2022 0

# Double ring polariton condensates with polariton vortices

1,2,3,4E. S. Sedov  3,5V. A. Lukoshkin  3,5V. K. Kalevich  6,4I. Yu. Chestnov
7Z. Hatzopoulos  1,2,6,8P. G. Savvidis  1,2,3,9A. V. Kavokin  1Key Laboratory
for Quantum Materials of Zhejiang Province, School of Science, Westlake
University, 18 Shilongshan Rd, Hangzhou 310024, Zhejiang, China
2Institute of Natural Sciences, Westlake Institute for Advanced Study, 18
Shilongshan Road, Hangzhou, Zhejiang Province 310024, China
3Spin Optics Laboratory, St. Petersburg State University, Ulyanovskaya 1, St.
Petersburg 198504, Russia
4Vladimir State University, 87 Gorky str. 600000, Vladimir, Russia
5Ioffe Institute, Russian Academy of Sciences, 26 Politechnicheskaya, 194021
St-Petersburg, Russia
6Department of Physics and Technology, ITMO University, St. Petersburg,
197101, Russia
7FORTH-IESL, P.O. Box 1527, 71110 Heraklion, Crete, Greece
8 Department of Materials Science and Technology, University of Crete, P.O.
Box 2208, 71003 Heraklion, Crete, Greece
9Moscow Institute of Physics and Technology, Institutskiy per., 9,
Dolgoprudnyi, Moscow Region, 141701, Russia<EMAIL_ADDRESS>
###### Abstract

We study formation of persistent currents of exciton polaritons in annular
polariton condensates in a cylindrical micropillar cavity under the spatially
localised nonresonant optical pumping. Since polariton condensates are
strongly nonequilibrium systems, the trapping potential for polaritons, formed
by the pillar edge and the reservoir of optically induced incoherent excitons,
is complex in general case. Its imaginary part includes the spatially
distributed gain from the pump and losses of polaritons in the condensate. We
show that engineering the gain-loss balance in the micropillar plane gives
access to the excited states of the polariton condensate. We demonstrate both
theoretically and experimentally formation of vortices in double concentric
ring polariton condensates in the complex annular trap potential.

###### pacs:

71.36.+c, 03.75.Lm

###### keywords:

polariton, exciton-polariton condensate, persisten current, micropillar,
vortex

††volume: 0

##  1 Introduction

Exciton polaritons are hybrid bosonic quasiparticles emerging under the strong
coupling of photons and excitons. The coupling can be provided in specially
designed stratified structures, semiconductor optical microcavities with
embedded quantum wells [1]. Microcavities provide localization of light along
one spatial direction, reducing the problem to two dimensions. Quantum wells
are holders of excitons close to the resonance with the microcavity photons.

Cavity polaritons are able to form Bose-Einstein condensates. Such states are
known for their superfluid properties [2, 3]. Superfluidity implies the
existence of persistent polariton currents in the system [4]. The
configuration of the current field considerably depends on the geometry of the
localizing potential for polaritons. Azimuthal persistent currents have been
predicted and observed in polariton condensated trapped in annular potentials
[5, 6, 7, 8, 9]. Such currents can exhibit both vortex and non-vortex nature.
To distinguish them, one needs to pay attention to their phase. For vortices,
phase rotates around the trap by $2\pi m$, where the integer number $m$ is
known as the winding number, which plays the role of the topological charge of
the vortex [10]. The second distinctive feature of vortices, which is the
vanishing density of the superfluid at the core, is not relevant for the
annular geometry of the potential.

In the above mentioned papers [5, 6, 7, 8], the polariton condensates
supporting circular polariton currents have been characterized by a single
ring shape occupying the ground radial state of the annular trap. As one has
shown [11, 12], due to the nonequilibrium nature of polariton condensates,
which can exist only under the external (optical) pumping, they are able to
occupy the excited states, including those in the form of concentric rings.
Herewith circular polariton currents are also allowed in such condensates
[13].

In this manuscript, we consider the system of a cylindrical micropillar cavity
excited by a non-resonant laser pump focused close to its center. The pump
creates a reservoir of incoherent excitons, which feeds the exciton polariton
condensate through stimulated scattering processes. We study the conditions
needed for exciting condensates in the form of concentric rings in the annular
trapping potential by analysing the balance of gain of the condensate from the
optical pump and losses of polaritons. We predict formation of double
concentric ring polariton vortices and confirm our predictions by experimental
observations. We note that double ring polariton condensates may be used as
building blocks in polariton quantum networks [14].

##  2 Exciton polariton condensates in an annular trap

###  2.1 Gross-Pitaevskii equation for describing polariton condensates

The traditional approach to characterizing the exciton polariton condensate is
based on using the generalized Gross-Pitaevskii equation for the polariton
wave function (WF) $\Psi(t,\mathbf{r})$ [15, 16, 17, 13]:

$i\hbar\partial_{t}\Psi=\left\\{[i\eta
n_{\text{R}}-1]\frac{\hbar^{2}}{2M}\nabla^{2}\right.+V(r)\left.+\alpha|\Psi|^{2}+\alpha_{\text{R}}n_{\text{R}}+\frac{i\hbar}{2}\left[Rn_{\text{R}}-\gamma\right]\right\\}\Psi,$
(1)

where $M$ is the effective polariton mass in the microcavity plane,
$V(r)=V_{0}\delta(r-d/2)$ is the stationary potential of the micropillar,
taken in the form of cylinder of a diameter $d$ and height $V_{0}$. $\alpha$
and $\alpha_{\text{R}}$ are the polariton-polariton and polariton-exciton
interaction constants, respectively. The rightmost imaginary term in Eq. (1)
characterizes non-conservative processes of polariton losses and gain from the
incoherent reservoir of optically induced excitons.
$n_{\text{R}}(t,\mathbf{r})$ is the density of the reservoir, $R$ is the
stimulated scattering rate from the reservoir to the condensate, $\gamma$ and
$\gamma_{\text{R}}$ are the decay rates of polaritons and excitons,
respectively. In Eq. (1) we also take into account the energy relaxation of
polaritons during their propagation [15, 16]. $\eta$ is the energy relaxation
constant. This latter term distinguishes our model from one proposed in [7].

Due to larger effective masses and higher losses of excitons, the reservoir
dynamics is significantly faster than one of polaritons. This allows us to
treat the change of the reservoir as the adiabatic adjustment to the evolution
of the condensate:

$n_{\text{R}}\simeq\frac{P(\mathbf{r})}{\gamma_{\text{R}}+R|\Psi|^{2}}\approx\frac{P(\mathbf{r})}{\gamma_{\text{R}}}-\frac{P(\mathbf{r})R|\Psi|^{2}}{\gamma_{\text{R}}^{2}}.$
(2)

The reservoir is excited by the optical pump of the intensity
$P(\mathbf{r})=P_{\text{s}}(r)+\delta P(\mathbf{r})$, where
$P_{\text{s}}(r)\propto\exp\left[-r^{2}/2w^{2}\right]$ is the Gaussian
function of width $w$ centered in the micropillar, and $\delta P(\mathbf{r})$
describes perturbation due to the displacement of the pump spot or deformation
of its shape.

###  2.2 Gain-loss balance for emergence of the polariton condensate

Let us start our consideration with the rotationally symmetric ($\delta
P(\mathbf{r})\rightarrow 0$) linear (${\alpha|\Psi|^{2}\rightarrow 0}$)
system. The polariton WF then can be factorized as follows:
$\Psi_{m,n}=\Upsilon_{n}(r)\exp\left[-i(E_{m,n}t/\hbar-m\theta)\right]$, where
$\Upsilon_{n}(r)$ is the radial component of WF with the quantum number $n$.
$m$ is the azimuthal quantum number, that is also the winding number
characterizing circulation of polaritons around the trap. $E_{n,m}$ is the
complex eigenvalue of the problem, obeying the equation

$\left(E_{n,m}+i\frac{\hbar}{2}\gamma\right)\Upsilon_{n}=\left\\{\frac{\hbar^{2}}{2M}\left(i\frac{\eta
P_{\text{s}}(r)}{\gamma_{\text{R}}}-1\right)\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{m^{2}}{r^{2}}\right)\right.\\\
\left.+V(r)+\frac{P_{\text{s}}(r)}{\gamma_{\text{R}}}\left[\alpha_{\text{R}}+i\frac{\hbar
R}{2}\right]\right\\}\Upsilon_{n}.$ (3)

The emergence of the condensate is determined by the overall balance of gain
and loss in the system. The gain-loss balance for the state $\Psi_{n,m}$ at a
given pump power is determined by the imaginary part of the eigenvalue
$E_{n,m}$. The pump power, for which the condition ${\text{Im}[E_{n,m}]=0}$ is
satisfied, is the threshold power for the state $\Psi_{n,m}$ [18]. Those
states, for which gain exceeds losses (${\text{Im}[E_{n,m}]>0}$), are
supported by the system, and they can be occupied by the polariton condensate.

The gain and loss terms can be included in the effective potential of
polaritons, which then acquires a complex form. Tuning the parameters of the
complex potential, one can manipulate by the emerging polariton condensate
modes. We have numerically studied the effect of the width of the annular
potential and the pump power on the condition of emergence of excited
polariton condensate states, see illustrations in Fig. 1. For numerical
simulations, we take the following values of the parameters. The effective
mass of polaritons is $M=3\cdot 10^{-5}m_{\text{e}}$, where $m_{\text{e}}$ is
the free electron mass. The height of the stationary potential is
$V_{0}=1\,\text{meV}$. The polariton and exciton decay rates are $\gamma=0.1$
ps-1 and $\gamma_{\text{R}}=0.33$ ps-1, respectively. The stimulated
scattering rate is $\hbar R=0.1\,\text{meV}\,\mu\text{m}^{2}$. The
nonlinearity coefficients are
$\alpha=\alpha_{\text{R}}/2=3\,\mu\text{eV}\,\mu\text{m}^{2}$. The fitting
parameter of the energy relaxation is $\eta=0.01\,\mu\text{m}^{2}$. The pump
width is $w=3\,\mu\text{m}$. The pump power simultaneously contributes both
the real part of the trapping potential, affecting height of the potential
hill from the exciton reservoir, and the imaginary part, affecting gain of
polaritons in the condensate. We would also like to note that although in this
manuscript we consider the effective trapping potential which includes the
stationary potential of the micropillar, our discussion can be expanded to the
case of a fully optically induced potential [19, 20].

The dependence of the imaginary parts of the eigenvalues $E_{n,0}$ on the
width $d$ of the potential ring at the fixed pump power is shown in Fig. 1(a).
One can see that at a small radius, the system supports only the ground
polariton condensate state with ${n=1}$ (see Fig. 1(c)), characterized by a
single ring shape. This is realized, e.g., in 25 $\mu\text{m}$ wide
micropillars (solid vertical line on the panel) studied in Refs. [6, 7, 8]. At
some $d$ the eigenspectrum of the trap is supplemented with the first excited
states with ${n=2}$ (see Fig. 1(d)), which, however, remains the only
eigenstate of the system with further increase in $d$ (when
$\text{Im}[E_{1,0}]$ becomes negative). These states characterized by a double
concentric ring shape in a 30 $\mu\text{m}$ wide micropillar (dashed vertical
line in the panel (a)) are in the focus of our present consideration. The
further increase in $d$ brings states with larger $n$ into play, while the
states with smaller $n$ are eliminated. Similar turnover of the eigenstates
takes place with the change of the pump power. In Fig. 1(b) the dependencies
of $E_{n,0}$ on the pump power for $d=25\,\mu\text{m}$ and $30\,\mu\text{m}$
wide micropillars are shown.

Figure 1: The dependence of the imaginary part of the eigenvalues $E_{n,0}$
for several lower states on the diameter of the pillar, $d$, at the fixed pump
power (a) and on the pump power at the fixed diameter (b). (c–e) Examples of
the density distribution of the three lowest eigenstates with $n=1,2,3$. The
vertical solid and dashed lines in (a) as well as the solid and dashed curves
in (b) correspond to $d=25\,\mu\text{m}$ and $30\,\mu\text{m}$, respectively.
The vertical dashed line in (b) indicates the pump power used for (a). The
gray shaded regions in (c–e) indicate the effective trapping potential for
polaritons. The colors in all panels indicate the number $n$ of the energy
level.

###  2.3 Polariton voirtices in concentric ring condensates

We now return to the general case of the system and bring the pump
perturbation, $\delta P(\mathbf{r})$, and polariton-polariton interactions,
$\alpha|\Psi|^{2}$, back into consideration. We use the approach developed in
[7] and project Eq. (1) onto the solution $\Upsilon_{2}(r)$ of Eq. (3). We
substitute the decomposition
$\Psi(t,\mathbf{r})=\Upsilon_{2}(r)\Phi(t,\theta)$ in Eq. (1) and after
averaging over $r$ we obtain the following equation for the azimuthal WF
component $\Phi(t,\theta)$:

$i\partial_{t}\Phi=\left\\{\left[A_{1}(\theta)-A_{2}(\theta)|\Phi|^{2}\right]\frac{\partial^{2}}{\partial\theta^{2}}+U_{1}(\theta)+U_{2}(\theta)|\Phi|^{2}\right\\}\Phi,$
(4)

where

$\displaystyle
A_{1}(\theta)=-\frac{\hbar}{2M}\left\langle\frac{1}{r^{2}}\left(1-i\frac{\eta
P(\mathbf{r})}{\gamma_{\text{R}}}\right)\right\rangle,$ (5a) $\displaystyle
A_{2}(\theta)=-i\frac{\hbar\eta
R}{2M\gamma_{\text{R}}^{2}}\left\langle\frac{1}{r^{2}}P(\mathbf{r})|\Upsilon|^{2}\right\rangle,$
(5b) $\displaystyle
U_{1}(\theta)=\frac{i\hbar\eta}{2M\gamma_{\text{R}}}\left\langle\delta
P(\mathbf{r})\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}\right)\right\rangle+\frac{1}{\gamma_{\text{R}}}\left(\frac{\alpha_{\text{R}}}{\hbar}+i\frac{R}{2}\right)\left\langle\delta
P(\mathbf{r})\right\rangle,$ (5c) $\displaystyle
U_{2}(\theta)=-\frac{i\hbar\eta R}{2M\gamma_{\text{R}}^{2}}\left\langle
P(\mathbf{r})|\Upsilon|^{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\right)\right\rangle$
$\displaystyle\,\phantom{U_{2}(\theta)=}-\frac{R}{\gamma_{\text{R}}^{2}}\left(\frac{\alpha_{\text{R}}}{\hbar}+i\frac{R}{2}\right)\left\langle
P(\mathbf{r})|\Upsilon|^{2}\right\rangle+\frac{1}{\hbar}\alpha\left\langle|\Upsilon|^{2}\right\rangle.$
(5d)

In (5), $\langle...\rangle=\int_{0}^{\infty}...|\Upsilon|^{2}rdr$. We omitted
the subscript “2” at $\Upsilon$ for brevity. The functions of angle
$U_{1}(\theta)$ and $U_{2}(\theta)$ contribute to the symmetry breaking due to
the pump perturbation and interaction-induced nonlinearity, respectively.

As one has shown in [7], breaking the rotational symmetry of a single ring
polariton condensate, e. g., due to weak ellipticity of the pump spot and its
small displacement from the center of the micropillar, can lead to the
emergence of azimuthal polariton currents. We apply similar approach to the
concentric ring geometry of the condensate. We take the pump in the form
$P(\mathbf{r})\propto\exp\left[\left.-(\mathbf{r}-\mathbf{r}_{\text{p}})^{2}\right/(1+s\cos^{2}\theta)w^{2}\right]$,
where $s$ is responsible for deformation of the pump spot,
$\mathbf{r}_{\text{p}}$ describes the displacement of the pump spot. Keeping
$s$ fixed and solving Eqs. (3)–(4) with different $\mathbf{r}_{\text{p}}$, we
simulate concentric ring polariton condensates with and without azimuthal
polariton currents.

Figure 2: Simulated (a) and experimentally observed (b) density distribution
of the polariton condensate in the shape of double concentric rings. The
magenta dashed circle in (a) indicates the edge of the micropillar.

Figure 2(a) shows the simulated density distribution of the double ring
polariton condensate. Weak submicrometer displacement of the pump spot hardly
affects its shape, so it remains the same in all simulations discusses below.
We can judge the emergence of azimuthal polariton currents by the phase
variation of the condensate around the micropillar. In the upper panels in
Fig. 3 we show three examples of the phase distribution of the simulated
concentric ring polariton condensates. In panel (c) the phase does not vary in
the azimuthal direction, while in (a) and (e) the phases change by $2\pi$ with
one turn around the pillar in the counterclockwise and clockwise directions in
both condensate rings, respectively. For quantifying polariton vortices, we
use the winding number $m$. Following the definition, given in the
Introduction, we obtain $m=+1,0$ and $-1$ for panels (a), (b) and (c),
respectively. The lower panels in Fig. 3 show the phases of the condensates
relative to the phase of the spherical wave. These figures are helpful for
comparing the simulation results with the experimental observations discussed
below.

Figure 3: Examples of the phase components of the simulated polariton
condensates with ($m=\pm 1$) and without ($m=0$) azimuthal polariton currents.
Upper panels show phases of the concentric ring polariton condensates, lower
panels show corresponding phases relative to the phase of the reference
spherical wave.

##  3 Experimental observation of double ring polariton vortices.

Vortex and non-vortex states of double concentric ring polariton condenstes
have been observed by us in the experiment. We excited the polariton
condensates in a cylindrical pillar of diameter of $30\,\mu\text{m}$, etched
in the GaAs microcavity with a set of embedded quantum wells. The excitation
was performed at temperature of 4 K by a nonresonant laser pump beam focused
close to the center of the micropillar. We measured photoluminescence from the
condensate to reveal the density distribution of polaritons in the
micropillar. To get access to the phase of the polariton condensate, we
measured interference of the condensate photoluminescence with the spherical
reference wave using the Mach-Zehnder interferometer. The spherical wave was
obtained by magnifying a small peripheral area of the condensate image. More
details on the experimental setup and the sample are given in [5, 13].

Figure 2(b) shows photoluminescence from the double ring exciton polariton
condensate in the micropillar. Upper panels in Fig. 4 show interferometry
images of three observed double ring condensates. The interference fringes in
(a) and (e) have the shape of spirals, diverging in counterclockwise and
clockwise directions, respectively. The fringes in (c) represent concentric
rings. Brightness of the fringes fades beyond the condensate, including the
slit between the condensate rings.

Figure 4: Observation of concentric ring polariton condensates with azimuthal
polariton currents. Upper panels show images of interference of the polariton
condensates with the reference spherical wave. Lower panels show corresponding
distribution of the phases of the condensates relative to the phase of the
reference wave.

We use the Fourier-transform method adapted for closed interference fringes
[5] for extracting phase of the condensate from the interferograms. The lower
panels in Fig. 4 show the phases of the observed polariton condensates
relative to the phase of the spherical reference wave. One can see that the
phase patterns repeat the spiral or concentric ring shape of the
interferometry images. The comparison of the restored relative phases with the
results of numerical simulations (lower panels in Fig. 3) allows to conclude
that among the observed condensates, the two of them, (a,b) and (e,f), are in
the vortex states with $m=+1$ and $m=-1$, respectively, while the remaining
state (c,d) is in the non-vortex state with $m=0$.

The observed condensates with different vorticity were obtained by slightly
displacing the pump spot near the center of the micropillar. Along with the
inevitable inhomogeneities of the localizing potential landscape, the
displacement led to breaking the rotational symmetry of the system and
selection of the preferred direction for azimuthal polariton currents. We also
would like to note that at the chosen pump power in the micropillar under
consideration we hadn’t observed the polariton condensates with the radial
quantum number other than $n=2$. The vortex states were highly stable and
existed as long as the optical excitation was present. The double concentric
ring shape of the condensates also did not change during their observation.

##  4 Conclusion

In the manuscript, we have analyzed the conditions for exciting concentric
ring exciton polariton condensates in cylindrical micropillars under the
nonresonant optical pumping. We have shown that manipulating by the imaginary
part of the trapping potential, responsible for the spatial distribution of
gain and loss in the micropillar, one can select the radial quantization
states of the polariton condensate. Based on numerical analysis, we have
predicted excitation of azimuthal polariton current states in concentric ring
polariton condensates. Our predictions have been supported by experimental
observation of stable double ring polariton vortices. The results of our study
have prospects for application in developing new interferometric devices and
in coding and transferring optical information using the orbital degree of
freedom of light, as well as for implementation of double-qubit logic gates.

## Acknowledgements

This work was done under the support of the Saint-Petersburg State University
(Grant No. 91182694). A.K. and P.S. acknowledge the support of Westlake
University, Project 041020100118 and Program 2018R01002 funded by Leading
Innovative and Entrepreneur Team Introduction Program of Zhejiang Province of
China. A.K. acknowledges support from the Moscow Institute of Physics and
Technology under the Priority 2030 Strategic Academic Leadership Program.
Numerical simulations were supported by the RF Ministry of Science and Higher
Education under Agreement No. 0635-2020-0013 and by the RF Presidential Grants
for state support of young scientists No. MK-4729.2021.1.2 and No.
MK-5318.2021.1.2.

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# QUANTITATIVE DESTRUCTION OF INVARIANT CIRCLES

Lin Wang  School of Mathematics and Statistics, Beijing Institute of
Technology, Beijing 100081, China<EMAIL_ADDRESS>
###### Abstract.

For area-preserving twist maps on the annulus, we consider the problem on
quantitative destruction of invariant circles with a given frequency $\omega$
of an integrable system by a trigonometric polynomial of degree $N$
perturbation $R_{N}$ with $\|R_{N}\|_{C^{r}}<\epsilon$. We obtain a relation
among $N$, $r$, $\epsilon$ and the arithmetic property of $\omega$, for which
the area-preserving map admit no invariant circles with $\omega$.

###### Key words and phrases:

invariant circle, minimal configuration, Peierls’s barrier, trigonometric
polynomial

###### 2010 Mathematics Subject Classification:

37J50, 37E40.

##  1\. Introduction and main result

Area-preserving twist maps on the annulus first appeared in the work of
Poincaré on the three-body problem. They served as suitable prototypes for the
study of a complicated Hamiltonian system. As two dimensional discrete
dynamical models, they describe the behavior of area preserving surface
diffeomorphisms in the neighborhood of a generic elliptic periodic point. The
study of such maps was initiated by Birkhoff in the 1920s. Since then, this
class of maps have offered many opportunities for the rigorous analysis of
aspects of Hamiltonian systems. As a highlight, Moser proved the first
differentiable version of the KAM theorem in the context of twist maps.

The smoothness of the perturbation in the KAM theorem can be reduced. By the
efforts of Moser, Takens, Rüssman and Herman et al., it was proved that
certain invariant circle with constant type frequency can be persisted under
arbitrarily small perturbations in the $C^{3}$ topology, where the invariant
circle (also referred to as an essential curve) is an invariant curve that is
not homotopic to a point.

The works on non-existence of the invariant circles belong to converse KAM
theory. It was shown by Herman [H2] that the invariant circle with a given
frequency can be destroyed by $C^{3-\delta}$ arbitrarily small $C^{\infty}$
perturbations. Following the ideas and techniques developed by Mather in the
1980s, a variational proof of Herman’s result was provided in [W1]. As a
complement, it was considered in [W3] for Gevrey-$\alpha$ ($\alpha>1$) systems
to destroy the invariant circles with given frequencies. For Hamiltonian
systems with multi-degrees of freedom, the corresponding results were obtained
by [CW] and [Po]. Moreover, it was obtained that all of the Lagrangian tori of
an integrable positive definite Hamiltonian system with $d$ ($d\geq 2$)
degrees of freedom can be destroyed by an arbitrarily small $C^{\omega}$
perturbation in the $C^{d-\delta}$ topology [W2].

For certain special frequencies, it was obtained by Mather (resp. Forni) in
[M4] (resp. [F]) that the invariant circles with those frequencies can be
destroyed by small perturbations in finer topology respectively. More
precisely, Mather considered Liouvillian frequencies and the topology of the
perturbation induced by $C^{\infty}$ metric. Forni was concerned more about
more special frequencies which can be approximated by rational ones
exponentially and the topology of the perturbation induced by the supremum
norm of real-analytic function. Recently, Chen and Cheng [CC] gave an open and
dense property about the destruction of invariant circles by using regular
dependence of the Peierls barriers on perturbations. Roughly speaking, there
is a balance between the arithmetic property of the frequency, the regularity
of the perturbation and its topology.

From physical point of view, it is more natural to consider real analytic
perturbations, e.g. trigonometric polynomials instead of $C^{\infty}$ ones.
Consider a completely integrable system with the generating function

$h_{0}(x,x^{\prime})=\frac{1}{2}(x-x^{\prime})^{2},\quad
x,x^{\prime}\in\mathbb{R}.$

Based on Poincaré’ pioneering work, it is well known that if
$\omega\in\mathbb{Q}$, then the invariant circles with frequency $\omega$
could be easily destroyed by an analytic perturbation arbitrarily close to $0$
in the topology induced by the supremum norm of real-analytic function.
Therefore it suffices to consider the irrational $\omega$. Herman proved in
[H2] that the invariant circle with a given irrational frequency is unique.

An irrational number $\omega\in\mathbb{R}$ is called $\mu$-well approximable
if there exist infinitely many integers $q_{n}\in\mathbb{N}$ such that

$|q_{n}\omega-p_{n}|<{q_{n}}^{-1-\mu},$ (1.1)

for some integer $p_{n}$. It follows from Dirichlet approximation that any
irrational number is 0-well approximable. $\omega$ is called a Liouvillian
number if it is $\mu$-well approximable for all $\mu>0$. Otherwise, it is
called a Diophantine number. Moreover, Jarník’s theorem shows that the set of
$\mu$-well approximable numbers has Hausdorff dimension $\frac{2}{2+\mu}$.

Compared to the aforementioned results above, one can ask the following
question from a quantitative of view:

> Question: Given an irrational frequency $\omega$ and $0<\epsilon\ll 1$. Let
> $R_{N}(x)$ be a trigonometric polynomial of degree $N$ which satisfies
> $\|R_{N}\|_{C^{r}}<\epsilon$. If the area-preserving map generated by
> $h_{0}(x,x^{\prime})+R_{N}(x^{\prime})$ admit no invariant circles with
> $\omega$, then what are the relation among $\epsilon$, $N$, $r$ and the
> arithmetic property of $\omega$?

Based on the KAM result ([H3]), we know that $r<4$ for a given badly
approximable frequency $\omega$ if the perturbation is taken among
$C^{\infty}$ functions. Nevertheless, it is not natural to expect the same
upper bound of $r$ for a trigonometric polynomial of degree $N$. Given $\mu$
and $\epsilon$, we are devoted to looking for a bigger $r$ and a smaller $N$.
First of all, we obtain a relation between $r$ and $\mu$. Given any $u\in
C^{r}(\mathbb{T})$. Let us recall the $C^{r}$-norm:

$\|u\|_{C^{r}}:=\max_{|\alpha|\leq r}\max_{x\in\mathbb{T}}|D^{\alpha}u(x)|,$

where $\alpha:=(\alpha_{1},\ldots,\alpha_{n})$,
$|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$.

######  Theorem 1.1.

Let $\omega$ be a $\mu$-well approximable frequency. Given $0<\epsilon\ll 1$,
there exists a trigonometric polynomial $R_{N}(x)$ which satisfies
$\|R_{N}\|_{C^{r}}<\epsilon$ with $r<3+\mu$, such that the area-preserving map
generated by $h_{0}(x,x^{\prime})+R_{N}(x^{\prime})$ admit no invariant
circles with frequency $\omega$.

Theorem 1.1 provides an analytic version of Mather’s result on non-existence
of the invariant circle with a Liouvillian frequency in [M4]. It implies that
both of small $C^{\infty}$ and trigonometric polynomial perturbations play the
same role in the $C^{\infty}$ topology for destroying invariant circles with
given Liouvillian frequencies.

It is well known that the rigidity of the invariant circle with a $\mu$-well
approximable frequency increases as $\mu$ decreases. In particular, $\omega$
is called badly approximable if it is exactly $0$-well approximable. Based on
Jarník’s theorem, the set of badly approximable numbers has Hausdorff
dimension $1$ and Lebesgue measure $0$.

In order to find a smaller $N$, we provide a relation between $N$ and
$\epsilon$ for the the invariant circle with a badly approximable frequency.

######  Theorem 1.2.

Let $\omega$ be a badly approximable frequency. Given $0<\epsilon\ll 1$ and
$r\in[0,3)$, there exists $R_{N}(x)$ of degree

$N\leq C\epsilon^{-\frac{3}{2(3-r)}}$

with $\|R_{N}\|_{C^{r}}<\epsilon$ such that
$h_{0}(x,x^{\prime})+R_{N}(x^{\prime})$ admit no invariant circles with
frequency $\omega$.

It is worth noting that we can not conclude the optimality of $r$ and $N$ in
Theorem 1.1 and Theorem 1.2. Note that for badly approximable frequencies, we
obtain $r<3$, which is smaller than $r=4$ in the KAM result in [H3]. To fill
the gap, some further developments of (converse) KAM theory are needed.

##  2\.  Preliminaries

###  2.1. Minimal configuration

Let $F$ be a diffeomorphism of $\mathbb{R}^{2}$ denoted by
$F(x,y)=(X(x,y),Y(x,y))$. Let $F$ satisfy:

  * •

Periodicity: $F\circ T=T\circ F$ for the translation $T(x,y)=(x+1,y)$;

  * •

Twist condition: the map $\psi:(x,y)\mapsto(x,X(x,y))$ is a diffeomorphism of
$\mathbb{R}^{2}$;

  * •

Exact symplectic: there exists a real valued function $h$ on $\mathbb{R}^{2}$
with $h(x+1,y)=h(x,y)$ such that

$YdX-ydx=dh.$

Then $F$ induces a map on the cylinder denoted by $f$:
$\mathbb{T}\times\mathbb{R}\mapsto\mathbb{T}\times\mathbb{R}$
($\mathbb{T}=\mathbb{R}/\mathbb{Z}$). $f$ is called an exact area-preserving
twist map. The function $h$: $\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is
called a generating function of $F$, namely $F$ is generated by the following
equations

$\begin{cases}y=-\partial_{1}h(x,x^{\prime}),\\\
y^{\prime}=\partial_{2}h(x,x^{\prime}),\end{cases}$

where $F(x,y)=(x^{\prime},y^{\prime})$.

The function $F$ gives rise to a dynamical system whose orbits are given by
the images of points of $\mathbb{R}^{2}$ under the successive iterates of $F$.
The orbit of the point $(x_{0},y_{0})$ is the bi-infinite sequence

$\\{...,(x_{-k},y_{-k}),...,(x_{-1},y_{-1}),(x_{0},y_{0}),(x_{1},y_{1}),...,(x_{k},y_{k}),...\\},$

where $(x_{k},y_{k})=F(x_{k-1},y_{k-1})$. The sequence

$(...,x_{-k},...,x_{-1},x_{0},x_{1},...,x_{k},...)$

denoted by $(x_{i})_{i\in\mathbb{Z}}$ is called a stationary configuration if
it satisfies the identity

$\partial_{1}h(x_{i},x_{i+1})+\partial_{2}h(x_{i-1},x_{i})=0,\ \text{for\
every\ }i\in\mathbb{Z}.$

Given a sequence of points $(z_{i},...,z_{j})$, we can associate its action

$h(z_{i},...,z_{j})=\sum_{i\leq s<j}h(z_{s},z_{s+1}).$

A configuration $(x_{i})_{i\in\mathbb{Z}}$ is called minimal if for any
$i<j\in\mathbb{Z}$, the segment $(x_{i},...,x_{j})$ minimizes
$h(z_{i},...,z_{j})$ among all segments $(z_{i},...,z_{j})$ of the
configuration satisfying $z_{i}=x_{i}$ and $z_{j}=x_{j}$. It is easy to see
that every minimal configuration is a stationary configuration. There is a
visual way to describe configurations. A configuration
$(x_{i})_{i\in\mathbb{Z}}$ is a function from $\mathbb{Z}$ to $\mathbb{R}$.
One can interpolate this function linearly and obtain a piecewise affine
function $\mathbb{R}\rightarrow\mathbb{R}$ denoted by $t\mapsto x_{t}$. The
graph of this function is sometimes called the Aubry diagram of the
configuration. By [B], minimal configurations satisfy a group of remarkable
properties as follows:

  * •

Two distinct minimal configurations seen as the Aubry diagrams cross at most
once, which is so called Aubry’s crossing lemma.

  * •

For every minimal configuration $\mathbb{x}=(x_{i})_{i\in\mathbb{Z}}$, the
limit

$\rho(\mathbb{x})=\lim_{n\rightarrow\infty}\frac{x_{i+n}-x_{i}}{n}$

exists and doesn’t depend on $i\in\mathbb{Z}$. $\rho(\mathbb{x})$ is called
the frequency of $\mathbb{x}$.

  * •

For every $\omega\in\mathbb{R}$, there exists a minimal configuration with
frequency $\omega$. Following the notations of [B], the set of all minimal
configurations with frequency $\omega$ is denoted by $M_{\omega}^{h}$, which
can be endowed with the topology induced from the product topology on
$\mathbb{R}^{\mathbb{Z}}$. If $\mathbb{x}=(x_{i})_{i\in\mathbb{Z}}$ is a
minimal configuration, considering the projection $pr:\
M_{\omega}^{h}\rightarrow\mathbb{R}$ defined by $pr(\mathbb{x})=x_{0}$, we set
$\mathcal{A}_{\omega}^{h}=pr(M_{\omega}^{h})$.

  * •

If $\omega\in\mathbb{Q}$, say $\omega=p/q$ (in lowest terms), then it is
convenient to define the rotation symbol to detect the structure of
$M_{p/q}^{h}$. If $\mathbb{x}$ is a minimal configuration with frequency
$p/q$, then the rotation symbol $\sigma(\mathbb{x})$ of $\mathbb{x}$ is
defined as follows

$\sigma(\mathbb{x})=\left\\{\begin{array}[]{ll}p/q+,&\text{if}\
x_{i+q}>x_{i}+p\ \text{for\ all\ }i,\\\ p/q,&\text{if}\ x_{i+q}=x_{i}+p\
\text{for\ all\ }i,\\\ p/q-,&\text{if}\ x_{i+q}<x_{i}+p\ \text{for\ all\
}i.\\\ \end{array}\right.$

Moreover, we set

$\displaystyle M_{{p/q}^{+}}^{h}=\\{\mathbb{x}\text{\ is a minimal
configuration with rotation symbol}\ p/q\text{\ or\ }p/q+\\},$ $\displaystyle
M_{{p/q}^{-}}^{h}=\\{\mathbb{x}\text{\ is a minimal configuration with
rotation symbol}\ p/q\text{\ or\ }p/q-\\},$

then both $M_{{p/q}^{+}}^{h}$ and $M_{{p/q}^{+}}^{h}$ are totally ordered.
Namely, every two configurations in each of them (seen as Aubry diagrams) do
not cross. We denote $pr(M_{{p/q}^{+}}^{h})$ and $pr(M_{{p/q}^{-}}^{h})$ by
$\mathcal{A}_{{p/q}^{+}}^{h}$ and $\mathcal{A}_{{p/q}^{-}}^{h}$ respectively.

  * •

If $\omega\in\mathbb{R}\backslash\mathbb{Q}$ and $\mathbb{x}$ is a minimal
configuration with frequency $\omega$, then $\sigma(\mathbb{x})=\omega$ and
$M_{\omega}^{h}$ is totally ordered.

  * •

$\mathcal{A}_{\omega}^{h}$ is a closed subset of $\mathbb{R}$ for every
rotation symbol $\omega$.

###  2.2. Peierls’s barrier

In [M3], Mather introduced the notion of Peierls’s barrier and gave a
criterion of existence of invariant circle. Namely, the exact area-preserving
twist map generated by $h$ admits an invariant circle with frequency $\omega$
if and only if the Peierls’s barrier $P_{\omega}^{h}(\xi)$ vanishes
identically for all $\xi\in\mathbb{R}$. The Peierls’s barrier is defined as
follows:

  * •

If $\xi\in\mathcal{A}_{\omega}^{h}$, we set $P_{\omega}^{h}(\xi)$=0.

  * •

If $\xi\not\in\mathcal{A}_{\omega}^{h}$, since $\mathcal{A}_{\omega}^{h}$ is a
closed set in $\mathbb{R}$, then $\xi$ belongs to some complementary interval
$(\xi^{-},\xi^{+})$ of $\mathcal{A}_{\omega}^{h}$ in $\mathbb{R}$. By the
definition of $\mathcal{A}_{\omega}^{h}$, there exist minimal configurations
with rotation symbol $\omega$, $\mathbb{x^{-}}=(x_{i}^{-})_{i\in\mathbb{Z}}$
and $\mathbb{x^{+}}=(x_{i}^{+})_{i\in\mathbb{Z}}$ satisfying
$x_{0}^{-}=\xi^{-}$ and $x_{0}^{+}=\xi^{+}$. For every configuration
$\mathbb{x}=(x_{i})_{i\in\mathbb{Z}}$ satisfying $x_{i}^{-}\leq x_{i}\leq
x_{i}^{+}$, we set

$G_{\omega}(\mathbb{x})=\sum_{I}(h(x_{i},x_{i+1})-h(x_{i}^{-},x_{i+1}^{-})),$

where $I=\mathbb{Z}$, if $\omega$ is not a rational number, and
$I=\\{0,...,q-1\\}$, if $\omega=p/q$. $P_{\omega}^{h}(\xi)$ is defined as the
minimum of $G_{\omega}(\mathbb{x})$ over the configurations
$\mathbb{x}\in\Pi=\prod_{i\in I}[x_{i}^{-},x_{i}^{+}]$ satisfying $x_{0}=\xi$.
Namely

$P_{\omega}^{h}(\xi)=\min_{\mathbb{x}}\\{G_{\omega}(\mathbb{x})|\mathbb{x}\in\Pi\
\text{and}\ \ x_{0}=\xi\\}.$

By [M3], $P_{\omega}^{h}(\xi)$ is a non-negative periodic function of the
variable $\xi\in\mathbb{R}$ with the modulus of continuity with respect to
$\omega$ and its modulus of continuity with respect to $\omega$ can be bounded
from above. Due to the periodicity of $P_{\omega}^{h}(\xi)$ with respect to
$\xi$, we only need to consider it in the interval $[0,1]$.

For simplicity, we don’t distinguish the constant $C$ in the following
different estimate formulas unless it is necessary. In the following sections,
we will prove Theorem 1.1 and Theorem 1.2, which will be achieved by the
detailed analysis on the regularity of Peierls’s barrier and an approximation
from trigonometric polynomials to $C^{\infty}$ functions.

##  3\.  Construction of the generating functions

We construct the perturbation of $h_{0}(x,x^{\prime})$ as follows. The first
one is

$u_{n}(x)=\frac{1}{n^{a}}(1-\cos(2\pi x)),\quad x\in\mathbb{R},$ (3.1)

where $n\in\mathbb{N}$ and $a$ is a positive constant independent of $n$.

Let $\bar{h}_{n}(x,x^{\prime})=h_{0}(x,x^{\prime})+u_{n}(x^{\prime})$, we have

######  Lemma 3.1.

Let $(x_{i})_{i\in\mathbb{Z}}$ be a minimal configuration of $\bar{h}_{n}$
with rotation symbol $0^{+}$, then

$x_{k+1}-x_{k}\geq C(n^{-\frac{a}{2}}),\quad\text{for}\quad
x_{k}\in\left[\frac{1}{4},\frac{3}{4}\right].$

The proof of Lemma 3.1 is similar to [W1, Lemma 4.1]. For the sake of
completeness, we will give it in Appendix A.

We construct the second part of the perturbation in the following. Let
$p_{N}(x)$ be a trigonometric polynomial of degree $N$. By Hadamard’s three-
circle theorem (see [F, Page 286-287] for more details), one has that for any
$r>0$,

$||p_{N}(x)||_{r}\leq e^{rN}||p_{N}(x)||,$ (3.2)

where $||p_{N}(x)||_{r}$ denotes the maximum of $|p_{N}(z)|$ in the strip
$S_{r}=\\{z\in\mathbb{C}|\,|\text{Im}z|\leq r\\}$ of width $2r$ in the complex
plane and $||p_{N}(x)||$ denotes the maximum of $|p_{N}(x)|$ on the real line.
Without loss of generality, we take $r=1$, namely

$||p_{N}(x)||_{1}\leq e^{N}\max|p_{N}(x)|.$ (3.3)

Then, by the Cauchy estimates, for any fixed $s\geq 0$, we have

$||p_{N}(x)||_{C^{s}}\leq C_{s}e^{N}\max|p_{N}(x)|,$ (3.4)

where $C_{s}$ is a constant depending on $s$ only.

Based on Lemma 3.1, we need to construct a real analytic function with a
“bump” in correspondence with the interval $\Lambda_{n}$ satisfying

$\mathcal{L}(\Lambda_{n})\sim
n^{-\frac{a}{2}}\quad\text{and}\quad\Lambda_{n}\subset\left[\frac{1}{4},\frac{3}{4}\right],$
(3.5)

where $\mathcal{L}(\Lambda_{n})$ denotes the Lebesgue measure of $\Lambda_{n}$
and $f\sim g$ means that $\frac{1}{C}g<f<Cg$ holds for some constant $C>0$.

The “bump” will be accomplished by using Jackson’s approximation theorem (see
[Z, Theorem 13.6, p115]). Let $\phi(x)$ be a $k$-times differentiable periodic
function on $\mathbb{R}$, then for every $N\in\mathbb{N}$, there exists a
trigonometric polynomial $p_{N}(x)$ of degree $N$ such that

$\max|p_{N}(x)-\phi(x)|\leq A_{k}N^{-k}||\phi(x)||_{C^{k}},$

where $A_{k}$ is a constant depending on $k\in\mathbb{N}$ only.

We take a $C^{\infty}$ bump function $\phi$ supported on the interval
$\Lambda_{n}$, whose maximum is equal to $2$. By (3.5), the length of
$\Lambda_{n}$ is bounded by $Cn^{-\frac{a}{2}}$. Thus, one can choose
$\phi(x)$ such that

$||\phi(x)||_{C^{k}}\sim\left(n^{\frac{a}{2}}\right)^{k}=n^{\frac{ak}{2}},$
(3.6)

where $k$ is determined by (4.7) below. Then, chose $N$ large enough to
achieve

$\sigma:=A_{k}N^{-k}||\phi(x)||_{C^{k}}\ll 1,$ (3.7)

where $\sigma$ is determined by (4.1) below. By Jackson’s approximation
theorem, we can construct a trigonometric polynomial $p_{N}(x)$ of degree $N$
such that:

$\left\\{\begin{array}[]{ll}\max p_{N}(x)\geq 1,&\text{attained on}\
\Lambda_{n},\\\ |p_{N}(x)|\leq\sigma,&\text{on}\
[0,1]\backslash\Lambda_{n}.\\\ \end{array}\right.$ (3.8)

By (3.7), we have

$N\sim\sigma^{-\frac{1}{k}}n^{\frac{a}{2}}.$ (3.9)

Finally, we consider the normalized trigonometric polynomial

$\tilde{p}_{2N}(x)=e^{-2N}\left(\frac{p_{N}(x)}{\max p_{N}(x)}\right)^{2}.$
(3.10)

From (3.4), $\tilde{p}_{N}(x)$ satisfies:

$\left\\{\begin{array}[]{ll}\tilde{p}_{2N}(x)\geq 0,\\\
||\tilde{p}_{2N}(x)||_{C^{s}}\leq C,\\\
\max\tilde{p}_{2N}(x)=e^{-2N},&\text{attained on}\ \Lambda_{n},\\\
|\tilde{p}_{2N}(x)|\leq\sigma^{2}e^{-2N},&\text{on}\
[0,1]\backslash\Lambda_{n}.\\\ \end{array}\right.$ (3.11)

Based on preparations above, we can construct the second part of the
perturbation as follow

$v_{n}(x)=u_{n}(x)\tilde{p}_{2N}(x)=\frac{1}{n^{a}}(1-\cos 2\pi
x)\tilde{p}_{2N}(x).$ (3.12)

It is easy to see $v_{n}$ satisfies the following properties:

$\left\\{\begin{array}[]{ll}v_{n}(x)\geq 0,&\\\ ||v_{n}(x)||_{C^{s}}\leq
Cn^{-a},&\\\ \max v_{n}(x)\geq e^{-2N}n^{-a},&\text{attained on}\
\Lambda_{n},\\\ |v_{n}(x)|\leq C\sigma^{2}e^{-2N}n^{-a},&\text{on}\
[0,1]\backslash\Lambda_{n}.\end{array}\right.$ (3.13)

So far, we complete the construction of the generating function of the nearly
integrable system,

$h_{n}(x,x^{\prime})=h_{0}(x,x^{\prime})+u_{n}(x^{\prime})+v_{n}(x^{\prime}),$
(3.14)

where $n\in\mathbb{N}$.

##  4\.  Proof of Theorem 1.1

First, we prove the non-existence of invariant circles with a small enough
frequency. More precisely, we have the following Lemma:

######  Lemma 4.1.

For $\omega\in\mathbb{R}\backslash\mathbb{Q}$ and $n$ large enough, the exact
area-preserving twist map generated by $h_{n}$ admits no invariant circle with
the frequency satisfying

$|\omega|<n^{-a-\delta},$

where $\delta$ is a small positive constant independent of $n$.

Proof First of all, we estimate the lower bound of $P_{0^{+}}^{h_{n}}$. Let
$(\xi_{i})_{i\in\mathbb{Z}}$ be a minimal configuration of $h_{n}$ defined by
(3.14) with rotation symbol $0^{+}$ satisfying $\xi_{0}=\eta$, where $\eta$
satisfies $v_{n}(\eta)=\max v_{n}(x)$ and let $(x_{i})_{i\in\mathbb{Z}}$ be
the minimal configuration of
$\bar{h}_{n}(x_{i},x_{i+1})=h_{0}(x_{i},x_{i+1})+u_{n}(x_{i+1})$ with rotation
symbol $0^{+}$, then

$\displaystyle\sum_{i\in\mathbb{Z}}(h_{n}($
$\displaystyle\xi_{i},\xi_{i+1})-h_{n}(\xi_{i}^{-},\xi_{i+1}^{-}))$
$\displaystyle\geq
v_{n}(\eta)+\sum_{i\in\mathbb{Z}}\bar{h}_{n}(\xi_{i},\xi_{i+1})-\sum_{i\in\mathbb{Z}}h_{n}(\xi_{i}^{-},\xi_{i+1}^{-}),$
$\displaystyle\geq
v_{n}(\eta)+\sum_{i\in\mathbb{Z}}\bar{h}_{n}(x_{i},x_{i+1})-\sum_{i\in\mathbb{Z}}h_{n}(x_{i},x_{i+1}),$
$\displaystyle=v_{n}(\eta)-\sum_{i\in\mathbb{Z}}v_{n}(x_{i+1}).$

By [M4, Page 208, (4.2)], there holds

$P_{0^{+}}^{h_{n}}(\eta)=\sum_{i\in\mathbb{Z}}(h_{n}(\xi_{i},\xi_{i+1})-h_{n}(\xi_{i}^{-},\xi_{i+1}^{-})).$

Therefore, we have shown:

$\displaystyle P_{0^{+}}^{h_{n}}(\eta)\geq
v_{n}(\eta)-\sum_{i\in\mathbb{Z}}v_{n}(x_{i+1}).$

By (3.13), we have

$v_{n}(\eta)\geq e^{-2N}n^{-a}.$

It follows from (A.1) that

$\sum_{i\in\mathbb{Z}}v_{n}(x_{i+1})\leq\sigma^{2}e^{-2N}\sum_{i\in\mathbb{Z}}u_{n}(x_{i+1})\leq\sigma^{2}e^{-2N}\sum_{i\in\mathbb{Z}}\frac{1}{4}(x_{i+1}-x_{i-1})^{2}\leq\sigma^{2}e^{-2N}.$

Hence,

$P_{0^{+}}^{h_{n}}(\eta)\geq e^{-2N}(n^{-a}-\sigma^{2}),$

we choose then $\sigma$ (consequently $N$) in such a way that

$\sigma^{2}=\frac{1}{4}n^{-a},$

which means

$\sigma=\frac{1}{2}n^{-\frac{a}{2}}.$ (4.1)

By (3.9), it follows that

$N\sim n^{\frac{a}{2}+\frac{a}{2k}},$ (4.2)

from which we have

$P_{0^{+}}^{h_{n}}(\eta)\geq\frac{3}{4}n^{-a}\exp\left(-Cn^{\frac{a}{2}+\frac{a}{2k}}\right).$
(4.3)

Secondly, following a similar argument as in [W1], we have

$|P_{\omega}^{h_{n}}(\xi)-P_{0^{+}}^{h_{n}}(\xi)|\leq
C\exp\left(-2n^{\frac{a}{2}+\frac{\delta}{2}}\right).$ (4.4)

where $\xi\in\Lambda_{n}$ and $\delta$ is a small positive constant
independent of $n$. Here, $\Lambda_{n}$ is the same as in (3.5). For the sake
of completeness, we will prove (4.4) in Appendix B.

Based on the preparations above, it is easy to prove Lemma 4.1. We assume that
there exists an invariant circle with frequency $0<\omega<n^{-a-\delta}$ for
$h_{n}$, then $P_{\omega}^{h_{n}}(\xi)\equiv 0$ for every $\xi\in\mathbb{R}$.
By (4.4), we have

$|P_{0^{+}}^{h_{n}}(\xi)|\leq
C\exp\left(-2n^{\frac{a}{2}+\frac{\delta}{2}}\right),\quad\text{for}\quad\xi\in\Lambda_{n}.$
(4.5)

On the other hand, $(\ref{lowb})$ implies that there exists a point
$\eta\in\Lambda_{n}$ such that

$P_{0^{+}}^{h_{n}}(\eta)\geq\frac{3}{4}n^{-a}\exp\left(-Cn^{\frac{a}{2}+\frac{a}{2k}}\right).$

Hence, we have

$n^{-a}\exp\left(-Cn^{\frac{a}{2}+\frac{a}{2k}}\right)\leq
C\exp\left(-2n^{\frac{a}{2}+\frac{\delta}{2}}\right).$ (4.6)

To achieve the contradiction, it suffices to take

$k>\frac{a}{\delta},$ (4.7)

which implies

$\frac{a}{2k}<\frac{\delta}{2}.$

Note that $C$ is independent of $n$. Hence, for $n$ large enough

$n^{-a}\exp\left(-Cn^{\frac{a}{2}+\frac{a}{2k}}\right)>C\exp\left(-2n^{\frac{a}{2}+\frac{\delta}{2}}\right),$

which contradicts (4.6). Therefore, there exists no invariant circle with
frequency $0<\omega<n^{-a-\delta}$.

For $-n^{-a-\delta}<\omega<0$, by comparing $P_{\omega}^{h_{n}}(\xi)$ with
$P_{0^{-}}^{h_{n}}(\xi)$, the proof is similar. We omit the details. This
completes the proof of Lemma 4.1. $\hfill{\Box}$

The case with a given irrational frequency can be easily reduced to the one
with a small enough frequency. More precisely,

######  Lemma 4.2.

Let $h_{P}$ be a generating function as follows

$h_{P}(x,x^{\prime})=h_{0}(x,x^{\prime})+P(x^{\prime}),$

where $P$ is a periodic function of periodic $1$. Let
$Q(x)=q^{-2}P(qx),q\in\mathbb{N}$, then the exact area-preserving twist map
generated by $h_{Q}(x,x^{\prime})=h_{0}(x,x^{\prime})+Q(x^{\prime})$ admits an
invariant circle with frequency $\omega\in\mathbb{R}\backslash\mathbb{Q}$ if
and only if the exact area-preserving twist map generated by $h_{P}$ admits an
invariant circle with frequency $q\omega-p,p\in\mathbb{Z}$.

We omit the proof and for more details, see [H2]. For the sake of simplicity
of notations, we denote $Q_{q_{n}}$ by $Q_{n}$ and the same for
$u_{q_{n}},v_{q_{n}}$ and $h_{q_{n}}$. Let

$Q_{n}(x)={q_{n}}^{-2}(u_{n}(q_{n}x)+v_{n}(q_{n}x)),$

where $(q_{n})_{n\in\mathbb{N}}$ is a sequence satisfying (1.1)

$|q_{n}\omega-p_{n}|<\frac{1}{q^{1+\mu}_{n}},$ (4.8)

where $p_{n}\in\mathbb{Z}$ and $q_{n}\in\mathbb{N}$. Since
$\omega\in\mathbb{R}\backslash\mathbb{Q}$, we have $q_{n}\rightarrow\infty$ as
$n\rightarrow\infty$. Let
$\tilde{h}_{n}(x,x^{\prime})=h_{0}(x,x^{\prime})+Q_{n}(x^{\prime})$, we prove
Theorem 1.1 for $(\tilde{h}_{n})_{n\in\mathbb{N}}$ as follow:

From the constructions of $u_{n}$ and $v_{n}$, it follows that

$\begin{split}||\tilde{h}_{n}&(x,x^{\prime})-h_{0}(x,x^{\prime})||_{C^{r}}\\\
&=||Q_{n}(x^{\prime})||_{C^{r}},\\\
&\leq{q_{n}}^{-2}(||u_{n}(q_{n}x^{\prime})||_{C^{r}}+||v_{n}(q_{n}x^{\prime})||_{C^{r}}),\\\
&\leq{q_{n}}^{-2}({q_{n}}^{-a}(2\pi)^{r}{q_{n}}^{r}+C_{1}{q_{n}}^{-a}{q_{n}}^{r}),\\\
&\leq C_{2}{q_{n}}^{r-a-2},\end{split}$ (4.9)

where $C_{1},C_{2}$ are positive constants depending on $r$ only.

Hence, it is enough to make $r-a-2<0$. Based on Lemma 4.1 and the Dirichlet
approximation (4.8), it suffices to take $a:=1+\mu-\delta$. Moreover, we
choose

$r<3+\mu,\quad\delta:=\frac{3+\mu-r}{2}.$

Then

$r-a-2=\frac{r-(3+\mu)}{2}<0.$

This completes the proof of Theorem 1.1.

##  5\.  Proof of Theorem 1.2

Given a badly approximable frequency $\omega$. Let $p_{n}\in\mathbb{Z}$ and
$q_{n}\in\mathbb{N}$ satisfy (4.8). Let $a\in(0,1)$ and

$R_{N^{\prime}}(x):=\frac{1}{q_{n}^{2+a}}\left(1-\cos(2\pi
q_{n}x)\right)\left(1+\tilde{p}_{2N}(q_{n}x)\right).$

It is clear to see that $R_{N^{\prime}}(x)$ is a trigonometric polynomial of
degree $N^{\prime}=(2N+1)q_{n}$. Next, we give an estimate on the upper bound
of $N^{\prime}$ in terms of $\epsilon$ and $r$.

By (4.9), we have

$\|R_{N^{\prime}}(x)\|_{C^{r}}\leq C_{1}\left(\frac{1}{q_{n}}\right)^{2+a-r},$

where $r\in[0,2+a)$. In order to achieve
$\|R_{N^{\prime}}(x)\|_{C^{r}}<\epsilon$, it suffices to require

$q_{n}>C_{2}\epsilon^{-\frac{1}{2+a-r}}.$ (5.1)

In order to get a smaller $N^{\prime}$, we assume

$q_{n-1}\leq C_{2}\epsilon^{-\frac{1}{2+a-r}}.$

It is clear that $\omega$ is badly approximable if and only if it is constant-
type from continued fraction expansion point of view. Namely, there exists
$K:=K(\omega)$ such that $|a_{n}|\leq K(\omega)$, where $a_{n}$ denotes the
$n$th partial quotient of $\omega$. By virtue of [S, Lemma 5F], we have

$q_{n}\leq C_{3}q_{n-1},$

where $C_{3}$ is a positive constant independent of $n$. It follows that

$q_{n}<C_{4}\epsilon^{-\frac{1}{2+a-r}}.$ (5.2)

By (4.2), to ensure that the area-preserving map generated by
$h_{0}(x,x^{\prime})+R_{N^{\prime}}(x^{\prime})$ admit no invariant circles
with frequency $\omega$, we only need

$N\sim q_{n}^{\frac{a}{2}+\frac{a}{2k}},$

where $k$ satisfies (4.7). Combining with (5.1), we have

$N>C_{5}\epsilon^{-\frac{a(k+1)}{2k(2+a-r)}}.$

Moreover, it gives rise to

$N^{\prime}=(2N+1)q_{n}>C_{6}\epsilon^{-\frac{a(k+1)}{2k(2+a-r)}-\frac{1}{2+a-r}}+C_{2}\epsilon^{-\frac{1}{2+a-r}}.$
(5.3)

We denote

$\gamma:=\frac{a(k+1)}{2k(2+a-r)}+\frac{1}{2+a-r}=\frac{a(k+1)+2k}{2k(2+a-r)}.$

For each $a\in(0,1)$ and $r\in[0,2+a)$, a direct calculation implies

$\gamma=\frac{3}{2(3-r)}+\frac{(1-a)r}{2(2+a-r)(3-r)}+\frac{a}{2(2+a-r)}\frac{1}{k}.$
(5.4)

Let $\delta:=1-a$. By (5.4), $\gamma$ can be reformulated as

$\gamma=\frac{3}{2(3-r)}+\frac{\delta
r}{2(3-r-\delta)(3-r)}+\frac{1-\delta}{2(3-r-\delta)}\frac{1}{k}.$

In terms of the Dirichlet approximation (4.8), $\delta\in(0,1)$. We pursue the
largest value of $r$ and the smallest value of $\gamma$. Thus, we focus on the
case with $0<3-r\ll 1$. Moreover, one can take for $m\in\mathbb{N}_{+}$,

$\delta=\frac{3-r}{2m},$

It follows that

$\gamma=\frac{3}{2(3-r)}+\frac{r}{2(2m-1)(3-r)}+\frac{2m-3+r}{2(2m-1)(3-r)^{2}}\frac{1}{k}.$

Given $r$ with $0<3-r\ll 1$ and $0<\epsilon\ll 1$. One can take $k$ and $m$
large enough such that

$\gamma<\frac{3}{2(3-r)}+\epsilon.$

Note that $\epsilon^{\epsilon}\rightarrow 1$ as $\epsilon\rightarrow 0$. In
view of (5.2) and (5.3), one can find a a trigonometric polynomial of degree

$N^{\prime}\leq C\epsilon^{-\frac{3}{2(3-r)}},$

such that the area-preserving map generated by
$h_{0}(x,x^{\prime})+R_{N^{\prime}}(x^{\prime})$ admit no invariant circles
with frequency $\omega$, where $C$ is a positive constant independent of
$\epsilon$.

This completes the proof of Theorem 1.2.

##  Appendix A Proof of Lemma 3.1

Without loss of generality, we assume $x_{i}\in[0,1]$ for all
$i\in\mathbb{Z}$. By Aubry’s crossing lemma, we have

$0<...<x_{i-1}<x_{i}<x_{i+1}<...<1.$

Let $x_{k}\in\left[\frac{1}{4},\frac{3}{4}\right]$. We consider the
configuration $(\xi_{i})_{i\in\mathbb{Z}}$ defined by

$\xi_{i}=\left\\{\begin{array}[]{ll}x_{i},&i<k,\\\ x_{i+1},&i\geq k.\\\
\end{array}\right.$

Since $(x_{i})_{i\in\mathbb{Z}}$ is minimal, we have

$\sum_{i\in\mathbb{Z}}\bar{h}_{n}(\xi_{i},\xi_{i+1})-\sum_{i\in\mathbb{Z}}\bar{h}_{n}(x_{i},x_{i+1})\geq
0.$

By the definitions of $\bar{h}_{n}$ and $(\xi_{i})_{i\in\mathbb{Z}}$, we have

$\displaystyle 0$
$\displaystyle\leq\sum_{i\in\mathbb{Z}}\bar{h}_{n}(\xi_{i},\xi_{i+1})-\sum_{i\in\mathbb{Z}}\bar{h}_{n}(x_{i},x_{i+1})$
$\displaystyle=\bar{h}_{n}(x_{i-1},x_{i+1})-\bar{h}_{n}(x_{i-1},x_{i})-\bar{h}_{n}(x_{i},x_{i+1})$
$\displaystyle=(x_{i+1}-x_{i})(x_{i}-x_{i-1})-u_{n}(x_{i}).$

Moreover,

$u_{n}(x_{i})\leq(x_{i+1}-x_{i})(x_{i}-x_{i-1})\leq\frac{1}{4}(x_{i+1}-x_{i-1})^{2}.$

Therefore,

$x_{i+1}-x_{i-1}\geq 2\sqrt{u_{n}(x_{i})}.$ (A.1)

For $x_{k}\in[\frac{1}{4},\frac{3}{4}]$, $u_{n}(x_{k})\geq n^{-a}$, hence,

$x_{k+1}-x_{k-1}\geq 2n^{-\frac{a}{2}}.$ (A.2)

Since $(x_{i})_{i\in\mathbb{Z}}$ is a stationary configuration, we have

$\displaystyle x_{i+1}-x_{i}$
$\displaystyle=-\partial_{1}\bar{h}_{n}(x_{i},x_{i+1}),$
$\displaystyle=\partial_{2}\bar{h}_{n}(x_{i-1},x_{i}),$
$\displaystyle=x_{i}-x_{i-1}+u_{n}^{\prime}(x_{i}).$

Since $u_{n}^{\prime}(x)=\frac{2\pi}{n^{a}}\sin(2\pi x)$, it follows from
$(\ref{ls})$ that

$x_{k+1}-x_{k}\geq C(n^{-\frac{a}{2}}),\quad
x_{k}\in\left[\frac{1}{4},\frac{3}{4}\right].$

The proof of Lemma 3.1 is completed.$\hfill{\Box}$

##  Appendix B Proof of (4.4)

Following a similar argument as in [F, M3, W1], we give some details of the
proof of (4.4). In [W1], we obtain that for certain $g_{n}$ and $\xi$ and
every irrational rotation symbol $\omega$ satisfying
$0<\omega<n^{-\frac{a}{2}-\delta}$,

$|P_{\omega}^{g_{n}}(\xi)-P_{0^{+}}^{g_{n}}(\xi)|\leq
C\exp\left(-2n^{\frac{\delta}{2}}\right).$ (B.1)

where $\delta$ is a small positive constant independent of $n$. Here we aim to
prove that for every irrational rotation symbol $\omega$ satisfying
$0<\omega<n^{-a-\delta}$,

$|P_{\omega}^{h_{n}}(\xi)-P_{0^{+}}^{h_{n}}(\xi)|\leq
C\exp\left(-2n^{\frac{a}{2}+\frac{\delta}{2}}\right).$ (B.2)

where $\xi\in\Lambda_{n}$ and $\delta$ is a small positive constant
independent of $n$, and $\Lambda_{n}$ is given by (3.5).

To achieve (B.1), the key part is to verify that each of the intervals
$[0,\exp(-n^{\frac{\delta}{2}})]$ and $[1-\exp(-n^{\frac{\delta}{2}}),1]$
contains a large number of points of the minimal configuration of $g_{n}$ for
$n$ large enough.

Correspondingly, in order to prove (B.2), it suffices to show that each of the
intervals $[0,\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})]$ and
$[1-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),1]$ contains a large number of
points of the minimal configuration of $h_{n}$ for $n$ large enough. Based on
that, (4.4) can be obtained by a standard argument (see [W1]). In the rest of
the appendix, we are devoted to proving the following lemma.

######  Lemma B.1.

Let $(x_{i})_{i\in\mathbb{Z}}$ be a minimal configuration of $h_{n}$ with
rotation symbol $0<\omega<n^{-a-\delta}$, then there exist
$j^{-},j^{+}\in\mathbb{Z}$ such that

$\displaystyle
0<x_{j^{-}-1}<x_{j^{-}}<x_{j^{-}+1}\leq\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),$
$\displaystyle 1-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})\leq
x_{j^{+}-1}<x_{j^{+}}<x_{j^{+}+1}<1.$

To prove Lemma B.1, we need to do some preliminary work. First of all, we
count the number of the elements of a minimal configuration
$(x_{i})_{i\in\mathbb{Z}}$ with arbitrary rotation symbol $\omega$ in a given
interval. With the method of [F], we can conclude the following lemma.

######  Lemma B.2.

Let $(x_{i})_{i\in\mathbb{Z}}$ be a minimal configuration of $h_{n}$ with
rotation symbol $\omega>0$,
$J_{n}=\left[\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),\frac{1}{2}\right]$ and
$\Sigma_{n}=\\{i\in\mathbb{Z}|\,x_{i}\in J_{n}\\}$, then

$\sharp\Sigma_{n}\leq Cn^{a+\frac{\delta}{2}},$

where $\sharp\Sigma_{n}$ denotes the number of elements in $\Sigma_{n}$ and
$\delta$ is a small positive constant independent of $n$.

Proof Let
$x^{-}=\exp\left(-n^{\frac{a}{2}+\frac{\delta}{2}}\right),x^{+}=\frac{1}{2}$
and $\sigma=\left(\frac{x^{+}}{x^{-}}\right)^{\frac{1}{M}}$, hence,

$\ln\sigma=\frac{\ln(x^{+})-\ln(x^{-})}{M}.$

We choose $M\in\mathbb{N}$ such that $1\leq\ln\sigma\leq 2$, then $M\sim
n^{\frac{a}{2}+\frac{\delta}{2}}$.

We consider the partition of the interval $J_{n}=[x^{-},x^{+}]$ into the
subintervals $J_{n}^{k}=[\sigma^{k}x^{-},\sigma^{k+1}x^{-}]$ where $0\leq
k<M$. Hence, $J_{n}=\cup_{k=0}^{M-1}J_{n}^{k}$. We set
$S_{k}=\\{i\in\Sigma_{n}|(x_{i-1},x_{i+1})\subset J_{n}^{k}\\}$ and
$m_{k}=\sharp S_{k}$.

By the similar deduction as the one in Lemma 3.1, we have

$x_{i+1}-x_{i-1}\geq 2\sqrt{u_{n}(x_{i})+v_{n}(x_{i})}\geq
Cn^{-\frac{a}{2}}x_{i},\quad\text{for}\quad
x_{i}\in\left[0,\frac{1}{2}\right].$

For simplicity of notation, we denote $Cn^{-\frac{a}{2}}$ by $\alpha_{n}$.

If there exists $k$ such that $i\in S_{k}$ for $(x_{i})_{i\in\mathbb{Z}}$,
then $x_{i+1}-x_{i-1}\geq\alpha_{n}\sigma^{k}x^{-}$, moreover,

$m_{k}\alpha_{n}\sigma^{k}x^{-}\leq
2\mathcal{L}(J_{n}^{k})=2(\sigma-1)\sigma^{k}x^{-},$

where $\mathcal{L}(J_{n}^{k})$ denotes the length of the interval of
$J_{n}^{k}$. Hence $m_{k}\leq 2(\sigma-1)\alpha_{n}^{-1}$.

On the other hand, if $i\in\Sigma_{n}\backslash\cup_{k=0}^{M-1}S_{k}$ , then
there exists $l$ satisfying $0\leq l<M$ such that

$x_{i-1}<\sigma^{l}x^{-}<x_{i+1}.$

Hence,

$\sharp\\{i\in\Sigma_{n}|i\not\in S_{k}\ \text{for\ any}\ k\\}\leq 2M.$

Therefore,

$\displaystyle\sharp(\Sigma_{n})\leq 2M(\sigma-1)\alpha_{n}^{-1}+2M.$

Since $1\leq\ln\sigma\leq 2$ and $M\sim n^{\frac{a}{2}+\frac{\delta}{2}}$,
then we have

$\sharp\Sigma_{n}\leq Cn^{a+\frac{\delta}{2}}.$

The proof of Lemma B.2 is completed.$\hfill{\Box}$

Let $(x_{i})_{i\in\mathbb{Z}}$ be a minimal configuration of $h_{n}$ with
rotation symbol $\omega>0$, An argument as similar as the one in Lemma B.2
implies that

$\sharp\left\\{i\in\mathbb{Z}\bigg{|}\,x_{i}\in\left[\exp\left(-n^{\frac{a}{2}+\frac{\delta}{2}}\right),1-\exp\left(-n^{\frac{a}{2}+\frac{\delta}{2}}\right)\right]\right\\}\leq
Cn^{a+\frac{\delta}{2}}.$

It is easy to count the number of the elements of a minimal configuration with
irrational rotation symbol. More precisely, we have the following lemma.

######  Lemma B.3.

Let $(x_{i})_{i\in\mathbb{Z}}$ be a minimal configuration with frequency
$\omega\in\mathbb{R}\backslash\mathbb{Q}$. Then for every interval $I_{k}$ of
length $k$, $k\in\mathbb{N}$,

$\frac{k}{\omega}-1\leq\sharp\\{i\in\mathbb{Z}|x_{i}\in
I_{k}\\}\leq\frac{k}{\omega}+1.$

Proof For every minimal configuration $(x_{i})_{i\in\mathbb{Z}}$ with
frequency $\omega$, there exists an orientation-preserving circle
homeomorphism $\phi$ such that $\rho(\Phi)=\omega$, where
$\Phi:\mathbb{R}\rightarrow\mathbb{R}$ denotes a lift of $\phi$. Since
$\omega\in\mathbb{R}\backslash\mathbb{Q}$, thanks to [H1], $\phi$ has a unique
invariant probability measure $\mu$ on $\mathbb{T}$ such that
$\int_{x}^{\Phi(x)}d\mu=\omega$ for every $x\in\mathbb{R}$. We denote
$\int_{x}^{\Phi(x)}d\mu$ by $\mu([x,\Phi(x)])$. In particular,

$\mu([x_{i},x_{i+1}])=\omega,\quad\text{for\ every\ }i\in\mathbb{Z}.$

From $\mu(I_{k})=k$, it follow that

$\displaystyle\omega\cdot(\sharp\\{i\in\mathbb{Z}|x_{i}\in I_{k}\\}-1)\leq k,$
$\displaystyle\omega\cdot(\sharp\\{i\in\mathbb{Z}|x_{i}\in I_{k}\\}+1)\geq k,$

which completes the proof of Lemma B.3.$\hfill{\Box}$

Based on Lemma B.2 and Lemma B.3, if $0<\omega<n^{-a-\delta}$ and $\omega$ is
irrational, then for $n$ large enough,

$\sharp\\{i\in\mathbb{Z}|x_{i}\in I_{1}\\}\geq\frac{1}{\omega}-1\geq
C_{1}n^{a+\delta}>C_{2}n^{a+\frac{\delta}{2}},$ (B.3)

where $I_{1}$ denotes the closed interval of length $1$.

Based on two counting lemmas above, it is easy to prove Lemma B.1. By
contradiction, we assume that there exist at most two points of
$(x_{i})_{i\in\mathbb{Z}}$ in $[0,\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})]$,
say $x_{m}$ and $x_{m+1}$. It follows that $x_{m-1}<0$ and
$x_{m+2}>\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})$. Hence, among the intervals
$[x_{m-1},x_{m}]$, $[x_{m},x_{m+1}]$ and $[x_{m+1},x_{m+2}]$, there exists at
least one such that its length is not less than
$\frac{1}{3}\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})$. Without loss of
generality, say $[x_{m+1},x_{m+2}]$.

Since $(x_{i})_{i\in\mathbb{Z}}$ is a stationary configuration, we have

$x_{m+2}-x_{m+1}=x_{m+1}-x_{m}+u^{\prime}_{n}(x_{m+1}),$

where $u^{\prime}_{n}(x_{m+1})=\frac{2\pi}{n^{a}}\sin(2\pi x_{m+1})$. From
$x_{m+1}\in[-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})]$,
it follows that

$|u^{\prime}_{n}(x_{m+1})|\leq
Cn^{-a}\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),$

which implies there exists $K$ independent of $n$ such that
$[-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})]$
contains at most $K$ points of $(x_{i})_{i\in\mathbb{Z}}$.

On the other hand, by (B.3), we have that for $n$ large enough, the number of
points of $(x_{i})_{i\in\mathbb{Z}}$ in
$[-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})]$
is also large enough, which is a contradiction. Therefore, there exists
$j^{-}\in\mathbb{Z}$ such that

$\displaystyle 0\leq
x_{j^{-}-1}<x_{j^{-}}<x_{j^{-}+1}<\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}).$

Similarly, there exists $j^{+}\in\mathbb{Z}$ such that

$\displaystyle 1-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})\leq
x_{j^{+}-1}<x_{j^{+}}<x_{j^{+}+1}<1.$

The proof of Lemma B.1 is completed.

From the proof of Lemma B.1, it is easy to see that each of
$[0,\exp(-n^{\frac{a}{2}+\frac{\delta}{2}})]$ and
$[1-\exp(-n^{\frac{a}{2}+\frac{\delta}{2}}),1]$ contains a large number of
points of the minimal configuration $(x_{i})_{i\in\mathbb{Z}}$ for $n$ large
enough.

Acknowledgement The author would like to thank the referees for the careful
reading of the paper and invaluable comments which are very helpful in
improving this paper. This work is supported by NSFC Grant No. 11790273,
11631006.

## References

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11institutetext: Leibniz-Institut für Astrophysik Potsdam (AIP), An der
Sternwarte 16, 14482 Potsdam, Germany
email<EMAIL_ADDRESS>22institutetext: INAF, Astrophysics and Space Science
Observatory, Via Gobetti 93/3, I-40129 Bologna, Italy 33institutetext:
Dipartimento di Fisica, Sezione di Astronomia, Università di Trieste, via G.B.
Tiepolo 11, I-34131, Trieste, Italy 44institutetext: I.N.A.F. Osservatorio
Astronomico di Trieste, via G.B. Tiepolo 11, I-34131, Trieste, Italy
55institutetext: I.N.F.N. Sezione di Trieste, via Valerio 2, 34134 Trieste,
Italy 66institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, France

# Explaining the decrease in ISM lithium at super-solar metallicities in the
solar vicinity

G. Guiglion 11 C. Chiappini 11 D. Romano 22 F. Matteucci 33 4 4 5 5 F. Anders
11 M. Steinmetz 11 I. Minchev 11 P. de Laverny 66 A. Recio-Blanco 66

(Received 7 September 2018 / Accepted 7 February 2019)

We propose here that the lithium decrease at super-solar metallicities
observed in high-resolution spectroscopic surveys can be explained by the
interplay of mixed populations that originate in the inner regions of the
Milky Way disk. The lower lithium content of these stars is a consequence of
inside-out disk formation plus radial migration. In this framework, local
stars with super-solar metallicities would have migrated to the solar vicinity
and depleted their original lithium during their travel time. To obtain this
result, we took advantage of the AMBRE catalog of lithium abundances combined
with chemical evolution models that take into account the contribution to the
lithium enrichment by different nucleosynthetic sources. A large proportion of
migrated stars can explain the observed lower lithium abundance at super-solar
metallicities. We stress that no stellar model is currently able to predict
Li-depletion for these super-solar metallicity stars, and solar Li-depletion
has to be assumed. In addition, no solid quantitative estimate of the
proportion of migrated stars in the solar neighborhood and their travel time
is currently available. Our results illustrate how important it is to properly
include radial migration when chemical evolution models are compared to
observations, and that in this case, the lithium decrease at larger
metallicities does not necessarily imply that stellar yields have to be
modified, contrary to previous claims in the literature.

###### Key Words.:

Galaxy: abundances - Galaxy: stellar content - Stars: abundances

##  1 Introduction

The chemical evolution of the Milky Way (MW) lithium abundance is still a
matter of debate. Recent high-resolution spectroscopic surveys have reported
that the interstellar medium (ISM) lithium content of the MW disk in the solar
vicinity decreases at super-solar metallicities
($[\text{M}/\text{H}]>0.0\,$dex). Delgado Mena et al. (2015) were the first to
state that lithium tend to decrease for $[\text{M}/\text{H}]>0$, for which
result they used high-resolution spectra. This result has been confirmed by
Guiglion et al. (2016) using HARPS, FEROS, and UVES spectroscopic data from
the AMBRE project (de Laverny et al. 2012). These data cover a wider range in
metallicity, up to $[\text{M}/\text{H}]\sim 0.45\,$dex. The lithium decrease
at super-solar metallicity was then observed by Fu et al. (2018) using Gaia-
ESO spectrocsopic survey data and by Bensby & Lind (2018) based on high-
resolution data from various spectrographs (such as MIKE and FEROS).

This decrease is puzzling because standard chemical evolution models point
toward a monotonous increase in lithium with metallicity (Romano et al. 2001;
Prantzos 2012). Fu et al. (2018) suggested a possible connection with a
reduced nova outburst rate at super-solar metallicity. With the _ad hoc_
assumption of a significant reduction of the Li yields above solar
metallicity, Prantzos et al. (2017) reproduced such a decreaes in lithium
abundance at super-solar metallicity. However, previous stellar
nucleosynthesis studies pointed out an increase in lithium yields for
$[\text{M}/\text{H}]>0$ (see, e.g., Karakas & Lugaro 2016).

When we study the Galactic lithium evolution with metallicity, we have to keep
in mind that lithium is easily depleted in stellar interiors throughout the
life of the star. Moreover, some lithium can be produced at specific stages of
stellar evolution. The stellar lithium content cannot be assumed to be
representative of the Li abundance of the ISM material from which the star was
formed. Because of the broad spread in lithium for a given metallicity, it is
important to only consider dwarf stars with the highest lithium, corresponding
to those that have destroyed at least their photospheric lithium and have
synthesized none. In practice, when predictions from a Galactic chemical
evolution model are compared to the observations, the upper envelope of the
data for dwarf stars needs to be considered (this is believed to be closer to
the original lithium in the ISM) as a function of metallicity.

In this paper, we provide an explanation for the lithium decrease observed in
super-solar metallicity thin-disk stars based on the AMBRE lithium catalog of
Guiglion et al. (2016), the Galactic chemical evolution model for the thin
disk of Chiappini (2009), and our current understanding of the chemodynamic
evolution of our Galaxy (Minchev et al. 2013, 2014a; Kubryk et al. 2015). In
Sect 2 we present lithium patterns in different MW disk populations, and in
Sect 3 we explain the decrease in lithium boundary at high metallicities as
due to the presence of old stars ($\tau>5$ Gyr) that originate in the very
inner regions that have a lower lithium content than locally born solar-
vicinity stars.

##  2 ISM lithium abundance in the MW disk

We used the AMBRE catalog of lithium abundances from Guiglion et al. (2016).
This catalog contains chemical abundances of lithium and their non-local
thermal equilibrium (NLTE) corrections, derived homogeneously from ESO HARPS,
FEROS, and UVES spectra. The atmospheric parameters have been derived by
Worley et al. (2012, 2016) and De Pascale et al. (2014), and can be found in
the publicly available catalog from Guiglion et al. (2018), with detailed
chemical abundances of _s-_ and _r-_ process elements. In this study, we
selected dwarf stars ($\text{log}(\textit{g})>3.5$) for which lithium
abundance and NLTE corrections were available, excluding upper limits. We
focused on the best stellar parameters and lithium abundances, that is, on
__stars with AMBRE QUALITY_FLAG=0 (see Worley et al. (2012) for more details
on this parameter). We rejected the coolest dwarf stars
$(\textit{T}_{\text{eff}}<5\,300\,$K) with a deeper convective zone (which
results in higher lithium destruction). The median signal-to-noise ratio
$(S/N)$ of our final sample is 105.

As recently demonstrated by Anders et al. (2018), the use of the
dimensionality reduction algorithm t-SNE111t-distributedStochastic Neighbour
Embedding (Van der Maaten & Hinton 2008) method ( see Section 3) allows a more
robust definition of subpopulations in abundance space. For this technique to
work, it is crucial to confine the analysis to narrow regions in atmospheric-
parameter space to avoid spurious abundance trends induced by differences in
atmospheric parameters. Although we do not have all the detailed chemical
information to be able to compute the t-SNe subpopulations for our sample, we
can use those found in Anders et al. (2018) for guidance because the two
samples (the sample studied by Anders et al. and ours) are consistent in terms
of abundance range and atmospheric parameters (see their Figure 2, using data
from Delgado Mena et al. (2017)).

Therefore, we chemically identified the MW disk populations in the
$[\alpha/\text{Fe}]\,\text{versus}\,[\text{M}/\text{H}]$ chemical space (e.g.,
Fuhrmann 1998, 2011; Minchev et al. 2015; Martig et al. 2016; Kawata &
Chiappini 2016) using the same definitions as in Guiglion et al. (2016, see
their Fig. 10 and their Sect. 6.2 for more details). We find that the thin and
chemical thick disks are fully consistent with those of Anders et al. (2018).
The main difference is that through the t-SNE method, Anders et al. (2018)
also identified a metal-rich $\alpha$-rich population (so-called $mr\alpha r$,
Adibekyan et al. 2011), which we here now also include by defining as super-
solar metallicity stars all thin disk stars with
$[\text{M}/\text{H}]>0.2\,$dex and solar $\alpha$-element abundance. Once
again, this is possible because the atmospheric parameters and chemical
abundances in AMBRE are consistent with the study of Delgado Mena et al.
(2017) (see Guiglion et al. (2018) for more details). The resulting chemical
patterns are presented in the left panel of Figure 1.

After we defined the sub-populations and in order to trace the lithium
abundance of the ISM as a function of metallicity, we followed the same
procedure as Lambert & Reddy (2004) and Guiglion et al. (2016): we binned the
AMBRE data in metallicity, and computed an average lithium abundance for the
six stars with the highest lithium abundance in a given metallicity bin. The
NLTE lithium abundance as a function of $[\text{M}/\text{H}]$ of our sample is
shown in the right panel of Figure 1, with different colors corresponding to
the four populations identified in the left panel. The resulting trends in ISM
lithium abundance for the four populations are overplotted with the colored
lines, with error bars corresponding to the standard deviation of the six
stars in each metallicity bin.

The thick-disk lithium envelope first rises from
$\text{A}_{\text{Li}}^{\text{NLTE}}\sim 2\,$dex at
$[\text{M}/\text{H}]=-1\,$dex to $\text{A}_{\text{Li}}^{\text{NLTE}}\sim
2.4\,$dex at $[\text{M}/\text{H}]=-0.2\,$dex. The thin-disk lithium envelope
shows an increase of about 1 dex and reaches the meteoric abundance at solar
metallicity. The envelope for the $mr\alpha r$ stars reaches that of the thin
disk at solar metallicity and clearly decreases by about 0.20 dex at
$[\text{M}/\text{H}]\sim+0.3\,$dex. In the metal-rich solar-$\alpha$
population, the lithium content is clearly lower than in the $mr\alpha r$
stars and in the thin-disk stars with solar metallicity. The trends presented
here remain the same when the slope of the chemical separation is slightly
varied and the bin size of the metallicity axis is changed. In addition, low
statistics is unlikely to degrade the observed trends because the proportion
of stars ($4\,\%$) with $\text{A}_{\text{Li}}^{\text{NLTE}}>3.0\,$dex is the
same for both $-0.1<[\text{M}/\text{H}]<+0.1\,$dex and
$[\text{M}/\text{H}]>+0.1\,$dex. The lithium decrease is therefore not due to
a poor sampling of the metal-rich populations.

Figure 1: _Left_ : $[\alpha/\text{Fe}]\,vs.\,[\text{M}/\text{H}]$ patterns for
the thin-disk (blue), canonical thick-disk (red), metal-rich
$[\alpha/\text{Fe}]$-rich stars (green), and super-metal-rich solar-$\alpha$
stars (magenta). The black lines show how we identified these four MW
populations. _Right:_ AMBRE NLTE lithium abundance $vs.$ metallicity for the
corresponding four populations using the same color-code as in the left panel.
For a given population, we plot the maximum lithium abundance using the same
method as Guiglion et al. (2016). The meteoric lithium abundance, which is not
affected by from stellar evolution, is also shown ($\otimes$).

##  3 Explaining the lithium decrease at super-solar metallicity

The answer to why in the metal-rich $[\alpha/\text{Fe}]$-rich (green) and
super-metal-rich solar-$\alpha$ (magenta) populations Li decreases with
metallicity involves three different processes that are simultaneously at
work: (1) the Li-enrichment in the ISM by different stellar sources (e.g.,
Prantzos et al. 2017), (2) the Li depletion throughout the evolution of the
star (e.g., Lagarde et al. 2012), and (3) the fact that the more metal-rich
stars are not the youngest stars, as has been shown observationally (e.g.,
Casagrande et al. 2011; Trevisan et al. 2011; Anders et al. 2017a) and
explained theoretically by the process of radial migration (e.g., Roškar et
al. 2008; Schönrich & Binney 2009; Minchev et al. 2013; Bird et al. 2013;
Vera-Ciro et al. 2014; Martinez-Medina et al. 2017).

In Figure 2 we illustrate the points made above and investigate the
feasibility of our explanation quantitatively as well. Although the Li-
enrichment is mostly well understood, large uncertainties still prevent a
detailed quantification of the effects of Li-depletion and radial migration
from pure theoretical arguments. In the case of the two latter processes, we
therefore use empirical data to guide our estimates.

###  3.1 Lithium enrichment

In Figure 2 we show chemical evolution tracks for three different radial bins
centered on R = 2, 4, and 8 kpc and the data described in Figure 1. The
theoretical curves are color-coded by age, corresponding to the color bar.
These tracks were computed using the Galactic chemical evolution model of
Chiappini (2009) (see also Minchev et al. (2013) for more details). In this
model, the Galactic disk is divided into rings of 2 kpc width that evolve
independently (_i.e.,_ without radial gas flows) in an inside-out fashion
(Matteucci & Francois 1989). This allows us to track the contributors to the
ISM lithium enrichment with time and radius. The nucleosynthesis prescriptions
for lithium adopted in the Chiappini (2009) model are the same as in the best
model of Romano et al. (1999)222In the model of Romano et al. (1999), several
lithium sources in addition to Big Bang nucleosynthesis were considered:
cosmic rays impacting on ISM nuclei, Type II supernovae (through the $\nu-$
process), carbon stars, massive asymptotic giant branch stars (AGBs), and
novae. Lithium astration in stars of all masses was also taken into account.
It was found that a long-lived stellar source was needed to reproduce the
observations, with novae being recently confirmed as a critical ingredient of
the model (see Izzo et al. 2015).. We are not interested in a detailed
discussion of the Li nucleosynthesis here (see Prantzos et al. 2017; Romano et
al. 1999, 2001, where a discussion can be found), but in showing the general
behavior of a chemical evolution model that assumes an inside-out formation
(i.e., higher star formation efficiencies (SFE) toward the inner regions of
the Galaxy), and takes into account the several sources of Li nucleosynthesis.

The chemical evolution tracks clearly show that the onset of the strong Li
increase, mainly powered by the contribution of long-lived sources such as
AGBs and novae (depending on the details of the chemical evolution model
adopted) occurs at higher metallicities for the innermost regions333We here
take the 2 kpc curve as our innermost example, but more extreme models
assuming higher star formation efficiencies could be constructed in which the
Li knee is displaced to even higher metallicities. However, in this case, the
final metallicity would be beyond [Fe/H] = 0.6, which is the value already
reached for the R=2kpc curve, and therefore unrealistic given that no star has
so far been observed at such a high metallicity. (a consequence of the higher
star formation rates; see, e.g., Matteucci & Brocato 1990; Chiappini et al.
2001; Hou et al. 2000; Minchev et al. 2013 ; this is illustrated by the
horizontal arrow in Figure 2.).

###  3.2 Lithium depletion throughout the stellar evolution

In addition to lithium evolution in the ISM, lithium is depleted in stellar
interiors during the main-sequence phase. No stellar model is currently able
to reproduce the correct lithium depletion in the Sun or predict Li-depletion
in [Fe/H]$>$0 stars. In stellar models that include rotation and thermohaline
mixing, the lithium depletion does not appear to increase for more metal-rich
stars (see the models of Lagarde et al. 2012). In Figure 2 we therefore
assumed a typical solar depletion for the more metal-rich stars in populations
green and magenta as well. The arrow in Figure 2 at [Fe/H]$=0.5$ is then not
arbitrary, but indicates the difference between the pre-solar cloud value
measured in meteorities and the solar photospheric lithium value (both values
are also illustrated in Figure 2), which implies a $\sim 2.2\,$dex depletion
during 4.5 Gyr of evolution (illustrated by the dashed arrow at [Fe/H] = 0).

###  3.3 Effect of radial migration

As clearly shown by both observations (e.g., Casagrande et al. 2011; Trevisan
et al. 2011; Anders et al. 2017a) and chemodynamical models (Minchev et al.
(2013, 2014b), the most metal-rich stars (those with [Fe/H] $>$ 0.25) are
typically 5-10 Gyr old. Therefore all of them will have depleted their lithium
by large amounts (as discussed in 3.1). It follows that the lithium decrease
at super-solar metallicities that is observed in the solar neighborhood might
be explained by radial migration, that is, that these stars were not born
locally (Minchev et al. 2013, 2014b, 2018; Anders et al. 2018, Chiappini et
al. 2018 in prep). These stars will have systematically lower lithium than
young stars that were born at the solar radius (here illustrated by the R=8kpc
curve).

Figure 2: Lithium abundance $vs.$ metallicity for the thin-disk (blue dots), a
metal-rich $\alpha-$rich population (green dots) and a super-metal-rich
solar-$\alpha$ population (pink dots) characterized in Figure 1. We show a
Galactic chemical evolution model of the thin disk (Chiappini 2009), for three
different galactocentric bins of 2 kpc width (centered at 2, 4, and 8 kpc),
color-coded by age. First, inside-out chemical evolution models predict the
knee of the lithium tracks to occur at higher metallicities for the innermost
zones because star formation is more efficient (illustrated by the horizontal
thick arrow). Second, the internal depletion of lithiun in the Sun is
illustrated by the vertical dashed arrow, from the meteoric abundance to the
photospheric one. Third, current data suggest super-metal rich stars to be
older than $\sim 5\,$Gyr, which implies a further lithium decrease due to
depletion. These old ages can be understood in the context of radial
migration. However, because there is no lithium depletion model at super-solar
metallicities, we have assumed this depletion to be equal to the solar one
(vertical thick arrow).

##  4 Conclusion

We have chemically characterized four Milky Way disk populations using a
classical chemical criterion in the well-known
$[\alpha/\text{Fe}]\,vs\,[\text{M}/\text{H}]$ plane: a thin-disk, a chemical
thick-disk, a metal-rich $\alpha-$rich population, and a super-metal-rich
solar-$\alpha$ population ($[\text{M}/\text{H}]>0.2\,$dex). These populations
and chemical patterns are consistent with the recent analysis of Anders et al.
(2018) and Chiappini et al. (in prep). For these four distinct populations, we
took advantage of the AMBRE catalog of lithium abundances from Guiglion et al.
(2016) to trace the lithium abundance in the ISM as a function of metallicty.
We showed that stars migrating from the innermost parts of the Galactic disk
(birth radii ¡ 2kpc) will have low lithium. If confirmed by stellar age-
dating, this finding would rule out the need of modifying stellar yields at
super-solar metallicity.

This work illustrates the effect that more robust definitions of MW stellar
populations (allowed by methods such as the t-SNE; Anders et al. 2018,
Chiappini et al. in prep) can have on the recent claim that a Li decline in
super-solar metallicity stars found by many authors (Delgado Mena et al. 2015;
Guiglion et al. 2016; Fu et al. 2018) would imply a revision of the lithium
stellar yields at high metallicities. As shown in Anders et al. (2018) and
Chiappini et al. (in prep), the most metal-rich thin-disk stars have most
probably migrated from inner disk regions (as has been indicated by previous
observations such as Casagrande et al. (2011), Trevisan et al. (2011), and
Anders et al. (2017b); see the discussion in Chiappini 2015), and cannot be
modeled as part of a local thin disk without taking into account the fact that
stars move out of their birth places, as is clearly shown by chemodynamical
models (Minchev et al. 2013, 2014b and Minchev et al. 2018). It is also
consistent with the kinematics found in super-solar metallicity stars in the
solar neighborhood, see, _e.g.,_ Wojno et al. (2018).

###### Acknowledgements.

The authors acknowledge the anonymous referee for the comments and suggestions
that improved the readability of the paper. GG and CC thank Corinne Charbonnel
for useful discussions. CC acknowledge the DFG project number 283705981:
”Analysing the chemical fingerprints left by the first stars: chemical
abundances in the oldest stars”. ARB acknowledges financial support form the
ANR 14-CE33-014-01. IM acknowledges support by the Deutsche
Forschungsgemeinschaft under the grant MI 2009/1-1.

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