and fermion
mass plane obtained from the asymptotic form (31) and the nonuniversal
constants (33). The dashed lines $T=\pm\Delta,\,\Delta=r(m-m_{c})$ in the left
panel are the crossover lines that classify the phases effectively.

##  5 Summary and conclusion

In this paper, we have explored the phase diagram of the Schwigner model at
$\theta=\pi$ in the temperature and fermion mass plane. We first reviewed the
spontaneous CP symmetry breaking at zero temperature in the Schwinger model at
$\theta=\pi$ and revisited the phase diagram of the quantum Ising chain in the
temperature and coupling constant plane. There, we observed that the phase
diagram of the quantum Ising chain can be deduced from the explicit asymptotic
form of the correlation function (12). From the perspective of universality,
we then conjectured the phase diagram of the Schwinger model at $\theta=\pi$
near the QCP, which is entirely analogous to the quantum Ising chain. This
conjecture was subsequently confirmed by validating that the correlation
function of the electric field shares the same asymptotic form as the quantum
Ising chain through first-principle Monte Carlo simulations. We circumvented
the sign problem, which hinders the conventional Monte Carlo simulation at
$\theta\neq 0$, by using the lattice formulation of the bosonized Schwinger
model. Furthermore, we examined the range of the scaling behavior by comparing
the predictions based on universality to direct numerical results and
identified a surprisingly wide applicable region.

Our present study has demonstrated the power of universality in a gauge
theory. In fact, the data used to determine the nonuniversal constants (33)
were obtained through Monte Carlo simulations at a single finite temperature.
Nevertheless, through universality with the quantum Ising chain, we were able
to quantitatively predict the critical behavior of the electric field at zero
temperature, as shown in Fig. 9. This prediction closely matched the direct
numerical results obtained from a completely different formulation
Byrnes:2002nv . We hope that our work serves as a successful application
example of universality in a gauge theory and inspires similar research for
understanding critical behaviors, not only in the Schwinger model but also in
other gauge theories, including quantum chromodynamics.

###### Acknowledgements.

The author was supported by a Grant-in-Aid for JSPS Fellows (Grant
No.22KJ1662). The numerical simulations have been carried out on Yukawa-21 at
Yukawa Institute for Theoretical Physics (YITP), Kyoto University.

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11institutetext: Universidad Nacional de Ingeniería, Lima, Perú
http://www.uni.edu.pe 22institutetext: Universidad Tecnológica del Perú, Lima,
Perú
https://www.utp.edu.pe/

# Car Monitoring System in Apartment Garages by Small Autonomous Car using
Deep Learning

Leonardo León-Vera Felipe Moreno-Vera
Renato Castro-Cruz Jose Navío-Torres and Marco Capcha-Mansilla 1111111122
{lleonv, felipe.moreno.v, racastroc<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Currently, there is an increase in the number of Peruvian families living in
apartments instead of houses for the lots of advantage; however, in some cases
there are troubles such as robberies of goods that are usually left at the
parking lots or the entrance of strangers that use the tenants parking lots
(this last trouble sometimes is related to kidnappings or robberies in
building apartments). Due to these problems, the use of a self-driving mini-
car is proposed to implement a monitoring system of license plates in an
underground garage inside a building using a deep learning model with the aim
of recording the vehicles and identifying their owners if they were tenants or
not. In addition, the small robot has its own location system using beacons
that allow us to identify the position of the parking lot corresponding to
each tenant of the building while the mini-car is on its way. Finally, one of
the objectives of this work is to build a low cost mini-robot that would
replace expensive cameras or work together in order to keep safe the goods of
tenants.

###### keywords:

Localization Self-Driving Low energy License Plate Recognition Bluetooth Low
cost minirobot License plate detection.

##  1 Introduction

Peruvian families prefer to live in apartments (see Fig. 1), since the common
advantages are more comfortable spaces and less effort in cleaning unnecessary
rooms; however, the size of the apartments is reduced every year. Therefore,
families decide to keep their goods at parking lots next to their vehicles or
inside of them, as long as they are in an underground garage, but some of
these stuff are stolen. Also, people who do not live in the buildings use the
parking lots that are only for building users, sometimes thieves parked in the
building garage before a stealing inside the building.

A smart self-driving mini-car is proposed to supervise the vehicles in the
underground garage by identifying their owners using computer vision in their
respective parking lots during a schedule that does not bother the tenants and
cars. Using autonomous driving techniques and object detection, a
computational vision system is proposed in order to achieve the autonomous
navigation of the mini-car and the identification mainly of the cars owners
with an ALPR (Automatic License Plate Recognition) based on a deep learning
model that was developed by the open source organization OpenALPR. Also, the
mini-car has its own location system with Beacons that determine its relative
position in real time and it can verify if a vehicle would be at its correct
parking lot or not.

Vehicles of the future will be strongly dependent on the development of the
software algorithms controlling the autonomous navigation and how they can
process the massive amount of data generated, so this mini-car wants to be a
prototype of that type of vehicle that is able to be autonomous and at the
same time translate the license plates that recognize into an important
information for security purposes.

Figure 1: Growth of people living in apartments. Source: Private households of
Peruvians 2007-2016. INEI.

The navigation of the mini car in [14] is based on images recognition using
deep learning, developed for more complicated tracks than an underground
garage and it uses different sensors for better accuracy in indoor scene.

##  2 Structure of the System

Implementation of a mini-robot that runs through an underground garage and
verify that the tenants cars are in their respective parking lots. It is
proposed to solve the problem by detecting objects to identify the vehicle
position and its license plate to obtain information from the owner of the
car. The holistic system is made up of three main parts: The computer vision
system that allows the self-driving and the license plate recognition; the
location system that allows us to know the mini-car location and which parking
lots are near at the same time; and the system that describes how the mechanic
and electronic mini-car components are integrated.

Figure 2: Structure of the system joining each task.

###  2.1 Data collected from tenants of the building.

The first step to build a well-designed system that verifies if the vehicle
parked on its respective parking lot is having the owner information. This
data is collected from the tenants of the building such as the apartment
number, name, ID of parking lot, type of vehicle, license plate, objects they
stored in the parking lot, etc (See Fig. 3). The owners name and their license
plates will be compared with the data obtained by the mini-car license plate
recognition and shown to the caretaker so that he can draw his conclusions if
a bizarre event happens.

Figure 3: Data of tenants.

###  2.2 Autonomous Navigation of the small car

To face this task we use logitech C920 web cam which is cheaper than Lidars
and Radars. These are very used for self-driving training since they can
calculate the distance between objects with high precision. In contrast, it
turns out that web cams emulate the way in which people can see the
environment giving a better classification and interpretation of images
textures in comparison to the previous ones [4]. We are imitating Behavioral
Cloning method described in Udacity Self Driving Course because there is only
one way where the mini car can go. I mean, It only has to turn left when it is
going down and turn right when it is going up. It is one constraint about the
garage.

A Convolutional Neural Network based on the NVIDIA model[1] for regression is
used to predict the angle of rotation of the wheels of the mini robot only
obtaining an image of the path (See Fig. 9).

####  2.2.1 Data Collection

Being a supervised learning, it was necessary to collect a video recording the
route that the mini car must follow to obtain an optimal performance. Being a
flat floor, we did not get additional problems; however, when the garage walls
are very similar, it is needed to look for a correct camera position and cut
out the image so that it only focus on taking images that correctly identify
curves of straight roads. The input data are the images of three front cameras
(images from one camera and 2 images modified) separated from each other by
few centimeters to collect more data from the road and speed of the small car
each moment; and angles between 0 ° and 180 ° as output values for training
the model.

####  2.2.2 Neural Network Architecture

A deep sequence model of layers is used in the following manner, with its
respective number of filters: Conv24-Conv36-Dropout-Conv48-Dropout-
Conv64-Dropout-Conv64-Dropout-FC-FC-FC-FC with non - linear activation
function ReLU, which bring good results in computational vision tasks in the
Convolutional layers. Regularization methods such as Dropout with a
probability of 0.5 are added, due to the overfitting that occurs with the
NVIDIA model when iterations are increased considerably.

####  2.2.3 Neural Network Training and Testing

The training of the data was done with the board NVIDIA P4000 with a partition
of 20% validation set and 80% training set. The MSE loss function and the ADAM
optimizer method are used without the need to manually set the speed of the
learning rate.

Testing of the CNN is performed on the Udacity simulator, obtaining a clear
outstanding performance when driving autonomously as can be seen in the video
[5] and later applied to Jetson TX1 in an underground garage.

###  2.3 Vehicles Detection

The first attempt to find objects in each parking slot was cars detection. A
convolutional neuronal network is in charge of this task using Tiny YOLO
model[3] that is the model with the best performance between accuracy and
inference time in detection objects in real time such as cars (our priority),
bicycles and others objects (see Fig. 10) trained using VOC dataset of 2012.
The model also detects void spaces if probability to find any object is lower
than the threshold. Once the car is detected, its position is saved in
jetson’s memory with relative position brought by the beacons.

####  2.3.1 Neural Network Architecture

Tiny YOLO model consists in a convolutional neuronal network with 9
convolutional layers of 16, 32, 64, 128, 256, 512, 1024, 512, 425 filters each
one. This model is lightweight in comparison with ”YOLO9000: Better, Faster,
Stronger” [3] and accors.

Figure 4: Mini-Robot for testing in the garage

####  2.3.2 Neural Network Test

Test of the neural network was performed on an NVIDIA P4000 GPU server with
images taken in the garage and on the NVIDIA Jetson TX1 board which we used in
the mini-robot in the garage with a C920 camera obtaining 15 fps which works
without any problem with the tiny YOLO model.

###  2.4 License Plate Recognition

After trying cars detection and getting acceptable results, OpenALPR library
has been tried to detect the plate license of each car and later to compare
with SUNARP dataset.

The mini robot records videos on its way in the garage so that the license
plate recognition could be done.

After obtaining many candidates, we are going to select the license plate that
have six digits (Peruvian’s license plate rule) and the major often weighted.

After recognizing the characters (letter and digits) of the license plates,
the system must access the Peruvian public records office (SUNARP) online
query[10] to collect information related to the owner of the vehicle (see
Fig.5). Once important information such as name and license plate is obtained,
it will be compared with the database that the mini robot stores from
collecting data of the building tenants.

Figure 5: Query of the owner of license plate in SUNARP online
application[10].

###  2.5 Mini-robot mechanism

Mini-robot is built with base of a Monster Truck 1/10 which includes a motor
and 2 servos for the movement of front and rear tires. An Arduino UNO is added
that allows communication between the engine and the NVIDIA Jetson TX1 board,
as you can see in Fig 4. The Arduino sends instructions to the engine through
its GPIOs where indicates speed of the wheels and angle of rotation of the
front wheels. Communication of the board TX1 to the Arduino is serial through
a USB which allows to send integer values encoded in characters to indicate
the rotation of the front wheels and the instructions to go forward, stop and
rewind. Through Python program on the TX1 board, instructions are sent to the
Arduino and this sends to the motor which angle have to turn.

Figure 6: Hardware communication.

###  2.6 Tenant Parking Mapping by Beacons

####  2.6.1 Summary

In this section, we present the way that robot knows where is it in real time.
This action can be regulated with Beacons that is useful for indoor location
in places where the Internet is not enough to make a connection [7], even
allowing tracking in real-time applications [8].

(a) Detection of objects, walls and signals of beacons. See the colors: yellow
(beacons), red (object detection field), green (direction of route) and
celestial (Bluetooth signal).

(b) Positioning of the beacons in the parking lot. See the colors: yellow
(beacons), red (object detection field), green (direction of route) and
celestial (Bluetooth signal)

Figure 7: Environment and Beacon’s location map

####  2.6.2 Detection of Beacons

In the garage, there is a distance about 8 meters between walls and 2.45 or
7.35 meters between vehicles, as shown in the Fig. 7(a) and Fig. 7(b).

As it is seen, Beacons have been distributed efficiently (see Fig. 8) that
allows to calculate where it is located in real time, as well as identifies
who owns that area due to the match between image recognition and the current
position, based on previous work [9].

Figure 8: Locations of the beacons in the garage and variables to be taken for
the calculation and identification of the beacons.

####  2.6.3 Determination of the position of the mini-robot

This section is about the equations that define the system to solve, by
forming imaginary circles around each beacon, the respective radii are
calculated. Then we proceed to calculate the relative position with respect to
the global system of beacons based on distance equations forming a system of
equations with the following form:

$E_{i}:(x-x_{i})^{2}+(y-y_{i})^{2}=d_{1}^{2}$

for i=1, …, 3

Where (x, y) is the current position of the mini-robot and the indexes i
corresponding to the beacons and $r_{i},...,r_{3}$ to the respective vectors.

For this system formed proceeds to solve: Take $(x_{i},y_{i})$ as coordinates
of each beacon, we deduce $r_{i}=r_{c}+d_{i}$.

Where $r_{c}$ is the current position of the mini-robot, for all equations the
pair of indices i, j are not equal ($i\neq j$).

So, The module is taken:

$\|r_{i}\|^{2}=\|r_{c}\|^{2}+2(r_{c})(r_{i})+\|d_{i}\|^{2}$.

Calculating:

$\|r_{i}\|^{2}-\|r_{j}\|^{2}$

The result is:

$r_{c}(d_{i}-d_{j})=\|d_{j}\|^{2}-\|d_{i}\|^{2}+\|r_{i}\|^{2}-\|r_{j}\|^{2}=Y_{i}$

By which we would have:

$x_{c}(x_{i}-x_{j})+y_{c}(y_{i}-y_{j})=Y_{i}$

Which forms a new linear system that is solved by numerical methods and
represented as:

Y = AX.

Where:

$x=(x_{c},y_{c})^{t}$: is the column vector of the mini-robot positions.

A: is the matrix forms by row vectors.

A = $\begin{bmatrix}x_{1}-x_{2}&y_{1}-y_{2}\\\ x_{2}-x_{3}&y_{2}-y_{3}\\\
x_{3}-x_{1}&y_{3}-y_{1}\end{bmatrix}$

Y: is the column vector of substractions between beacons.

##  3 Results

###  3.1 Determination of the position of the mini robot

To solving the Linear Equation System, we compared different methods, like
iterative and directly. In directly methods we have the problem that the order
of the algorithm is almost O($n^{3}$) like Gauss Elimination and derivate
should be applied to special matrix (like symmetric, semi positive, positive,
dominant diagonal, etc).

Using Methods like Gauss Elimination and iterative methods like Jacobi and
Gauss-Seidel (first verify if is a diagonal dominant matrix to apply)
iterative methods), we parallel using OpenMP and CUDA. Using Montercalo
methods to get an execution average time and approximate a pseudo real
solution time.

Method | Serial time | Parallel time (n=2) | Parallel time (n=4) | Parallel time (n=8) | Parallel time CUDA
---|---|---|---|---|---
Gauss-Seidel | 0.0729176 | 0.0670215 | 0.0362759 | 0.0152195 | 0.0000512
Jacobi | 0.366089 | 0.358308 | 0.370612 | 0.0151156 | 0.0000586
Gauss | 68.8792 | 34.4715 | 17.9784 | 17.0431 | 2.321465
Table 1: Execution average time for 40000 times

###  3.2 Self-driving

After training about 150 epochs we saw that the curve of loss function won’t
be lower. We got 0.014 of Error in validation and training set using Udacity
simulator. Next weeks, we are going to get results of our data obtained.

Figure 9: Graphics of demand and training.

###  3.3 Results of Cars Detection

Detection of objects is very well identifying cars of general classes (See
Fig. 10), however when it is required to identify brand and model of the car
there are no results since there is no dataset of images of front or back of
the cars with annotations of their marks from which the neural network can
learn. The nearest set of images found is Stanford dataset[13], but the
majority of images are not taken from front or back view. The model of car
recognition was trained with this dataset, but the images used for test were
unlabeled. So, this dataset was discarded for the model.

Figure 10: Detection of cars by mini robot

###  3.4 Results of Automatic License Plate Recognition

OpenALPR library has a good performance in plate recognition, characters
segmentation, and characters recognition [11]. OpenALPR has been trained on
its own dataset, however there exists other datasets for this task such as
SSIG dataset [15], a commercial dataset or UFPR-ALPR dataset[2] recently made
with more fully annotated images and more vehicles in real-world scenarios for
academic purposes.

As it can be seen (See Fig. 12), the actual license plate was recognized with
a high total confidence. It turns out that the input is a video file that was
recorded by the mini-robot and its outputs in the console (See Fig. 11) are
lots of candidates to be considered as the real license plate. After it, three
candidates with the most high confidence are chosen to be verified in the
tenants database and the SUNARP online consultation.

Figure 11: License plate recognition of a video recorded by the mini-car
Figure 12: License Plates vs Total Confidence

##  4 Conclusions

We see that during the implementation of the robot car, it has to count on
many things that had not been foreseen to be able to handle the autonomous
car.

In the work procedure, an optimal performance is obtained for the object
detection and the automatic license plate recognition using Artificial
Intelligence algorithms.

For the determination of positions of the mini robot based on the terrain
delimited by columns and walls, it is necessary to specify the locations and
the numerical system that solves the system of equations generated in such a
way that a minimum error is obtained, which serves to determine and to
identify the place where it is when we detect cars and objects of a certain
car park, generating the relation object detected - location.

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