# ZZ Piscis Austrinus (ZZ PsA): A bright red nova progenitor and the
instability mass ratio of contact binary stars

Surjit S. Wadhwa,1 Ain De Horta,1 Miroslav D. Filipović,1 N. F. H. Tothill,1
Bojan Arbutina,2 Jelena Petrović3 and Gojko Djurašević3
1School of Science, Western Sydney University, Locked Bag 1797, Penrith, NSW
2751, Australia
2Department of Astronomy, Faculty of Mathematics, University of Belgrade,
Studentski trg 16, 11000 Belgrade, Serbia
3Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

ZZ PsA is a neglected bright southern contact binary system with maximum V
magnitude of 9.26. We present the first multi-band photometric analysis and
find the system to be in deep contact (>95%) with an extremely low mass ratio
of 0.078. The primary has a mass of 1.213 M☉ in keeping with its reported
spectral class of F6. In order to determine if ZZ PsA is a merger candidate we
outline the current status regarding the instability mass ratio and develop
new relationship linking the mass of the primary to the instability mass ratio
of the system and the degree of contact. We find that ZZ PsA along with two
other examples from the literature to be merger candidates while an additional
three require further observations to be confirmed as potential merger
candidates.

###### keywords:

binaries: eclipsing – stars: mass-loss – techniques: photometric

††pubyear: 2020††pagerange: ZZ Piscis Austrinus (ZZ PsA): A bright red nova
progenitor and the instability mass ratio of contact binary stars–ZZ Piscis
Austrinus (ZZ PsA): A bright red nova progenitor and the instability mass
ratio of contact binary stars

##  1 Introduction

Luminous red novae are rare transient events thought to be the result of the
merger of contact binary system components into a single star. So far, there
is only one confirmed observed event, that of V1309 Sco — Nova Sco 2008
(Tylenda et al., 2011). Unfortunately, that system was recognized as a contact
binary only after the merger event, so targeted observations to fully
elucidate the properties of the components and events leading up to the merger
itself could not be performed. Estimates suggest that 1 in 500 stars in the
galactic disk are contact binaries (Rucinski, 2006). Sky survey projects such
as All Sky Automated Survey (ASAS, Pojmanski, 1997), the Zwicky Transient
Facility (ZTF, Bellm et al., 2019), Asteroid Terrestrial-impact Last Alert
System (ATLAS, Heinze et al., 2018) and ASAS—SN (Shappee et al., 2014;
Jayasinghe et al., 2018), among many others, are continually adding to our
lists of contact binary systems. Kochanek et al. (2014) estimated the Galactic
rate of stellar mergers such as V1309 Sco to be once every 10 years.
Identification of candidates that are close to merging is critical to our
understanding of these systems.

Most contact binary systems are of low mass, with stars of spectral class F to
K (Stępień & Gazeas, 2012a). Theoretical modelling suggests that several
factors influence the rate at which any particular system moves towards
merger. Among the most critical appear to be the relationship between orbital
and spin angular momentum (Rasio, 1995), the degree of contact (Rasio &
Shapiro, 1995) and angular momentum loss (Stępień & Gazeas, 2012a).

The bright contact binary system ZZ Piscis Austrinus (ZZ PsA)
($\alpha_{2000.0}=21\ 50\ 35.17$, $\delta_{2000.0}=-27\ 48\ 35.48$) was
recognized as a variable star in 1967 (Strohmeier, 1967), “rediscovered” by
Demartino et al. (1996), and designated NSV 13890. Despite its brightness, the
system has remained largely unobserved apart from sky surveys. ASAS confirmed
the contact binary light curve with a period of 0.37388 days. A single time of
minimum was reported by Otero (2004) based on the only section of the ASAS
data of sufficient cadence to allow a rough estimation. Preliminary analysis
of the ASAS photometry (Wadhwa, 2006) revealed a very low mass ratio system
with a high degree of contact.

We present and analyse the first multi-band photometry of ZZ PsA and determine
its geometric and absolute parameters (Section 2). In Section 3 we explore the
problem of determining the orbital/spin angular momentum instability of
contact binary systems and propose some new relationships between parameters
determined by the light curve solution which shed new light on the subject of
the instability mass ratio of contact binary systems. We apply the
relationships to ZZ PsA and several other systems from the literature to
define a very small number of systems that we suspect are potential red nova
candidates. Finally, in Section 4 we summarise our results.

##  2 Observations, Light Curve and Analysis

ZZ PsA was observed over 5 nights during 2019 September–October with the
Western Sydney University 0.6 m telescope using standard Johnson $BVR$
filters. Images in all three bands were acquired during each observing
session. Aperture photometry was performed on calibrated images using the
American Association of Variable Star Observers (AAVSO) VPHOT engine. TYC
6959–1122-1 and TYC 6959–885–1 were used as comparison stars and 2MASS
21503231–2752141 was used as the check star.

###  2.1 Period, Time of Minima and Light Curve Characteristics

Our observations yielded a single time of primary minimum, determined to be
HJD 2458776.9669 (0.0002) using the method of Kwee & van Woerden (1956). A new
period estimate of 0.37405 (0.00003) days was also determined using the
PSearch V1.5 (Nelson 2006) software package available through the AAVSO. No
meaningful period change (O-C) analysis was possible given the lack of
previous observations as noted above.

The normalized and fitted light curves for all bands are shown in Figures 1, 2
and 3. The light curves were folded according to the elements stated above.
The maximum magnitudes at phase 0.25 and 0.75 in B were estimated as $9.83\pm
0.01$ and $9.84\pm 0.01$; in V $9.26\pm 0.01$ and $9.26\pm 0.01$ and in R
$8.71\pm 0.01$ and $8.72\pm 0.01$. Surprisingly, given the low mass ratio (see
below) no significant variation in the maxima (O’Connell effect, Milone, 1968)
was evident in any of the bands observed. The eclipses are of essentially
equal depth (variation < 0.02 mag) and last approximately 37 minutes. The $V$
band amplitude of the light curve is 0.29 mag which agrees with the coarse
ASAS survey amplitude of 0.27 mag. Rucinski (2001) developed a theoretical
relationship between maximum eclipse amplitude, fill-out factor and the mass
ratio of contact binary systems. Using values from Table 2 of that study we
deduce that the system is likely to have a mass ratio below 0.1.

Figure 1: $B$ band light curve for ZZ PsA: Observed ($\circ$) and Simultaneous
Fitted ($\bullet$) fluxes are shown. Figure 2: $V$ band light curve for ZZ
PsA: Observed ($\circ$) and Simultaneous Fitted ($\bullet$) fluxes are shown.
Figure 3: $R$ band light curve for ZZ PsA: Observed ($\circ$) and Simultaneous
Fitted ($\bullet$) fluxes are shown.

###  2.2 Photometric Analysis of the Light Curve

The system shows complete eclipses and therefore it is suitable for
photometric analysis with high confidence that the correct mass ratio can be
determined (Terrell & Wilson, 2005). The light curve was analyzed using the
2009 version of the Wilson-Devinney (WD) code (Wilson & Devinney, 1971). As
there is no appreciable O’Connell effect, only unspotted modelling was carried
out. The SIMBAD database records the spectrum of the system as F6V (Wenger et
al., 2000). Based on calibrations from Cox (2000) the effective temperature of
the primary component was fixed at 6514 K. The near equal depth of the
eclipses suggests similar temperatures of the components and a common
convective envelope, as such we fixed the gravity darkening coefficients
$g_{1}=g_{2}=0.32$ (Lucy, 1967), bolometric albedos $A_{1}=A_{2}=0.5$, and
applied simple reflection treatment (Rucinski, 1969). Limb darkening co-
coefficients for each passband were obtained from van Hamme (1993) and
logarithmic law for the coefficients was implemented as per Nelson & Robb
(2015).

No spectroscopic mass ratio is available, only the visual classification (F6V)
by Houk (1982) appears in the literature. The complete eclipses however mean
that the photometric mass ratio can be more reliably discovered and is likely
to be close to the actual mass ratio (Terrell & Wilson, 2005). Rucinski (1993)
showed that the brightness variations of the contact binary stars are
essentially completely dominated by geometrical elements, with reflection and
gravity/limb darkening playing a minimal role. He showed that the light curve
depends practically on only three parameters; the mass ratio $(q)$,
inclination $(i)$ and the degree of contact or fill-out $(f)$. Unfortunately,
there is a high degree of correlation between these parameters. The
correlation is most marked between mass ratio and inclination, and between
mass ratio and degree of contact. Reliable photometric solutions cannot be
obtained when either of these two combinations are treated as free. Russo &
Sollazzo (1982) described the “$q$” search or “grid” method whereby solutions
are obtained for various fixed values of $q$, with the potential (fill-out)
and inclination free and then compared to the observed light curve to find the
best solution. Wadhwa (2006) obtained a preliminary mass ratio of 0.08 based
on the analysis of ASAS data.

Using the WDWint package V5.6 (Nelson 2009) we analyzed the light curves of
all three passbands simultaneously over the mass ratio range 0.05 to 0.13 at
intervals of 0.01. Preliminary solution was found at $q=0.08$. To further
refine the mass ratio a finer search at intervals of 0.001 for the mass ratio
was made over the interval $0.075<q<0.085$. Throughout the analysis the
inclination, potential (fill-out), temperature of the secondary and relative
value of $L_{1}$ were adjusted for each fixed value of the mass ratio.
Iterations were carried out until the suggested variation was less than the
reported standard deviation. The normalised residual error (as reported by the
analysis software) between the observed and modelled light curve was plotted
against the mass ratio. The mass ratio with the smallest residual error, i.e.
the best fit between observed and modelled light curve, was selected as the
true mass ratio of the system. The “$q$” search grid is shown in Figure 4 and
the critical parameters from the WD solution are presented in Table 1. A 3D
representation of the system is shown in Figure 5.

###  2.3 Review of the Light Curve Solution and Determination of Absolute
Parameters

Our light curve solution, with an estimated fill-out of >95%, represents one
of the highest recorded degrees of contact for a binary system and suggests
probable near-merger. In addition to good physical contact, there is solid
thermal contact between the systems. The secondary, although smaller than the
primary, is hotter by less than 200 K and as such the system is similar to
other low mass ratio systems with hotter secondaries such as V857 Her (Qian &
Yang, 2005), ASAS J083241+2332.4 (Sriram et al., 2016), V1187 Her (Wadhwa,
2017), LO And (Nelson & Robb, 2015) and TY Pup (Sarotsakulchai et al., 2018).

Table 1: Key parameters from the WD solution for ZZ PsA. $T_{1}$ (K) (Fixed) | 6514
---|---
$T_{2}$ (K) | $6703\pm 11$
$q({M}_{2}$/${M}_{1})$ | $0.078\pm 0.002$
Inclination (∘) | $75.25\pm 0.34$
$\Omega_{\text{1}}=\Omega_{\text{2}}$ | $1.839\pm 0.001$
Fill-out% | $97\pm 2$

Apart from the mass ratio, determination of the absolute magnitude
$\text{M}_{\text{v}}$ of the primary is essential to accurately determine the
absolute parameters of the system. Based on previously published colour-period
and colour-period-$\text{M}_{\text{v}}$ relationships, Gazeas & Stȩpień (2008)
derived the relationship $\text{M}_{\text{v}}=-8.4\log P+0.31$ (where $P$ is
the period in days), giving $\text{M}_{\text{v}}=3.90$ for the ZZ PsA primary.
However, the distance to ZZ PsA is known from parallax to be $135.48\pm 0.99$
pc (Gaia Collaboration, 2018). During the eclipse at phase 0.5 (lasting about
37 minutes), only the light from the primary contributes to the light curve at
mid-eclipse. The apparent visual magnitude at mid-eclipse was estimated as
$9.53\pm 0.01$, giving a distance-based absolute magnitude of $3.97\pm 0.02$
for the primary, after applying the extinction correction of $E(B-V)=0.034$
(Schlegel et al., 1998), as described by Wadhwa (2019). The distance-based
results are used in all subsequent calculations.

From the primary $\text{M}_{\text{v}}$, we can find the primary luminosity. At
maximum brightness (phase 0.25 or 0.75), the system has $V=9.26\pm 0.01$ which
yields system absolute magnitude M${}_{V}=3.70\pm 0.02$, and hence the system
luminosity, secondary luminosity and secondary absolute magnitude. Rovithis-
Livaniou et al. (2001) found that the primary components of contact binary
systems follow in general the zero-age main sequence (ZAMS) mass-luminosity
relation. Using this relation (Demircan & Kahraman, 1991) for ZAMS stars with
mass $>$ 0.7$M_{\sun}$, we can derive the mass of the primary; from the mass
ratio we then estimate the mass of the secondary. The estimated $M_{\text{1}}$
of 1.213$M_{\sun}$ is consistent with the F6 spectral classification. The
separation of the components $A$ is then estimated from Kepler’s third law,
expressed as
$(A/R_{\sun})^{3}=74.5(P/\mathrm{days})^{2}((M_{1}/M_{\sun})+(M_{2}/M_{\sun}))$.

Figure 4: Coarse and fine (inset) mass ratio (q) search grid for ZZ PsA.
Figure 5: 3D Representation of ZZ PsA at phase 0.35. The broad neck between
the components is an indicator of the high fill-out.

As the two components are enclosed in a common envelope, their relative radii
are not independent, but dependent on the Roche geometry and mass ratio. The
WD program solution provides the fractional radii of the primary and secondary
in three orientations. From the geometric mean we estimate the fractional
radius of the primary $r_{1}=0.5958$, and that of the secondary
$r_{2}=0.2341$. From ${R}_{1}/{R}_{\sun}=r_{1}A$ and
${R}_{2}/{R}_{\sun}=r_{2}A$ (Awadalla & Hanna, 2005), we derive the primary
and secondary radii. The relative radius and luminosity of the secondary are
considerably higher than those predicted for stars either at ZAMS or terminal-
age main sequence (TAMS) stages of evolution. This is a common finding among
contact binary systems (Stępień & Gazeas, 2012b). The absolute parameters for
ZZ PsA are given in Table 2.

Table 2: Absolute parameters for ZZ PsA: ${M}_{1}/{M}_{\sun}$ | $1.213\pm 0.008$
---|---
${M}_{2}/{M}_{\sun}$ | $0.095\pm 0.001$
${L}_{1}/{L}_{\sun}$ | $2.20\pm 0.04$
${L}_{2}/{L}_{\sun}$ | $0.63\pm 0.04$
${R}_{1}/{R}_{\sun}$ | $1.422\pm 0.004$
${R}_{2}/{R}_{\sun}$ | $0.559\pm 0.004$
$\text{A}~{}({R}_{\sun})$ | $2.387\pm 0.008$
$\text{M}_{\text{V}1}$ | $3.97\pm 0.02$
$\text{M}_{\text{V}2}$ | $5.34\pm 0.02$

##  3 Instability mass ratio of Contact Binary Systems

Contact binary systems are thought to merge into a single, rapidly spinning
object when the total angular momentum ($J_{T}$) of the system is at a
critical ($dJ_{T}=0$) value such that tidal instability ensues (Darwin, 1879).
$J_{T}$ is the sum of the orbital angular momentum ($J_{O}$) of the system and
the combined spin angular momentum ($J_{S}$) of the components. The
instability occurs $\approx$ when $J_{O}\leq 3J_{S}$ (Hut, 1980). For contact
binaries, this equates to a critical mass ratio of 0.071–0.077 (Rasio, 1995),
depending on the dimensionless gyration radius $k$ of the component stars and
the degree of contact. Over the years, $k$ appears to have been chosen
empirically to match the observations. Rasio (1995) chose $k=0.245$, such that
the lowest mass ratio system known at the time, AW UMa with $q=0.075$
(Rucinski, 1992) would be stable, and did not take into account the spin
momentum of the secondary when arriving at the global critical mass ratio. As
more systems with mass ratios below the estimated critical value were
discovered, researchers tried to modify the calculations either to include the
spin momentum of the secondary (Li & Zhang, 2006) or to use different values
of $k$ (Jiang et al., 2010).

With differing values of $k$ being employed, there is no established
relationship to confidently say whether a contact binary system is unstable
regardless of its measured mass ratio. Rucinski (1988) and Claret (2019)
calculated $k$ for stars of different masses and radii (non-rotating), showing
that there is considerable variation in $k$ with changing stellar mass. In
both cases, stars with mass $>0.8M_{\sun}$ were modelled and no rotational or
tidal effects were included. Landin et al. (2009) determined gyration radii
for ZAMS stars, including effects of rotation and tidal distortion due to
binary interaction. We use these results to derive more robust and complete
criteria for the critical instability mass ratio of contact binary systems
with different masses of the primary component.

###  3.1 The Instability Mass Ratio ($q_{inst}$)

Our derivation of the instability mass ratio is similar to that of Arbutina
(2007), expanded to more clearly show the contribution of the secondary
component. We adopt the stability criterion $dJ_{T}=0$, assume that the
gyration radii of the primary and secondary are not equal, and further assume
that, as both stars are within a common envelope, their radii are not
independent, but the radius of the secondary is dependent on the separation,
mass ratio and the radius of the primary.

The orbital angular momentum of a binary can be written as

$J_{\mathrm{O}}=\mu A^{2}\omega=\frac{q\ \sqrt{GM^{3}A}}{(1+q)^{2}},$ (1)

where $M_{1}$ and $M_{2}$ are masses of the primary and secondary component,
respectively, total mass $M=M_{1}+M_{2}$, $\mu=M_{1}M_{2}/M$ and
$q=M_{2}/M_{1}<1$. $\omega$ is the orbital angular velocity. With
synchronization assumed, the spin angular momentum of a binary

$J_{\mathrm{S}}=k_{1}^{2}M_{1}R_{1}^{2}\omega+k_{2}^{2}M_{2}R_{2}^{2}\omega,$
(2)

where $R_{1}$ and $R_{2}$ are taken to be the volume radii. The fill-out
factor is defined as

$f=\frac{\Omega-\Omega_{\mathrm{IL}}}{\Omega_{\mathrm{OL}}-\Omega_{\mathrm{IL}}}\approx\frac{R-R_{\mathrm{IL}}}{R_{\mathrm{OL}}-R_{\mathrm{IL}}},$
(3)

where $\Omega$ is the dimensionless surface equipotential. We have adopted
linear dependence of $\Omega$ on the volume radius (Arbutina, 2007). Volume
radii for the inner Roche lobe are (Eggleton, 1983):

$\frac{R_{\mathrm{IL}i}}{A}=\Bigg{\\{}\begin{array}[]{ll}{\frac{0.49q^{-2/3}}{0.6q^{-2/3}+\ln(1+q^{-1/3})},}&i=1\\\
\\\ {\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})},}&i=2,\end{array}$ (4)

and for the outer Roche lobe (Yakut & Eggleton, 2005):

$\frac{R_{\mathrm{OL}i}}{A}=\Bigg{\\{}\begin{array}[]{ll}{\frac{0.49q^{-2/3}+0.15}{0.6q^{-2/3}+\ln(1+q^{-1/3})},}&i=1\\\
\\\
{\frac{0.49q^{2/3}+0.27q-0.12q^{4/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}.}&i=2,\end{array}$
(5)

where $i=1$ and $2$ refer to the primary and secondary star.

Since the surfaces of components in the contact system are at the same
potential (the same $f$), from Eq. 3:

$R_{2}=P(q)A+Q(q)R_{1},$ (6)

where

$Q(q)=\frac{R_{\mathrm{OL}2}-R_{\mathrm{IL}2}}{R_{\mathrm{OL}1}-R_{\mathrm{IL}1}}\\\
=\frac{(0.27q-0.12q^{4/3})({0.6q^{-2/3}+\ln(1+q^{-1/3})})}{0.15(0.6q^{2/3}+\ln(1+q^{1/3}))},$
(7)

and

$P(q)=\frac{R_{\mathrm{IL}2}}{A}-Q(q)\frac{R_{\mathrm{IL}1}}{A}=\frac{0.49q^{2/3}-3.26667q^{-2/3}(0.27q-0.12q^{4/3})}{0.6q^{2/3}+\ln(1+q^{1/3})}.$
(8)

The total angular momentum of a contact binary system
$J_{\mathrm{T}}=J_{\mathrm{O}}+J_{\mathrm{S}}$ can then be expressed as

$\begin{array}[]{@{\extracolsep{-3mm}} lll @{}}{J_{\mathrm{tot}}}&=&\frac{q\
\sqrt{GM^{3}R_{1}}}{(1+q)^{2}}\Big{(}\frac{A}{R_{1}}\Big{)}^{1/2}\bigg{[}1+\\\
&+&\frac{k_{1}^{2}(1+q)}{q}\Big{(}(1+q\frac{k_{2}^{2}}{k_{1}^{2}}{Q}^{2})\Big{(}\frac{R_{1}}{A}\Big{)}^{2}+2q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q}\Big{(}\frac{R_{1}}{A}\Big{)}+q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}\Big{)}\bigg{]},\end{array}$
(9)

From the condition $\frac{\mathrm{d}J_{\mathrm{T}}}{\mathrm{d}A}=0$, the
critical separation ($A_{inst}$) is given by

$\frac{A_{\mathrm{\scriptscriptstyle
inst}}}{R_{1}}=\frac{q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q}+\sqrt{(q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q})^{2}+3(1+q\frac{k_{2}^{2}}{k_{1}^{2}}{Q}^{2})(q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}+\frac{q}{(1+q)k_{1}^{2}})}}{q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}+\frac{q}{(1+q)k_{1}^{2}}}.$
(10)

If the angular momentum of the secondary has been neglected ($k_{2}=0$), the
above condition for critical separation reduces to (Rasio, 1995)

$\frac{A_{\mathrm{\scriptscriptstyle
inst}}}{R_{1}}=k_{1}\sqrt{\frac{3(1+q)}{q}}.$ (11)

By combining Eq. 3 for the primary

$\frac{R_{1}}{A}=f\Big{(}\frac{R_{\mathrm{OL}1}}{A}-\frac{R_{\mathrm{IL}1}}{A}\Big{)}+\frac{R_{\mathrm{IL}1}}{A}=\frac{0.49q^{-2/3}+0.15f}{0.6q^{-2/3}+\ln(1+q^{-1/3})}$
(12)

and Eq. 10, for a fixed fill-out factor ($0\leq f\leq 1$) one can also find
the instability mass ratio given in the implicit form by an algebraic equation
that needs to be solved numerically for $q\equiv q_{\mathrm{\scriptscriptstyle
inst}}$:

$\begin{split}\begin{array}[]{@{\extracolsep{-3mm}} lll
@{}}\frac{q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q}+\sqrt{(q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q})^{2}+3(1+q\frac{k_{2}^{2}}{k_{1}^{2}}{Q}^{2})(q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}+\frac{q}{(1+q)k_{1}^{2}})}}{q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}+\frac{q}{(1+q)k_{1}^{2}}}&-\\\
\\\
\frac{0.6q^{-2/3}+\ln(1+q^{-1/3})}{0.49q^{-2/3}+0.15f}=0.\end{array}\end{split}$
(13)

Given the low mass ratio of contact binary systems approaching merger, the
secondary is a very low mass star, for which the totally convective $n=1.5$
polytrope would be appropriate, with $k^{2}\approx 0.205$. The value is almost
identical to the value obtained for all ZAMS stars below 0.4 M☉ modelled by
Landin et al. (2009). Our relationships are derived by adopting this fixed
value of $k$ for the secondary. We use the values of $k$ for 0.5 M☉ to 1.6 M☉
from Landin et al. (2009) to derive the instability mass ratio ($q_{inst}$),
for a range of fill-out fractions, of contact binary systems with primary
component of spectral classes from M0 to F0. The results are summarised in
Table 3. The fill-out fraction $f$ has a significant effect on $q_{inst}$ for
systems with primary components below one solar mass. On the other hand, there
is minimal difference in $q_{inst}$ between $f=1$ and $f=0$ for systems with
primaries in the upper F spectral class. Figure 6 shows the relationship
between primary mass and $q_{inst}$ at $f=1$ and at $f=0$. Both can be fitted
with quadratic relationships (correlation coefficients >0.99):

$q_{inst}=0.1269M_{1}^{2}-0.4496M_{1}+0.4403(f=1)$ (14)

and

$q_{inst}=0.0772M_{1}^{2}-0.3003M_{1}+0.3237(f=0).$ (15)

Table 3: Instability mass ratio for systems with primary stars of $0.5M_{\sun}<M<1.6_{\sun}$ and various fill-out fractions. $q_{inst}$ as $M_{2}/M_{1}$ Mass | $q_{inst}$ | $q_{inst}$ | $q_{inst}$ | $q_{inst}$ | $q_{inst}$ | $q_{inst}$
---|---|---|---|---|---|---
($M_{\sun}$) | $f1.0$ | $f0.9$ | $f0.75$ | $f0.5$ | $f0.25$ | $f0$
0.5 | 0.253 | 0.246 | 0.236 | 0.220 | 0.207 | 0.196
0.6 | 0.213 | 0.208 | 0.200 | 0.189 | 0.178 | 0.169
0.7 | 0.182 | 0.178 | 0.172 | 0.163 | 0.155 | 0.148
0.8 | 0.160 | 0.156 | 0.152 | 0.145 | 0.138 | 0.132
0.9 | 0.136 | 0.134 | 0.130 | 0.125 | 0.120 | 0.115
1.0 | 0.126 | 0.124 | 0.121 | 0.116 | 0.111 | 0.107
1.2 | 0.087 | 0.086 | 0.084 | 0.081 | 0.079 | 0.076
1.4 | 0.054 | 0.053 | 0.052 | 0.051 | 0.050 | 0.049
1.6 | 0.047 | 0.046 | 0.046 | 0.045 | 0.044 | 0.043
Figure 6: Instability mass ratios for 0.5 M☉ to 1.6 M☉ stars at $f=1$ and
$f=0$

Table 3 shows a linear relationship between $f$ and $q_{inst}$ for stars of
different mass. One can therefore easily obtain the instability mass ratio for
any given fill-out, after obtaining the values for $f=1$ and $f=0$ using the
above quadratic relationships.

###  3.2 Instability Parameters of ZZ PsA and Other Merger Candidates

Using Eqs. 14 and 15, we estimate the instability mass ratio of ZZ PsA to be
0.0813. The photometric mass ratio is marginally below this level, suggesting
that the system has entered an unstable period. From Eq 10 the theoretical
instability separation can be expressed as a function of the mass ratio,
primary radius and the gyration radii. Using the photometric mass ratio for ZZ
PsA and Eq 12, we determine the theoretical fractional radius of the primary
as 0.630 which at the current separation would yield a radius for the primary
as 1.50 $R_{\sun}$. Using this estimate of the radius of the primary and
0.2432 as the value of $k$ from Landin et al. (2009) for a 1.2 $M_{\sun}$
star, from equation 10 we derive the theoretical instability separation as
2.487 $R_{\sun}$. Combining the theoretical instability separation with our
component mass estimates, we would expect that the system reach its
instability separation at a period of 0.397 days. ZZ PsA is well within the
instability parameters on a number of fronts and represents a very bright
potential red nova progenitor worthy of further study and regular monitoring.

Since the observed merger event of V1309 Sco, there has been considerable
interest in the analysis of contact binary light curves. We examined the list
of 46 deep low mass ratio contact systems from Yang & Qian (2015) and searched
the literature for more recent examples of potential merger candidates. We
used the reported values of the light curve solutions and mass of the primary
to exclude all systems where the reported value of $q$ was more than 5% above
$q_{inst}$ as calculated using the relationships above. For the remaining
examples the instability separation and estimated period at onset of
instability were compared to the current reported values. From an initial
sample of more than 60 systems we were left with two other potential merger
candidates and three others that were difficult to classify. We discuss these
briefly.

####  3.2.1 V857 Her

V857 Her was observed by Qian et al. (2005) and found to be a W-Type system
with mass ratio 0.0653, a fill-out of 83.8%, and the secondary hotter than the
primary (very similar to ZZ PsA). Yang & Qian (2015) reported the mass of the
primary as $1.3M_{\sun}$. The instability mass ratio for such a system would
be 0.069. Based on the reported period of 0.3822 days we estimate the current
separation as 2.473$R_{\sun}$. The fractional radius of the primary is
estimated as 0.6369 and from the current separation the radius of the primary
is estimated as 1.575$R_{\sun}$. Although Landin et al. (2009) did not
specifically report the gyration radius for a 1.3$M_{\sun}$ star, review of
the reported values suggest a linear decline from 0.6$M_{\sun}$ to
1.4$M_{\sun}$, so we estimate the gyration radius of the primary of V857 Her
as 0.216. Using Eq 10, we determine the separation distance at instability to
be 2.476$R_{\sun}$ and the period at onset of instability to be 0.3829 days.
Just like ZZ PsA V857 Her appears to be unstable, and a potential red nova
progenitor.

####  3.2.2 V1187 Her

Preliminary analysis of amateur red band photometry by Wadhwa (2017) suggested
a W-Type star with mass ratio 0.059 and only marginal fill-out of 20%. Multi-
band photometry analysis (Caton et al., 2019), however, yielded a system with
an extremely low mass ratio of 0.044 and fill-out exceeding 80%. Although no
component masses were givenn, the spectrum is reported to be F8.5,
corresponding to a primary mass of $1.1\,M_{\sun}$ (Cox, 2000). Hence we
derive $q_{inst}=0.097$ and, from other parameters in the light curve
solution, the separation $A=2.02\,R_{\sun}$. The system is so far below the
estimated critical mass ratio that it should have already merged. Photometric
mass ratios are not as reliable as spectroscopic observations particularly in
systems with low inclination, as stressed by Rucinski & Pribulla (2008) in
their analysis of GSC 1387:0475 and confirmed by Yang et al. (2010) who
derived a significantly different photometric solution for the same system.
V-1187 Her has a very low inclination (66∘). This system is very likely to be
unstable, but further spectroscopic studies are probably required to fully
elucidate the parameters.

####  3.2.3 EM Psc, FG Hya, and V870 Ara

EM Psc was observed by Yang et al. (2005) who found it be a W-Type contact
binary with a mass ratio of 0.1 and fill-out $>$90%. Qian et al. (2008)
observed the system and found it to be an A-Type with a mass ratio of 0.149
and fill-out $>$90%. Neither study estimated the mass of the primary. Yang &
Qian (2015), in their statistical analysis of 46 low mass ratio systems,
indicate the mass of the primary as 0.95$M_{\sun}$. At the reported mass of
the primary the instability mass ratio is estimated as 0.126. If we take the
mass ratio as 0.149 than the system would not be considered unstable. If,
however we were to adopt the mass ratio of 0.1 from Yang et al. (2005) than
the system would be well below the critical mass ratio and a potential merger
candidate. The system clearly needs further study.

FG Hya was observed by Qian & Yang (2005) who found mass ratio 0.11 with
$>$85% contact. The only available value for the mass of the primary
$1.41M_{\sun}$ is from Lu & Rucinski (1999), who stressed that the value was
likely to be erroneous. If this primary mass is adopted, the estimated
critical mass ratio is 0.059, and the system would not be classified as
unstable. The spectral classification of the system however is estimated as G0
(Qian & Yang, 2005), and if we assume that the primary stars of contact
binaries follow the ZAMS, this would yield a primary mass of $1.05M{\sun}$ and
a critical mass ratio 0f 0.106, placing the system on the border of
instability.

Lastly, a combined photometric and spectroscopic solution for V870 Ara (Szalai
et al., 2007) gives mass ratio 0.082 and a primary mass of $1.50M_{\sun}$. On
this basis, the system is stable. The spectral class, however, was determined
to be F8 — far later than the mass of the primary. The system would be
unstable if the mass of the primary based on spectral class was adopted.

####  3.2.4 Some Recent Reports

Since the early drafts of this paper, a recent study summarising light curve
solutions from survey photometry of the northern Catalina Sky Survey (Sun et
al., 2020) has been published. Physical parameters of over 2300 late type
contact binary systems were derived from the analysis of the survey
photometry. Our preliminary analysis suggests that $\sim 25$ candidates may
fit the mass ratio criteria for instability. As this analysis was carried out
on survey photometry, the identification of such systems opens the window for
dedicated multi-band photometry of targeted systems that may prove to be
merger candidates.

##  4 Summary and Conclusions

We present the first multi-band photometric analysis for the bright southern
contact binary ZZ PsA. We find that the system has an extremely low mass ratio
of 0.078 and high degree of contact, in excess of 95%. With the distance well-
established by parallax, we were able to accurately determine the absolute
parameters of the components, and of the system as a whole. The primary
exhibits properties similar to other ZAMS stars of the same spectral type,
while the secondary, as is typical in contact binary systems, has a larger
radius and luminosity than would be expected.

In order to determine if ZZ PsA is a potential merger candidate, we explore
the concept of the instability mass ratio of contact binary systems and find
it currently sub-optimal because of the empirical values used for certain
parameters. Based on several established models of the internal stellar
geometry, we propose a new concept of an instability mass ratio (as opposed to
the minimum mass ratio), which is dependent on the mass of the primary and
modelled with variable values of the gyration radius rather than the
empirically fixed values previously employed. Our relationships suggest that
within the spectral range F0 to M0 (and most contact binary systems fall
within this range) earlier spectral class systems are likely to become
unstable at extremely low mass ratios while the later types may become
unstable at mass ratios near 0.25. We show that ZZ PsA is likely a highly
unstable contact binary system and hence a potential merger (red nova)
candidate. We identify two other potential merger candidates in the literature
and further three which require ongoing study to confirm instability.

## Acknowledgements

JP and GD acknowledge the financial support of the Ministry of Education,
Science and Technological Development of the Republic of Serbia through
contract No 451-03-68/2020/14/20002. This research has made use of the SIMBAD
database, operated at CDS, Strasbourg, France.

## Availability of Data

The data underlying this article will be shared on reasonable request to the
corresponding author.

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††thanks: Author to whom all correspondence should be addressed

# Giant anisotropy and Casimir phenomena: the case of carbon nanotube
metasurfaces

Pablo Rodriguez-Lopez<EMAIL_ADDRESS>Área de Electromagnetismo and Grupo
Interdisciplinar de Sistemas Complejos (GISC), Universidad Rey Juan Carlos,
28933, Móstoles, Madrid, Spain  Laboratoire Charles Coulomb (L2C), UMR 5221
CNRS-University of Montpellier, F-34095 Montpellier, France  Dai-Nam Le
Department of Physics, University of South Florida, Tampa, Florida 33620, USA
Igor V. Bondarev  Department of Mathematics and Physics, North Carolina
Central University, Durham, NC 27707, USA  Mauro Antezza  Laboratoire Charles
Coulomb (L2C), UMR 5221 CNRS-University of Montpellier, F-34095 Montpellier,
France  Institut Universitaire de France, 1 rue Descartes, F-75231 Paris Cedex
05, France  Lilia M. Woods<EMAIL_ADDRESS>Department of Physics, University
of South Florida, Tampa, Florida 33620, USA

###### Abstract

The Casimir interaction and torque are related phenomena originating from the
exchange of electromagnetic excitations between objects. While the Casimir
force exists between any types of objects, the materials or geometrical
anisotropy drives the emergence of the Casimir torque. Here both phenomena are
studied theoretically between dielectric films with immersed parallel single
wall carbon nanotubes in the dilute limit with their chirality and collective
electronic and optical response properties taken into account. It is found
that the Casimir interaction is dominated by thermal fluctuations at sub-
micron separations, while the torque is primarily determined by quantum
mechanical effects. This peculiar quantum vs. thermal separation is attributed
to the strong influence of reduced dimensionality and inherent anisotropy of
the materials. Our study suggests that nanostructured anisotropic materials
can serve as novel platforms to uncover new functionalities in ubiquitous
Casimir phenomena.

##  I Introduction

The discovery of layered materials has elevated the importance of van der
Waals (vdW) interactions as they are responsible for keeping their inert
components together [1, 2]. Advanced computational schemes have been
implemented in state of the art density functional theory packages to take
into account the vdW energy when simulating various materials properties [3,
4, 5]. The materials aspects of the Casimir force, a retarded vdW interaction,
has also generated significant interest [6]. This ubiquitous interaction
governs not only the performance of micro and nanomachines, but it also probes
fundamental properties stemming from Dirac and topologically nontrivial
physics [7].

Another aspect of Casimir phenomena is the ability to generate Casimir torque
when optically anisotropic materials are involved [8, 9, 10]. Indeed, the
misalignment of the inequivalent optical axis of two bodies results in their
relative rotation when electromagnetic (EM) fluctuations are exchanged. This
type of motion has been recently demonstrated in the laboratory in
birefringent liquid crystal systems [11] giving further impetus of enhancing
our possibilities to study basic physics via EM interactions. The main
ingredient for a large Casimir torque is the materials strong anisotropy
ensuring a substantial effect at various separations and temperature ranges.
In this context, quasi-one dimensional structures, in which optical and
geometrical anisotropies are combined, provide excellent conditions for
angular dependence of the force and much enhanced Casimir torque [12, 11].

Theoretical studies of the Casimir energy and torque rely on the Lifshitz
formalism, where the relative orientation of the optical axis of materials
separated by a distance $D$ is taken into account in the EM boundary
conditions. Typically, the torque decays roughly as $D^{-3}$ and it has the
characteristic $\sin(2\varphi)$ behavior. Also, its sign and magnitude depend
on the optical properties of the materials [13, 14, 15, 16, 17]. Additionally,
the angular dependence of the vdW and Casimir force has been explored in
cylindrical quasi-one dimensional structures, for which optical levitation of
a nanorod above a birefringent crystal has been proposed [18, 19].

In order to further exploit rotations generated by Casimir torque for
manipulating micro and nano-machines, more studies are necessary to identify
materials with strong anisotropy. In this regard, carbon nanotubes and
metasurfaces containing single-wall carbon nanotubes (SWCNs) are quite
suitable due to their quasi-one dimensionality. Such materials offer new
applications in quantum electron transport, electron energy-loss spectroscopy,
and mechanical reinforcement [20, 21, 22]. Ultrathin films composed of
periodically arranged nanotubes have recently emerged as transdimensional
materials with extraordinary optoelectronic properties [23, 24, 25, 26]. SWCN
films can also support plasmon, exciton, and phonon-polariton eigenmodes. This
brings novel aspects in light-matter interactions with nanotube chiralities,
different mixtures, and dielectric background material as effective ”knobs” of
tunability [27, 28].

Figure 1:  Two identical ultrathin SWCN films separated by a distance $D$
along the $z$-axis. The nanotube radius is $R$ and the intertube separation is
$\Delta$. The SWCNs are imbedded in a solid dielectric layer with thickness
$d\sim 2R\ll D$ and effective background dielectric constant $\epsilon_{b}$.
Such films can be immersed in a liquid dielectric surroundings with a
dielectric constant $\epsilon_{s}$ or be free-standing in air. The relative
orientation of the nanotube axis from both structures is denoted by $\varphi$.

While much work has been devoted to properties of individual carbon nanotube
films, light-matter interactions between such films has not been explored yet.
Understanding how fluctuation-induced interactions occur, the factors that
control their strength and characteristic behavior are questions of
fundamental importance. In this paper, we investigate Casimir phenomena
between two identical nanotube films using the Lifshitz formalism. By taking
into account the nanotube optical response properties and the dielectric
environment we show how the interplay between anisotropy, quantum mechanical
and thermal fluctuations affect the ubiquitous Casimir force. The Casimir
torque, a direct consequence of the anisotropy from the nanotube quasi-one
dimensionality, is also investigated. Due to the reduced dimensionality and
properties of the film, strong thermal fluctuations are found to play a
dominant role in the interaction energy, while quantum mechanical effects are
much more pronounced in the Casimir torque in the studied nm-$\mu$m separation
limit.

##  II Properties of carbon nanotube thin films in the dilute regime

We consider an ultrathin single wall carbon nanotube (SWCN) film composed of
identical parallel aligned SWCNs embedded in a solid dielectric layer with
thickness $d$ and effective background dielectric permittivity $\epsilon_{b}$
being a real constant. Such a film can be immersed in liquid dielectric
surroundings with a dielectric constant $\epsilon_{s}$, or it can be free-
standing in air. The SWCN array, schematically shown in Fig. 1, is aligned
along the $y$ axis, and adjacent nanotubes are separated by a distance
$\Delta$ in the $x$-direction (bottom material). The basic properties of each
SWCN are captured by their chirality index $(n,m)$, which also determines the
SWCN radius $R=\frac{\sqrt{3}b}{2\pi}\sqrt{m^{2}+nm+n^{2}}$ ($b=1.42$ $\AA\,$
is the C–C interatomic distance) [29].

Since the SWCN radius ($R$ is in the nm range) is much smaller than its length
(typically in the $\mu$m range), the optical response along the $y$-direction
is essentially dominated by the collective longitudinal response of the array,
while the response along the $x$-axis is entirely due to the dielectric
medium. This in-plane anisotropic SWCN film can be treated as a quasi-2D
system, where the vertical confinement due to the film thickness can be taken
into account via an effective model using the Keldysh-Rytova potential [30],
as done in recent works for closely packed SWCN films [27, 28]. Casimir
interactions in densely packed SWCN film systems were recently studied in Ref.
[31]. The focus of our studies here is on the ’dilute SWCN film’ regime,
however, defined as $\Delta-2R\\!\gg\\!\epsilon_{b}d/(2\epsilon_{s})$ with
$d\\!\sim\\!2R$. In this case, the intertube electrostatic coupling is given
by a $d$-independent 2D Coulomb interaction potential with a screening
constant $\epsilon_{s}$ of the dielectric surrounding of the film [30]. As a
consequence, the low-frequency (quasi-static) response of the SWCN array in
the $y$-direction is predominantly due to individual SWCNs [32], while its
higher-frequency optical (dynamical) response comes from the collective inter-
tube exciton energy exchange due to induced dipole-dipole interactions [28].
The former can be represented by the properly normalized intraband surface
conductivity of the individual constituent SWCN found in Ref. [32]. The latter
can be described by the dynamical surface conductivity of the SWCN array
obtained from its collective excitonic response function reported recently in
Ref. [28].

Taking the above into account, the total surface conductivity of the SWCN
array along the $y$-direction is (Gaussian units)

$\displaystyle\sigma_{yy}^{\text{array}}(k_{y},\omega)=\frac{2\pi
R}{\epsilon_{s}\Delta}\sigma_{yy}^{\text{intra}}(k_{y},\omega)\hskip
71.13188pt$ (1) $\displaystyle+\frac{\epsilon_{b}d}{2\pi}\frac{i\omega
K(k_{y})\sigma_{yy}^{\text{inter}}(k_{y},\omega)}{i\omega+K(k_{y})\sigma_{yy}^{\text{inter}}(k_{y},\omega)}.$

The first term in the above expression is the intraband surface conductivity
contribution coming from the single-tube intraband conductivity per unit
surface,

$\displaystyle\sigma_{yy}^{\text{intra}}(k_{y},\omega)=-\dfrac{2\alpha
cv_{F}}{\pi^{2}R}\frac{i\omega-1/\tau}{(i\omega-1/\tau)^{2}+(v_{F}k_{y})^{2}},$
(2)

where $k_{y}$ is the absolute value of the electron quasi-momentum along the
SWCN axis, $\alpha\\!=\\!e^{2}/\hbar c$, $v_{F}\\!=\\!c/300$ is the electron
Fermi velocity in graphene, and $\tau$ is the phenomenological relaxation time
parameter. This can be obtained by dividing its analogue per unit length of
Ref. [32] by $2\pi R$. The dimensionless prefactor $2\pi R/\Delta$ represents
the surface fraction of the aligned SWCNs in the array.

Figure 2:  Scaled by $\sigma_{0}=\alpha c/4$ optical conductivity along the
$y$-axis for a carbon film composed of (12,0) SWCNs at different separations
$\Delta$ in (a) real frequency and (b) imaginary frequency domains. Here
$k_{y}=1/R$ and $\hbar/\tau=6.61$ meV corresponding to $\tau=100$ fs [33, 34].

The second term in Eq.(1) is the interband surface conductivity contribution
and it can be found from its 3D equivalence
$\sigma(\omega)=(-i\omega/4\pi))(\epsilon(\omega)-\epsilon)$ that relates the
conductivity per unit volume to the dynamical response function of bulk
isotropic material with static permittivity $\epsilon$ (see, e.g., Ref. [35]).
In this relationship, the right-hand side must be multiplied by $d$ to obtain
conductivity per unit surface, the in-plane collective optical response
$\epsilon(k_{y},\omega)$ must be used for $\epsilon(\omega)$ and
$\epsilon_{b}$ for $\epsilon$. The in-plane collective response function
$\epsilon(k_{y},\omega)$ in the alignment direction of the finite-thickness
periodically aligned SWCN films was obtained in Ref. [28] using the many-
particle Green’s function formalism in the Matsubara formulation combined with
the effective model based on the Keldysh-Rytova potential to include nonlocal
plasmonic effects due to strong vertical confinement in ultrathin films
systems [36, 37, 38, 39, 40, 41]. This yields the interband contribution to
the SWCN array surface conductivity in Eq.(1), where
$\sigma_{yy}^{\text{inter}}(k_{y},\omega)$ is the interband conductivity of
the individual SWCN (metallic or semiconducting) constituent, which can be
found numerically using the Kubo formula and $({\bf k}\cdot{\bf p})$ method of
the electronic band structure calculations [42], and the function

$K(k_{y})=f_{CN}\frac{m^{*}\omega_{p}^{2}(k_{y})d}{e^{2}N_{2D}R}$ (3)

with $f_{CN}\\!=\\!\pi R^{2}/(\Delta d)$ being the volume fraction of the
SWCNs in the film. In essence, $K(k_{y})$ captures the effect of collective
oscillations of the surface electron density along the SWCN alignment
direction with nonlocal plasma frequency

$\omega_{p}(k_{y})=\sqrt{\frac{4\pi e^{2}N_{\rm
2D}}{m^{\ast}\epsilon_{b}\,d}\frac{2k_{y}RI_{0}(k_{y}R)K_{0}(k_{y}R)}{1+2\epsilon_{s}/(\epsilon_{b}k_{y}d)}\,},$
(4)

where $m^{\ast}$ is the electron effective mass, $N_{\rm 2D}$ $(=\\!N_{\rm
3D}d)$ is the surface electron density, $k_{y}$ is the absolute value of the
quasi-momentum along the nanotube axis, and $I_{0}$ and $K_{0}$ are the
zeroth-order modified cylindrical Bessel functions responsible for the correct
normalization of the electron density distribution over cylindrical surfaces
[37].

The conductivity component $\sigma_{xx}^{\text{array}}$ along the
$x$-direction can similarly be obtained from the 3D expression as discussed
earlier. Given that the SWCN response is negligible in directions transverse
to the nanotube axis, one finds,

$\sigma_{xx}^{\text{array}}(\omega)=-d\frac{i\omega}{4\pi}(\epsilon_{b}-\epsilon_{s}).$
(5)

The above expression corresponds to the in-plane collective optical response
function along the $x$-direction of the finite-thickness periodically aligned
SWCNs.

With all of the above, the two-component in-plane surface conductivity tensor
of the diluted quasi-2D SWCN array takes the diagonal form,

$\hat{\sigma}=\begin{pmatrix}\sigma_{xx}^{\text{array}}(\omega)&0\\\
0&\sigma_{yy}^{\text{array}}(k_{y},\omega)\end{pmatrix}$ (6)

with its individual components defined by Eqs. (1)–(4).

In Fig. 2, we show the calculated optical conductivity along the $y$-axis for
a film composed of (12,0) SWCNs, taken as an example system. For the chosen
values of $\Delta$ it is assumed that the dilute regime
$\frac{\epsilon_{s}}{\epsilon_{b}}\Delta\gg 2R$ holds (see earlier
discussions). The optical transitions in the $\hbar\omega>1$ eV region result
from the interband transitions of the collective excitations as shown in the
above equations. We find that although smaller $\Delta$ moves the transitions
towards smaller frequencies, the shift is rather minor. The main role of the
intertube separation is much more pronounced in the strength of the
transitions, which can also be seen in the imaginary frequency domain. As
shown in Fig. 2b, the larger nanotube density per unit area results in a
stronger overall response, which is important for the Casimir phenomena as
discussed below.

Figure 3:  Density plots in $(\varphi,D)$ space of (a) Casimir energy $E$
normalized to $E_{M}=-\frac{\pi^{2}\hbar c}{720D^{3}}$ and (h) Casimir torque
$\overline{\mathcal{T}}=\mathcal{T}/E_{M}$ for a SWCN film composed of (12,0)
nanotubes with $\Delta=10R$. $E/E_{M}$ as a function of separation $D$ for (b)
$\Delta=10R$, (c) $\Delta=20R$, (d) $\Delta=50R$ for optical axis relative
orientation with $\varphi=\\{0,\frac{\pi}{4},\frac{\pi}{2}\\}$. $E/E_{M}$ as a
function of angle $\varphi$ for (e) $\Delta=10R$, (f) $\Delta=20R$, (g)
$\Delta=50R$ for separation film between $D=\\{32,100,316\\}$ nm.
$\overline{\mathcal{T}}$ as a function of separation $D$ for (b) $\Delta=10R$,
(c) $\Delta=20R$, (d) $\Delta=50R$ for optical axis relative orientation with
$\varphi=\\{\frac{\pi}{8},\frac{\pi}{4},\frac{3\pi}{8}\\}$.
$\overline{\mathcal{T}}$ as a function of angle $\varphi$ for (e)
$\Delta=10R$, (f) $\Delta=20R$, (g) $\Delta=50R$ for film separation
$D=\\{32,100,316\\}$ nm.

##  III Casimir effects at the quantum and thermal limits

For the Casimir phenomena we consider two identical SWCN films (each with
thickness $d=2R$) separated by a distance $D$ along the $z$-axis as displayed
in Fig. 1 as each film is composed of the equally spaced (12,0) nanotubes. In
addition to the Casimir energy per unit area $E$, a Casimir torque
$\mathcal{T}=-\partial_{\varphi}E$ per unit area is also possible due to the
optical anisotropy in this system set by the nanotube axis. Within the
Lifshitz formalism the interaction energy and torque are found as

$\displaystyle E=$ $\displaystyle
k_{B}T\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}\int\frac{\mathrm{d}^{2}\bm{k}_{\perp}}{\left(2\pi\right)^{2}}$
(8)
$\displaystyle\times\ln\left|\mathbbm{1}-e^{-2D\sqrt{\kappa_{n}^{2}+\bm{k}_{\perp}^{2}}}\mathbb{R}_{0}\mathbb{R}_{\varphi}\right|,$
$\displaystyle\mathcal{T}=$ $\displaystyle-
k_{B}T\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}\int\frac{d^{2}\bm{k}_{\perp}}{(2\pi)^{2}}\times$
$\displaystyle\times\text{tr}\left[\left(\mathbbm{1}e^{2D\sqrt{\kappa_{n}^{2}+\bm{k}_{\perp}^{2}}}-\mathbb{R}_{0}\mathbb{R}_{\varphi}\right)^{-1}\mathbb{R}_{0}\frac{d\mathbb{R}_{\varphi}}{d\varphi}\right].$

The above expressions are obtained in imaginary Matsubara frequencies
$\xi_{n}=c\kappa_{n}=2\pi nk_{B}T/\hbar$ and the prime in the summation
corresponds to the $n=0$ term multiplied by $1/2$. The response properties of
the materials, also taken in Matsubara frequencies, are captured in the
Fresnel reflection matrices found from standard electromagnetic boundary
conditions [17]

$\displaystyle\mathbb{R}_{\varphi}=\frac{2\pi}{\delta_{\varphi}}\begin{pmatrix}r^{xx}_{\varphi}&r^{xy}_{\varphi}\\\
r^{yx}_{\varphi}&r^{yy}_{\varphi}\end{pmatrix},$ (9)
$\displaystyle\left\\{\begin{array}[]{lll}r^{xx}_{\varphi}&=&-2\pi\left(\frac{\sigma_{yy}^{\varphi}}{c\lambda_{n}}+\frac{2\pi}{c}\left|\sigma^{\varphi}\right|\right)\\\
r^{xy}_{\varphi}&=&-\frac{2\pi}{c}\sigma_{yx}^{\varphi}\\\
r^{yx}_{\varphi}&=&\frac{2\pi}{c}\sigma_{xy}^{\varphi}\\\
r^{yy}_{\varphi}&=&2\pi\left(\lambda_{n}\frac{\sigma_{xx}^{\varphi}}{c}+\frac{2\pi}{c}\left|\sigma^{\varphi}\right|\right)\\\
\delta_{\varphi}&=&1+\frac{2\pi}{c}\left(\lambda_{n}\sigma_{xx}^{\varphi}+\frac{\sigma_{yy}^{\varphi}}{\lambda_{n}}\right)+\frac{4\pi^{2}}{c^{2}}\left|\sigma^{\varphi}\right|,\end{array}\right.,$
(15)

where $\lambda_{n}=\sqrt{\bm{k}_{\perp}^{2}c^{2}/\kappa_{n}^{2}+1}$ with
$\bm{k}_{\perp}=(k_{x},k_{y})$ being the two-dimensional wave vector and
$\left|\sigma^{\varphi}\right|=\sigma_{xx}^{\varphi}\sigma_{yy}^{\varphi}-\sigma_{xy}^{\varphi}\sigma_{yx}^{\varphi}$.
The conductivity tensor components of the rotated film can be obtained from
$\sigma^{\varphi}=R_{\varphi,\hat{z}}^{-1}\sigma R_{\varphi,\hat{z}}$ where
$\sigma$ is taken from Eq. 6 and
$R_{\varphi,\hat{z}}=\begin{pmatrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{pmatrix}$ is the rotation matrix around the
$z$-axis by an angle $\varphi$. The Fresnel matrix $\mathbb{R}_{0}$ correspond
to $\mathbb{R}_{\varphi=0}$.

In Eqs. 8 and 8 we distinguish between the quantum mechanical limit, in which
the summation over Matsubara frequencies
$k_{B}T\sideset{}{{}^{\prime}}{\sum}\limits_{n=0}^{\infty}$ is replaced by the
integral $\dfrac{\hbar c}{2\pi}\int\limits_{0}^{\infty}d\kappa$, and the
purely thermal limit, found from the $n=0$ Matsubara term. Both of these
limits are examined in what follows.

The quantum mechanical Casimir energy $E_{qm}$ and torque $\mathcal{T}_{qm}$
are calculated numerically using the Lifshitz expressions by taking the
optical response model for the SWCN films, as discussed earlier. In Fig.
3(a,h), we show $(\varphi,D)$ density maps with $\Delta=10R$ for both
properties. Clear signatures of the nanotube anisotropy are noted in Fig.
3(a). At larger separations, where thermal effects dominate, the interaction
is strongest when the optical films are aligned ($\varphi=\\{0,\pi,2\pi\\}$)
and weakest when the optical axis of the fims are perpendicular to each other
($\varphi=\\{\frac{\pi}{2},\frac{3\pi}{2}\\}$). At smaller $D$, however, this
trend is reversed: the strongest coupling occurs for perpendicular optical
axis, which was also recently found for the quantum Casimir force in densely
packed SWCN films [31]. To better understand the behavior of $E_{qm}$, we
further show its dependence upon distance separation for several angles
$\varphi$ and parameter $\Delta$.

We find that as $D$ becomes larger and
$\varphi\neq\\{\frac{\pi}{2},\frac{3\pi}{2}\\}$, $E_{qm}$ approaches the limit
of interacting metals with the characteristic $\frac{1}{D^{3}}$ scaling law,
as shown in Fig. 3(b-d). This is not surprising since the interaction in this
range is dominated by the Drude-like response of the SWCN film. As $D$ becomes
smaller, however, the energy experiences a transition to a
$E_{qm}\sim\frac{1}{D^{4}}$ marking the onset dominance of the interband
optical response in Eq. 1. The parameter $\Delta$ and the particular distance
at which the scaling transition happens have a positive correlation as shown
in Fig. 3(b-d). While this is the case for $\varphi=\\{0,\frac{\pi}{4}\\}$,
for $\varphi=\frac{\pi}{2}$ the energy is markedly different. In this case,
the $E_{qm}\sim\frac{1}{D^{4}}$ behavior (typical for the Casimir metal-
dielectric interaction in 2D [43, 44]) is found in the entire distance range.
Such orientation-dependent Casimir-Polder scaling laws have also been found in
systems involving anisotropic particles [45], but to our knowledge they have
not been reported in two-dimensional anisotropic materials. The angular
dependence of the Casimir energy is also shown in Fig. 3(e, g). The
oscillatory-like features are more pronounced for SWNT films with smaller
$\Delta$ and smaller separations between the films.

The density plot in Fig. 3(h) shows that the Casimir torque in the quantum
mechanical regime displays the characteristic $\sin(2\varphi)$ oscillations
whose phase changes to $-\sin(2\varphi)$ at a certain distance. This can also
be seen explictly in Fig. 3(i-k). Our results indicate that the distance at
which the torque experiences this phase change is closely related to the
distance at which the energy changes its $D$ dependence (described earlier).
The trend that the phase change occurs at smaller $D$ for smaller $\Delta$ is
also observed in the Casimir torque. It appears that the interband-intraband
terms and their relevance at different separation regimes are the main driving
factor behind this effect. The $\sin(2\varphi)$ oscillations are explictly
given in Fig. 3(i-k), which correlate with the oscillatory features of the
energy in Fig. 3(e-g). We also find that smaller $\Delta$ results in larger
$E_{qm}$ and $\overline{\mathcal{T}}_{qm}$ as expected since denser nanotube
arrays have stronger response properties.

Casimir phenomena are also affected by temperature as thermal fluctuations may
become prominent even at sub-micron sepations. This is the case especially for
materials with reduced dimensions [46, 47, 48, 49, 50, 51]. To capture the
role of temperature in the Casimir energy and torque of the films, we first
begin by considering the $T$-dependence in the optical response properties.
Many dielectric substances, such as teflon, polymers, or different types of
glass experience very weak temperature dependence in their dielectric
properties [52, 53, 54], thus here we assume that $\epsilon_{b}$ is
$T$-independent. The intraband conductivity of the nanotube, however, is
modified according to

$\displaystyle\sigma_{yy}^{\text{intra}}(k_{y},\omega,T)$ $\displaystyle=$
$\displaystyle\sigma_{yy}^{\text{intra}}(k_{y},\omega,0)$ (16)
$\displaystyle\times$ $\displaystyle\dfrac{k_{B}T}{\hbar
v_{F}k_{y}}\ln\left(\dfrac{e^{(\mu-\hbar
v_{F}k_{y})/k_{B}T}-1}{e^{\mu/k_{B}T}-1}\right),$

where $\sigma_{yy}^{\text{intra}}(k_{y},\omega,0)$ is the intraband
conductivity at $T=0$ (considered in the quantum limit calculations discussed
earlier) and the chemical potential is taken to be $\mu=0.5$ eV [55]. It is
noted, however, that the dependence upon $\mu$ in the
$\sigma_{yy}^{\text{intra}}(k_{y},\omega,T)$ is relatively weak. The above
expression can be obtained by using the Maldague formula [56]. On the other
hand, the interband conductivity of the nanotube film is not significantly
affected by temperature, as also shown in [38, 57] where the combined effect
of temperature and exciton-plasmon coupling in the individual nanotubes was
considered.

The $T-$dependent Casimir properties can subsequently be calculated by using
the Lifhsitz formalism with the explicit Matsubara summation in the energy and
torque expressions from Eqs. 8 and 8, respectively. The special $n=0$ term
corresponds to the completely classical thermal regime and here it is also
evaluated separately. We find that in this case one has
$\mathbb{R}_{\varphi}(\kappa_{n}=0)=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$
regardless of the relative angle $\varphi$. As a result, the thermal Casimir
energy is found as $E_{T}=-\frac{\zeta(3)k_{B}T}{16\pi D^{2}}$. It appears
that in this classical thermal regime, the materials properties (including
their anisotropy) are absent, and the Casimir energy is the same as the one
for isotropic Drude metals.

In Fig. 4(a,b), we give the $(\varphi,D)$ density plots of the ratio $E_{T}/E$
at $T=30$ and $T=300$ K, where $E$ is calculated from Eq. 8. These results
show that temperature has a very strong effect on the Casimir energy. At lower
$T$, quantum mechanical effects dominate the interaction energy at separation
less than 100 nm, but for larger $D$ thermal effects become much more
prominent. At higher temperatures, $E_{T}$ is much stronger even at $D<100$
nm. Another feature found here is that the strength of thermal effects
compared to the quantum mechanical interaction depend on $\varphi$. For
$\varphi=\\{\frac{\pi}{2},\frac{3\pi}{2}\\}$, quantum mechanical effects are
strong at smaller $D$ and thermal effects are strong at larger $D$ as can be
seen even at $T=30$ K (4(a)). This transition shifts not only shifts towards
smaller separations as $T=300$K, but the strength of thermal fluctuations
become more prominent (4(b)). For $\varphi=\pi$, however, this trend does
appear: thermal fluctuations have a diminished role at larger $D$, but they
appear more prominent at intermediate separations. This type of non-uniform
$\varphi$ dependence is associated with the phase change behavior in the
energy entangled with the optical response of the film, as discussed earlier
in this particular angular axis orientation.

Figure 4:  Density plots in ($\varphi,D$) space of the Casimir energy ratio
$E_{T}/E_{qm}$ of the (12,0) CNT films with $\Delta=10R$ at: (a) $T=30$ K and
(b) $T=300$ K. The Casimir energy ratio $E/E_{M}$ (obtained from Eq. 8) as a
function of CNT film separation $D$ for (c) $\varphi=\pi/8$ and (d)
$\varphi=\pi/2$ at different temperatures. The dashed lines correspond to the
thermal limit from the $n=0$ Matsubara term. The Casimir torque ratio
$\overline{\mathcal{T}}=\mathcal{T}/E_{M}$ (obtained from Eq. 8) as a function
of separation $D$ for (e) $\varphi=\pi/8$ and (f) $\varphi=\pi/4$ at different
temperatures.

These trends are shown in more details in Fig. 4(c,d), where the Casimir
energy is shown for $\varphi=\pi/8$ and $\pi/2$ but at several temperatures.
One can see that at small $T$ the energy deviates significantly from $E_{T}$
in the displayed separation range for $\varphi=\pi/8$, however, for
$\varphi=\pi/2$ the energy tracks $E_{T}$ at larger $D$. As $T$ increases,
both orientations exhibit similar onset of thermal fluctuations as a function
of distance separation. We further find that due the weak $T$-dependence of
the optical response of the SWCN film, the Casimir energy at any temperature
can be represented simply by adding the quantum mechanical $E_{qm}$ and the
$n=0$ Matsubara term: $E\approx E_{qm}+E_{T}$. In fact, the results shown in
Fig. 4(c,d) obtained via Eq. 8 completely overlap, which makes the Matsubara
summation redundant. The distance for quantum-thermal effects transition in
the Casimir energy is found by taking $E_{qm}=E_{T}$ and it is also displayed
in Fig. 4(a,b) (dashed black curves).

For the Casimir torque, we find that the classical $n=0$ term vanishes
indicating the absense of classical thermal fluctuations. This unusual result
is directly connected with the peculiar form of the Fresnel reflection matrix
at zero Matsubara frequency $\mathbb{R}_{\varphi}(\kappa_{n}=0)$. For this
special term, the anisotropy of the SWCN films is washed away meaning that the
Casimir torque from purely thermal fluctuations is zero. In Fig. 4(e,f),
results are given for the Casimir torque as a function of separation for
$\varphi=\\{\frac{\pi}{8},\frac{\pi}{4}\\}$ obtained via Eq. 8 for several
temperatures. One finds that the thermal effect is rather different than in
the case of the Casimir energy. For small distances, ($D<200$ nm in Fig.
4(e,f)), the torque is completely determined by quantum fluctuations, while
for larger distances, the torque can be approached by the $n=1$ Matsubara
term, becoming exponentially suppresed ($\mathcal{T}\propto
e^{-\frac{k_{B}T}{\hbar c}D}$) for distances $D>200$ nm and larger
temperatures (4(e,f)).

##  IV Conclusions

In this study, we have investigated the Casimir interaction between ultrathin
SWCN films in the dilute regime reporting on a system where materials
properties, dimensionality, and temperature have unexpected consequences. SWCN
films are inherently anisotropic: when immersed in dielectric layers the
quasi-one dimensionality of individual nanotubes asserts the dominance of the
response along their lengths. It is thus expected that the Casimir interaction
is strongly dependent on the relative optical axis orientation $\varphi$ of
two interacting films, as also recently studied in densely packed SWCN films
[31]. This giant anisotropy in composite quasi-one dimensional materials then
leads us to the notion that Casimir torque in quasi-2D materials is also
possible.

We find that, indeed, the Casimir energy depends on $\varphi$, which drives
the emergence of Casimir torque. The interplay between the optical anisotropy
and temperature leads to a peculiar separation of quantum mechanical and
thermal contributions in the energy and torque. It turns out that thermal
fluctuations are especially strong dominating $E$ at sub-$\mu$m separations.
The main reason is the reduced dimensionality of the system, which also shows
that the particular optical properties (especially the interband terms) play a
secondary role in the Casimir interaction. This is consistent with previous
studies, which have shown that the reduced dimensionality elevates the
importance of thermal fluctuations at smaller separations making the
properties of the materials much less important [49, 51]. This is unlike the
case of double wall CNTs with inter-tube separations $\sim 3-4$ Å, where the
interaction is quantum mechanical and controlled by the specific structure of
the nanotube intra and interband optical response contributions [58].

While thermal fluctuations determine the energy, the Casimir torque, on the
other hand, is mostly a quantum mechanical phenomenon. The main reason is
attributed to the disappearance of the special $n=0$ Matsubara term in
$\mathcal{T}$, a consequence of the quasi-1D anisotropy of the system. We find
that the torque in the sub-$\mu$ range is controlled primarily by the
intraband contributions in the SWNT optical response and it is exponentially
screened by the temperature.

Our results show that in the dilute limit the anisotropic Drude response
arising from the quasi-1D SWCN dimensionality is the main reason for the
$\varphi$ dependence in the Casimir energy resulting in a relatively strong
torque. Qualitatively similar results can be found for other nanotube fims
with metallic chiralities. This quantum vs. thermal separation in probing
fluctuation induced interactions is a unique feature in metallic nanotube
films. We suggest that this peculiar delineation can be studied
experimentally, as measurements of the Casimir energy and torque are also
possible. For example, our calculations show that, at $T=10$ K, $D=50$ nm and
$\varphi=\frac{\pi}{8}$, $|E|\sim 10.89$ $nJ\cdot m^{-2}$ and
$|\mathcal{T}|\sim 8.49$ $nN.m\cdot m^{-2}$, while at $T=300$ K we have $E\sim
9.19$ $nJ\cdot m^{-2}$ and $\mathcal{T}\sim 7.57$ $nN.m\cdot m^{-2}$, which is
achievable in the laboratory [11, 59]. At smaller separations the magnitudes
of the Casimir energy and torque are expected to increase due to their scaling
laws discussed earlier, which may also be beneficial for potential
experimentaion. Our study further shows that investigations of other
anisotropic systems at the nanoscale are needed to further understand the
interplay between dimensionality, temperature, and materials properties in
Casimir phenomena.

###### Acknowledgements.

We acknowledge support from the US Department of Energy under Grant No. DE-
FG02-06ER46297. P. R.-L. acknowledges support from AYUDA PUENTE 2023, URJC and
the hospitality of the Theory of Light-Matter and Quantum Phenomena group at
the Laboratoire Charles Coulomb, University of Montpellier, where part of this
work was done. I.V.B. is supported by the U.S. Army Research Office under
award No. W911NF2310206. I.V.B., L.M.W. and M.A. gratefully acknowledge
support from the Kavli Institute for Theoretical Physics (KITP), UC Santa
Barbara, under U.S. National Science Foundation Grant No. PHY-1748958, where
this collaborative work was started. I.V.B. acknowledges KITP hospitality
during his invited visit as a KITP Fellow 2022–23 made possible by the
Heising-Simons Foundation. M.A. acknowledges the grant ”CAT”, No.
A-HKUST604/20, from the ANR/RGC Joint Research Scheme sponsored by the French
National Research Agency (ANR) and the Research Grants Council (RGC) of the
Hong Kong Special Administrative Region.

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