11institutetext:  Centre for Nonlinear Science and Engineering, School of
Electrical and Electronics Engineering, SASTRA Deemed University, Thanjavur -
613401, Tamilnadu, India 22institutetext: School of Physics, Indian Institute
of Science Education and Research, Thiruvananthapuram - 695551, India
33institutetext: Department of Nonlinear Dynamics, School of Physics,
Bharathidasan University, Tiruchirappalli 620 024, India

# Symmetry breaking induced tipping to aging

I. Gowthaman 11 V. K. Chandrasekar 11 D. V. Senthilkumar 22 M. Lakshmanan 33

###### Abstract

The competing effect of heterogeneity and symmetry breaking coupling on the
emerging dynamics in a system of $N$ globally coupled Stuart-Landau
oscillators is investigated. Increasing the heterogeneity, using the standard
deviation of the Hopf bifurcation parameter, favors the macroscopic
oscillatory state for low values of the symmetry breaking coupling and
inhomogeneous steady state for larger values of the coupling. There is also a
transition, tipping, to homogeneous steady state (aging state) from the
macroscopic oscillatory state. The limiting factor in the diffusive coupling
favors the macroscopic oscillatory state even in the presence of a large
fraction of inactive oscillators in the network thereby increasing the
robustness of the network. The globally coupled oscillators are reduced to a
system of two evolution equations for the macroscopic order parameters,
corresponding to the mean-field and the shape parameter, using the self-
consistent field approach. The bifurcation diagrams obtained from the mean-
field variables elucidate various bifurcation scenarios responsible for the
dynamical transitions observed in $N$ globally coupled Stuart-Landau
oscillators. In particular, tipping to the aging state is found to occur via
the Hopf and pitchfork bifurcations illustrating the phenomenon of bifurcation
induced tipping. Analytical stability (critical) curves of these bifurcations,
deduced from the mean-field variables, are found to fairly well agree with the
simulation results.

##  1 Introduction

Coupled oscillator network constitutes an excellent framework to understand a
plethora of complex collective behavior. The phenomenon of aging is such an
intriguing collective behavior, introduced to understand the effect of local
deterioration/failures on the macroscopic dynamical state. Originally, it was
introduced by Daido et al hdkn2004 , by increasing the fraction of inactive
oscillators in an ensemble of active oscillators to find out the threshold at
which the macroscopic oscillation ceases to exist. Such a threshold has also
been identified as a measure of the dynamical robustness of the coupled
networks, which has actually been rigorously deduced ranging from regular to
complex network topologies hd2009 ; hd2011 ; whxz2014 ; gtkm2014 ; hd2008 ;
btds2014 ; hdnk2007 ; tgmk2012 ; mktg2011 . Initially, aging phenomenon has
been reported in globally coupled oscillator networks hdkn2004 , but later
extended to locally coupled oscillator networks by introducing local disorders
hd2009 ; hd2011 ; whxz2014 , heterogeneous oscillator networks with various
topologies gtkm2014 ; hd2008 , and also by introducing time-delay in the
coupling btds2014 . It has also been extended to complex networks hdnk2007 ;
tgmk2012  and multilayered networks mktg2011 . Interestingly, a
desynchronization horn has been associated with the onset of the aging
hdnk2007 .

Recent activities on aging are centered at enhancing the robustness of the
macroscopic oscillatory state despite the presence of a large fraction of
inactive oscillators. For instance, several methods such as introducing
auxiliary systems in the coupling sksm2018 , introducing a simple limiting
factor that limits the diffusive interaction ylwz2016 , incorporating large
variance of the asymmetry parameter sksm2018 , employing adaptive coupling
such as low-pass filters usks2020 , introducing asymmetry coupling among the
populations sksm2018 , and allowing normal and uniform random errors in the
Hopf bifurcation parameters ssnm2017 , etc. have been shown to increase the
robustness of the macroscopic oscillatory state over a large range of coupling
strength and even shown to prevent the onset of aging in several scenarios
despite the presence of a large fraction of the inactive oscillators.

Further, the onset of inhomogeneous (asymmetric) dynamical states in coupled
oscillator networks has been known to arise largely due to symmetry breaking
coupling, which breaks the rotational symmetry of the entire system of
equations. In particular, the distinct oscillation quenching states, namely
amplitude and oscillation deaths, chimera death and so on, have been shown to
be induced by breaking the rotational symmetry of the coupled oscillator
networks. Notably, the dynamical transition from amplitude death to
oscillation death via Turing bifurcation has been shown to get mediated by the
spontaneous symmetry breaking phenomenon gsap2012 ; akev2013 ; ismk2015 .
Further, the loss of rotational symmetry also leads to breaking of either
translational or permutation symmetry of the coupled oscillator networks.
Recent research reports unveiled that breaking the prevailing
permutation/translational symmetry of the coupled oscillators facilitates the
emergence of diverse macroscopic dynamical states such as chimera, solitary
state, clustering, etc. alev2013 ; azis2013 ; azmk2014 ; tb2015 ; kpvkc2015 ;
kpvkc2016 ; kssk2017 ; ksvkc2018 .

Despite the growing body of literature on the phenomenon of aging, the effect
of symmetry breaking coupling on the same and in particular the role of the
competing interaction between the symmetry breaking coupling and the
heterogeneity of the coupled oscillator networks on the aging has not yet been
explored in great detail to the best of our knowledge. In particular, the
nature of bifurcation that leads to the transition from the macroscopic
oscillatory state to the macroscopic quiescent state, namely tipping to the
aging state, of the coupled oscillator networks has not yet been identified.
In this work, using the coupled oscillator network comprising of active and
inactive oscillators, we unravel the competing interaction between the
symmetry breaking coupling and the degree of heterogeneity in understanding
the dynamical robustness of the considered network. In particular, we find
that increasing the standard deviation of the Hopf bifurcation parameter, that
is increasing the heterogeneity of the network, favors the macroscopic
oscillatory state and inhomogeneous dynamical states over a large range of the
parameter space. Further, the nature of the dynamical transition largely
depends on the system frequency leading to the state dependent aging. Limiting
the diffusive interaction using the simple limiting factor favors macroscopic
oscillatory state even in the presence of a large fraction of inactive
oscillators thereby increasing the robustness of the network. We deduce the
amplitude of $N$ globally coupled oscillator network in terms of the global
order parameters using the mean-field approximation. The global order
parameters quantify the macroscopic oscillatory state. The stability
conditions for the onset of the aging state from the macroscopic oscillatory
state and the inhomogeneous steady state are deduced from the evolution
equations of the global order parameters, which exactly mimics the dynamical
transitions obtained numerically from the $N$ globally coupled oscillator
network. The bifurcation diagrams, obtained using XPPAUT be2002 , from the
evolution equations of the global order parameters also clearly elucidate the
dynamical transitions. Specifically, we show that the onset of tipping to
aging state occurs via the Hopf and inverse pitch-fork bifurcations thereby
illustrating bifurcation induced tipping to the aging state.

The organization of the paper is as follows. In Sec. 2, we introduce a
heterogeneous network of $N$ globally coupled Stuart-Landau limit-cycle
oscillators with symmetry breaking coupling. We discuss bifurcation induced
tipping to the aging state using two-parameter bifurcation diagrams in Sec. 3.
We deduce the evolution equations corresponding to the mean-field global order
parameters from the system of $N$ globally coupled Stuart-Landau limit-cycle
oscillators under the self-consistent field approach in Sec. 4 and unravel the
bifurcations mediating the tipping to the aging state from the evolution
equations. The effect of the limiting factor on the observed dynamics is also
discussed in Sec. 4. Finally, we summarize our results and provide conclusion
in Sec. 5.

##  2 Model

We consider a heterogeneous network of $N$ globally coupled paradigmatic model
of Stuart-Landau oscillators with symmetry breaking coupling represented by
the set of dynamical equations,

$\displaystyle\dot{z}_{j}$
$\displaystyle\,=(a_{j}+i\omega-|z_{j}|^{2})z_{j}+\frac{k}{N}\sum_{l=1}^{N}Re\left({z_{l}}-\alpha{z_{j}}\right),\qquad
i=1,2,\ldots,N$ (1)

where $z_{j}$ represents the position of the $j^{th}$ oscillator in the
complex plane. In Eq. (1), $\omega$ is the intrinsic frequency, $a_{j}$ is the
parameter specifying the distance of the $j^{th}$ oscillator from the Hopf
bifurcation value, which is drawn from a distribution with mean $a$ and
variance $\sigma^{2}$, $k$ is the coupling strength and $\alpha$ is the
limiting factor wzdv2015 ; dgtb2015 , which controls the rate of diffusion
when $0\leq\alpha\leq 1$. The actual diffusive coupling is retrieved for
$\alpha=1$ and direct feedback coupling for $\alpha=0$. The coupling among the
real variables only results in breaking the rotational symmetry
$z_{j}\rightarrow z_{j}e^{i\theta}$ of the globally coupled network. The
Stuart-Landau oscillator represents the normal form of the Hopf bifurcation
and hence a large collection of nonlinear oscillators exhibiting the Hopf
bifurcation can be approximated as the Stuart-Landau oscillators. Hence, an
ensemble of Stuart-Landau oscillators serves as an excellent framework to
unravel and understand a plethora of macroscopic dynamical states that can be
displayed by several other nonlinear oscillators. In addition, Stuart-Landau
oscillators can also open up the feasibility of rigorous analytical treatment
in several circumstances. Stuart-Landau oscillators exhibit limit-cycle
oscillation for $a_{j}>0$ and stable homogeneous steady state at the origin
for $a_{j}<0$. In the following section, we will discuss the dynamical
transitions and spread of the dynamical states in the two-parameter phase
diagrams by directly integrating Eq. (1) numerically.

Figure 1: (color online) The spatiotemporal behavior and the order parameter
$X$, respectively, for $\omega=0.50$. (a, d) Limit cycle oscillations for
$k=0.30$, (b, e) homogeneous steady state (aging) for $k=1.5$ and (c, f)
inhomogeneous steady state (oscillation death) for $k=4.00$. The mean of the
distribution of the Hopf bifurcation parameters is chosen as $a=-0.50$ with
the standard deviation $\sigma=0.60$.

##  3 Bifurcation induced tipping to aging: Simulation results

The evolution equations corresponding to $N=100$ globally coupled Stuart-
Landau oscillators are solved using the Runge-Kutta 4th order integration
scheme with the time step $h=0.01$. The limiting factor is taken as
$\alpha=1$. The Hopf bifurcation parameter $a_{j}$ is uniformly distributed
with $\left<a_{j}\right>=a=-0.5$ and standard deviation $\sigma=0.7$ so that a
large fraction of oscillators are in their stable homogeneous steady state.
The spatiotemporal plot and the order parameter $X=Re(Z)$, where
$Z=|\frac{1}{N}\sum_{j=1}^{N}z_{j}|$, as a function of time are depicted in
Figs. 1 for $\omega=0.5$ and for different values of the coupling strength
$k$. The globally coupled Stuart-Landau oscillators exhibit limit-cycle
oscillations for $k=0.3$ despite the presence of a large fraction of inactive
oscillators (see Fig. 1(a)). The oscillatory nature of the order parameter $X$
in Fig. 1(d) corroborates the limit-cycle oscillations for $k=0.3$. Further
increase in the coupling strength results in tipping to the homogeneous steady
state (HSS), which is the aging state, as depicted in Fig. 1(b) for $k=1.5$,
while the corresponding order parameter reaches the null value after initial
transients confirming the homogeneous steady state at the origin. However, the
oscillators populate the branches of the inhomogeneous steady state (IHSS)
randomly for $k=4$ as depicted in Fig. 1(c) and the corresponding order
parameter acquires a finite value as illustrated in Fig. 1(f). Note that the
order parameter $X$ in Fig. 1(f) indeed corresponds to OD and not to
nontrivial AD dgtb2014 .

Figure 2: (color online) Two-parameter phase diagrams in the $(k,\omega)$
parameter space for different standard deviation $\sigma$ and for $a=-0.5$.
(a) $\sigma=0.6$, (b) $\sigma=0.7$, and (c) $\sigma=0.8$.

The two-parameter phase diagrams as a function of $k$ and $\omega$ are
depicted in Fig. 2 for a global perspective of the dynamical transitions for
different values of the standard deviation with $a=-0.5$. The parameter space
with limit-cycle oscillations, inhomogeneous steady state (oscillation death)
and homogeneous steady state (aging) are indicated by LC (unshaded region),
IHSS (region with diagonal lines) and HSS (shaded region), respectively. For
$\sigma=0.6$, there is a transition from limit-cycle oscillations to IHSS as a
function of the coupling strength $k$ for low values of the natural frequency
$\omega\in[0,0.14)$ (see Fig. 2(a)). There is also a transition from LC to
IHSS via HSS in a rather narrow range of $\omega\in[0.14,0.37)$ as a function
of $k$. Finally, there is a direct transition (tipping) from limit-cycle
oscillations to aging state for $\omega\geq 0.37$. Such a frequency dependent
tipping to aging is also known as state dependent aging igks2020 . Similar
dynamical transitions are depicted in Figs. 2(b) and 2(c) for $\sigma=0.7$ and
$0.8$, respectively. It is to be noted that as the fraction of active
oscillators are increased for large values of the standard deviation $\sigma$,
the spread of the aging state decreases while that of the limit-cycle
oscillations and IHSS increases. These results corroborate the fact that
increasing the value of standard deviation favors the macroscopic oscillatory
state in a large range of the coupling strength $k$, thereby increasing the
robustness of the coupled network. Furthermore, the threshold (minimum) value
of the natural frequency for the tipping to the aging also increases
appreciably with $\sigma$. Due to the large heterogeneity for increasing
$\sigma$, the spread of IHSS also increases to a large extent for large values
of the coupling strength.

Figure 3: (color online) Two-parameter phase diagrams in the $(k,\sigma)$
parameter space for different mean of the Hopf bifurcation parameter for
$\omega=0.25$. (a) $a=-0.30$ (b) $a=-0.50$, and (c) $a=-0.70$.

Two-parameter phase diagrams in the $(k,\sigma)$ parameter space are depicted
in Fig. 3 for different mean values of the Hopf bifurcation parameters
$a=\left<a_{j}\right>$ for $\omega=0.25$. Only the aging state is observed for
$\sigma<0.3$ for $a=-0.3$ in the entire explored range of the coupling
strength $k$ (see Fig. 3(a)) because for $\sigma<0.3$, the value of the Hopf
bifurcation parameter of all the oscillators are less than zero attributing to
the stable homogeneous steady state of the oscillators. Similar scenario
prevails in Figs. 3(b) and 3(c) for $a=-0.5$ and $-0.7$, respectively. For
negative values of $a$, when the value of $\sigma$ increases beyond $|a|$,
there exist some oscillators with $a_{j}>0$ resulting in the macroscopic
oscillatory state for small values of the coupling strength $k$. However, for
even slightly larger values of the symmetry breaking coupling $k$, the
heterogeneous nature of the oscillators plays a dominant role in facilitating
the inhomogeneous steady state in a large range of $k$ and $\sigma$.
Nevertheless, there is a transition from macroscopic oscillatory state to IHSS
via the aging state in a narrow range of $\sigma>|a|$.

##  4 Mean-field approximation

The system of globally coupled Stuart-Landu oscillators, Eq. (1), cannot be
solved exactly. However, it can be reduced to a system of two evolution
equations for the macroscopic order parameters, corresponding to the mean-
field and the shape parameters, within the self-consistent field approach
under the strong coupling limit sdm2002 ; gmp2016 . To be specific, we adopt a
mean-field approach proposed by Monte and Ovidio sdm2002  under a narrow
distribution of the Hopf bifurcation parameter and in the strong coupling
limit, which is based on a moment expansion and a closure assumption sdm2002 .
Let $Z=\left<z_{j}\right>$ denotes the mean field
$\frac{1}{N}\sum_{j=1}^{N}z_{j}$. As we can see $Z$ is the centroid of the $N$
oscillators, we can represent the position of the oscillators as the distance
from the centroid, $z_{j}=Z+\varepsilon_{j}$, where $\varepsilon_{j}$ is the
distance of the $j^{th}$ oscillator from the centroid $Z$. Since,
$Z=\left<z_{j}\right>$, therefore $\left<\varepsilon_{j}\right>=0$. By
definition, $<a_{j}>=a$. It is also necessary to account for the dispersion of
the $z_{j}$ about $Z$ by introducing the shape parameter, denoted by
$W=\left<a_{j}\varepsilon_{j}\right>$. Now, it is straightforward to arrive at
a class of macroscopic variables, corresponding to the higher orders, as the
equations of motion of the centroid. Nevertheless, many macroscopic properties
can be deduced even from the evolution equation of the macroscopic variables
corresponding to the first order approximation sdm2002 ; gmp2016 . However, we
have also verified the emergence of similar results for the second order
approximation in the current investigation.

The evolution equations corresponding to the centroid $Z$ and the shape
parameter $W$ can be deduced, restricting to the first order approximation,
under the narrow distribution of the Hopf bifurcation parameter $a_{j}$ and in
the strong coupling limit as

$\displaystyle\dot{Z}$
$\displaystyle\,=(a+i\omega+(k/2)(1-\alpha)-|Z|^{2})Z+(k/2)(1-\alpha)Z^{*}+W,$
(2)

which has the same functional form as that of individual uncoupled elements in
Eq. (1), except for the term $W$. Now, the time derivative of the shape
parameter $W$ can be written as

$\displaystyle\dot{W}$
$\displaystyle\,=\left<a_{j}\dot{z}_{j}\right>-\left<a_{j}\right>\dot{Z},$ (3)

where, neglecting higher order terms, $\left\langle
a_{j}\dot{z}_{j}\right\rangle$ can be deduced as

$\displaystyle\left\langle a_{j}\dot{z}_{j}\right\rangle$
$\displaystyle\,=\left\langle a_{j}^{2}\right\rangle Z+\left\langle
a_{j}^{2}\epsilon_{j}\right\rangle+i\omega Z\left\langle
a_{j}\right\rangle+i\omega\left\langle\epsilon_{j}a_{j}\right\rangle-|Z|^{2}Z\left\langle
a_{j}\right\rangle-2|Z|^{2}\left\langle\epsilon_{j}a_{j}\right\rangle-Z^{2}\left\langle\epsilon_{j}^{*}a_{j}\right\rangle$
$\displaystyle\,+\frac{k}{2}(1-\alpha)Z\left\langle
a_{j}\right\rangle-\frac{k}{2}\alpha\left\langle\epsilon_{j}a_{j}\right\rangle+\frac{k}{2}(1-\alpha)Z^{*}\left\langle
a_{j}\right\rangle-\frac{k}{2}\alpha\left\langle\epsilon_{j}^{*}a_{j}\right\rangle.$
(4)

Substituting Eqs. (4) and (2) in the evolution equation for the shape
parameter (3), the latter can be rewritten, neglecting
$\left\langle(a_{j}-a)^{2}\epsilon_{j}\right\rangle$ due to the narrow
distribution of the Hopf bifurcation parameter $a_{j}$, as

$\displaystyle\dot{W}$ $\displaystyle\,=\sigma^{2}Z+(a-(\alpha
k/2)-2|Z|^{2}+i\omega)W-(Z^{2}-(\alpha k/2))W^{*},$ (5)

where $\sigma^{2}=\left\langle a_{j}^{2}\right\rangle-a^{2}$. Equations (2)
and (5) govern the dynamics of the oscillators around the centroid.
Specifically, dispersion of the oscillators around their mean is governed by
Eq. (5). It is also clear that the standard deviation $\sigma$ of the Hopf
bifurcation (distance) parameter $a_{j}$, the coupling strength $k$ and the
limiting factor $\alpha$ determine the macroscopic dynamical states.

###  4.1 Bifurcation induced tipping to aging from mean-field equations

We have depicted the real part of $Z$ in the bifurcation diagrams, plotted
using XPPAUT be2002  from the evolution equations corresponding to the
centroid $Z$, Eqs. (2) and (5), as a function of $k$ for two different natural
frequencies in Figs. 4. Stable limit-cycle oscillations, IHSS and HSS are
indicated by filled circles, yellow/light grey solid lines and red/dark grey
solid line, respectively. Open circles correspond to unstable limit-cycle
oscillations and dashed line corresponds to the unstable steady states. The
transition from limit-cycle oscillations (filled circles) to IHSS (indicated
by yellow/light grey solid lines) is shown in Fig. 4(a) for $\omega=0.1$,
which is mediated by the saddle-node bifurcation. The stable limit-cycle
oscillations obtained from Eqs. (2) and (5), just before the saddle-node
bifurcation, is depicted in Fig. 5(a), where the unstable steady state at the
origin is indicated by an open circle. Trajectories starting from different
initial states eventually settle on to the stable limit-cycle. The dynamics
near the saddle-node bifurcation is depicted in Fig. 5(b), where stable nodes
are indicated by the filled circles and saddle points are by open circles.
Trajectories starting from the initial states in the basin of the stable nodes
reach them, whereas the trajectories emerging near the saddle points slip away
from them and settle at the nearby stable nodes as depicted in Fig. 5(b). The
saddle points collide with each other resulting in an unstable steady state
though the pitchfork bifurcation at a large value of the coupling strength
away from the saddle-node bifurcation as can be observed from Fig. 4(a), and
the dynamics after the pitchfork bifurcation is indicated in Fig. 5(d).

Figure 4: (color online) Bifurcation diagrams as a function of the coupling
strength using XPPAUT for $a=-0.50$ and for two different values of the
natural frequency. (a) $\omega=0.1$, (b) $\omega=0.2$. Filled circles and
solid lines correspond to stable limit-cycle oscillations and stable steady
states, whereas unfilled circles and dashed lines correspond to unstable
limit-cycle oscillations and unstable steady states.

Figure 5: (color online) Phase space trajectories near (a) Limit-cycle
oscillation, (b) IHSS with saddles (open circles) and nodes (filled circles),
(c) HSS and (d) Two stable nodes (filled circles) along with a unstable node
at origin (open circle) near pitch-fork bifurcation.

Transition from macroscopic oscillatory state to IHSS via aging is depicted in
Fig. 4(b) for $\omega=0.2$ as a function of the coupling strength. The
transition to the aging state (red/dark grey solid line) from the macroscopic
oscillatory state (filled circles), that is tipping to the aging, occurs via
the Hopf bifurcation. Similarly, the transition from the aging to the IHSS
(yellow/light grey solid lines) is mediated by the pitch-fork bifurcation (see
Fig. 4(b)). The stable limit-cycle oscillation (filled circle) looses its
stability and becomes unstable (open circles) at the Hopf bifurcation, which
then coexists with the aging state. The unstable limit-cycle oscillation
prevails even beyond the pitchfork bifurcation and coexists with the unstable
steady state (origin) and the IHSS. The dynamics near the Hopf bifurcation is
shown in Figs. 5(a) and 5(c). All the trajectories emanating from the
neighborhood of stable HSS reaches it as depicted in Fig. 5(c). The dynamics
just before and after the pitchfork bifurcation is captured in Figs. 5(c) and
5(d), respectively,

Now, the stability of the homogeneous steady state (aging state) can be
deduced, following the Routh-Hurwitz criterion, from the characteristic
equation corresponding to Eqs. (2) and (5). The critical curves corresponding
to the Hopf and pitch-fork bifurcations can be obtained as

$\displaystyle f_{HB}(k,a,\omega,\sigma,\alpha)$
$\displaystyle\,=\omega^{2}\beta_{1}^{2}((2a-\alpha
k)\beta_{2}-4\sigma)+2a\beta_{3}\left(\beta_{2}^{2}(\alpha
k-2a)^{2}-\sigma\beta_{1}^{2}\right)=0,$ (6)

and

$\displaystyle f_{PB}(k,a,\omega,\sigma,\alpha)$
$\displaystyle\,=\omega^{2}\left(a\beta_{3}+2\sigma^{2}\right)$ (7)
$\displaystyle\,+(a^{2}-\sigma^{2})\left((a-\alpha k)(a-\alpha
k+k)-\sigma^{2}\right)+\omega^{4}=0,$

where $\beta_{1}=4a-2\alpha k+k$, $\beta_{2}=2a-\alpha k+k$,
$\beta_{3}=2a-2\alpha k+k$. For $\alpha=1$, from the condition
$f_{PB}(k,a,\omega,\sigma,\alpha)=0$, one can explicitly deduce the critical
curve corresponding to the pitch-fork bifurcation as

$\displaystyle
k_{PB}=\frac{-2a^{2}\sigma^{2}+2a^{2}\omega^{2}+a^{4}+2\sigma^{2}\omega^{2}+\sigma^{4}+\omega^{4}}{a\left(a^{2}-\sigma^{2}+\omega^{2}\right)}$
(8)

Figure 6: (color online) Analytical two parameters phase diagrams for $a=-0.5$
in the (a) $(k,\omega)$ space for $\sigma$=0.60, and (b) $(k,\sigma)$ space
for $\omega$=0.25.

Analytical critical curves demarcating the dynamical regions are shown as two-
parameter phase diagrams in the $(k,\omega)$ space in Fig. 6(a) and in the
$(k,\sigma)$ space in Fig. 6(b) for $a=-0.5,\omega=0.25$ and $\sigma=0.60$.
The critical stability curve corresponding to the pitchfork bifurcation, Eq.
(8), is depicted as a dashed-dotted line across which there is a tipping to
the aging state. Similarly, there is also tipping to aging from the limit-
cycle oscillations via Hopf bifurcation curve (indicated by a dotted line)
obtained from Eq. (6). The saddle-node bifurcation curve, indicated by a solid
line, mediating the transition from the macroscopic oscillatory state to IHSS
is actually obtained from XPPAUT. The dynamical transitions and their spread
in Figs. 6(a) and 6(b) fairly well agree with that in the numerical figures in
Figs. 2(a) and 3(b), respectively.

###  4.2 Effect of the limiting factor

In this section, we will unravel the effect of the limiting factor $\alpha$ on
the observed dynamical regions. Two parameter phase diagrams in the
$(k,\alpha)$ space for $a=-0.5$, and $-0.7$ are shown in Figs. 7(a) and 7(b),
respectively. The standard deviation is chosen as $\sigma=0.60$ and the
natural frequency of oscillation is fixed as $\omega=0.25$. The analytical
critical curve (Hopf bifurcation curve) is shown as a dashed line, whereas the
saddle-node bifurcation curve obtained from XPPAUT is depicted as a solid
line. Shaded and unshaded regions are obtained by numerically solving the
dynamical equation (1) of the globally coupled Stuart-Landau limit-cycle
oscillators. As the limiting factor is decreased from the unity value, the
spread of the aging state starts to decrease first at large values of $k$ and
this behavior gradually extends to smaller values of $k$. Below a threshold
value of $\alpha$, the aging state (HSS) completely disappears for $a=-0.5$
(see Fig. 7(a)) leading to the macroscopic oscillatory state thereby
increasing the robustness of the globally coupled Stuart-Landau limit-cycle
oscillators. On the other hand, a finite but narrow range of aging state
persists even at $\alpha=0$ for $a=-0.7$ (see Fig. 7(b)) as there exists
oscillators only with $a_{j}<0$. Decreasing the limiting factor favors the
spread of the IHSS to a large region of the coupling strength. Limit-cycle
oscillations emerge in between the homogeneous and inhomogeneous steady
states. Thus it is evident that the limiting factor favors the macroscopic
oscillatory state (see Fig. 7(a)) even in the presence of a large fraction of
inactive oscillators, whereas it favors largely IHSS for all inactive
oscillators in addition to the macroscopic oscillatory state (see Fig. 7(b)).

Figure 7: (color online) Two parameters phase diagrams in the $(k,\alpha)$
space for (a) $a=-0.5$, and (b) $a=-0.7$. The other parameters are
$\sigma$=0.60, and $\omega$=0.25.

##  5 Summary and Conclusions

We considered a heterogeneous network of $N$ globally coupled Stuart-Landau
limit-cycle oscillators along with the symmetry breaking coupling. The
competing interaction between the heterogeneity, which is increased by
increasing the standard deviation $(\sigma)$ of the Hopf bifurcation parameter
of the Stuart-Landau oscillator about a mean, and the symmetry breaking
coupling is explored in considerable detail. The globally coupled oscillators
exhibit limit-cycle oscillations, homogeneous (aging) and inhomogeneous steady
states. Transitions among these dynamical states depend on the natural
frequency of oscillations resulting in the state dependent aging. Increasing
the heterogeneity is found to favor the limit-cycle oscillations in the entire
range of the natural frequency for low values of coupling, whereas strong
symmetry breaking coupling is found to favor the IHSS state. The threshold
value of $\omega$ for the tipping to aging from the oscillatory state is also
increased for increasing heterogeneity of the coupled oscillator network. The
aging state predominates most of the parameter space for the values of the
standard deviation of the Hopf bifurcation parameter below its mean value as
there exist only inactive oscillators in the entire network. The limiting
factor in the diffusive coupling favors the macroscopic oscillatory state even
in the presence of a large fraction of inactive oscillators in the network
thereby increasing the robustness of the network. Surprisingly, a finite
region of macroscopic oscillatory state is also found even when all the
oscillators are inactive upon limiting the diffusive interaction.

The globally coupled limit-cycle oscillators are reduced to a system of two
evolution equations for the macroscopic order parameters, corresponding to the
mean-field and the shape parameter, using the self-consistent field approach
in the strong coupling limit and narrow distribution of the heterogeneous
parameter. The bifurcation diagrams obtained from the macroscopic order
parameters elucidate different bifurcation scenarios responsible for the
dynamical transitions observed in $N$ globally coupled Stuart-Landau limit-
cycle oscillators. In particular, Hopf and pitchfork bifurcations are found to
induce the tipping to the aging state. The dynamical states demarcated by the
critical stability curves (bifurcation curves), deduced from the
characteristic equation corresponding to the macroscopic order parameters, are
found to agree fairly well with the two-parameter phase diagrams obtained by
numerically solving the $N$ globally coupled Stuart-Landau limit-cycle
oscillators. The bifurcation diagrams obtained from the evolution equations of
the macroscopic order parameters clearly elucidate the bifurcation induced
tipping to the aging state in addition to the other dynamical transitions as a
result of the competing interactions between the heterogeneity and the
symmetry breaking coupling.

Acknowledgement: IG wishes to thank SASTRA Deemed University for research fund
and extending infrastructure support to carry out this work. The work of VKC
forms part of a research project sponsored by CSIR Project under Grant No.
03(1444)/18/EMRII and SERB-DST- MATRICS Grant No. MTR/2018/000676. The work of
ML is supported by the Department of Science and Technology Science and
Engineering Research Board Distinguished Fellowship (Grant No.
SERB/F/6717/2017-18).

Author contribution statement: VKC and DVS formulated the problem. IG carried
out the simulations and calculations. All the authors discussed the results
and drafted the manuscript.

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# Quantum graphs: self-adjoint, and yet exhibiting a nontrivial
$\mathcal{PT}$-symmetry

Pavel Exner<EMAIL_ADDRESS>Miloš Tater<EMAIL_ADDRESS>Doppler Institute
for Mathematical Physics and Applied Mathematics, Czech Technical University,
Břehová 7, 11519 Prague, Czechia  Department of Theoretical Physics, Nuclear
Physics Institute, Czech Academy of Sciences, 25068 Řež near Prague, Czechia

###### Abstract

We demonstrate that a quantum graph exhibits a $\mathcal{PT}$-symmetry
provided the coefficients in the condition describing the wave function
matching at the vertices are circulant matrices; this symmetry is nontrivial
if they are not invariant with respect to transposition. We also illustrate
how the transport properties of such graphs are significantly influenced by
the presence or absence of the non-Robin component of the coupling.

###### keywords:

quantum graph , $\mathcal{PT}$-symmetry , vertex coupling , lattice transport
properties

###### MSC:

[2010] 81Q35 , 35J10

††journal: Physics Letters A

Having two research areas, each based on a strong concept, one is naturally
inspired to look for connecting links between them. This applies, in
particular, to the concepts of quantum graphs [6] and $\mathcal{PT}$-symmetry
[7]. Neglecting the prehistory of the former associated with the idea of Linus
Pauling [23], they came to life roughly at the same time and experienced three
decades of intense development. Quantum graphs were rediscovered as a
versatile model of nanostructurs, but proved themselves a useful tool to
investigate fundamental properties of quantum systems; as an example one can
mention their role in analysis of quantum chaos [21]. $\mathcal{PT}$-symmetry,
on the other hand, started from the observation that Schrödinger operators
with complex potentials can have a real spectrum [5], and while the importance
of this fact for quantum mechanics remains a matter of dispute for those who
are not $\mathcal{PT}$-proselytes, the idea found a number of applications in
various areas.

Speaking of $\mathcal{PT}$-symmetry the focus was understandably on the
nontrivial situations when neither parity nor the time-reversal invariance
were preserved but their composition was. This motivated the use of complex
potentials having in mind that Schrödinger operators with real ones are
naturally time-reversal invariant. Sometimes, of course, a real potential
emerges in this context, as is the case of the negative quartic one, but even
then the ways in which it is treated may be very different [4, 26].

Various ways of introducing $\mathcal{PT}$-symmetry to quantum graphs have
been proposed, all of them having in common that they go beyond the class of
self-adjoint Hamiltonians. Mostly the authors extended the class of boundary
conditions determining the vertex coupling, either globally [1, 14] or in a
particular subset of vertices. For instance, in [20] the coupling is supposed
to be Kirchhoff in the internal vertices while at the endpoints of the ‘loose
edges’ complex Robin boundary conditions are imposed, in fact generalizing the
very simple and elegant example from [17] in which the graph was just a line
segment; we note that in these situations the Robin conditions have a natural
interpretation as a source or drain. Another approach is to begin with a
modification of the scalar product in the state Hilbert space [22]; it is
known that operator facilitating the appropriate similarity transformation may
or may not exist [18].

The main observation in this letter is that self-adjoint quantum _may be_
$\mathcal{PT}$-symmetric and parity violating at the same time, even if (a)
there is no potential, that is, the only interaction comes from wave function
matching at the graph vertices, and (b) the coefficients describing the latter
are _real_. To be more specific, we are going to demonstrate such a symmetry –
nontrivial whenever the vertex coupling violates the time-reversal invariance
– applies to graphs in which those coefficients are _circulant matrices_. In
addition, we are going to provide some insights concerning spectral and
transport properties of quantum graph vertices underlying the role played by
the balance between the three component of the coupling, the Dirichlet,
Neumann, and Robin one [6, Thm. 1.4.4]. We will also give an example how an
arbitrarily small change of coupling parameters can completely change
transport properties of a lattice in terms of the probability introduced in
[2].

As usual the state Hilbert of a quantum graph is the direct sum,
$\mathcal{H}=\sum_{j}^{\oplus}L^{2}(e_{j})$, over the graph edges and the
Hamiltonian acts on the $j$th edge as
$\psi_{j}\mapsto-\psi_{j}^{\prime\prime}$. Without loss of generality we may
discuss a star graph, $N$ halflines meeting at a single vertex, because (a)
the vertex coupling is local, and (b) more complicated graphs can be replaced
by a graph with one vertex only, joining a certain number of halflines and
loops, and the boundary conditions with matrix coefficients having a block
structure reflecting the topology of the original graph. We use the symbols
$\Psi=(\psi_{1}(0),\ldots,\psi_{n}(0))^{T}$ and
$\Psi^{\prime}=(\psi_{1}^{\prime}(0),\ldots,\psi_{n}^{\prime}(0))^{T}$,
respectively, for the vectors of boundary values of the wave function
components and their derivatives, conventionally taken in the outward
direction, at the vertex. The boundary conditions $A\Psi+B\psi^{\prime}=0$
ensure the self-adjointness provided the matrix $(A|B)$ has maximum rank and
$B^{*}A$ is Hermitean [15]; alternatively we can write them in the unique
form,

$(U-I)\Psi+i\ell(U+I)\Psi^{\prime}=0,$ (1)

where $U$ is an $N\times N$ unitary matrix [12, 13]. The parameter $\ell>0$
fixes the length scale; keeping it fixed, we denote the Hamiltonian determined
by the condition (1) by $H_{U}$.

As usual in quantum mechanics, a symmetry is described by an operator
$\mathcal{H}\to\mathcal{H}$ with respect to which the Hamiltonian is
invariant. Since the symbol of our operator is real and invariant with respect
to the orientation of the edges, the nontrivial part of the task concerns the
condition (1): a particular symmetry is associated with an invertible map in
the space of the boundary values, $\Theta:\,\mathbb{C}^{n}\to\mathbb{C}^{n}$,
such that $(U-I)\Theta\Psi(0)+i(U+I)\Theta\Psi^{\prime}(0)=0$ holds for all
$\Psi$ satisfying (1), which is equivalent to the fact that $U$ obeys the
identity

$\Theta^{-1}U\Theta=U.$ (2)

The question is now which operators can be associated with the parity and time
reversal transformations.

The latter is simpler. The operator $\Theta_{\mathcal{T}}$ is antilinear and
idempotent, and since the particles we consider have no internal degrees of
freedom such as spin, it is just the complex conjugation. Using the unitarity,
$U^{T}\bar{U}=\bar{U}U^{T}=I$ we find easily that $\bar{\Psi}$ satisfies the
condition (1) with the transposed matrix, that is,

$\Theta_{\mathcal{T}}^{-1}U\Theta_{\mathcal{T}}=\Theta_{\mathcal{T}}U\Theta_{\mathcal{T}}=U^{T},$
(3)

and consequently, the $H_{U}$ is $\mathcal{T}$-invariant if and only the
matrix defining the coupling is transposal-symmetric, $U=U^{T}$. This also
immediately implies that a (self-adjoint) quantum graph _is
$\mathcal{PT}$-symmetric if and only if the mirror transformation acts in the
same way,_

$\Theta_{\mathcal{P}}^{-1}U\Theta_{\mathcal{P}}=\Theta_{\mathcal{P}}U\Theta_{\mathcal{P}}=U^{T}.$
(4)

Asking under which circumstances can the condition (4) be satisfied, we have
to note first that – while the concept of quantum graph _per se_ does not need
an ambient space – investigation of spatial reflections forces us to think of
the graph as embedded in the Euclidean space. For the sake of simplicity we
regard our star graph as planar, but the conclusion will certainly extend to
more general situations.

Let us first indicate which operators cannot serve as $\Theta_{\mathcal{P}}$.
One might be tempted to reverse the edge orientation leading to flipping the
sign of $\Psi^{\prime}$, however, the edges are all parametrized in the same
outward direction, which does not change when viewed in a mirror; moreover,
such a change would turn, say, an attractive $\delta$ coupling to a repulsive
one. Having a planar star graph, one can also think of reversing the edge
numeration (as considered in [11]), but the corresponding candidate for the
role of $\Theta_{\mathcal{P}}$, the matrix with one on the main antidiagonal
and zeros elsewhere, leads to a double transpose of $U$, both with respect to
the diagonal and antidiagonal, meaning just renaming the edges.

To see which linear operator can facilitate the similarity between $U$ and
$U^{T}$, we use the unitarity of the matrix: there is a unitary $V$ such that
$VUV^{*}$ is diagonal, and as such equal to its transpose. It follows that the
matrix $\Theta$ satisfying $\Theta U\Theta=U^{T}$ is of the form
$\Theta=V^{T}V$. We know, however, how $V$ looks like: the $j$th column of
$V^{*}$ coincides with $\phi_{j}^{T}$, where $\phi_{j}$ is the $j$th
normalized eigenvector of $U$. Consequently, we have

$\Theta_{ij}=(\bar{\phi}_{i},\phi_{j}),\quad i,j=1,\ldots,n\,;$ (5)

note the complex conjugation in the left entry which makes these expressions
nontrivial. Denoting by $\\{\nu_{j}\\}$ the basis in the boundary value space
referring to the direct sum form of $\mathcal{H}$,
$\nu_{1}=(1,0,\ldots,0)^{T}$ etc., we see that $\Theta$ maps $\nu_{j}$ to
$((\bar{\phi}_{1},\phi_{j}),\ldots(\bar{\phi}_{n},\phi_{j}))^{T}$, so it
general one cannot expect that such a $\Theta$ to be associated with a mirror
transformation.

The situation changes, however, when we restrict our attention to vertex
couplings referring to the subset of _circulant_ matrices. They are of the
form

$U=\left(\begin{array}[]{ccccc}c_{1}&c_{2}&\cdots&c_{n-1}&c_{n}\\\
c_{n}&c_{1}&c_{2}&&c_{n-1}\\\ \vdots&c_{n}&c_{1}&\ddots&\vdots\\\
c_{3}&&\ddots&\ddots&c_{2}\\\
c_{2}&c_{3}&\cdots&c_{n}&c_{1}\end{array}\right)\,;$ (6)

the unitarity requires that

$c_{j}=\frac{1}{n}\left(\lambda_{1}+\lambda_{2}\omega^{-j}+\lambda_{3}\omega^{-2j}+\cdots+\lambda_{n}\omega^{-(n-1)j}\right),\quad
j=1,\ldots,n,$ (7)

where $\lambda_{j},\>j=1,\ldots,n$, are the eigenvalues of $U$ and
$\omega:=\mathrm{e}^{2\pi i/n}$. The corresponding normalized eigenvectors are
independent of the choice of the $c_{j}$’s being

$\phi_{j}=\frac{1}{\sqrt{n}}\left(1,\omega^{j},\omega^{2j},\ldots,\omega^{(n-1)j}\right)^{T},\quad
j=1,\ldots,n.$ (8)

Furthermore, the eigenvalues can be written in terms of the matrix entries as
$\lambda_{j}=\sum_{k=1}^{n}c_{k}\omega^{j(k-1)}$. The diagonalization is
achieved in this case by the discrete Fourier transformation,

$V^{*}=\frac{1}{\sqrt{n}}\left(\begin{array}[]{cccccc}1&1&1&1&\ldots&1\\\
1&\omega&\omega^{2}&\omega^{3}&\ldots&\omega^{(n-1)}\\\
1&\omega^{2}&\omega^{4}&\omega^{6}&\ldots&\omega^{2(n-1)}\\\
\vdots&\vdots&\vdots&\vdots&&\vdots\\\
1&\omega^{n-1}&\omega^{2(n-1)}&\omega^{3(n-1)}&\ldots&\omega^{(n-1)^{2}}\\\
\end{array}\right)\,.$

Using relations (5) and (8) we easily find that now we have

$\Theta_{\mathcal{P}}=\left(\begin{array}[]{ccccccc}1&0&0&\cdots&0&0&0\\\
0&0&0&\cdots&0&0&1\\\ 0&0&0&\cdots&0&1&0\\\ \vdots&&&\iddots&&&\vdots\\\
0&0&1&\cdots&0&0&0\\\ 0&1&0&\cdots&0&0&0\end{array}\right)\,;$ (9)

Now we may add the subscript $\mathcal{P}$ to $\Theta$ because the
transformation has the properties expected from the parity transformation. It
preserves the edge $e_{1}$, as well as $e_{k+1}$ if $n=2k$, and among the
remaining ones it switches $e_{j}$ with $e_{n+2-j}$, and moreover, the same
will be true if we renumber the edges.

One may ask what is the meaning of the vertex couplings characterized by
matrices of the form (6). It is clear that the corresponding operator $H_{U}$
is invariant with respect to a cyclic renumbering of the edges meeting at the
vertex. This becomes illustrative if we think of our graph as embedded in the
Euclidean space, for simplicity again as a planar star. If the angles between
the edges are the same, each such renumbering is equivalent to a discrete
rotations; in this sense we can regard such a system as _isotropic_.

In this way we have identified a class of vertex couplings for which the graph
exhibits a $\mathcal{PT}$-symmetry. It depends on $n$ real parameters, out of
the number $n^{2}$ which characterize an arbitrary self-adjoint coupling.
Among them, a subset depending on $\big{[}\frac{n}{2}\big{]}+1$ parameters is
separately symmetric with respect to the time inversion and mirror
transformation, while in the $\big{[}\frac{n-1}{2}\big{]}$-parameter
complement the $\mathcal{PT}$-symmetry is nontrivial.

The former contains some well-known coupling such as $\delta$ or
$\delta^{\prime}_{s}$, or more generally, all the permutation-invariant ones
which correspond to $U$ of the form $U=uI+vJ$, where $I$ is the unit matrix
and $J$ denotes the matrix with all the entries equal to one; the parameters
have to fulfil the conditions $|u|=1$ and $|u+nv|=1$. Among vertex couplings
which violate the time-reversal symmetry, a simple example was proposed in
[10] being inspired by an attempt to use quantum graphs to model the anomalous
Hall effect [24]. The indicated coupling is determined by the matrix

$U=R:=\begin{pmatrix}0&1&0&0&\cdots&0&0\\\ 0&0&1&0&\cdots&0&0\\\
0&0&0&1&\cdots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&\cdots&0&1\\\ 1&0&0&0&\cdots&0&0\\\ \end{pmatrix},$ (10)

which is obviously circulant and transposition-asymmetric. The same is true
for a one-parameter class of couplings introduced in [11] which interpolates
between the $\delta$ coupling, to which the matrix $U=-I+\frac{2}{n+i\alpha}J$
corresponds making it separately $\mathcal{P}$\- and $\mathcal{T}$-symmetric,
and the one referring to (10).

Spectral properties of a self-adjoint star graph do not depend on the
particular form of the matrix $U$, only on its spectrum. The essential
spectrum of $H_{U}$ is always the halfline $\mathbb{R}^{+}$, being absolutely
continuous, because any of these Hamiltonians has a common symmetric
restriction with $H_{\mathrm{D}}:=H_{-I}$ describing the graph with Dirichlet-
decoupled edges. Its deficiency indices are at most $n$; this also means that
the number of negative eigenvalues of $H_{U}$ does not exceed $n\>$ [25, Corr.
to Thm 8.19]. In fact, we can find the eigenvalues explicitly using the fact
that they are preserved by unitary equivalence. Choosing the orthonormal basis
in $\mathcal{H}$ referring to the eigenvectors of $U$, we get a decomposition
of $H_{U}$ into a direct sum, the components of which are operators
$-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}$ on the halfline with the boundary
condition
$(\mathrm{e}^{i\gamma_{j}}-1)\phi_{j}(0)+i\ell(\mathrm{e}^{i\gamma_{j}}+1)\phi^{\prime}_{j}(0)=0$,
where $\lambda_{j}=\mathrm{e}^{i\gamma_{j}},\>j=1,\ldots,n$, with
$\gamma_{j}\in[0,2\pi)$ are the eigenvalues of $U$. Using the Ansatz
$\phi_{j}(x)=\mathrm{e}^{-\kappa_{j}x}$ we see that
$\\#\sigma_{\mathrm{disc}}(H_{U})$ _coincides with the number of eigenvalues
of $U$ is the upper complex halfplane_ and they are of the form
$-\kappa_{j}^{2}$, where

$\kappa_{j}=\frac{1}{\ell}\,\tan\frac{\gamma_{j}}{2},\quad\gamma_{j}\in(0,\pi).$
(11)

Likewise, the eigenvalues of $U$ in the lower halfplane give rise to antibound
states.

On the other hand, transport properties of such a star graph depend on the
form of $U$, not just on its spectrum. In [10] an interesting topological
property of the coupling (10) was observed: in the high-energy asymptotic
regime the transport is nontrivial if the vertex degree is even, while for an
odd degree the vertex becomes effectively decoupled. The origin of this
property is clear from the eigenvalue structure of (10): in the notation of
(7) its discrete spectrum consists of the numbers
$\omega^{j}:=\mathrm{e}^{2\pi ij/n},\>j=0,\ldots,n-1$. In particular,
$-1\not\in\sigma(U)$ for $n$ odd, in which case the on-shell S-matrix,

$S(k)=\frac{(k\ell-1)I+(k\ell+1)U}{(k\ell+1)I+(k\ell-1)U},$ (12)

satisfies $\lim_{k\to\infty}S(k)=I$; in the even case $-1$ is an eigenvalue
and the limit looks differently on the corresponding eigenspace. This
difference has various important consequences, in particular, for the band and
gap structure of periodic lattices and chains [3, 10], bulk-edge differences
in transport on lattices with a boundary [9], as well as for high-energy
properties of finite graphs spectra [8].

This example illustrates the importance of the decomposition of the vertex
coupling into the Dirichlet (the eigenspace referring to eigenvalue -1),
Neumann (the eigenvalue 1 eigenspace) and Robin (the rest) parts [6, Thm.
1.4.4]. One can find other examples involving circulant matrices. A very
simple one refers to the replacement of (10) with $U=-R$ in case of
$n=3,\,\ell=1$. Introducing $\eta:=\frac{1-k}{1+k}$, we have $S(k)=\frac{-\eta
I+U}{I-\eta U}$ which is easily calculated to be

$S(k)=\frac{1}{1-\eta+\eta^{2}}\begin{pmatrix}-\eta&\eta-1&\eta(1-\eta)\\\
\eta(1-\eta)&-\eta&\eta-1\\\ \eta-1&\eta(1-\eta)&-\eta\end{pmatrix}.$ (13)

This time the Dirichlet component is present but the Neumann one is not, and
consequently, we have $\lim_{k\to 0}S(k)=I$, while the transport at high
energies is nontrivial,

$\lim_{k\to\infty}S(k)=\frac{1}{3}\left(\begin{array}[]{rrr}1&-2&-2\\\
-2&1&-2\\\ -2&-2&1\end{array}\right),$ (14)

differing just by the sign of the off-diagonal elements from the Kirchhoff
coupling S-matrix.

On the other hand, the vertex coupling can have a purely Robin character. To
illustrate this claim, we replace the matrix (10) with

$U=\epsilon
R,\quad\epsilon=\mathrm{e}^{i\mu},\quad\mu\in\big{(}0,\textstyle{\frac{2\pi}{n}}\big{)}\,;$
(15)

in components the matching condition then reads

$\epsilon\psi_{j+1}-\psi_{j}+i\ell(\epsilon\psi^{\prime}_{j+1}+\psi^{\prime}_{j})=0\quad(\mathrm{mod}\,n)$
(16)

and its $\mathcal{PT}$-symmetry is obvious. Putting
$\eta:=\frac{1-k\ell}{1+k\ell}$ we find for the S-matrix elements the
expression

$S_{ij}(k)=\frac{1}{1-\epsilon^{n}\eta^{n}}\Big{(}-\eta(1-\epsilon^{n}\eta^{n-2})\delta_{ij}+(1-\delta_{ij})(1-\eta^{2})\epsilon(\epsilon\eta)^{(j-i-1)(\mathrm{mod}\,n)}\Big{)},$
(17)

which reduces for $\epsilon=1$ to the appropriate formula from [10]. In
distinction to that case, however, we have now $\lim_{k\to\infty}S(k)=I$
because of the factor $1-\eta^{2}$ which cancels out with the prefactor only
if $\epsilon=1$.

To see how the presence of the phase factor in (15) influences the spectrum,
let us revisit an example discussed in [10], a periodic _square lattice_ of
edge length $\ell$. To make the comparison easier, we put the length-scale
parameter in (16) equal to one, which we can do without loss of generality,
because such a choice is by the natural scaling transformation equivalent to
keeping the $\ell$ and setting the lattice spacing to one. The way to find the
spectrum is the same as in the said paper: we use the Ansatz

$\displaystyle\psi_{1}(x)$
$\displaystyle=a_{1}\mathrm{e}^{ikx}+b_{1}\mathrm{e}^{-ikx},$
$\displaystyle\psi_{2}(x)$
$\displaystyle=a_{2}\mathrm{e}^{ikx}+b_{2}\mathrm{e}^{-ikx},$
$\displaystyle\psi_{3}(x)$
$\displaystyle=\omega_{1}\left(a_{1}\mathrm{e}^{ik(x+\ell)}+b_{1}\mathrm{e}^{-ik(x+\ell)}\right),$
$\displaystyle\psi_{4}(x)$
$\displaystyle=\omega_{2}\left(a_{2}\mathrm{e}^{ik(x+\ell)}+b_{2}\mathrm{e}^{-ik(x+\ell)}\right),$

for wave functions in the elementary cell of the square lattice having the
cross-shaped form with four edges of length $\frac{1}{2}\ell$. We substitute
the corresponding boundary values into (1) remembering that the derivatives
have to be taken in the outward direction, and require the values at the
‘loose’ ends of the cross to differ by the Bloch factors
$\omega_{j}=\mathrm{e}^{i\theta_{j}}$, where $\theta_{j},\,j=1,2$, are the
quasimomentum components. This yields a system of four linear equations for
the coefficients $a_{j},\,b_{j}$ which is solvable provided the determinant

$D\equiv
D(\eta,\omega_{1},\omega_{2})=\left|\begin{array}[]{cccc}-1&-\eta&\epsilon\eta&\epsilon\\\\[1.99997pt]
\epsilon\omega_{1}\xi^{2}&\epsilon\omega_{1}\bar{\xi}^{2}\eta&-1&-\eta\\\\[1.99997pt]
-\omega_{1}\xi^{2}\eta&-\omega_{1}\bar{\xi}^{2}&\epsilon\omega_{2}\xi^{2}&\epsilon\omega_{2}\bar{\xi}^{2}\eta\\\\[1.99997pt]
\epsilon\eta&\epsilon&-\omega_{2}\xi^{2}\eta&-\omega_{2}\bar{\xi}^{2}\end{array}\right|$
(19)

with $\xi=\mathrm{e}^{ik\ell}$ and $\epsilon=\mathrm{e}^{i\mu}$ with
$\mu\in\big{(}0,\frac{1}{2}\pi\big{)}$ vanishes. Passing from $\eta$ to the
original momentum variable $k$ and using the trigonometric expressions for
$\epsilon^{4}\pm 1$ we arrive at the condition

$8i\epsilon^{2}\,\frac{\mathrm{e}^{i(\theta_{1}+\theta_{2})}}{(k+1)^{4}}\,\sum_{j=0}^{4}c_{j}k^{j}=0,$
(20)

where

$\displaystyle c_{0}$ $\displaystyle=c_{4}=-\sin 2\mu\,\sin^{2}k\ell,$
$\displaystyle c_{2}$ $\displaystyle=\sin 2\mu\,(1+3\cos 2k\ell),$
$\displaystyle c_{1}$ $\displaystyle=2\big{(}2\cos 2\mu\,\cos
k\ell-\cos\theta_{1}-\cos\theta_{2}\big{)}\sin k\ell,$ $\displaystyle c_{3}$
$\displaystyle=2\big{(}2\cos 2\mu\,\cos
k\ell+\cos\theta_{1}+\cos\theta_{2}\big{)}\sin k\ell;$

for the negative spectrum we set $k=i\kappa$ with $\kappa>0$ and the spectral
condition becomes $\sum_{j=0}^{4}c_{j}\kappa^{j}=0$, where

$\displaystyle c_{0}$ $\displaystyle=c_{4}=\sin 2\mu\,\sinh^{2}\kappa\ell,$
$\displaystyle c_{2}$ $\displaystyle=-\sin 2\mu\,(1+3\cosh 2\kappa\ell),$
$\displaystyle c_{1}$ $\displaystyle=-2\big{(}2\cos
2\mu\,\cosh\kappa\ell-\cos\theta_{1}-\cos\theta_{2}\big{)}\sinh\kappa\ell,$
$\displaystyle c_{3}$ $\displaystyle=2\big{(}2\cos
2\mu\,\cosh\kappa\ell+\cos\theta_{1}+\cos\theta_{2}\big{)}\sinh\kappa\ell;$

This shows that the presence of the phase factor influences the spectrum
significantly. If $\mu=0$ the even coefficients are zero and the corresponding
solution to (23) given in [10] shows that the positive spectrum is dominated
by spectral bands the widths of which grow linearly with band index, while the
gap widths are asymptotically constant. Moreover, the factor $\sin k\ell$ in
$c_{1}$ and $c_{3}$ means that the spectrum contains infinitely degenerate
eigenvalues, or flat bands.

This changes once we have $\mu\neq 0$, within the indicated interval. First of
all, the form of $c_{2}$ does not allow to factorize a $\theta$-independent
term from (23) which means that _there is no infinite series of flat bands_
such as the one at $k^{2}=\big{(}\frac{\pi
n}{\ell}\big{)}^{2},\,n\in\mathbb{Z}$, appearing for $\mu=0$. This does not
mean that the point spectrum is void, though. Choosing $k=1$, the spectral
condition reduces to $\sum_{j=0}^{4}c_{j}=0$ where the $\theta$-dependent
terms cancel and the sum vanishes provided

$\cot 2\mu=\frac{-1-3\cos 2\ell+2\sin^{2}\\!\ell}{4\sin 2\ell}=-\cot 2\ell,$
(23)

that is, $\mu=\frac{\pi}{2}-\ell\;\big{(}\mathrm{mod}\,\frac{\pi}{2}\big{)}$.
Furthermore, while we lack now the nice graphical way in which we were able to
solve the spectral problem in [10], it is not difficult to determine the high-
energy asymptotic behavior. Since $c_{4}\neq 0$ for $k\neq\frac{\pi n}{\ell}$,
it is clear from (23) that – while $\big{(}\frac{\pi
n}{\ell}\big{)}^{2}\not\in\sigma(H_{U})$ – the spectral bands may exist only
in the vicinity of those points. Up to higher-order terms, $k=\frac{\pi
n}{\ell}+\delta$ may correspond to a spectral point only for $\delta$ within
the range of the function $(\theta_{1},\theta_{2})\mapsto\frac{\pi n(2\cos
2\mu+(-1)^{n}Q)}{(\pi^{2}n^{2}+6\ell^{2})\sin 2\mu}$, where
$Q:=\cos\theta_{1}+\cos\theta_{2}$ runs through the interval $[-2,2]$. For
large $n$ we thus get $|\delta_{n}|\lesssim\frac{2}{\pi n}\cot\mu$ as
$n\to\infty$, and consequently, the width of the $n$th band on the energy
scale is _for a fixed_ $\mu\in\big{(}0,\textstyle{\frac{\pi}{2}}\big{)}$
asymptotically constant,

$\Delta_{n}\lesssim\frac{8}{\ell}\,\cot\mu.$ (24)

We stress the fixed value of $\mu$. The band width is decreasing as $\mu$
grows from zero, however, it is not monotonous over the whole interval
$\big{[}0,\textstyle{\frac{\pi}{2}}\big{]}$; as we are approaching the right
endpoint, it starts growing again, because the corresponding matrix $U=iR$ has
again $-1$ as its eigenvalue which means that the spectrum is anew dominated
by the bands.

This non-uniform character of the asymptotical behavior is also manifested in
another quantity, the probability of finding a randomly chosen value of the
energy in the spectrum,

$P_{\sigma}(H_{U}):=\lim_{K\to\infty}\frac{1}{K}\left|\sigma(H_{U})\cap[0,K]\right|=0,$
(25)

introduced by Band and Berkolaiko [2]. It is clear from (24) that
$P_{\sigma}(H_{U})=0$ for $\mu\in\big{(}0,\textstyle{\frac{\pi}{2}}\big{)}$,
while for both the real-valued $U=R$ and the purely imaginary $U=iR$ the
probability is equal to one.

One can also inspect the negative spectrum of $H_{U}$, which has at most two
bands. In particular, for large $\ell$ they are expected to be narrow and to
shrink to the eigenvalues (11) referring to $\kappa=\tan\frac{\mu}{2}$ and
$\tan\big{(}\frac{\mu}{2}+\frac{\pi}{4}\big{)}$ as $\kappa\to\infty$. This is
indeed the case: for large $\ell$ the leading behavior in the spectral
condition with the coefficients (LABEL:coeff-) is determined by the terms
containing $\mathrm{e}^{2\kappa\ell}$, which yields

$0=\kappa^{4}+1-6\kappa^{2}+4\kappa(\kappa^{2}-1)\cot
2\mu+\mathcal{O}(\mathrm{e}^{-2\kappa\ell}),$ (26)

where only the error term depends on the quasimomentum. If we neglect it and
look for solutions of the resulting equation, it us useful to put
$c:=\cos\frac{\mu}{2}$ and $s:=\sin\frac{\mu}{2}$, so that $\cos
2\mu=1-8c^{2}s^{2}$ and $\sin 2\mu=4cs(c^{2}-s^{2})$; by a straightforward
computation one can then check that $\kappa=\frac{s}{c}$ and
$\kappa=\frac{c+s}{c-s}$ are the roots.

Figure 1: The spectrum of $H_{U}$ as a function of $\mu$ for
$\ell=\frac{3}{2}$. The dot at $(\mu,k)=\big{(}\frac{1}{2}(\pi-3),1\big{)}$ is
the flat band.

These results are illustrated in Fig. 1 in which we plot the spectrum of
$H_{U}$ in dependence of the parameter $\mu$ with the choice
$\ell=\frac{3}{2}$, including the indication of the flat band. Fig. 2 shows
the same spectrum for a larger value of the length parameter, $\ell=10$. This
choice makes the bands narrow, in particular, it is clear that the negative
ones are centered around the corresponding star-graph eigenvalues (11)
obtained as solutions to (26). The flat band at $(\mu,k)=(0,-1)$ becomes
absolutely continuous for $\mu>0$ and escapes to negative infinity as
$\mu\to\frac{\pi}{2}-$ while the next one shrinks to $-1$ in this limit;
recall that at that point the matrix $U$ has eigenvalue $-1$ again.

Figure 2: The same for $\ell=10$.

In Fig. 3 we plot a higher part of the same spectrum. The picture illustrates
that for a fixed $\mu\in\big{(}0,\frac{\pi}{2}\big{)}$ the positive spectral
bands get narrower as the energy grows, while at the endpoints of the interval
they dominate the spectrum; as expected the limit (25) is nonuniform with
respect to $\mu$.

Figure 3: High energy spectrum of $H_{U}$ as a function of $\mu$ for
$\ell=10$.

In addition, we see a number of points where the gaps are closing. In
distinction to the point $(\mu,k)=\big{(}\frac{\pi}{2}-\ell,1)$ mentioned
above, however, those are not flat bands but true band crossings occurring
either in the center of the Brillouin zone or its corners. Two examples are
shown in Fig. (4); one clearly sees the Dirac cones at the touching points.

Figure 4: Fermi surfaces in the momentum variable for the lattice with
$\ell=10$ at the points of closing gaps, left at the values
$(\mu,k)=(1.55068665,10.07328547)$, right at
$(\mu,k)=(1.55190524,10.38681556)$

## Acknowledgements

The research was supported by the Czech Science Foundation within the project
21-07129S and by the EU project CZ.02.1.01/0.0/0.0/16_019/0000778.

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On a Ramanujan-type series associated with the Heegner number 163

John M. Campbell

###### Abstract

Using the Wolfram NumberTheory package and the Recognize command, together
with numerical estimates involving the elliptic lambda and elliptic alpha
functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type
series related to the class number $h(-d)=1$ for a quadratic form with
discriminant $d=163$. This conjectured series is of level one and has positive
terms, and recalls the Chudnovsky brothers’ alternating series of the same
level, given the connection between the Chudnovsky–Chudnovsky formula and the
Heegner number $d=163$ such that $\mathbb{Q}\left(\sqrt{-d}\right)$ has class
number one. We prove Bagis and Glasser’s conjecture by proving evaluations for
$\lambda^{\ast}(163)$ and $\alpha(163)$, which we derive using the Chudnovsky
brothers’ formula together with the analytic continuation of a formula due to
the Borwein brothers for Ramanujan-type series of level one. As a byproduct of
our method, we obtain an infinite family of Ramanujan-type series for
$\frac{1}{\pi}$ generalizing the Chudnovsky algorithm.

_MSC:_ 11R29, 11Y60

_Keywords:_ Chudnovsky algorithm, Ramanujan-type series, class number, Heegner
number, elliptic

lambda function

##  1 Introduction

Bagis and Glasser, in 2013 [1], obtained the following infinite family of
Ramanujan-type series:

$\frac{3}{\pi\sqrt{r}\sqrt{1-J_{r}}}=\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{5}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}}{(n!)^{3}}(J_{r})^{n}(6n+1-T_{r}),$
(1)

where

$J_{r}:=1728j_{r}^{-1}=4\beta_{r}(1-\beta_{r})$ (2)

and

$T_{r}:=\frac{2j_{r}^{1/3}\sigma(r)G_{r}^{8}}{\sqrt{r}\sqrt{j_{r}-1728}},$ (3)

referring to [1] for details. Following Bagis and Glasser’s work in [1], the
_class number_ $h(-d)$ of a quadratic form of discriminant $d$ is such that

$h(-d)=-\frac{w(d)}{2d}\sum_{n=1}^{d-1}\left(\frac{-d}{n}\right)n,$ (4)

where $\left(\frac{n}{m}\right)$ denotes the Jacobi symbol and $w(3)=6$,
$w(4)=4$, and $w(d)=2$ otherwise. Since (4) reduces to $1$ for $d=163$, it was
suggested by Bagis and Glasser that this could be indicative of a greater
likelihood of determining relatively simple closed forms for the coefficients
involved in the Ramanujan-type series in (1). Bagis and Glasser [1] applied
the Wolfram Mathematica NumberTheory package and the Recognize command to
determine conjectured closed forms for $J_{163}$ and $T_{163}$, based on
numerical estimates of $J_{163}$ and $T_{163}$. This led Bagis and Glasser [1]
to provide a conjectured Ramanujan-type series, given by setting the
conjectured values for $J_{163}$ and $T_{163}$ into (1). This purported
Ramanujan-type series would provide a natural companion to the famous
Chudnovsky–Chudnovsky formula [6]

$\frac{711822400}{\sqrt{10005}\,\pi}=\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}\left(\frac{5}{6}\right)_{n}}{\left(1\right)_{n}^{3}}\left(-\frac{1}{53360}\right)^{3n}\left(\frac{13591409}{6}+90856689n\right).$
(5)

The Chudnovsky brothers’ formula in (5) is closely related to the same Heegner
number $d=163$. Since this Heegner number gives the value of $1$ in (4), the
quadratic field $\mathbb{Q}\left(\sqrt{-163}\right)$ has class number $1$.

The connection between the $r=163$ case of (1) and the Chudnovsky brothers’
formula in (5) has not previously been investigated, and the conjectured
values for $J_{163}$ and $T_{163}$ have not previously been proved. The
connection between (5) and the Ramanujan-type series obtained by setting
$r=163$ in (1) is of a nontrivial nature. This is evidenced by the very
unwieldy algebraic coefficients involved in Bagis and Glasser’s conjecture, in
contrast to the rational series in (5).

In this article, we succeed in proving Bagis and Glasser’s conjectured
Ramanujan-type series, by proving closed forms for $\lambda^{\ast}(163)$ and
$\alpha(163)$, referring to Section 2 for details. As a byproduct of the
technique we introduce to prove closed forms for the coefficients involved in
the $r=163$ case of (1), we obtain an infinite family of generalizations of
the famous Chudnovsky–Chudnovsky formula in (5). This infinite family is given
explicitly in terms of the elliptic lambda and elliptic alpha functions.

The value of Ramanujan’s class invariant $G_{163}=2^{-1/4}x$, where
$x^{3}-6x^{2}+4x-2=0$, was given in Part V of Ramanujan’s Notebooks [2, p.
194]. However, this cannot be applied in any direct way to prove Bagis and
Glasser’s conjecture. This becomes apparent if we consider the family of
Ramanujan-type series of level $1$ given by the Borwein brothers [4, p. 365]
[5, p. 183] and reproduced below (see also [7]):

$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}\left(\frac{5}{6}\right)_{n}}{\left(n!\right)^{3}}f_{n}(N)\left(J_{N}^{-1/2}\right)^{2n+1},$
(6)

where

$\displaystyle\begin{split}f_{n}(N):=&\frac{1}{3\sqrt{3}}[\sqrt{N}\sqrt{1-G_{N}^{-24}}+2(\alpha(N)-\sqrt{N}k_{N}^{2})(4G_{N}^{24}-1)]\\\
&+n\sqrt{N}\frac{2}{3\sqrt{3}}[(8G_{N}^{24}+1)\sqrt{1-G_{N}^{-24}}].\end{split}$
(7)

The values for $\lambda^{\ast}(163)$ and $\alpha(163)$ have not been given or
used in the Chudnovsky brothers’ derivation of (5) [6] or in subsequent work.
Bagis and Glasser’s techniques [1] do not provide any way of proving any
evaluation for the required value $\alpha(163)$.

##  2 Bagis and Glasser’s conjecture

The formula in (1) given by Bagis and Glasser [1] can be shown to be
equivalent to Equation (5.5.18) from the seminal _Pi and the AGM_ text [5].
This equivalent formula is given in (6). We apply a closely related formula to
prove Bagis and Glasser’s conjecture concerning the Heegner number 163.

The _elliptic lambda function_ $\lambda^{\ast}(r)$ [5, p. 67, eq. (3.2.2)] is
such that $\lambda^{\ast}(r):=k_{r}$ for $0<k_{r}<1$, where
$\frac{\mathbf{K}^{\prime}}{\mathbf{K}}(k_{r})=\sqrt{r}$. The _elliptic alpha
function_ [5, p. 152, eq. (5.1.2)] is such that

$\alpha(r)=\frac{\pi}{4\mathbf{K}^{2}}-\sqrt{r}\left(\frac{\mathbf{E}}{\mathbf{K}}-1\right).$
(8)

The complete elliptic integrals of the first and second kinds are,
respectively, such that

$\mathbf{K}(k):=\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}\
\ \ \text{and}\ \ \
\mathbf{E}(k):=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\theta}\,d\theta.$ (9)

The argument $k$ in (9) is referred to as the _modulus_.

A _Ramanujan-type series_ is of the form

$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}\left(\frac{1}{s}\right)_{n}\left(1-\frac{1}{s}\right)_{n}}{\left(1\right)_{n}^{3}}z^{n}(a+bn),$
(10)

where $s\in\\{2,3,4,6\\}$ and where $z$, $a$, and $b$ are real and algebraic.
The level of (10) is $\ell=4\sin^{2}\frac{\pi}{s}$ and refers to the level of
the modular form parametrizing (10). Equation (5.5.18) from _Pi and the AGM_
[5] may be formulated as follows. If we set

$x=4\left(\left(\lambda^{\ast}(r)\right)^{2}-\left(\lambda^{\ast}(r)\right)^{4}\right),$
(11)

then the following Ramanujan-type series evaluation holds true:

$\displaystyle\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}\left(\frac{5}{6}\right)_{n}}{\left(1\right)_{n}^{3}}z^{n}(a+bn),$
(12)

where

$\displaystyle z$ $\displaystyle=\frac{27x^{2}}{(4-x)^{3}},$ (13)
$\displaystyle a$
$\displaystyle=\frac{2(4-x)\alpha(r)+\left(x-4+4\sqrt{1-x}\right)\sqrt{r}}{\sqrt{(4-x)^{3}}},$
(14) $\displaystyle b$
$\displaystyle=\frac{2(x+8)\sqrt{1-x}\sqrt{r}}{\sqrt{(4-x)^{3}}},$ (15)

provided that $|z|<1$. As noted by the Borwein brothers [4, p. 368], Ramanujan
gave two series for $\frac{1}{\pi}$ of level $1$ with a positive convergence
rate, and these two series are given by special cases of (12), subject to the
relations in (13)–(15). The formulation of (1) by the Borwein brothers in both
[4] and [5] only admits positive convergence rates, but it was stated in [4,
p. 368] that this formulation may be extended, via analytic continuation, to
obtain a corresponding family of Ramanujan-type series for $\ell=1$ with
negative convergence rates. An explicit evaluation for the coefficients and
the convergence rates for this family is given as follows, and has not
appeared in [4, 5] or in subsequent literature. Let $r>1$ and let $x$ be as in
(11). Then the formula in (12) holds for $z=-\frac{27x}{(1-4x)^{3}}$ and
$b=\frac{(1+8x)\sqrt{1-x}\sqrt{r}}{\sqrt{(1-4x)^{3}}}$ and

$a=\frac{2(1-4x)\alpha(r)+\left(4x-1+\sqrt{1-x}\right)\sqrt{r}}{2\sqrt{(1-4x)^{3}}}$
(16)

provided that $|z|<1$.

By inputting expressions as in

    Recognize[N[J163,1500],16,x]

and

    Recognize[N[T163,1500],16,x]

into Mathematica, for numerical estimates of $J_{163}$ and $T_{163}$ derived
from (2) and (3), this led Bagis and Glasser [1] to conjecture that the
following evaluations hold for $J_{163}$ and $T_{163}$:

$J_{163}=\frac{4\left(C_{1}-C_{2}\frac{1}{\sqrt[3]{-A_{1}+\sqrt{489}\text{B1}}}+30591288\sqrt[3]{-A_{1}+\sqrt{489}\text{B1}}\right)}{10792555251621895860488211571345343375},$

where

$\displaystyle
A_{1}=12737965652562547164590026038483234248161827096523072256574968383,$
$\displaystyle
B_{1}=229038073182066825378006485964950394558349727761749294205546402325349,$
$\displaystyle C_{1}=8808429913332498766352891,$ $\displaystyle
C_{2}=902206261147132595923169636910570558029813352485594880,$

and

$T_{163}=\frac{5\left(12948195754365757115+8\sqrt[3]{A_{2}-B_{2}\sqrt{489}}+8\sqrt[3]{A_{2}+B_{2}\sqrt{489}}\right)}{83470787671093501833},$

where

$\displaystyle A_{2}=3802386862487392962897493239274992371253057854289262,$
$\displaystyle B_{2}=3865464212119923579732688315287754932290919450.$

Numerical evidence suggests that

$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}\left(\frac{5}{6}\right)_{n}}{\left(n!\right)^{3}}\mathcal{J}^{n}\left(\frac{\sqrt{1-\mathcal{J}}\sqrt{163}(1-\mathcal{T})}{3}+2\sqrt{1-\mathcal{J}}\sqrt{163}n\right),$
(17)

where $\mathcal{J}$ and $\mathcal{T}$ respectively denote Bagis and Glasser’s
conjectured closed forms for $J_{163}$ and $T_{163}$ [1]. The purpose of the
current section is to prove Bagis and Glasser’s conjectured formula in (17).
As suggested by Bagis and Glasser [1], a main source of interest in this
Ramanujan-type series is due to how it provides about 32 digits per term.

######  Lemma 1.

The elliptic lambda function is such that

$\lambda^{\ast}(163)=\frac{\sqrt{\frac{1}{2}\left(-1-\frac{4270934400}{\sqrt[3]{1+557403\sqrt{489}}}+80040\sqrt[3]{1+557403\sqrt{489}}\right)}}{2\sqrt{-1-\sqrt{1+\frac{1}{4}\left(-1-\frac{4270934400}{\sqrt[3]{1+557403\sqrt{489}}}+80040\sqrt[3]{1+557403\sqrt{489}}\right)}}}.$

Writing
$x=4(\left(\lambda^{\ast}(163)\right)^{2}-\left(\lambda^{\ast}(163)\right)^{4})$,
the elliptic alpha function is such that

$\alpha(163)=\frac{13591409\sqrt{1-4x}}{426880\sqrt{10005}}+\frac{\sqrt{163}\sqrt{1-x}}{8x-2}+\frac{\sqrt{163}}{2}.$

###### Proof.

We normalize the Chudnovsky–Chudnovsky formula shown in (5) and proved in [6]
so that

$\displaystyle\begin{split}\frac{1}{\pi}=&\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}\left(\frac{5}{6}\right)_{n}}{\left(n!\right)^{3}}\left(-\frac{1}{53360}\right)^{3n}\times\\\
&\ \ \ \
\left(\frac{13591409}{426880\sqrt{10005}}+\frac{272570067}{213440\sqrt{10005}}n\right).\end{split}$
(18)

By the specialization of Bailey’s transformation [5, p. 181, eq. (5.5.9)]

$\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}^{3}}{\left(1\right)_{n}^{3}}x^{n}=\frac{1}{\sqrt{1-4x}}\sum_{n=0}^{\infty}\frac{\left(\frac{1}{6}\right)_{n}\left(\frac{1}{2}\right)_{n}\left(\frac{5}{6}\right)_{n}}{\left(n!\right)^{3}}\left(-\frac{27x}{(1-4x)^{3}}\right)^{n},$
(19)

we may obtain an equivalent version of the normalized Chudnovsky formula in
the following way. By applying the generating function evaluation (cf. [5, p.
180, Theorem 5.7(a), eq. (i)])

$\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}^{3}}{\left(1\right)_{n}^{3}}x^{n}=\frac{4\mathbf{K}^{2}\left(\sqrt{\frac{1-\sqrt{1-x}}{2}}\right)}{\pi^{2}}$
(20)

and the term-by-term derivatives of the series in (20) to rewrite (19) and the
term-by-term derivatives of (19), this gives us that the normalized Chudnovsky
series is equal to the right-hand side of (12) for the $a$-value in (16) and
for $z=-\frac{27x}{(1-4x)^{3}}$ and
$b=\frac{(1+8x)\sqrt{1-x}\sqrt{r}}{\sqrt{(1-4x)^{3}}}$, and for the unique
value of $r$ such that

$\left(-\frac{1}{53360}\right)^{3}=-\frac{108\left(\left(\lambda^{\ast}(r)\right)^{2}-\left(\lambda^{\ast}(r)\right)^{4}\right)}{\left(1-16\left(\left(\lambda^{\ast}(r)\right)^{2}-\left(\lambda^{\ast}(r)\right)^{4}\right)\right)^{3}},$
(21)

referring to [7] for details. We then take the unique real value $x$ such that
$-\frac{1}{53360^{3}}=-\frac{27x}{(1-4x)^{3}}$. By then rewriting the
coefficient of $n$ in the polynomial factor in (18) as
$\frac{(1+8x)\sqrt{1-x}\sqrt{r}}{\sqrt{(1-4x)^{3}}}$, this gives us the
desired value $r=163$. So, we obtain the desired closed form for
$\lambda^{\ast}(r)$ from (21). By then rewriting the constant term in the
polynomial factor in (18) as in (16), we obtain the desired closed form for
$\alpha(163)$. ∎

######  Theorem 1.

Bagis and Glasser’s Ramanujan-type series evaluation in (17) holds true, for
the conjectured values of $J_{163}$ and $T_{163}$.

###### Proof.

This follows in a direct way by setting the values for $\lambda^{\ast}(163)$
and $\alpha(163)$ given in Lemma 1 into (11), for the values of $z$, $a$, and
$b$ indicated in (13)–(15). ∎

The above Theorem may be regarded as providing a series acceleration, since
Bagis and Glasser’s formula gives about 32 digits per term, compared to the
Chudnovsky brothers’ algorithm giving about 14 digits per term.

##  3 Conclusion

Our infinite family of Ramanujan-type series involving (16) has led us to
discover a new Ramanujan-type series with a cubic convergence rate that was
not provided in [3]. This new series corresponds to the $r=243$ case of (16),
noting that $243=3^{5}$, so that this non-squarefree case does not agree with
the class number three cases covered in [3], and noting that we can use known
recursions for Ramanujan’s class invariants [5, p. 145] and for the elliptic
alpha function [5, p. 160] to prove the following result, given by a
Ramanujan-type series as in (12) for the following values of $z$, $a$, and
$b$.

    z = Root[6561 + 7046099782711303104000*#1 - 8241190499340288000000*#1^2 +
    4245232549888000000000*#1^3 & , 1, 0]
    a = -Root[-529386137841972399112323 + 5193712075415612200704000*#1^2 +
    279610488520114176000000*#1^4 + 271694883192832000000000*#1^6 & , 1, 0]
    b = -Root[-43766201502243114733034506827 +
    194827240993239940764096000*#1^2 - 1092165237528662016000000*#1^4 +
    4245232549888000000000*#1^6 & , 1, 0]

This gives about 18 digits per term. We encourage the systematic exploration
of Ramanujan-type series derived by analogy with our proof of Theorem 1, to
yield faster versions of the Chudnovsky algorithm.

### Acknowledgements

The author is grateful to acknowledge support from a Killam Postdoctoral
Fellowship. The author is sincerely thankful to Shane Chern for many useful
discussions. The Reviewer feedback that has been provided has been very
helpful and has led to considerable improvements of this article.

## References

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  * [4] Borwein, J. M., Borwein, P. B. (1988). More Ramanujan-type series for $1/\pi$. In: _Ramanujan revisited (Urbana-Champaign, Ill., 1987)_. Boston, MA: Academic Press.
  * [5] Borwein, J. M., Borwein, P. B. (1987). _Pi and the AGM_ , Canadian Mathematical Society Series of Monographs and Advanced Texts. New York: John Wiley & Sons, Inc.
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  * [7] Milla, L. (2018). A detailed proof of the Chudnovsky formula with means of basic complex analysis–Ein ausführlicher Beweis der Chudnovsky-Formel mit elementarer Funktionentheorie. arXiv:1809.00533.

Department of Mathematics and Statistics

Dalhousie University

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