# Reduction of the Twisted Bilayer Graphene Chiral Hamiltonian into a $2\times
2$ matrix operator and physical origin of flat-bands at magic angles

Gerardo G. Naumis, Leonardo A. Navarro-Labastida, Enrique Aguilar-Méndez
<EMAIL_ADDRESS>Depto. de Sistemas Complejos, Instituto de Física,
Universidad Nacional Autónoma de México (UNAM)
Apdo. Postal 20-364, 01000, CDMX, México.

(February 2021)

###### Abstract

The chiral Hamiltonian for twisted graphene bilayers is written as a $2\times
2$ matrix operator by a renormalization of the Hamiltonian that takes into
account the particle-hole symmetry. This results in an effective Hamiltonian
with an average field plus and effective non-Abelian gauge potential. The
action of the proposed renormalization maps the zero-mode region into the
ground state. Modes near zero energy have an antibonding nature in a
triangular lattice. This leads to a phase-frustration effect associated with
massive degeneration, and makes flat-bands modes similar to confined modes
observed in other bipartite lattices. Suprisingly, the proposed Hamiltonian
renormalization suggests that flat-bands at magic angles are akin to floppy-
mode bands in flexible crystals or glasses, making an unexpected connection
between rigidity topological theory and magic angle twisted two-dimensional
heterostructures physics.

††preprint: APS/123-QED

Introduction. The superconductivity found at low temperatures in condensed
systems leads to an efficient electronic transport without loss of energy.
Reaching this superconducting state is very difficult and requires very strict
laboratory parameters [1]. High $T_{c}$ superconductors uses cuprates, which
are well-ordered structures of atoms combined in three-dimensional
arrangements [2, 3, 4, 5, 6, 7]. At high temperatures, the existence of
superconductivity cannot yet be explained because the mechanism that
counteracts the Colombian repulsion force generating the appearance of cooper
states is not exactly known, for this reason these unconventional states are
referred to as strongly correlated [8, 9]. Recently, it has been discovered
that twisted bilayer graphene exhibits superconducting states at certain
rotation angles [10, 11] where the electron interactions are maximized [12].
This rotated graphene bilayer model generates a Moire pattern as a function of
the rotation angle, and then we can define a Moire Brillouin zone (mBZ). These
special angles are called ”magic” and were predicted as a possible consequence
of flat-bands that were observed in a previous theoretical work [13]. In the
work of Cao et.al. [10], a Mott insulating state appears in the middle of
these superconducting phases. The study of other interactions has also been
reported, such as, for example, electron-electron interaction through a
Hubbard potential [14, 15]. Therefore, the manipulation of heterostructures
has received a lot of interest, because these, through stacking, deformation,
and more recently rotation, have been found to generate more efficient
conductive properties [16, 17]. The study of the electronic properties of
rotated graphene over graphene started before the discovery of
superconductivity at magic angles. In the work of J. Santos [18] and A.
Macdonald [13], a continuous Hamiltonian model was presented; however, due to
the presence of an interlayer amplitude AA coupling, this model did not
present chiral symmetry. In a recent work by G. Tarnopolsky et. al. [19], a
chiral continuum model was studied, and only the AB and BA inter-layer
couplings are different from zero. Perhaps, so far, it is the simplest model
that best captures the nature of magic angles; at these angles the dispersion
energy becomes flat and has a recurrence behavior. At these magic angles the
Fermi velocity also goes to zero. Due to its chiral symmetry, the Hamiltonian
of this model also produces an intra-valley inversion symmetry [20], so the
energy dispersion is inversion symmetric at all twist angles. Also, we can
distinguish between different magic angles when zero modes occurs, and thus
the inter-valley inversion classifies the topology of the twist angle. The
zero-mode flat-band solutions have some resemblance to the ground state of a
quantum Hall effect wave function on a torus [19, 21], and, therefore, the
solution is of the harmonic oscillator type, where Landau levels arise [22,
23]. Another interesting mathematical insight is that flat-band modes are
constructed from a complex analytic functions that are ratios of the theta
Jacobi functions [19, 21].

The mechanism that causes the appearance of these magic angles is still not
known, however, many investigations suggest that it is a topological aspect of
the band structure [24, 25, 26, 27, 28, 29, 30]. This is due to the appearance
of flat bands and, these in turn, are related to the appearance of edge
states, which suggests a topological study of the invariants of the system.
Furthermore, due to the quantized recurrence of the magic angles it appears
that there is an adiabatic change in integers of this topological invariant.
The aim of this work its to clarify the physical behavior of this model and to
develop an effective equation for the Hamiltonian matrix in order to study all
other non flat band states. In particular, here we renormalize the Hamiltonian
to take into account the particle-hole symmetry which results in a folding of
the spectrum around zero-energy. We found that the effective system is made
from an average potential plus a non-Abelian potential. Then we discuss the
physical picture that arises from the renormalization

Twisted Bilayer Graphene Effective Model. Tarnopolsky et. al. derived a chiral
Hamiltonian for electrons-holes in twisted bilayer graphene. It captures the
”true magic” of the magic angle physics [19],

$\begin{split}\mathcal{H}&=\begin{pmatrix}0&D^{\ast}(-r)\\\
D(r)&0\end{pmatrix}\\\ \end{split}$ (1)

where the zero-mode operator is,

$\begin{split}D(r)&=\begin{pmatrix}-i\bar{\partial}&\alpha U(r)\\\ \alpha
U(-r)&-i\bar{\partial}\end{pmatrix}\\\ \end{split}$ (2)

and,

$\begin{split}D^{*}(-r)&=\begin{pmatrix}-i\partial&\alpha U^{*}(-r)\\\ \alpha
U^{*}(r)&-i\partial\end{pmatrix}\\\ \end{split}$ (3)

with $\bar{\partial}=\partial_{x}+i\partial_{y}$,
$\partial=\partial_{x}-i\partial_{y}$. The potential is,

$U(\bm{r})=e^{-i\bm{q}_{1}\cdot\bm{r}}+e^{i\phi}e^{-i\bm{q}_{2}\cdot\bm{r}}+e^{-i\phi}e^{-i\bm{q}_{3}\cdot\bm{r}}$
(4)

Figure 1: a)Moiré mBrillouin zone, $\bm{b_{1,2}}$ are the base vectors. b)
Real space Moiré unit cell,$\bm{a_{1,2}}$ are two Moiré lattice vectors. Point
$\bm{r}_{0}=(\bm{a}_{1}-\bm{a}_{2})/3$ is the AB stacking point where all
components of the wave function vanishes at magic $\alpha$.

For this Hamiltonian, the parameters are $\phi=\frac{2\pi}{3}$ and
$\bm{q}_{1}=k_{\theta}(0,-1)$,
$\bm{q}_{2}=k_{\theta}(\frac{\sqrt{3}}{2},\frac{1}{2})$ and
$\bm{q}_{3}=k_{\theta}(-\frac{\sqrt{3}}{2},\frac{1}{2})$, the Moire modulation
vector is $k_{\theta}=2k_{D}\sin{\frac{\theta}{2}}$ with
$k_{D}=\frac{4\pi}{3a_{0}}$ is the magnitud of the Dirac wave vector and
$a_{0}$ is the lattice constant of monolayer graphene. The physics of this
model is captured by the parameter $\alpha$, defined as
$\alpha=\frac{w_{1}}{v_{0}k_{\theta}}$. Here $w_{1}$ is the interlayer
coupling of stacking AB and BA, take the value $w_{1}=110$meV and $v_{0}$ is
the Fermi velocity, with value $v_{0}=\frac{19.81eV}{2k_{D}}$.At magic angles
$\alpha=0.586,2.221,3.751,5.276,6.795,8.313,9.829,11.345,...$, flat-bands
appear. Magic $\alpha$’s follow a remarkable $3/2$ cuantization rule [19] for
$\alpha>0.586$ .

This Hamiltonian is difficult to tackle and in fact most of the studies have
been restricted to the zero-mode operator solutions at energy zero [20, 21].
Here, instead of solving the Schrödinger equation with $\mathcal{H}$ we first
propose to reduce the dimensionality of the problem. Starting with the
Schrödinger equation $\mathcal{H}\Phi=E\Phi$, where
$\Phi(r)=\begin{pmatrix}\psi_{1}(r),\psi_{2}(r),\chi_{1}(r),\chi_{2}(r)\end{pmatrix}^{T}$
are the four components of the twisted graphene bilayer, and the index $1,2$
represent each graphene layer, we consider the squared Hamiltonian
$\mathcal{H}^{2}$,

$\begin{split}\mathcal{H}^{2}&=\begin{pmatrix}D^{\ast}(-r)D(r)&0\\\
0&D(r)D^{\ast}(-r)\end{pmatrix}\end{split}$ (5)

This transforms the off-diagonal blocks into zeros. We can understand such
transformation as a removal of the particle-hole symmetry that is an anti-
unitary anti-commuting symmetry. Therefore, we obtain a decoupled equation
$\mathcal{H}^{2}\Phi(r)=E^{2}\Phi(r)$, where the eigenvalues are the squares
of the original energies. States at $E=0$ are ground states. As for arbitray
$\alpha$ there are always two zero-mode solutions in the $K$ and $K^{\prime}$
points [19], the signature of a magic angle in $\mathcal{H}^{2}$ is that the
Fermi velocity at $\bm{k}=K,K^{\prime}$ points of the Moiré Brillouin
(mBrillouin) zone approaches zero and a degenerate ground state.

We now define a $2\times 2$ effective Hamiltonian $H^{2}=D^{\ast}(-r)D(r)$. As
detailed in the supplementary material, the resulting effective Hamiltonian
is,

$\begin{split}H^{2}&=\begin{pmatrix}-\nabla^{2}+\alpha^{2}|U(\bm{-r})|^{2}&\alpha
A^{\dagger}(\bm{r})\\\ \alpha
A(\bm{r})&-\nabla^{2}+\alpha^{2}|U(\bm{r})|^{2}\end{pmatrix}\end{split}$ (6)

The norm of the potential is,

$|U(\bm{r})|^{2}=3+2\cos(\bm{b}_{1}\cdot\bm{r}-\phi)+2\cos(\bm{b}_{2}\cdot\bm{r}+\phi)+2\cos(\bm{b}_{3}\cdot\bm{r}+2\phi)$
(7)

where $\bm{b}_{1,2}=\bm{q}_{2,3}-\bm{q}_{1}$ are the mBrillouin zone Moiré
vectors and $\bm{b}_{3}=\bm{q}_{3}-\bm{q}_{2}$. The off-diagonal terms are,

$\begin{split}A(\bm{r})&=-i\sum_{\mu=1}^{3}e^{i\bm{q}_{\mu}\cdot\bm{r}}(2\bm{\hat{q}}_{\mu}^{\perp}\cdot\bm{\nabla}-k_{\theta})\\\
\end{split}$ (8)

and,

$\begin{split}A^{\dagger}(\bm{r})&=-i\sum_{\mu=1}^{3}e^{-i\bm{q}_{\mu}\cdot\bm{r}}(2\bm{\hat{q}}_{\mu}^{\perp}\cdot\bm{\nabla}+k_{\theta})\\\
\end{split}$ (9)

where $\bm{\nabla}^{\dagger}=-\bm{\nabla}$ with
$\bm{\nabla}=(\partial_{x},\partial_{y})$ and $\mu=1,2,3$. This is an
essential point as eigenvalues must be reals (notice that
$-A^{\dagger}(\bm{-r})=A(\bm{r})$). Also, $\bm{\hat{q}}_{\mu}^{\perp}$ is a
set of unitary vectors perpendicular to the set $\bm{q}_{\mu}$ (see
supplementary). The term $\bm{\hat{q}}_{\mu}^{\perp}\cdot\bm{\nabla}$ is the
directional gradient along the triangle defined by the Moire vectors.

Any eigenfunction of the original Hamiltonian is an eigenfunction of
$\mathcal{H}^{2}$, therefore, the spinor
$\Psi(\bm{r})=(\psi_{1}(\bm{r}),\psi_{2}(\bm{r}))$ is a solution of $H^{2}$.
In a similar way, we can proceed to use $D(r)D^{\ast}(-r)$ with solutions
$(\chi_{1}(\bm{r}),\chi_{2}(\bm{r}))$. These solutions are easily obtained
from $(\psi_{1}(\bm{r}),\psi_{2}(\bm{r}))$ using symmetry operations, and thus
here we only study $H^{2}$. It is wise to mention that $\mathcal{H}^{2}$
eigenfunctions can be any linear combination of both solutions due to double
degeneracy, as expected for a $\mathcal{H}$ chiral Hamiltonian. Clearly,
$\mathcal{H}^{2}$ eigenfunctions are made from a superposition of pseudo-spin
polarized states of $\mathcal{H}$. Next we write the renormalized twisted
graphene bilayer Hamiltonian in terms of the Pauli matrix vector
$\hat{\bm{\sigma}}$ as follows.

$H^{2}=h_{0}(\bm{r})\sigma_{0}+\bm{h}(\bm{r})\cdot\hat{\bm{\sigma}}$ (10)

where the vector $\bm{h}(\bm{r})$ is,

$\bm{h}(\bm{r})=(\alpha h_{x}(\bm{r}),\alpha
h_{y}(\bm{r}),\alpha^{2}h_{z}(\bm{r}))$ (11)

with,

$\begin{split}h_{x}(\bm{r})&=-\sum_{\mu}[k_{\theta}\sin{(\bm{q}_{\mu}\cdot\bm{r})}+2i\cos{(\bm{q}_{\mu}\cdot\bm{r})}\bm{\hat{q}}_{\mu}^{\perp}\cdot\bm{\nabla}]\\\
h_{y}(\bm{r})&=\sum_{\mu}[k_{\theta}\cos{(\bm{q}_{\mu}\cdot\bm{r})}-2i\sin{(\bm{q}_{\mu}\cdot\bm{r})}\bm{\hat{q}}_{\mu}^{\perp}\cdot\bm{\nabla}]\\\
h_{z}(\bm{r})&=\frac{|U(\bm{-r})|^{2}-|U(\bm{r})|^{2}}{2}\end{split}$ (12)

The operator in front of the identity $\sigma_{0}$ is,

$h_{0}(\bm{r})=-\nabla^{2}+\bar{V}(\bm{r})$ (13)

$\begin{split}\bar{V}(\bm{r})=\frac{|U(\bm{r})|^{2}+|U(\bm{-r})|^{2}}{2}\end{split}$
(14)

We observe that $h_{0}(\bm{r})$ corresponds to a Hamiltonian with an average
potential. The term $\Delta V(\bm{r})$ is a Zeeman-like coupling term. Plus,
it is tempting to identify $A(\bm{r})$ as a space-dependent magnetic field
vectorial potential, however, as the conmmutator
$[\bm{A}(\bm{r}),\bm{A}(\bm{r^{\prime}})]\neq 0$, this is a non-Abelian gauge
potential similar to the one found in Ref. [31]. Observe that the term
$\bm{h}(\bm{r})$ contains all the topological properties of the operator. Let
us further simplify this Hamiltonian. We define a new set of wavefunctions
$\psi_{\pm}(\bm{r})=\psi_{1}(\bm{r})\pm\psi_{2}(\bm{r})$. Using this, the
Schrödinger equation is transformed into
$H_{\text{eff}}(\psi_{+}(\bm{r}),\psi_{-}(\bm{r}))^{T}=E^{2}(\psi_{+}(\bm{r}),\psi_{-}(\bm{r}))^{T}$.
Finally, the stated effective $2\times 2$ Hamiltonian is,

$\begin{split}H_{\text{eff}}&=\begin{pmatrix}-\nabla^{2}+V_{\text{eff}}(\bm{r})&A_{\text{eff}}^{\dagger}(\bm{r})\\\
A_{\text{eff}}(\bm{r})&-\nabla^{2}+V_{\text{eff}}(\bm{r})\end{pmatrix}\\\
\end{split}$ (15)

Here we defined the effective potentials as,

$V_{\text{eff}}(\bm{r})=\alpha^{2}\bar{V}(\bm{r})+\alpha h_{x}(\bm{r})$ (16)

and,

$A_{\text{eff}}(\bm{r})=\alpha^{2}h_{z}(\bm{r})+i\alpha h_{y}(\bm{r})$ (17)

Effective Hamiltonian in triangular coordinates. Now we proceed to show the
explicit form of each component of the effective potential in a suitable
coordinate system. Due to the symmetry, it is convenient to introduce the
triangular coordinates $u_{j}$ with
$u_{1}=\bm{b}_{1}\cdot\bm{r},u_{2}=-\bm{b}_{2}\cdot\bm{r},u_{3}=\bm{b}_{3}\cdot\bm{r}$.
This transformation has the important property that $\sum_{j}u_{j}=0$ and
gives the distance to the edges of a triangle as seen in Fig. 2. Putting this
transformation in the potential we get $\bar{V}(\bm{r})=3-\sum_{j}\cos{u_{j}}$
and $h_{z}(\bm{r})=-\sqrt{3}\sum_{j}\sin{u_{j}}$.

Figure 2: Triangular coordinates.

A second change of variable is useful to reduce the Hamiltonian by taking into
account mirror symmetries. We use, $\zeta=u_{1}$ and $\eta=u_{2}-u_{3}$, where
$0\leq\eta\leq 4\pi$ and $0\leq\zeta\leq 4\pi$. $\zeta$ and $\eta$ are
coordinates that indicate the departure from or along the median of the
triangle. The inverse transformation is $u_{2}=(\eta-\zeta)/2$ and
$u_{3}=-(\eta+\zeta)/2$. The average and Zeeman like terms are,

$\bar{V}(\bm{r})=3-\cos{\zeta}-2\cos{\frac{\zeta}{2}}\cos{\frac{\eta}{2}}$
(18)

$h_{z}(\bm{r})=-\sqrt{3}\left(\sin{\zeta}-2\sin{\frac{\zeta}{2}}\cos{\frac{\eta}{2}}\right)$
(19)

and the Laplacian operator is,

$\nabla^{2}=3k_{\theta}^{2}(\partial^{2}_{\zeta}+3\partial^{2}_{\eta})$ (20)

To reduce the other terms, we introduce an intermediate change of variable
$\bm{q}_{\mu}\cdot\bm{r}=s_{\mu}$,

$\begin{split}h_{x}(\bm{r})&=-\sum_{\mu}k_{\theta}\sin{s_{\mu}}-2i[\cos{s_{1}}-\frac{\cos{s_{2}}+\cos{s_{3}}}{2}]\partial_{x}\\\
&-i\sqrt{3}[\cos{s_{2}}-\cos{s_{3}}]\partial_{y}\end{split}$ (21)

while,

$\begin{split}h_{y}(\bm{r})&=\sum_{\mu}k_{\theta}\cos{s_{\mu}}-2i[\sin{s_{1}}-\frac{\sin{s_{2}}+\sin{s_{3}}}{2}]\partial_{x}\\\
&-i\sqrt{3}[\sin{s_{2}}-\sin{s_{3}}]\partial_{y}\end{split}$ (22)

Then, we use the transformation into the $\zeta$ and $\eta$ coordinates by
first observing that $u_{1}=s_{2}-s_{1},u_{2}=s_{1}-s_{3},u_{3}=s_{3}-s_{2}$
and finally,

$\begin{split}s_{1}=\frac{\eta-3\zeta}{6},\ s_{2}=\frac{\eta+3\zeta}{6},\
s_{3}=-\frac{\eta}{3}\end{split}$ (23)

where $s_{1}=-(s_{2}+s_{3})$ and,

$\partial_{x}=\frac{\sqrt{3}}{2}k_{\theta}(\partial_{\zeta}-3\partial_{\eta}),\partial_{y}=\frac{3}{2}k_{\theta}(\partial_{\zeta}+\partial_{\eta})$
(24)

In $\zeta$ and $\eta$ coordinates we have,

$\begin{split}\frac{h_{x}(r)}{k_{\theta}}&=\left(\sin{\frac{\eta}{3}}-2\cos{\frac{\zeta}{2}}\sin{\frac{\eta}{6}}\right)\\\
&-i2\sqrt{3}\left(\cos{\frac{\zeta}{2}}\cos{\frac{\eta}{6}}-\cos{\frac{\eta}{3}}\right)\partial_{\zeta}\\\
&+i6\sqrt{3}\sin{\frac{\zeta}{2}}\sin{\frac{\eta}{6}}\partial_{\eta}\end{split}$
(25)

and

$\begin{split}\frac{h_{y}(r)}{k_{\theta}}&=\left(\cos{\frac{\eta}{3}}+2\cos{\frac{\zeta}{2}}\cos{\frac{\eta}{6}}\right)\\\
&-i2\sqrt{3}\left(\cos{\frac{\zeta}{2}}\sin{\frac{\eta}{6}}+\sin{\frac{\eta}{3}}\right)\partial_{\zeta}\\\
&-i6\sqrt{3}\sin{\frac{\zeta}{2}}\cos{\frac{\eta}{6}}\partial_{\eta}\end{split}$
(26)

Therefore, we have all expressions for our effective Hamiltonian is terms of
the triangular coordinates, i.e., $H_{\text{eff}}=H_{\text{eff}}(\zeta,\eta)$.
We end up this section by observing how the results indicate that we can scale
the operator $H^{2}$ by defining $\alpha^{\prime}=\alpha/k_{\theta}$ and
$\epsilon=E/k_{\theta}$, in such a way that $k_{\theta}$ is no longer in the
Hamiltonian, while we make the replacement $\alpha\rightarrow\alpha^{\prime}$
and $E\rightarrow\epsilon$. In fact, such effective parameters can be found
right away from the start by defining a scaled set of coordinates,

$\bm{r^{\prime}}=k_{\theta}\bm{r},\
\bm{q^{\prime}}_{\mu}=\frac{\bm{q}_{\mu}}{k_{\theta}}$ (27)

This transformation maps the $k_{\theta}$ dependent basis vectors of the
unitary cell into the $k_{\theta}$ independent basis vectors
$\bm{a}^{\prime}_{1,2}=4\pi/3(\pm\sqrt{3},1/2)$. Although is possible to make
this scaling from the full Hamiltonian Eq. (1), in our development we decided
to stick with the notation used by others in previous works, and keep this
argument as a cross-check point.

Physical interpretation of the effective model and flat-bands The
renormalization into a $2\times 2$ matrix not only simplifies the mathematics
but has a profound physical meaning. What we achieved is the decoupling of the
$A$ and $B$ bipartite lattices for each graphene layer. In graphene, such
squared Hamiltonian is equivalent to renormalize the honeycomb lattice into a
triangular lattice with renormalized interactions and a self-energy [32, 33],
here Eq. (18) shows that this is also the case but with an added non-Abelian
potential. As shown in Fig. 3, the transformation of $E$ to $E^{2}$ produces a
fold in the spectrum such that bonding states with the lowest energy are
mapped into higher energies. In other words, the transformation deletes the
alternating sign of one of the bipartite sublattices as explained in Fig. 3.

Figure 3: Density of states (DOS) in monolayer graphene ($\rho(E)$) and of
$\rho(E^{2})$ corresponding to the states of the squared Hamiltonian. This
shows how the transformation to $\mathcal{H}^{2}$ leads to a folding of the
spectrum. The signs are a sketch of the relative signs and amplitudes on each
bipartite sublattice. Arrows indicate how the states are mapped under such
transformation. Zero modes and close-by states are in the antibonding limit on
a triangular lattice (indicated by the triangle inside the hexagon), in which
frustration effects are important as tend to increase energy. The frustration
is seen here in the bond that joins the $+$ and $+$ sites in the triangle.
Zero-modes are pinned in this transformation.

The zero-modes at $E=0$, which are the flat-band modes for $\alpha$ at magic
angles, are the ground states of $H^{2}$. This gives a glimpse of the
observation made by Tarnpoksy et. al. that zero-modes resemble the lowest
Landau state in the quantum Hall effect [19], a fact recently confirmed [20].
We end up having a clear picture of nearby states around zero-energy modes,
they correspond to the antibonding limit in a triangular lattice. Anti-bonding
states in non-bipartite lattices are frustrated as they cannot achieve a phase
difference of $\pi$ between sites as odd-rings are present, and this cost
energy [34, 35]. For disordered systems, states are localized in regions of
lower frustration and a kind of Lifshitz tail appears [35]. Moreover,
frustration is always associated with massive degeneration [34, 35]; it leads
to Van Hove singularites or if possible, in a condensation of confined states.
These confined states appear in chiral models of the Penrose lattice [36, 37,
38], where they form beautiful fractal patterns [37], in random binary alloys
[39, 40] and in graphene with defects [32, 41]. Strictly confined states have
the property of having zero amplitude in one of the sublattices and the sum of
amplitudes in the other sublattice is always zero [39, 37]. As expected, this
requires a fine-tuning of the lattice topology and/or parameters. In this
particular case, Tarnpoksy et. al. showed that at magic angles [19], flat
bands have the property that $\psi_{\bm{K},1}(\bm{r}_{0})=0$ where
$\psi_{\bm{k},1}(\bm{r})$ is the wave function with momentum at the $\bm{K}$
point (here at $\bm{k}=0$) and $\bm{r}_{0}=(\bm{a}_{1}-\bm{a}_{2})/3$. Thus, a
node is seen at the center of the triangle in agreement with the confined
states picture. Notice that as these modes are degenerate, any linear
combination of them is also solution.

Quite surprisingly, for large angles, our renormalization is akin to a phonon
problem. Thus, flat-band states seem to have a resemblance to floppy modes in
an equivalent rigidity-phonon problem, i.e., the flat-band can be interpreted
as a massive zero frequency vibrational band, since $E^{2}$ is analogous to a
frequency [42]. These floppy modes occur in flexible crystals and are well
known in rigidity theory of glasses [43, 44].

Conclusions We showed that the chiral Hamiltonian for twisted garphene
bilayers can be written into a $2\times 2$ matrix operator, in which the
electron feels an average trigonal field plus an effective non-Abelian
potential. The action of the proposed renormalization, maps the zero-modes
into ground states. Modes next to zero energy have an antibonding nature in a
triangular lattice, and it seems that flat-bands modes are similar to confined
modes observed in many other bipartite systems [39, 38, 40]. Finally, a
surprising result is that our renormalization suggests that flat-bands are
somehow analogous to floppy modes in rigidity phonon models [42, 40,
Dasgupta], which also are reminiscent of the protected electronic boundary
modes that occur in the quantum Hall effect and in topological insulators
[42].

We thank UNAM-DGAPA project IN102620 and CONACyT project 1564464. L. Navarro
and E. Aguilar thank CONACyT SNI III schoolarship for providing critical
support during the COVID emergency. We thank Saul Herrera, Abdiel Espinosa
Champo and Elias Andrade for useful comments and suggestions.

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# Chaotic Dynamics and Zero Distribution: Implications for Yitang Zhang’s
Landau–Siegel Zero Theorem

Zeraoulia Rafik
University of Batna 2, Algeria
Department of Mathematics
Yabous, Khenchela
<EMAIL_ADDRESS>
&Alvaro Humberto Salas
University of Bogota, Colombia
Department of Mathematics
Bogota, Colombia
<EMAIL_ADDRESS>

###### Abstract

This study delves into the realm of chaotic dynamics derived from Dirichlet
L-functions, drawing inspiration from Yitang Zhang’s groundbreaking work on
Landau–Siegel zeros. The dynamic behavior reveals a profound chaos,
corroborated by the calculated Lyapunov exponents and entropy, which attest to
the system’s inherent unpredictability. The examination of stability further
uncovers the instability of fixed points. Most notably, these findings unveil
a compelling connection between the studied dynamics and the intricate
distribution of zeros, thereby offering indirect support for Zhang’s
groundbreaking theorem concerning Landau–Siegel zeros.

_K_ eywords New discrete dynamics $\cdot$ Yitang Zhang latest result $\cdot$
Landau-Siegel zeros $\cdot$ Chotic dynamics

##  1 Introduction

In recent years, the landscape of number theory has been reshaped by
remarkable advancements, with Yitang Zhang’s pioneering work on Landau–Siegel
zeros standing out as a testament to the field’s dynamic evolution. Zhang’s
contributions have taken center stage in the mathematical community,
culminating in virtual talks at Shandong University and Peking University,
where he presented his latest findings. Additionally, Zhang’s preprint [8] ,
which we explore in this study, introduces a pivotal result that asserts the
existence of Landau–Siegel zeros. Specifically, for a real primitive character
$\chi$ modulo $D$, Zhang’s work reveals an intriguing inequality:

$L(1,\chi)>c_{1}(\log D)^{-2022}$

where $c_{1}>0$ is an absolute, effectively computable constant. This result,
assuming its correctness, raises significant questions about its implications
in number theory.

In this context, our study explores a novel avenue: the dynamics of discrete
maps derived from Dirichlet L-functions. We investigate the chaotic behavior
of these dynamics, compute Lyapunov exponents, and determine the system’s
entropy. Additionally, we examine the stability of fixed points using two
distinct methods.

The intriguing aspect of our findings is their potential connection to the
distribution of zeros of the Dirichlet L-function. While chaotic dynamics do
not inherently imply a uniform distribution of zeros, we present evidence that
these dynamics may indirectly support the assumption of Landau–Siegel zeros,
bolstering Zhang’s groundbreaking theorem.

Our results contribute to the growing body of research that seeks to bridge
the gap between chaotic dynamics and the distribution of zeros in the
captivating landscape of number theory.

##  2 Main Results

###  2.1 Chaotic Dynamics of Yitang Dynamics 1

The Yitang Dynamics 1, defined by the equation:

$x_{n+1}=\frac{\beta}{\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=-1$
(1)

shows both stability and chaotic behavior. The derivative at fixed points
remains positive and equal to 1, indicating stability. However, the
eigenvalues of the Jacobian matrix have a real part of approximately 196 and
an imaginary part of approximately -3.58, suggesting potential chaotic
behavior. The calculated entropy for this system is 0.3365, indicating high
unpredictability and complexity.

###  2.2 Chaotic Dynamics of Yitang Dynamics 2

The Yitang Dynamics 2, defined by the equation:

$x_{n+1}=\frac{\beta\log(|\epsilon|)}{\pi\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=1$
(2)

also exhibits both stability and chaotic behavior. The derivative at fixed
points remains positive and equal to 1, indicating stability. Similar to
Yitang Dynamics 1, the eigenvalues of the Jacobian matrix suggest potential
chaotic behavior. The calculated entropy for this system is consistent at
0.3365, indicating high unpredictability.

###  2.3 Unimodal Distribution and Entropy

The approximate solutions for both Yitang Dynamics 1 and Yitang Dynamics 2
show a unimodal distribution, with solutions clustering around a central value
of approximately -601.0938. This unimodal distribution indicates that the
systems tend to converge towards a specific equilibrium state. The entropy
values of 0.333546 for both systems reveal their high predictability and
orderliness.

These results provide insights into the behavior of Yitang Dynamics 1 and
Yitang Dynamics 2 and their implications for the distribution of zeros in the
Dirichlet L-function.

##  3 Chaotic Dynamics for Yitang Zhang’s Latest Results on Landau–Siegel
Zeros

In this section, we will provide an explicit dynamics for Yitang Zhang’s
latest results on Landau–Siegel zeros, incorporating both Dirichlet’s
definition, which utilizes both odd and even characters $\chi$ modulo $m$, and
Theorem 4 of Zhang. We will also examine the behavior of this dynamics,
demonstrating a weak transition to chaos for certain parameter values within
this new discrete dynamics

###### Corollary 1

For special values $s=1$, let $\chi$ be a primitive real character modulo $m$.
Define $\xi_{m}=\exp\left(\frac{2\pi i}{m}\right)$,
$K=\mathbb{Q}\left(\sqrt{\chi(-1)m}\right)$, $h$ is its class number of roots
of unity in it, and $\epsilon$ is its fundamental unit. The Dirichlet L
Function may be defined as follows:

$L(1,\chi)=\begin{cases}\frac{2\pi h}{w\sqrt{m}}&\text{if}\ \chi(-1)=-1\\\
\frac{2h\log{|\epsilon|}}{w\sqrt{m}}&\text{if}\ \chi(-1)=1\\\ \end{cases}$ (3)

Yitang’s discrete dynamics may be derived from the definition of the Dirichlet
L function in (3) and the following Theorem [7]

###### Theorem 2

There exists $A>0$ and effective $C_{1}>0$ such that $L(1,\chi)>C_{1}(\log
q)^{-A}$.

. We will start with the first case, $\chi(-1)=-1$ in (3). Using both
Theorem$2$ and (3), the 1-D discrete dynamics can be reformulated as:

$x_{n+1}=\frac{\beta}{\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=-1$
(4)

with $x_{n}=m$, $n=0,1,\ldots$ and $\beta=\frac{2\pi h}{w}$, where $c$ is the
constant defined in Theorem$2$ .

For the second case, $\chi(-1)=1$, Yitang Dynamics can be written as:

$x_{n+1}=\frac{\beta\log(|\epsilon|)}{\pi\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=1$
(5)

To explore chaotic behavior, we iterated dynamic (10) and dynamic (11) 50,000
times, yielding behavior similar to what we will show in the following
section.

##  4 Analysis and Discussion

For $\alpha$ in the range $[0;10]$ and $c$ in the range $[0.0005;0.007]$, all
Lyapunov exponents
$(\lambda_{1}<\lambda_{2}<\lambda_{3},\ldots>\lambda_{10}=-0.348<0$) of both
dynamics are negative, indicating that the Yitang dynamics are non-chaotic.
However, we noticed a weak transition to chaos in the range $\alpha\in(3;4)$
(see Figure 1).

Figure 1: Lyapunov exponents for Dynamics 1 with $c\in[0.0005;0.007]$,
$\alpha\in[0;10]$.

By increasing the values of $\alpha$ up to $2022$ and keeping
$c\in[0.0005;0.007]$, we observe that the Lyapunov exponents remain
approximately constant at around $-0.39$ (see Figure 2).

Figure 2: Lyapunov exponents for Dynamics 1 with $c\in[0.0005;0.007]$,
$\alpha\in[0;2022]$.

Transition to chaos becomes evident for $c\in(10.007;11.07)$ and
$\alpha\in(3;5)$, where the Lyapunov exponents take positive values
$(0\leq\lambda\leq 0.21)$, with the maximum Lyapunov exponent reaching
$\lambda=0.21$ (see Figure 3).

Figure 3: Lyapunov exponents for Dynamics 1 with $c\in[10.007;11.007]$,
$\alpha\in(0;5)$.

By increasing the values of $c$, a large range for the transition to chaos in
Yitang dynamics can be achieved. A bifurcation diagram of 1-D Yitang dynamics
for some values of $c$ has been plotted (see Figure 4).

Figure 4: Bifurcation diagram for Yitang dynamics.

We have also discovered period doubling bifurcations by varying the parameter
$\alpha$ in the chaotic range and plotting $x_{n},y_{n}$ versus the iterations
$n$ of the map. We then printed the values of the points to distinguish period
cycles.

For one-dimensional systems of the form $x_{n+1}=f(x_{n})$, period-$2$ cycles
occur when the system:

$x_{1}=f(x_{2}),\quad x_{2}=f(x_{1})$

has a unique solution. It appears that $0$ is the fixed point for Yitang
dynamics (see Table 1) with an initial point $x_{0}=0.4$. In other words, the
trajectory "jumps" between $x_{1}$ and $x_{2}$" so while iterating the map
they keep appearing (until a certain value of the parameter of the system).
See Figure 5.

Figure 5: Yitang dynamics Fixed Points Plot versus Iterations, $c=100$, $\mu=\alpha\in(0;5)$. Point | x
---|---
86 | 3.55687
87 | 0.265123
88 | 0.971062
89 | $4.05719\times 10^{76}$
90 | $2.4823161470320785\times 10^{-39}$
91 | $1.00356\times 10^{19}$
92 | $1.5783353198977849\times 10^{-10}$
93 | 39798.8
94 | 0.00250631
95 | 9.9874
96 | 0.158214
97 | 1.25704
98 | $1.07455\times 10^{32}$
99 | $4.823440210703131\times 10^{-17}$
100 | $7.19932\times 10^{7}$
101 | 0.0000589283
Table 1:  Fixed Points values of Yitang dynamics with iteration of $500$
times.

We have undertaken a comprehensive study of the Yitang dynamics, which
involves the calculation of Lyapunov exponents to gain insights into the
system’s behavior. We have computed a total of 600 Lyapunov exponents for the
dynamics, revealing intricate details about its stability and predictability.

The list of these Lyapunov exponents is available for download via the
following link: Lyapunov exponents list. These numerical values serve as a
critical resource for researchers and practitioners interested in analyzing
the Yitang dynamics.

To provide a visual representation of our findings, we have generated a plot
that interprets the Lyapunov exponents. In Figure 6, you can observe the
Lyapunov exponents for Dynamics 1, where $\alpha$ ranges from 0 to 600. The
plot offers a comprehensive view of the system’s behavior, highlighting
trends, patterns, and key insights into the dynamics’ stability and chaos. The
interpretation of this plot is central to our understanding of the Yitang
dynamics and its significance in various scientific and engineering
applications.

Figure 6: Lyapunov exponents for Dynamics 1, $\alpha\in[0;600]$

##  5 Exploring Zero-Free Regions near $s=1$ of Dirichlet L-Function

In this section, we delve into the zero-free regions of the Dirichlet
L-function in the vicinity of $s=1$. Our analysis aims to shed light on the
behavior of this function as it approaches $s=1$, with particular relevance to
the Landau-Siegel zeros.

The Dirichlet L-function is defined as:

$L(1,\chi)=\begin{cases}\frac{2\pi h}{w\sqrt{m}}&\text{if }\chi(-1)=-1\\\
\frac{2h\log|\epsilon|}{w\sqrt{m}}&\text{if }\chi(-1)=1\end{cases}$

Our primary interest lies in the region around $s=1$, which may be associated
with the presence or absence of Landau-Siegel zeros.

###  5.1 Approximate Analytical Solutions

We employ the Newton-Raphson method to compute approximate analytical
solutions for the Dirichlet L-function near $s=1$. The method involves
iteratively updating the approximate solution for a range of initial guesses.
These initial guesses are represented on the x-axis of the plot, and the
corresponding approximate solutions are plotted on the y-axis.

Figure 7: Approximate Analytical Solutions near $s=1$ for the Dirichlet
L-function.

Recall the dynamics (11) under investigation are represented by the formula:

$x_{n+1}=\frac{\beta\log(|\epsilon|)}{\pi\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=1$
(6)

This dynamic equation is essential in understanding the behavior of the
Dirichlet L-function near $s=1$.

###  5.2 Analysis and Implications

The plot in Figure 7 presents a compelling view of the approximate solutions
as $s$ approaches 1. Notably, the behavior near $s=1$ has the potential to
influence the distribution of zeros, especially Landau-Siegel zeros. The
transition from non-zero to zero regions in this context is of particular
interest.[4]

Intriguingly, the regions where solutions cluster around zero might suggest
the presence of zeros of the Dirichlet L-function near $s=1$. On the other
hand, regions where solutions deviate from zero could indicate zero-free
regions. This insight is vital for understanding the distribution of zeros,
including the possibility of Landau-Siegel zeros near $s=1$.

The findings in this section contribute to the broader exploration of the
distribution of zeros in number theory and its connection to Yitang Zhang’s
latest results on Landau-Siegel zeros.

In the subsequent sections, we will further investigate the consequences of
these findings [3], particularly their impact on the Prime Number Theorem and
related number-theoretical problems.

##  6 Exploring Zero-Free Regions near $s=1$ for Yitang Dynamics 1

In this section, we turn our attention to Yitang Dynamics 1, a key component
of our investigation into the Dirichlet L-function near $s=1$. Our analysis
delves into the zero-free regions of this dynamic system, particularly in the
vicinity of $s=1$.

Yitang Dynamics 1 is described by the formula:

$x_{n+1}=\frac{\beta}{\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=-1$
(7)

###  6.1 Approximate Analytical Solutions

Similar to our previous analysis, we employ the Newton-Raphson method to
compute approximate analytical solutions for Yitang Dynamics 1. The method
involves iteratively updating the approximate solution for a range of initial
guesses. These initial guesses are represented on the x-axis of the plot, and
the corresponding approximate solutions are plotted on the y-axis.

Figure 8: Approximate Analytical Solutions near $s=1$ for Yitang Dynamics 1.

###  6.2 Analysis and Implications

The plot in Figure 9 provides valuable insights into the behavior of Yitang
Dynamics 1 as $s$ approaches 1. It is within this region that we seek to
understand the zero-free nature of the dynamics, which may have ramifications
for the broader study of number theory and the distribution of zeros in the
Dirichlet L-function.[5]

As with our previous analysis, we look for regions where solutions cluster
around zero, indicating zero-free regions, and regions where solutions deviate
from zero. This analysis is vital for understanding the distribution of zeros
and its implications for Yitang Zhang’s latest results on Landau-Siegel zeros.

In the upcoming sections, we will explore further consequences of these
findings, including their potential impact on number theory and chaos theory.

##  7 Comparing Zero Behavior of Dirichlet L-Function in Two Dynamics

In this section, we present a comparative analysis of the zero behavior of the
Dirichlet L-function in two distinct dynamics: Yitang Dynamics 1 and Yitang
Dynamics 2. These dynamics are critical in understanding the distribution of
zeros near $s=1$ and their implications for Landau-Siegel zeros.

###  7.1 Approximate Analytical Solutions for Yitang Dynamics 1

We start by examining the behavior of Yitang Dynamics 1, which is defined by
the formula:

$x_{n+1}=\frac{\beta}{\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=-1$
(8)

Using the Newton-Raphson method, we compute approximate analytical solutions
for this dynamic system. The plot in Figure 9 illustrates the behavior of
these solutions near $s=1$.

Figure 9: Approximate Analytical Solutions near $s=1$ for Yitang Dynamics 1.

###  7.2 Approximate Analytical Solutions for Yitang Dynamics 2

Now, let’s shift our focus to Yitang Dynamics 2, defined by the formula:

$x_{n+1}=\frac{\beta\log(|\epsilon|)}{\pi\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=1$
(9)

Once again, we employ the Newton-Raphson method to compute approximate
analytical solutions, specifically in the region near $s=1$. The plot in
Figure 10 displays the behavior of these solutions.

Figure 10: Approximate Analytical Solutions near $s=1$ for Yitang Dynamics 2.

###  7.3 Comparative Analysis

Comparing the two plots in Figure 9 and Figure 10, we observe intriguing
differences in the behavior of approximate solutions for Yitang Dynamics 1 and
Yitang Dynamics 2 near $s=1$. These differences may provide essential insights
into the distribution of zeros in the Dirichlet L-function, particularly in
the context of Landau-Siegel zeros.

Our ongoing exploration aims to further analyze and interpret these
distinctions, leading to a deeper understanding of the behavior of zeros of
the Dirichlet L-function near $s=1$.

##  8 Stability and Chaotic Behavior of Fixed Points

In this section, we explore the stability and potential chaotic behavior of
the fixed points within the context of two dynamic systems: Yitang Dynamics 1
and Yitang Dynamics 2. We consider the derivatives of the dynamic functions at
these solutions and also examine the eigenvalues of the Jacobian matrix to
gain further insights.

###  8.1 Stability in Yitang Dynamics 1

For Yitang Dynamics 1, described by the equation:

$x_{n+1}=\frac{\beta}{\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=-1$
(10)

We have computed approximate solutions for various initial guesses, and the
derivative at each solution remains positive and equal to 1. This consistent
value of the derivative suggests that the fixed points in Yitang Dynamics 1
are stable, which is indicative of the system’s behavior near these
points.[10]

Table 2 presents the values of the initial guesses, the corresponding
solutions, and the derivative of the dynamic function at these solutions.

Initial Guess | Solution (x) | Derivative
---|---|---
2 | $-599.194$ | 1
4 | $-501.094$ | 1
6 | $-501.196$ | 1
8 | $-501.094$ | 1
10 | $-501.094$ | 1
12 | $-601.194$ | 1
14 | $-601.094$ | 1
16 | $-601.094$ | 1
18 | $-601.194$ | 1
20 | $-601.094$ | 1
22 | $-601.094$ | 1
24 | $-401.194$ | 1
26 | $-601.094$ | 1
28 | $-601.094$ | 1
30 | $-601.094$ | 1
Table 2:  Derivative Values of Yitang Dynamics 1 Solutions.

While the derivative values confirm stability, further analysis of the
eigenvalues of the Jacobian matrix provides an intriguing insight into the
potential for chaotic behavior. The eigenvalues for this dynamic system were
found to have a real part of approximately 196 and an imaginary part of
approximately -3.58. This real part being positive and the positive entropy
value of 0.3365 indicate potential chaotic behavior.

The entropy value of 0.3365 was calculated using 600 values of Lyapunov
exponents, reinforcing the high degree of unpredictability and complexity in
this system.

###  8.2 Stability in Yitang Dynamics 2

Now, let’s shift our focus to Yitang Dynamics 2, governed by the equation:

$x_{n+1}=\frac{\beta\log(|\epsilon|)}{\pi\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=1$
(11)

Our analysis for this dynamic system has revealed similar behavior, with the
derivative at the solutions remaining positive and equal to 1. This implies
stability in the vicinity of these fixed points.

Table 3 shows the values of the initial guesses, corresponding solutions, and
the derivative of the dynamic function at these solutions.

Initial Guess | Solution (x) | Derivative
---|---|---
2 | $-601.094$ | 1
4 | $-601.094$ | 1
6 | $-601.094$ | 1
8 | $-601.094$ | 1
10 | $-601.094$ | 1
12 | $-601.094$ | 1
14 | $-601.094$ | 1
16 | $-601.094$ | 1
18 | $-601.094$ | 1
20 | $-601.094$ | 1
22 | $-601.094$ | 1
24 | $-601.094$ | 1
26 | $-601.094$ | 1
28 | $-601.094$ | 1
30 | $-601.094$ | 1
Table 3:  Derivative Values of Yitang Dynamics 2 Solutions with Initial Guess.

Similar to Yitang Dynamics 1, Yitang Dynamics 2 exhibits stability at its
fixed points. However, a striking similarity is observed in the eigenvalues of
the Jacobian matrix, with a real part of approximately 196 and an imaginary
part of approximately -3.58, indicating potential chaotic behavior.

The consistent real part of the eigenvalues in both dynamic systems warrants
further investigation into the chaotic properties of these systems. The
positive entropy values in both cases support the notion of chaotic behavior,
indicating high unpredictability and complexity.

These findings are significant for understanding the behavior of the dynamics
and their implications for Landau-Siegel zeros and the distribution of zeros
in the Dirichlet L-function.

##  9 Unimodal Distribution and Entropy in Yitang Dynamics

In this section, we delve into the analysis of our Yitang dynamics, focusing
on the unimodal distribution of approximate solutions and its relation to
system entropy. Understanding the dynamics of our system is crucial for making
informed decisions and predictions.

###  9.1 Understanding the Unimodal Distribution

The histogram of the approximate solutions for our Yitang dynamics illustrates
an intriguing characteristic—it displays a unimodal distribution. A unimodal
distribution implies that the data is centered around a prominent peak,
indicating that a substantial number of solutions cluster around a specific
value. In our case, this central value is approximately -601.0938.

The unimodal nature of this distribution is a critical observation. It
signifies that our dynamical system is not only predictable but also exhibits
a tendency to converge towards a particular equilibrium state. This
equilibrium state represents the most likely solution for the system given
various initial conditions, which is a fundamental concept in dynamical
systems theory.[9]

###  9.2 Entropy and Predictability

The concept of entropy plays a pivotal role in understanding the behavior of
dynamical systems. Entropy quantifies the degree of disorder or uncertainty in
a system. In our case, the entropy of the Yitang dynamics is calculated to be
0.333546. This value provides valuable insights into the predictability of our
system.

Low entropy, as indicated by the calculated value of 0.333546, implies that
our system is highly ordered and predictable. In other words, our system has a
strong tendency to converge to a specific solution, as seen in the unimodal
distribution, and there is relatively low variability among the approximate
solutions. This observation aligns with our findings from the histogram.[11]

###  9.3 Zero Dirichlet Function Behavior

Furthermore, the concept of entropy can be related to the behavior of the zero
Dirichlet function, a mathematical tool used to study solutions of partial
differential equations (PDEs). In our context, we can consider the zero
Dirichlet function as a representation of the boundary conditions and
constraints of our Yitang dynamics.

A low entropy value implies that the zero Dirichlet function of our dynamics
is highly structured and less noisy. This structured behavior indicates that
the solutions are influenced by specific underlying factors or constraints,
leading to a more deterministic and unimodal distribution of approximate
solutions.

###  9.4 Histograms of Dynamics 1 and Dynamics 2

To further illustrate the behavior of our system, we generated histograms
using the solutions of both Dynamics 1 and Dynamics 2. These histograms
provide additional insights into the distribution of solutions. Below, you can
observe the resulting histograms for Dynamics 1 and Dynamics 2:

The histograms confirm the unimodal distribution of solutions in both
dynamics, emphasizing the system’s propensity to converge to specific
equilibrium states.

In summary, the unimodal distribution observed in our histogram is a
reflection of the highly ordered and predictable behavior of our Yitang
dynamics. The low entropy value further underscores the system’s propensity to
converge to a specific solution. These findings provide valuable insights into
the stability and behavior of our system, which can have significant
implications in various applications, from physics to engineering and beyond.

##  10 Conclusion

In this study, we have delved into the fascinating realm of Yitang dynamics, a
mathematical system inspired by the groundbreaking work of Yitang Zhang in
number theory. Our exploration has encompassed the analysis of the dynamics
through both theoretical and computational approaches, shedding light on its
behavior, stability, and intricate properties.

One of the pivotal findings of our research is the derivation of a set of 600
Lyapunov exponents, providing a comprehensive overview of the Yitang dynamics
across a wide range of parameter values. These exponents serve as a valuable
resource for understanding the system’s sensitivity to initial conditions,
chaotic behavior, and the emergence of complex phenomena.

Moreover, our investigations have revealed a zero-free region, demonstrating
the resilience of the Yitang dynamics to factor into the rigorous bounds
provided by Zhang in number theory. The system’s behavior in relation to
Zhang’s Theorem 4 highlights its mathematical and scientific significance,
which extends beyond its initial conception.

Our results have been further illustrated through a carefully constructed
plot, emphasizing the significance of these Lyapunov exponents. This visual
representation serves as an accessible tool for researchers and practitioners,
aiding in the interpretation of the dynamics’ behavior and the identification
of relevant patterns.

Yitang, in his work, established a profound relationship between the existence
of the Landau-Siegel zero of $L(s,\chi)$ and the distribution of zeros of the
Dirichlet $L$-function $L(s,\psi)$, within a specific region. He demonstrated
that if the Landau-Siegel zero exists (equivalently, $L(1,\chi)$ is small), it
has far-reaching implications. Now, assuming the correct reformulation of our
dynamics, our computations have shown that the dynamics are chaotic.
Remarkably, this numerical evidence implies that the analytical solutions of
Discrete Dynamic 1 (Yitang Dynamics) correspond closely to the zeros of
$L(1,\chi)$, with a fixed point $\sigma$ that is proximate to, but not
precisely at, $0$ as $\alpha$ grows large. In this scenario, where the system
is non-chaotic for sufficiently large $\alpha$, we may deduce that
$1-\beta=L(1,\chi)/L^{\prime}(\sigma,\chi)$, as implied by the mean value
theorem due to the system’s non-linearity. The application of the classical
bound $L^{\prime}(\sigma,\chi)=O(\log^{2}q)$ and Zhang’s result (Theorem 2)
yields the zero-free region $1-\beta>C_{2}(\log q)^{-A-2}$.

This insight may have far-reaching consequences on the zeros of other
$L$-functions, compelling them to reside on the critical line as anticipated
and exhibit a highly regular spacing. For large values of $\alpha$, the
dynamics exhibit non-chaotic behavior, resulting in a uniform distribution of
zeros for $L(s,\chi)$.

In conclusion, the Yitang dynamics have proven to be a captivating subject of
mathematical exploration, offering a unique bridge between number theory and
dynamical systems. The implications of our findings extend to various
scientific and engineering applications, as well as a deeper understanding of
the interconnectedness between different fields of mathematics. As we look to
the future, further investigations into the Yitang dynamics promise to yield
more insights and applications that will undoubtedly enrich the world of
mathematics and its real-world counterparts

## Data Availability

The main basis of this paper’s main results relies on the work of Yitang Zhang
in ’On the Landau-Siegel Zeros Conjecture’ [6], which is available on arXiv at
https://arxiv.org/abs/0705.4306. This paper does not introduce new data but
rather builds upon and extends the findings and concepts presented in Zhang’s
work. Readers interested in the foundational data and results should refer to
the original source, which is publicly accessible through the provided link.

## Conflict of Interest

The authors declare that there is no conflict of interest regarding the
publication of this paper. We confirm that this research was conducted in an
unbiased and impartial manner, without any financial, personal, or
professional relationships that could be perceived as conflicting with the
objectivity and integrity of the research or the publication process.

.

## References

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  * [2] Yitang Zhang,Discrete mean estimates and the Landau-Siegel zero, arXiv:2211.02515 , https://arxiv.org/abs/2211.02515
  * [3] Consequences resulting from Yitang Zhang’s latest claimed results on Landau-Siegel zeros, https://mathoverflow.net/q/433949/51189
  * [4] Guégan, D. (1997). Nonparametric Methods for Time Series and Dynamical Systems. In: Babu, G.J., Feigelson, E.D. (eds) Statistical Challenges in Modern Astronomy II. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1968-2_17
  * [5] Paul Voutier,Introduction to L-functions: Dirichlet L-functions,CIMPA-ICTP Research School, Nesin Mathematics Village June 2017
  * [6] Yitang Zhang ,On the Landau-Siegel Zeros Conjecture, arXiv:0705.4306 , https://arxiv.org/abs/0705.4306
  * [7] zeraoulia rafik ,https://mathoverflow.net/q/444189/51189
  * [8] zeraoulia rafik ,New Chaotic dynamics for Yitang Zhang latest results on Landau-Siegel zero https://arxiv.org/abs/2304.12862
  * [9] Chen, L., and Zhang, Q. (2014). "Unimodal Distributions in Probability Theory." Probability and Mathematical Statistics, 40(2), 145-162.
  * [10] 1\. Wang, H., and Li, C. (2023). "Stability Analysis of Chaotic Systems with Applications." Nonlinear Dynamics, 30(3), 327-341.
  * [11] Lee, H., and Kim, S. (2011). "Entropy and Predictability in Dynamical Systems with Unimodal Distributions." Chaos, Solitons and Fractals, 18(5), 1019-1031.