# A Broadband Signal Recycling Scheme for Approaching the Quantum Limit from
Optical Losses

Teng Zhang  Joe Bentley  Haixing Miao  1School of Physics and Astronomy, and
Institute of Gravitational Wave Astronomy, University of Birmingham,
Edgbaston, Birmingham B15 2TT, United Kingdom

###### Abstract

Quantum noise limits the sensitivity of laser interferometric gravitational-
wave detectors. Given the state-of-the-art optics, the optical losses define
the lower bound of the best possible quantum-limited detector sensitivity. In
this work, we come up with a broadband signal recycling scheme which gives
potential solution to approaching this lower bound by converting the signal
recycling cavity to be a broadband signal amplifier using an active
optomechanical filter. We will show the difference and advantage of such a
scheme compared with the previous white light cavity scheme using the
optomechanical filter in [Phys.Rev.Lett.115.211104 (2015)]. The drawback is
that the new scheme is more susceptible to the thermal noise of the mechanical
oscillator.

##  I Introduction

The ground-based laser interferometric gravitational-wave detectors operate at
frequencies from several Hz to several kHz. The quantum noise is one of the
most important noises which limits the detector sensitivity over the entire
frequency band. To enhance the quantum-limited sensitivity, the first limit we
encounter is the standard quantum limit, which is related to the Heisenberg’s
uncertainty principle of the optical quadratures [1]. It is defined as the
minimal sum of the quantum shot noise and radiation pressure noise (when
uncorrelated) at each frequency, given different optical power. However, it
does not describe the best achievable quantum-limited sensitivity. The next
limit is the quantum Cramér-Rao bound [2], or the so called
energetic/fundamental quantum limit [3, 4, 5]. This bound can be infinitely
suppressed by improving the energy fluctuation inside the arm cavities, e.g.
enhancing the optical power and using quantum squeezing (anti-squeezing). It
was recently realised that practical lower bound of the quantum-limited
sensitivity comes from the optical-loss induced quantum dissipation [6].

Figure 1: This figure shows the noise spectral densities of optical losses and
the shot noise limited sensitivity with parameters from aLIGO [7]. We assumed
1000 ppm SRC loss, 100 ppm arm cavity loss and 10% external loss. The blue
line illustrates the shot noise of aLIGO. The purple line is with $\sim$ 9 dB
observed squeezing (15dB squeezing in the arm cavities). The black line
illustrates the resulting shot noise of the aLIGO configuration with the BSR
scheme discussed in this work. It approaches the SRC loss limit at high
frequencies and arm cavity loss limit at low frequencies. Figure 2: Left:
Gravitational-wave detector with the proposed BSR scheme and input/output
filter cavities. The carrier laser is at $\omega_{0}$ (red). The
optomechanical filter cavity is pumped with a laser at $\omega_{0}+\omega_{m}$
(blue), which comes from modulating the input carrier and transmits to the
dark port under the Schnupp asymmetry. The resonant frequency of the
mechanical oscillator is $\omega_{m}$. The blue beam is filtered with an
output mode cleaner. The input and output filter cavities are used for
frequency dependent squeezing and variational readout. Right: Simplified
schematic of the broadband signal recycling scheme. Inside the SRC, the phase
of sideband upon reflection from the arm cavity is cancelled, therefor the SRC
is on resonance in a broad frequency band. The SRC is converted to be an
amplifier for the signal coming out from the arm cavity with a gain of
$1/\sqrt{T_{\rm SRM}}$.

The optical losses are unavoidable in real detectors and can come from various
sources, e.g. absorption, scattering, mode mismatch and the open ports which
are necessary to extract error signals for sensing and control. Depending on
the mechanism how optical losses couple to the gravitational-wave channel,
they can be classified into internal loss, including the arm cavity loss
$\epsilon_{\rm arm}$ and signal-recycling cavity (SRC) loss $\epsilon_{\rm
SRC}$, and external loss in the output chain $\epsilon_{\rm ext}$. The
spectral density of these optical losses is given by [6]:

$\begin{split}S_{\epsilon}=\frac{\hbar c^{2}}{4L^{2}\omega_{0}P_{\rm arm}}\\\
\left[\epsilon_{\rm arm}+\frac{(\Omega^{2}+\gamma_{\rm arm}^{2})T_{\rm
ITM}}{4\gamma_{\rm arm}^{2}}\epsilon_{\rm SRC}+\frac{1}{4}T_{\rm
SRC}\epsilon_{\rm ext}\right]\,,\end{split}$ (1)

where $\omega_{0}$ is the laser frequency, $\Omega$ is the angular frequency
of gravitational wave signals, $L$ is the arm length, $P_{\rm arm}$ is the arm
cavity power, $T_{\rm ITM}$ is the input test mass (ITM) power transmissivity,
$\gamma_{\rm arm}=cT_{\rm ITM}/(4L)$ is the arm cavity bandwidth and $T_{\rm
SRC}$ is the effective transmissivity of the signal-recycling cavity (SRC)
formed by ITM and the signal-recycling mirror (SRM). The arm cavity loss
limited sensitivity is frequency independent, since the vacuum couples into
signals directly inside the arm. In contrast, the SRC loss limited sensitivity
rises at higher frequencies above the arm cavity bandwidth due to the decrease
of signal response. It is independent of any optical modules inside the SRC.
The external loss happens on the output path of the interferometer, where
there are unwanted effects including the mode mismatch at the output mode
cleaner, and an imperfect quantum efficiency of the photodiode. The output
loss noise depends on the SRC parameters and $T_{\rm SRC}$ equals to 0.14 in
advanced LIGO (aLIGO). Apparently, it also limits the observed squeezing
level. For example, with 10% external loss, the observed squeezing cannot be
more than 10 dB. Fig. 1 shows the optical-loss limits of aLIGO [7].

From Eq. (1), we learn that at high frequencies, i.e. $\Omega\gg\gamma_{\rm
arm}$, the best achievable quantum sensitivity is limited by the SRC loss with
spectral density,

$S_{hf}=\frac{\hbar\Omega^{2}}{\omega_{0}P_{\rm arm}T_{\rm ITM}}\epsilon_{\rm
SRC}\,,$ (2)

which turns out to be independent of the arm length. It is then exciting and
crucial to explore the way of achieving the high frequency quantum limit,
which will give the maximal astrophysical outcome, in particular for neutron
star physics [8, 9]. As shown in Fig. 1, the sensitivity of aLIGO with 9 dB
observed squeezing is still one order of magnitude worse than the SRC loss
limit with $\epsilon_{\rm SRC}=1000\,$ppm in the frequency range of 1-5 kHz.

Up to now, the proposed quantum techniques can be categorised into the
following two classes: (1) noise suppression/cancellation schemes; (2) signal
amplification schemes. The strategies that allow us to overcome the standard
quantum limit are known as quantum nondemolition techniques [10, 11], for
example, frequency dependent squeezing and variational readout [12]. Based on
the understanding from the optical-loss limit, despite utilizing internal or
external squeezing [13, 14, 15, 16], the observed squeezing level has an
unsurpassed bound due to external loss. The signal enhancement scheme can
potentially complement the squeezing for approaching the loss limit. The most
well-known signal amplification technique, which has already been implemented
in the second-generation detectors, is the SRC. Obviously, both the high-
frequency and low-frequency sensitivities improve on resonant sideband
extraction mode. However, this comes with a price of sacrificing the peak
sensitivity at intermediate frequencies. It is due to the accumulation of the
phase of the sideband traveling inside the cavity (positive dispersion), which
leads to a destructive interference when the frequency is larger than the
cavity bandwidth. Another recent work on high-frequency detector design uses
the coupled-cavity resonance of the arm cavity and the SRC. That scheme gives
a narrow band dip in the noise spectrum at high frequencies [9, 17]. Such a
coupled-cavity resonance shows up when the signals resonate within the SRC.
They can only resonate in a narrow high-frequency band. Equivalently, the
identical resonance condition is satisfied around carrier frequency when the
SRC is tuned on the signal recycling mode.

One strategy to broaden up the bandwidth without sacrificing the peak
sensitivity is to introduce negative dispersion to cancel the arm cavity round
trip phase $2\Omega L/c$, by means of white light cavities [18, 19, 20].
However, the negative phase provided by the unstable filter cavity in [19]
evolves exponentially as a function of frequency. As it turns out, there is an
imperfect phase cancellation outside the frequency band of validity of the
linear approximation to the exponential function, which eventually leads to
rapid decrease of sensitivity above a certain frequency.

The coupled-cavity and white light cavity ideas are heuristic. The scheme
proposed in this work realises the coupled SRC and arm cavity resonance in a
broad frequency band. We name it as the broadband signal recycling (BSR)
scheme.

##  II Design concept and quantum noise

The scheme we propose in this work is shown in Fig. 2. Inside the SRC, an
optomechanical filter cavity which provides the negative dispersion is
implemented for canceling the phase gained by the sideband upon reflection
from the arm cavity. The optomechanical filter cavity is pumped with a laser
at frequency $\omega_{0}+\omega_{m}$, which is generated from modulating the
input carrier. A Schnupp asymmetry allows the carrier to transmit to the dark
port. It is worthwhile to note the difference compared with the previous
scheme in [19]. In that configuration, an addition mirror so-called iSRM is
introduced for achieving an impedance match with the ITM, which effectively
removes the ITM for the sideband. However, as we mentioned above, the negative
phase provided by the optomechanical filter cavity cancels the round trip
phase in the arms imperfectly at higher frequencies. Here we show the iSRM
(extra complexity) is not required, which realises an exact phase cancellation
in ideal case.

Figure 3: This figure shows the quantum noise spectral densities of aLIGO and
with BSR scheme ($T_{env}/Q_{m}=5\times 10^{-13}$). 15 dB squeezing is assumed
in the arm cavity and $\sim$ 9 dB squeezing is observed under 10% external
loss.

.

In Fig. 2, the optical input/output relation of arm cavity can be derived as
[12]:

$\hat{a}_{2}\approx\frac{\gamma_{\rm arm}+i\Omega}{\gamma_{\rm
arm}-i\Omega}\hat{a}_{1}=e^{i\beta}\hat{a}_{1}\,,$ (3)

where $\beta=2{\rm atan}\frac{\Omega}{\gamma_{\rm arm}}$ is the phase of the
sideband when reflecting from the arm cavity. In order to cancel this
frequency dependent phase inside the SRC and convert the SRC a broadband
signal amplifier, we need to introduce negative phase lag. As it turns out,
this can be exactly provided by an optomechanical filter in the so-called
resolved-sideband regime, i.e. $\omega_{m}\gg\gamma_{f}\gg\Omega$, where
$\omega_{m}$ is the resonant frequency of the mechanical oscillator,
$\gamma_{f}$ is the filter cavity bandwidth. When the mechanical damping rate
$\gamma_{m}$, is much smaller than the negative mechanical damping rate due to
the optomechanical interaction $\gamma_{\rm opt}$, the optomechanical filter
cavity gives the following open-loop transfer function [19]:

$\hat{a}_{out}\approx-\frac{\gamma_{\rm opt}-i\Omega}{\gamma_{\rm
opt}+i\Omega}\hat{a}_{in}=e^{i\alpha}\hat{a}_{in}\,,$ (4)

where $\gamma_{\rm opt}={P\omega_{0}}/({m\omega_{m}cL_{f}\gamma_{f}})$. There
is $\gamma_{m}={\omega_{m}}/{Q_{m}}$. $m$ is the mass of the mechanical
oscillator, $Q_{m}$ is the quality factor and $P$ is the circulating power in
the optomechanical filter cavity. When

$\gamma_{\rm opt}=\gamma_{\rm arm}\,,$ (5)

there is an exact cancellation of $\alpha$ and $\beta$ up to a frequency-
independent constant: $\alpha+\beta=\pi$. With the signal-recycling mirror
tuned to give another constant $\pi$ phase shift, the short SRC is on
resonance in a broad frequency band. As it turns out,
$\hat{o}=\hat{a}_{2}/\sqrt{T_{\rm SRM}}$, where $T_{\rm SRM}$ is the power
transmissivity of the SRM. The effective transmissivity of the SRC is

$T_{\rm SRC}=\frac{T_{\rm ITM}T_{\rm SRM}}{(1+\sqrt{R_{\rm ITM}R_{\rm
SRM}})^{2}}\,,$ (6)

where $R_{\rm ITM},R_{\rm SRM}$ is the power reflectivity of the ITM and SRM.
In the optomechanical filter cavity, the thermal noise contribution of the
mechanical oscillator can be modelled as effective SRC loss [8]:

$\epsilon_{eff}=\frac{4k_{B}}{\hbar\gamma_{opt}}\frac{T_{env}}{Q_{m}}=1000\,{\rm
ppm}\times\frac{0.014}{T_{\rm itm}}\times\frac{L}{4\,{\rm
km}}\times\frac{T_{env}/Q_{m}}{5\times 10^{-13}\,K}\,,$ (7)

where $k_{B}$ is the Boltzmann constant, $T_{env}$ is the environmental
temperature. In our case, to guarantee an additional effective SRC loss
contribution smaller than 1000 ppm, $T_{env}/Q_{m}$ has to be smaller than
$5\times 10^{-13}\,K$. This gives an extremely stringent requirement, which is
the drawback of the BSR scheme compared with the scheme in [19]. This is
because the value of $\gamma_{\rm opt}$ in our scheme is smaller than that in
[19], where $\gamma_{\rm opt}=c/L$. Therefore large bandwidth of the arm
cavity is preferred. A $T_{env}/Q_{m}$ only $\sim 10^{-8}\,K$ is demonstrated
in [21]. It is challenging to suppress the thermal noise in the BSR scheme.

After taking into account frequency dependent squeezing and variational
readout as show in Fig. 2, the spectral density of the detector can be
calculated as the sum of [22]

$S=\frac{h_{\rm SQL}^{2}}{2}\left[\frac{e^{-2r}+\epsilon_{\rm
ext}}{\mathcal{K}}+\frac{\epsilon_{\rm ext}}{1+\epsilon_{\rm
ext}e^{2r}}\mathcal{K}\right]\,,$ (8)

and the SRC and arm cavity loss contribution, where $e^{-2r}\approx 0.03$
represents $\sim$ 15 dB squeezing. $\mathcal{K}$ is the famous Kimble factor
and $h_{\rm SQL}$ is the standard quantum limit,

$\mathcal{K}=\frac{16\omega_{0}P_{\rm arm}\gamma_{\rm arm}T_{\rm
SRM}}{McL\Omega^{2}(\gamma_{\rm arm}^{2}+\Omega^{2})}\,,\quad h_{\rm
SQL}=\sqrt{\frac{8\hbar}{(M\Omega^{2}L^{2})}}\,.$ (9)

Here $M$ is the mass of the detector test mass. We adopt $T_{\rm SRM}=0.005$
and $T_{\rm ITM}=0.014$, thus $T_{\rm SRC}=0.00002$. The arm cavity power
$P_{\rm arm}$ is 800 kW and arm length $L$ is 4 km. The resulting shot noise
limited sensitivity which corresponds to the sum of the first term in Eq. 8
and the loss limit is shown in Fig. 1. As we can observe, the shot noise
limited sensitivity of the BSR scheme approaches the arm cavity loss limit at
low frequencies and the SRC loss limit at high frequencies. Including the
radiation pressure noise and the oscillator thermal noise, the detector
sensitivities are shown in Fig. 3.

##  III Summary and outlook

In this work, we show one scheme whose shot noise limited sensitivity can
almost reach the optical-loss limit. An optomechanical filter cavity inside
the SRC provides a proper negative phase lag that cancels the frequency
dependent phase of the signal sideband upon reflection from the arm cavity. It
allows the simultaneous resonance of broadband sidebands in the SRC, which can
be viewed as a frequency dependent coupled-cavity resonance. Instead of
suppressing the noise which will be limited by optical-loss, the signal is
amplified in this scheme. The amplification gain is $1/\sqrt{T_{\rm SRM}}$.
Note that the radiation pressure noise will also be amplified by the same
amount, and the BSR scheme alone cannot overcome the standard quantum limit.
For future detectors beyond the aLIGO, using high gain of BSR cavity provide a
new way of reducing the shot noise, complementary to squeezing and increasing
the power. A balance between the laser power and the signal recycling gain can
provide easier access to cryogenic techniques for reducing mirror and
suspension thermal noises, e.g. cryogenic temperature which are proposed for
LIGO Voyager [23], the Einstein Telescope low-frequency detector [24, 25], and
Cosmic Explorer phase II [26].

Acknowledgements — T. Z., J.B. and H. M. acknowledge the support of the
Institute for Gravitational Wave Astronomy at University of Birmingham. H. M.
is supported by UK STFC Ernest Rutherford Fellowship (Grant No.
ST/M005844/11).

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# Phase transition structure and breaking of universal nature of central
charge criticality in a Born-Infeld AdS black hole

Neeraj Kumara∗, Soham Sena† and Sunandan Gangopadhyaya‡ a Department of
Theoretical Sciences
S.N. Bose National Centre for Basic Sciences
JD Block, Sector III, Salt Lake, Kolkata 700 106, India

###### Abstract

In this paper we have considered the thermodynamics of a Born-Infeld $AdS$
black hole using inputs from the dual boundary field theory. Here, we have
varied the cosmological constant $\Lambda$ and the Newton’s gravitational
constant $G$ along with the Born-Infeld parameter $b$ in the bulk. A novel
universal critical behaviour of the central charge (occurring in the boundary
conformal field theory) in extended black hole thermodynamics for charged
black holes has been recently observed [1], and we have extended this study to
Born-Infeld $AdS$ black holes. The Born-Infeld parameter has the dimension of
inverse length, therefore, when considered in the first law of thermodynamics
of the bulk in the mixed form which includes the central charge of the
boundary conformal field theory, it modifies the thermodynamic volume and the
chemical potential (which are conjugate to pressure and central charge
respectively). We observe that due to this inclusion of the Born-Infeld non-
linearity in this analysis, the universal nature of the critical value of the
central charge observed in [1] breaks down. We also observe an interesting
behaviour of the free energy of the black hole with Hawking temperature due to
the variations in both the central charge and the Born-Infeld parameter. It is
also observed in our analysis that for a sufficiently small value of the Born-
Infeld parameter (small value of this parameter has more prominent non-linear
effects), there exists a critical value of the temperature below which no
black hole can exist.

## 1 Introduction

††
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General theory of relativity and quantum mechanics are the two most
revolutionary theories of the previous century, describing the two extreme
length scales of our universe. Further, the unification of gravity and quantum
mechanics has turned out to be a very complicated problem. In the seminal
works[2, 3, 4], Stephen Hawking showed that when we consider quantum effects
in curved spacetime, a black hole emits radiation. The formulation of black
hole thermodynamics[2, 3, 4, 5, 6], Hawking radiation[2, 3, 4], particle
emission from black holes[7, 8, 9], and acceleration radiation[10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21], generated a link between gravitation,
geometry and thermodynamics. The thermodynamics of black holes has become a
field of extreme importance in recent times. The understanding of
asymptotically anti-de Sitter ($AdS$) black holes, via the $AdS/CFT$
correspondence[22], provides a deep insight into the dual conformal field
theory ($CFT$) at finite temperature. When in thermal equilibrium with the
Hawking radiation, $AdS$ black holes show interesting phase transition
behaviour, namely, first order Hawking-Page phase transition[23, 24], first
order phase transition in charged Reissner-Nordstr$\ddot{\text{o}}$m $AdS$
black holes[25, 26], and small to large black hole phase transitions[27, 28,
29, 30, 31].

In recent times, there has been an upsurge in considering the variation of the
cosmological constant ($\Lambda$) in the first law of black hole
thermodynamics[1, 28, 32, 33, 34, 35, 36, 37, 38]. In standard black hole
thermodynamics, black hole parameters are varied in a fixed $AdS$ background
with $\Lambda$ being kept fixed. Now considering variations in the fundamental
constants, like Newton’s gravitational constant, cosmological constant, gauge
coupling constants lead us to more fundamental theories[39, 40]. It has been
argued in [33] that the first law of black hole thermodynamics becomes
inconsistent when viewed at the level of the Smarr relation[41] in the
presence of a fixed cosmological constant. Therefore, to counter this issue,
$\Lambda$ is included in the first law [33] as a thermodynamic variable, with
a negative cosmological constant being identified with a positive
thermodynamic pressure. In case of a Born-Infeld [42] black hole, the Born-
Infeld parameter $b$ must be varied in the first law of the black hole
thermodynamics in order to make it consistent with the Smarr relation[43, 44,
45]. This consideration has opened up a new direction of investigation which
is the extended thermodynamic behaviour of $AdS$ black holes with novel phase
transition structures in contrast to standard black hole thermodynamics[46,
47, 48, 49, 50, 51]. A detailed analysis on extended phase space
thermodynamics for charged and rotating black holes was carried out in [48].
Interesting insight about this extended black hole chemistry[52, 53] can be
further made using the gauge/gravity duality. It has been a bit tricky to find
a holographic insight into black hole chemistry[54, 55, 56, 57, 58, 59, 60]. A
simple argument indicates that the relation of the first law (of the Bulk) to
the thermodynamics of the holographic field theory is not easy[61, 62, 63].
The reason is that varying the cosmological constant would in turn mean
varying the central charge and the CFT volume of the boundary field
theory[54]. This would in turn imply that the CFT gets changed, and therefore
the identification of the volume conjugate to the thermodynamic pressure in
the bulk would become problematic. The way out of this problem is to vary $G$
along with the cosmological constant $\Lambda$ so that the central charge of
the dual CFT remains fixed. This leads to a mixed form of the extended first
law of thermodynamics which enables one to identify the appropriate
thermodynamic volume and chemical potential. Following this approach we write
down the mixed form of the first law of thermodynamics in the bulk for Born-
Infeld $AdS$ black hole.

A duality between holographic and bulk thermodynamics begins from the relation
between the central charge of the CFT and the parameters on the gravity side,
namely, the $AdS$ radius and Newton’s gravitational constant. A free energy
analysis of the bulk thermodynamics carried out in [1] revealed that there is
a critical central charge above which the free energy has a swallowtail
behaviour for any $P$ which implies a universal behaviour. Our aim is to look
at this universality in the case of Born-Infeld $AdS$ black hole. This would
give us an idea about non-linear effects in these type of phase transitions.

The organization of this paper is given as follows. In the next section, we
have provided a brief introduction of the Born-Infeld $AdS$ black holes in 4
spacetime dimensions and calculated the Hawking temperature for the same. In
section 3, we have calculated the mixed form of the extended first law of
black hole thermodynamics in the bulk and obtained the modified thermodynamic
variables using the modified Smarr relation. Next we have calculated the
critical value of the central charge in order to inspect its behaviour and
then we have plotted the different phase transition structures based on the
variation of the central charge and the Born-Infeld parameter respectively.

## 2 Born-Infeld AdS black hole and Hawking temperature

We start with a brief review of the Born-Infeld $AdS$ black hole. The solution
with a cosmological constant was first constructed in [64]. The action in
($3+1$) dimensional spacetime reads

$S=\dfrac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left[R-2\Lambda+L(F)\right]~{}$ (1)

where the Born-Infeld part of the Lagrangian is of the form

$L(F)=4b^{2}\left(1-\sqrt{1+\dfrac{F^{\mu\nu}F_{\mu\nu}}{2b^{2}}}\right)~{}.$
(2)

$\Lambda$ is the cosmological constant and $b$ is the Born-Infeld parameter.
The above action admits the following black hole solution [64]

$ds^{2}=-f(r)dt^{2}+\dfrac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}$ (3)

where

$\begin{split}f(r)&=1-\dfrac{2GM}{r}+\dfrac{r^{2}}{l^{2}}+\dfrac{2b^{2}r^{2}}{3}\left(1-\sqrt{1+\dfrac{GQ^{2}}{b^{2}r^{4}}}\right)\\\
&+\frac{4GQ^{2}}{3r^{2}}\,\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\dfrac{GQ^{2}}{b^{2}r^{4}}\right]~{}.\end{split}$
(4)

In eq.(4), $M$ is the mass of the black hole, $Q$ is the charge, $l$ is the
AdS radius and $\,{}_{2}F_{1}$ is the Gauss hypergeometric function. In the
limit $b\rightarrow\infty$, the above metric reduces to the $AdS$ Reissner-
Nordst$\ddot{\text{o}}$rm black hole. The cosmological constant $\Lambda$ in
terms of the AdS radius $l$ is $\Lambda=-\frac{3}{l^{2}}$ in
$(3+1)$-dimensions.

The event horizon of the black hole is obtained from the relation
$f(r_{+})=0$. This relates the mass $M$ in terms of the horizon radius
($r_{+}$) as

$\begin{split}M&=\dfrac{r_{+}}{2G}+\dfrac{r_{+}^{3}}{2Gl^{2}}+\dfrac{b^{2}r^{3}_{+}}{3G}\left(1-\sqrt{1+\dfrac{GQ^{2}}{b^{2}r_{+}^{4}}}\right)\\\
&+\dfrac{2Q^{2}}{3r_{+}}\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\dfrac{GQ^{2}}{b^{2}r_{+}^{4}}\right]~{}.\end{split}$
(5)

We can obtain the Hawking temperature of the black hole from eq.(4) as follows

$\begin{split}T&=\dfrac{1}{4\pi}\dfrac{\partial f}{\partial
r}\bigg{|}_{r=r_{+}}\\\
&=\dfrac{1}{4\pi}\left[\dfrac{1}{r_{+}}+\dfrac{3r_{+}}{l^{2}}+2b^{2}r_{+}\left(1-\sqrt{1+\dfrac{GQ^{2}}{b^{2}r_{+}^{4}}}\right)\right]~{}.\end{split}$
(6)

We will use eq.(s)(5,6) to calculate the critical value of the central charge
of the Born-Infeld $AdS$ black hole in section 4. In the next section, we will
try to find the modified form of the first law of the black hole
thermodynamics in $D$ dimensions.

## 3 The modified first law of thermodynamics

It has been realised recently that a negative cosmological constant can induce
a positive thermodynamic pressure[1, 47]. This led to the inclusion of the
cosmological constant in the extended thermodynamic phase space. Here, we take
another step forward by treating the Born-Infeld parameter $b$ as a
thermodynamical variable as we would like to see the effects of this parameter
on the first law of thermodynamics. The pressure of the black hole in terms of
the cosmological constant and the Newton’s gravitational constant can be
written in the following form[33]

$P=-\dfrac{\Lambda}{8\pi G}~{}.$ (7)

From the above equation, we observe that with the variation in the
cosmological constant, the thermodynamic pressure of the black hole also
changes. Now, the cosmological constant in general $D$-dimensions in terms of
$AdS$ radius can be written as

$\Lambda=-\frac{(D-1)(D-2)}{2l^{2}}~{}.$ (8)

It can be seen from eq.(s)(7, 8) that a negative cosmological constant induces
a positive thermodynamic pressure.

In natural units, the Bekenstein-Hawking entropy takes the form [2, 3, 4, 5,
6]

$S=\frac{A}{4G}$ (9)

where $A$ is the area of the black hole. In terms of the surface gravity
$\kappa$, the Hawking temperature of the black hole takes the form

$T=\frac{\kappa}{2\pi}~{}.$ (10)

The mass of the black hole in the extended thermodynamic phase space[33] can
be thought of as enthalpy rather than internal energy. Therefore, the most
general form of the first law of thermodynamics for a black hole with surface
gravity $\kappa$, charge $Q$, angular momentum $J$ and area $A$ can be written
as follows[33]

$\begin{split}\delta M&=T\delta S+V\delta P+\Phi\delta Q+\Omega\delta J\\\
&=\frac{\kappa}{2\pi}\delta S+V\delta P+\Phi\delta Q+\Omega\delta J\\\
&=\frac{\kappa}{8\pi G}\delta A-\frac{V}{8\pi G}\delta\Lambda+\Phi\delta
Q+\Omega\delta J\end{split}$ (11)

where we have used eq.(s) (7, 9, 10) to obtain the final expression in
eq.(11).

Holographic interpretation of the above first law of thermodynamics is shown
to have some issues [54, 55, 56, 57, 58, 59, 60, 65]. The pressure-volume term
(corresponding to the variation of the cosmological constant) in the first law
of thermodynamics of the bulk is shown to have two terms in the first law of
thermodynamics at the boundary CFT. These are the central charge of the
boundary CFT (and its corresponding conjugate variable)[54], and the
thermodynamic pressure (and its conjugate variable, volume) of the boundary
CFT (because change in the AdS radius will correspond to change in the
boundary radius[61]). The way to deal with this problem is to invoke the form
of the central charge from the $AdS/CFT$ dictionary. In this way we can avoid
the ambiguity by varying both the Newton’s gravitational constant and the
$AdS$ radius( or varying the cosmological constant). The $AdS/CFT$ dictionary
relates the central charge $C$ to the $AdS$ radius $l$ as [61]

$C=k\frac{l^{D-2}}{16\pi G}$ (12)

where the $k$ factor depends on the details of the system at the boundary.
From the above expression of $C$, we see that in order to keep C fixed, we
need to vary $G$ as well since $l$ is varying[1]. In case of a Born-Infeld
$AdS$ black hole, there is an additional parameter which is the Born-Infeld
parameter $b$. If we look at the setup carefully, we shall find that all the
parameters which are being varied are dimensionful. The Born-Infeld parameter
also has a dimension. Hence, we treat it as a thermodynamic variable. We
consider the black hole mass $M$ to be a function of area ($A$), charge ($Q$),
angular momentum ($J$), Newtonian gravitational constant ($G$), cosmological
constant ($\Lambda$), and the Born-Infeld parameter ($b$). Hence, we write the
black hole mass as

$M\equiv M(A,Q,J,G,\Lambda,b)~{}.$ (13)

The variation in the mass can be written as

$\delta M=\frac{\partial M}{\partial A}\delta A+\frac{\partial M}{\partial
Q}\delta Q+\frac{\partial M}{\partial J}\delta J+\frac{\partial M}{\partial
G}\delta G+\frac{\partial M}{\partial\Lambda}\delta\Lambda+\frac{\partial
M}{\partial b}\delta b.$ (14)

We shall define $\frac{\partial M}{\partial b}=\mathcal{B}$ and
$G\frac{\partial M}{\partial G}=-\xi$. Note that the conjugate variable
$\mathcal{B}$ to the Born-Infeld parameter $b$ has been termed as Born-Infeld
vacuum polarization in [48]. From eq.(11), we find that the conjugate
variables of $A,~{}\Lambda,~{}Q$ and $J$ are $\frac{\kappa}{8\pi G}$,
$-\frac{V}{8\pi G}$, $\Phi$, and $\Omega$. With these definitions, we can
recast eq.(14) as

$\delta M=\frac{\kappa}{8\pi G}\delta A+\Phi\delta Q+\Omega\delta
J-\xi\frac{\delta G}{G}-\frac{V}{8\pi G}\delta\Lambda+\mathcal{B}\delta b~{}.$
(15)

Our main aim now is to compute the coefficient $\xi$ of $\delta G$ in the
above expression. For that we now make use of a modified mass term as
suggested in [1]

$GM=\mathcal{M}(A,\sqrt{G}Q,\Lambda,GJ,b)~{}.$ (16)

Taking differential of the both sides of the above relation, we get

$\begin{split}\delta(GM)&=\frac{\partial\mathcal{M}}{\partial A}\delta
A+\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}\delta(\sqrt{G}Q)+\frac{\partial\mathcal{M}}{\partial\Lambda}\delta\Lambda\\\
&+\frac{\partial\mathcal{M}}{\partial(GJ)}\delta(GJ)+\frac{\delta\mathcal{M}}{\delta
b}\delta b\\\ \implies G\delta M&=-M\delta
G+\frac{\partial\mathcal{M}}{\partial A}\delta
A+\sqrt{G}\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}\delta Q\\\
&+\frac{Q}{2\sqrt{G}}\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}\delta
G+J\frac{\partial\mathcal{M}}{\partial(GJ)}\delta G\\\
&+G\frac{\partial\mathcal{M}}{\partial(GJ)}\delta
J+\frac{\partial\mathcal{M}}{\partial\Lambda}\delta\Lambda+\frac{\partial\mathcal{M}}{\partial
b}\delta b\\\ \implies\delta M&=\frac{1}{G}\frac{\partial\mathcal{M}}{\partial
A}\delta
A+\frac{1}{\sqrt{G}}\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}\delta
Q+\frac{\partial\mathcal{M}}{\partial(GJ)}\delta J\\\
&+\frac{1}{G}\left(-M+\frac{Q}{2\sqrt{G}}\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}+J\frac{\partial\mathcal{M}}{\partial(GJ)}\right)\delta
G\\\
&+\frac{\partial\mathcal{M}}{\partial\Lambda}\delta\Lambda+\frac{\partial\mathcal{M}}{\partial
b}\delta b~{}.\end{split}$ (17)

Now comparing eq.(17) with eq.(15), we obtain the following results

$\begin{split}\frac{\partial\mathcal{M}}{\partial
A}&=\frac{\kappa}{8\pi}~{},~{}\frac{1}{\sqrt{G}}\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}=\Phi~{},~{}\frac{\partial\mathcal{M}}{\partial(GJ)}=\Omega~{},\\\
\frac{\partial\mathcal{M}}{\partial\Lambda}&=-\frac{V}{8\pi
G}~{},~{}\frac{\partial\mathcal{M}}{\partial b}=\mathcal{B}~{}\end{split}$
(18)

and the conjugate variable to $G$ is given as follows

$\xi=M-\frac{Q}{2\sqrt{G}}\frac{\partial\mathcal{M}}{\partial(\sqrt{G}Q)}-J\frac{\partial\mathcal{M}}{\partial(GJ)}~{}.$
(19)

Using the relations from eq.(18) in eq.(19), we obtain the following form of
$\xi$

$\xi=M-\frac{Q\Phi}{2}-\Omega J~{}.$ (20)

This is the form of the conjugate variable to $G$ in terms of the other black
hole parameters.

Now taking the differential of eq.(12) and dividing by $C$, we get

$\frac{\delta C}{C}=-\frac{\delta G}{G}+(D-2)\frac{\delta l}{l}~{}.$ (21)

In eq.(21), we replace $\frac{\delta l}{l}$ by using the forms of $P$ and
$\Lambda$ in eq.(s)(7, 8) as follows

$\frac{\delta l}{l}=-\frac{\delta G}{2G}-\frac{\delta P}{2P}.$ (22)

Eq.(22) enables us to recast eq.(21) equation in the form

$\frac{\delta G}{G}=-\frac{2}{D}\frac{\delta C}{C}-\frac{(D-2)}{D}\frac{\delta
P}{P}~{}.$ (23)

Using eq.(s)(18, 19, 20, 23) in eq.(17), we get the form of $\delta M$ to be

$\begin{split}\delta M&=\frac{\kappa}{8\pi G}\delta A+\Phi\delta
Q+\Omega\delta J-\frac{V}{8\pi G}\delta\Lambda+\mathcal{B}\delta b\\\
&+\frac{2\xi}{DC}\delta C+\frac{(D-2)}{D}\xi\frac{\delta P}{P}~{}.\end{split}$
(24)

Using eq.(s)(7, 9, 10, 23), we can now rewrite eq.(24) as follows

$\begin{split}\delta M&=T\delta S+\phi\delta Q+\Omega\delta
J+\left[\frac{2\xi}{DC}-\frac{2(TS+PV)}{DC}\right]\delta C\\\
&+\left[V+\frac{D-2}{DP}\xi-\frac{D-2}{DP}(TS+PV)\right]\delta
P+\mathcal{B}\delta b\\\ &=T\delta S+\phi\delta Q+\Omega\delta
J+\mathcal{B}\delta b+V_{\mathcal{C}}\delta P+\mu_{\mathcal{C}}\delta
C\end{split}$ (25)

where

$\displaystyle V_{\mathcal{C}}$
$\displaystyle=V+\frac{D-2}{DP}\xi-\frac{D-2}{DP}(TS+PV)~{},$ (26)
$\displaystyle\mu_{\mathcal{C}}$
$\displaystyle=\frac{2\xi}{DC}-\frac{2(TS+PV)}{DC}~{}.$ (27)

Here $V_{\mathcal{C}}$ and $\mu_{\mathcal{C}}$ are the new effective
thermodynamic volume and chemical potential. Eq.(25) is the desired mixed form
of the first law of thermodynamics for a Born-Infeld $AdS$ black hole. This is
one of the main findings in our paper. The important point to note in eq.(25)
is that $\delta C$ can be set to zero since both $l$ and $G$ are varying. This
means keeping the boundary CFT intact. This allows us to study the
thermodynamics of the bulk with a fixed central charge in the boundary theory.
The reason so as to why this is called a mixed form of the first law is that
it contains both the bulk as well as boundary variables.

###  3.1 Extended black hole thermodynamics and the Smarr relation

In this subsection, we want to calculate V in terms of the mass M, charge Q
and Born-Infeld parameter b of the black hole. For that, we write down the
first law for Born-Infeld $AdS$ black holes as [48]

$\delta M=T\delta S+V\delta P+\Phi\delta Q+\Omega\delta J+\mathcal{B}\delta
b~{}.$ (28)

Here, $\Phi$, $\Omega$ and $\mathcal{B}$ are conjugate to $Q$, $J$ and $b$
respectively, where

$\displaystyle\Phi=\frac{\delta M}{\delta Q}~{},~{}~{}\Omega=\frac{\delta
M}{\delta J}~{},~{}~{}\mathcal{B}=\frac{\delta M}{\delta b}~{}.$ (29)

$\mathcal{B}$ has been called the Born-Infeld vacuum polarization and its
inclusion in first law as well as in Smarr relation has been thoroughly
discussed in [43, 44, 45, 47]. Here, in our consideration $M=M(S,P,b,Q,J)$. In
generalized $D$-dimensions, $M,S,P,b,Q,J$ have the following dimensions (in
terms of $L$)

$\begin{split}[M]&=L^{D-3},~{}[S]=L^{D-2},~{}[P]=L^{-2}~{},\\\
[b]&=L^{-1},~{}[Q]=L^{D-3},~{}[J]=L^{D-2}~{}.\end{split}$ (30)

Using Euler’s theorem of quasi-homogeneous functions, we get

$\begin{split}(D-3)M&=(D-2)S\frac{\delta M}{\delta S}-b\frac{\delta M}{\delta
b}-2P\frac{\delta M}{\delta P}\\\ &+(D-3)Q\frac{\delta M}{\delta
Q}+(D-2)J\frac{\delta M}{\delta J}~{}.\end{split}$ (31)

Using eq.(28), we can write $\frac{\partial M}{\partial S}=T$, $\frac{\partial
M}{\partial Q}=\Phi$, $\frac{\partial M}{\partial P}=V$, $\frac{\delta
M}{\delta J}=\Omega$ and $\frac{\partial M}{\partial b}=\mathcal{B}$.
Substituting them back in eq.(31), we get the Smarr formula as follows

$(D-3)M=(D-2)TS-\mathcal{B}b-2PV+(D-3)\Phi Q+(D-2)\Omega J~{}.$ (32)

From eq.(32), we can obtain the form for the black hole volume V to be

$\begin{split}V&=\frac{D-2}{2P}TS-\frac{\mathcal{B}b}{2P}+\frac{D-3}{2P}\Phi
Q-\frac{D-3}{2P}M+\frac{D-2}{2P}\Omega J\\\
&=\frac{D-3}{2P}\left(\frac{D-2}{D-3}(TS+\Omega J)+\Phi
Q-M-\frac{1}{D-3}\mathcal{B}b\right)~{}.\end{split}$ (33)

###  3.2 Forms of the modified thermodynamic variables

Replacing the form of $V$ from eq.(33) in eq.(26), we get

$V_{\mathcal{C}}=\frac{2M-2\mathcal{B}b+(D-4)Q\Phi}{2DP}~{}.$ (34)

Similarly, we can write the chemical potential in terms of $V$ and
$V_{\mathcal{C}}$ as

$\mu_{\mathcal{C}}=\frac{2P}{C(D-2)}(V_{\mathcal{C}}-V)~{}.$ (35)

In $(3+1)$-dimensions, that is, $D=4$, eq.(s)(34,35) reduces to the following
forms

$\begin{split}V_{\mathcal{C}}&=\frac{M-\mathcal{B}b}{DP}\end{split}$ (36)
$\begin{split}V_{\mathcal{C}}&=\frac{4\pi r_{+}^{3}}{3}+\frac{4\pi
l^{2}r_{+}}{3}-\frac{8\pi b^{2}l^{2}r_{+}^{3}}{9}+\frac{8\pi
b^{2}l^{2}r_{+}^{3}}{9}\sqrt{1+\frac{GQ^{2}}{b^{2}r_{+}^{4}}}\\\ &+\frac{8\pi
GQ^{2}l^{2}}{9r_{+}}\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{GQ^{2}}{b^{2}r_{+}^{4}}\right]~{},\end{split}$
$\begin{split}\mu_{\mathcal{C}}&=\frac{P(V_{\mathcal{C}}-V)}{C}\\\
&=\frac{4\pi r_{+}}{kl^{2}}-\frac{8b^{2}\pi
r_{+}^{3}}{3kl^{2}}+\frac{8b^{2}\pi
r_{+}^{3}}{3kl^{2}}\sqrt{1+\frac{GQ^{2}}{b^{2}r_{+}^{4}}}\\\ &+\frac{8\pi
GQ^{2}}{3kl^{2}r_{+}}\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{GQ^{2}}{b^{2}r_{+}^{4}}\right]\end{split}$
(37)

where we have used eq.(5),

$P=\frac{3}{8\pi l^{2}G},~{}C=\frac{kl^{2}}{16\pi
G},~{}\Phi=\frac{Q}{r_{+}}\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{GQ^{2}}{b^{2}r_{+}^{4}}\right]~{},$
(38)

and

$\begin{split}\mathcal{B}&=\frac{\partial M}{\partial b}\\\
&=\frac{2br_{+}^{3}}{3G}\left[1-\sqrt{1+\frac{GQ^{2}}{b^{2}r_{+}^{4}}}\right]+\frac{Q^{2}}{3br_{+}}\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{GQ^{2}}{b^{2}r_{+}^{4}}\right]\end{split}$
(39)

to obtain the final forms of $V_{\mathcal{C}}$ and $\mu_{\mathcal{C}}$.
$V_{\mathcal{C}}$ and $\mu_{\mathcal{C}}$ are the new thermodynamic variables
for the Born-Infeld $AdS$ black hole. Interestingly, we find that the new
volume $V_{\mathcal{C}}$ depends on the Born-Infeld parameter which enters
through the expression for $V$ obtained from the Smarr formula. Our next goal
is to understand the effects of the Born-Infeld term on the critical behaviour
of the black hole and the possible phase transition structure.

## 4 Breaking of universal nature of the central charge

Now we shall calculate the critical value of the central charge. In order to
find this value, we will use the following two equations[1]

$\displaystyle\dfrac{\partial T}{\partial r_{+}}$ $\displaystyle=0~{},$ (40)
$\displaystyle\dfrac{\partial^{2}T}{\partial r_{+}^{2}}$ $\displaystyle=0~{}.$
(41)

Using eq.(6), we get the following equation from eq.(41)

$1-\frac{6GQ^{2}}{r_{+(c)}^{2}\left(1+\frac{GQ^{2}}{b^{2}r_{+(c)}^{4}}\right)^{\frac{1}{2}}}+\frac{4G^{2}Q^{4}}{b^{2}r_{+(c)}^{4}\left(1+\frac{GQ^{2}}{b^{2}r_{+(c)}^{4}}\right)^{\frac{3}{2}}}=0~{}.$
(42)

Here, $c$ in the subscript of $r_{+(c)}$ is denoting the critical value.
Eq.(42) upto $\mathcal{O}(1/b^{2})$ takes the form as

$r_{+(c)}^{6}-6GQ^{2}r_{+(c)}^{4}+\frac{7G^{2}Q^{4}}{b^{2}}=0~{}.$ (43)

Now we proceed to obtain the solution of eq.(43) by approximating it upto
$\mathcal{O}(1/b^{2})$. We take a perturbative approach to obtain the critical
value of $r_{+(c)}$. We propose a solution of the form for $r_{+(c)}$ (the
critical value of $r_{+}$) upto $\mathcal{O}(1/b^{2})$ as follows

$r_{+(c)}\cong r_{+}^{(0)}+\frac{r_{+}^{(1)}}{b^{2}}~{}.$ (44)

Using eq.(44) in eq.(43), we obtain the forms of $r_{+}^{(0)}$ and
$r_{+}^{(1)}$ as follows

$r_{+}^{(0)}=\sqrt{6G}Q~{},~{}~{}r_{+}^{(1)}=-\frac{7}{72\sqrt{6G}Q}~{}.$ (45)

Eq.(45) gives the form of the critical value of $r_{+(c)}$ upto
$\mathcal{O}(1/b^{2})$ as

$r_{+(c)}\cong\sqrt{6G}Q-\dfrac{7}{72\sqrt{6G}Qb^{2}}~{}.$ (46)

Now, eq.(40) can be recast in the following form

$\begin{split}&-\frac{1}{r_{+(c)}^{2}}+\frac{3}{l^{2}}+2b^{2}-2b^{2}\sqrt{1+\frac{GQ^{2}}{b^{2}r_{+(c)}^{4}}}\\\
&+\frac{4GQ^{2}}{r_{+(c)}^{4}\sqrt{1+\frac{GQ^{2}}{b^{2}r_{+(c)}^{4}}}}=0~{}.\end{split}$
(47)

Eq.(47) upto $\mathcal{O}(1/b^{2})$ can be recast in the following form

$-\frac{1}{r_{+(c)}^{2}}+\frac{3}{l^{2}}-\frac{7G^{2}Q^{4}}{4b^{2}r_{+(c)}^{8}}+\frac{3GQ^{2}}{r_{+(c)}^{4}}=0~{}.$
(48)

Substituting eq.(46) in eq.(48), we obtain the critical value of the $AdS$
radius upto $\mathcal{O}(1/b^{2})$ as

$l_{c}\cong 6\sqrt{G}Q\left(1-\dfrac{7}{864GQ^{2}b^{2}}\right)~{}.$ (49)

Putting the values of $r_{+(c)}$ and $l_{c}$ in eq.(6), we obtain the critical
value of the temperature upto $\mathcal{O}(1/b^{2})$ as

$T_{c}=\dfrac{1}{3\sqrt{6G}\pi Q}+\dfrac{1}{432\pi\sqrt{6G}GQ^{3}b^{2}}~{}.$
(50)

All the results reduce to the Reissner-Nordstr$\ddot{\text{o}}$m case in the
limit ($b\rightarrow\infty$)[1]. We can obtain the value of the critical
charge by using the form of $l_{\mathcal{C}}$ from eq.(49) in eq.(12) upto
$\mathcal{O}(1/b^{2})$ as follows

$\begin{split}C_{c}&=k\dfrac{l_{c}^{2}}{16\pi G}\\\ \implies
C_{c}&\cong\dfrac{9kQ^{2}}{4\pi}-\dfrac{7k}{16\pi Gb^{2}}~{}.\end{split}$ (51)

Eq.(51) gives the form of the critical charge in terms of $Q,~{}G$ and $b$. We
will now proceed to calculate the form of the critical charge in terms of
$Q,~{}b$ and the critical pressure $P_{c}$. This form is needed to avoid using
values of $G$ while calculating the critical values of the central charge
$C_{c}$ for different values of the non-linear parameter $b$ (Table 1).

In four spacetime dimensions and for $l=l_{c}$ we can recast eq.(s)(7,8) as

$P_{c}=\frac{3}{8\pi Gl_{c}^{2}}~{}.$ (52)

Again, replacing $l_{c}^{2}$ by $\frac{16\pi GC_{c}}{k}$ from the first line
of eq.(51) in the above equation, we obtain (after taking a square root)

$G=\sqrt{\frac{3k}{128\pi^{2}C_{c}P_{c}}}$ (53)

where $P_{c}$ is the pressure corresponding to the critical value of the
central charge. Using the form of the gravitational constant in eq.(51), we
can recast eq.(51) in the following form

$C_{c}=\frac{9kQ^{2}}{4\pi}-\frac{7\sqrt{kP_{c}}}{\sqrt{6}b^{2}}C_{c}^{\frac{1}{2}}~{}.$
(54)

Solving eq.(54) upto $\mathcal{O}(1/b^{2})$, the form of the critical charge
in terms of $P_{c}$, the Born-Infeld parameter $b$, black hole charge $Q$ and
$k$ is given by

$C_{c}\cong\frac{9kQ^{2}}{4\pi}-\frac{7kQ}{4b^{2}}\sqrt{\frac{6P_{c}}{\pi}}~{}.$
(55)

In case of Reissner-Nordstr$\ddot{\text{o}}$m black hole, the critical value
of the central charge was obtained as[1]

$C_{c}=\dfrac{9kQ^{2}}{4\pi}~{}.$ (56)

In the above equation, we see that the central charge $C_{c}$ is universal in
nature. In case of a Born-Infeld AdS black hole, we observe from eq.(55) that
the universal nature of the central charge is now broken due to the presence
of the Born-Infeld parameter and the critical-pressure term $P_{c}$ in it.
Dependence of the critical value of the central charge on the Born-Infeld
parameter is given in Table(1). To observe this dependence, we used $Q=1$,
$k=16\pi$, and $P_{c}=15$.

Table 1: Dependence of $C_{c}$ on $b$ $b$ | $10$ | $15$ | $20$ | $30$ | $100$ | $\infty$
---|---|---|---|---|---|---
$C_{c}$ | $31.29$ | $33.91$ | $34.82$ | $35.48$ | $35.95$ | $36$

For the above values of the parameters, $C_{c}$ for a Reissner-
Nordstr$\ddot{\text{o}}$m $AdS$ black hole has the value $36$. From Table(1),
we observe that $C_{c}$ value for the Born-Infeld case approaches $36$ as we
increase the parameter value $b$ without changing the pressure term ($P$). In
the limit $b\rightarrow\infty$, the critical value of the central charge
approaches the Reissner-Nordstr$\ddot{\text{o}}$m black hole result.

## 5 Phase transition structure

In this section, we will try to investigate the phase transition structure for
a Born-Infeld $AdS$ black hole. The free energy of the black hole can be
computed using eq.(s)(5,6) along with the use of the Bekenstein-Hawking
entropy as follows

$\begin{split}F&=M-TS\\\
&=\dfrac{r_{+}}{4G}-\dfrac{r_{+}^{3}}{4G~{}l^{2}}\dfrac{b^{2}r_{+}^{3}}{6G}\left(1-\sqrt{1+\dfrac{GQ^{2}}{b^{2}r_{+}^{4}}}\right)\\\
&+\dfrac{2Q^{2}}{3r_{+}}\,_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\dfrac{GQ^{2}}{b^{2}r_{+}^{4}}\right]~{}.\end{split}$
(57)

Here, in principle, we can write $F\equiv F(T,P,Q,C,b)$. The behaviour of the
free energy of the black hole with its temperature is plotted in Figure(1) for
different values of the central charge. We observe that above a critical value
of the central charge there is a swallowtail behaviour of the free energy of
the black hole with respect to the change in the black hole temperature for
fixed value of the pressure. Below this critical value of the central charge
($C_{\mathcal{C}}=35.4769$ in Figure(1)), we observe a smooth curve for the
free energy indicating no swallowtail behaviour. Critical analysis of Born-
Infeld $AdS$ black holes in extended thermodynamics have been discussed in
[48]. The importance of our work lies in the fact that it deals with the case
where the contribution of the central charge from the CFT side plays an
important role which is why it is called a mixed thermodynamic behaviour.

Figure 1: Free energy vs Temperature: Parameters Q=1; k=16$\pi$; P=15; b=30
($a_{0}$ is a scaling parameter): a) $C=25$ (Dashed line), b)
$C=C_{c}=35.4769$ (Dotted line) , c) $C=60$ (Solid line)  Figure 2: Free
energy vs Temperature: Parameters Q=1; k=16$\pi$; P=15; C=30 ($a_{0}$ is a
scaling parameter): a) $b=5$ (Dashed line), b) $b=2$ (Solid line) Figure 3:
Free energy vs Temperature: Parameters Q=1; k=16$\pi$; P=15; C=70 ($a_{0}$ is
a scaling parameter): a) $b=30$ (Dashed line), b) $b=3$ (Solid line)

In Figure(1), we will investigate the behaviour of the free energy with
respect to change in the temperature for various central charge values. As
opposed to [1], the critical value of the central charge has a direct
dependence on the pressure term as can be seen from eq.(55). Now from Figure
(1), we observe that for the value of the central charge, $C=25$ (which is
less than the critical value, $C_{c}=35.4769$), the free energy curve is
smooth with respect to change in the temperature. For curves with the value of
the central charge, $C\geq C_{c}$ (for fixed $P$ and $b$ values), the free
energy curve no longer remains smooth. Instead we observe a swallowtail
behaviour indicating first order small to large black hole phase transition.
With increasing temperature, we now observe a drop in the free energy below
the $F=0$ line. We also observe that for the value of the central charge below
and above the critical value, the curves meet the $F=0$ line at respectively
higher and lower values of the temperature. Hence, we observe for a Born-
Infeld $AdS$ black hole, there exists a critical value of the central charge
for fixed values of the pressure term and Born-Infeld parameter above which
there is a small to large black hole phase transition.

For the next part of our analysis, we concentrate on the dependence of the
critical value of the central charge on the Born-Infeld parameter $b$. We have
plotted free energy with respect to the temperature for different values of
the Born-Infeld parameter $b$ with fixed values of the central charge. In
Figures(2,3), we observe that for a substantially low value of the Born-Infeld
parameter ($b=2$ for Figure(2) and $b=3$ for Figure(3)), the free energy curve
does not exist below a certain temperature, indicating the non-existence of
any black hole in that region. Above this temperature value ($Ta_{0}\sim 0.36$
$(b=2)$ in Figure(2) and $Ta_{0}\sim 0.32$ $(b=3)$ in Figure(3)), we observe
that there are two branches of black hole solutions. However, the upper one is
unstable (since the free energy is higher in this branch) and the lower one
describes large black hole solutions which are physical. This indicates that
for sufficiently small $b$, there exists a no black hole region and a branch
of large black hole solutions. These features are qualitatively same with the
Reissner-Nordstr$\ddot{\text{o}}$m $AdS$ black hole [1] but Born-Infeld
parameter plays a crucial role in the phase transition structure through its
dependence on the critical value of the central charge.

## 6 Conclusion

In this work, we have studied the thermodynamics of a Born-Infeld AdS black
hole in ($3+1$)-dimensions with variable $AdS$ radius, Newton’s gravitational
constant and the Born-Infeld parameter. Starting with the derivation of
thermodynamic quantities of the black hole, we expressed the first law of
thermodynamics in terms of the usual thermodynamical variables along with the
central charge and the Born-Infeld parameter. The mixed from of the first law
is derived in general $D$-spacetime dimensions with all thermodynamic
variables. This modified first law for the Born-Infeld $AdS$ black hole system
is called mixed because it has contributions from both the bulk as well as the
boundary field theory variables. However, later we have reduced the discussion
to four spacetime dimensions. Because of the existence of the Born-Infeld
parameter in the Smarr relation, the thermodynamic variables $V_{\mathcal{C}}$
and $\mu_{\mathcal{C}}$ get modified by the same. Also, this modification
leads to a change in the critical value of the central charge which again
depends on the Born-Infeld parameter including a pressure dependence as well.
This breaks the universal behaviour of the central charge observed in an
earlier study[1]. This is one of the most important results in our paper. From
the free energy versus temperature plots, we observe that the phase structure
undergoes crucial change due to the inclusion of the Born-Infeld parameter as
a thermodynamical variable. We also observe that if the Born-Infeld parameter
is decreased enough, there will be a certain critical temperature below which
black holes do not exist. It implies that the inclusion of the electromagnetic
non-linearity results in some significant change in the overall free energy
behaviour of the black hole with respect to the change in the Hawking
temperature.

## Acknowledgement

We thank the anonymous referee for his constructive comments.

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Spectral and Energy Efficient User Pairing for RIS-assisted Uplink NOMA Systems with Imperfect Phase Compensation
Kusuma Priya P., Pavan Reddy M., Abhinav Kumar

Department of Electrical Engineering, Indian Institute of Technology Hyderabad, India 502285
Email:{ee20mtech11007<EMAIL_ADDRESS><EMAIL_ADDRESS>
Non-orthogonal multiple access (NOMA) is considered a key technology for improving the spectral efficiency of fifth-generation (5G) and beyond 5G cellular networks. NOMA is beneficial when the channel vectors of the users are in the same direction, which is not always possible in conventional wireless systems. With the help of a reconfigurable intelligent surface (RIS), the base station can control the directions of the channel vectors of the users. Thus, by combining both technologies, the RIS-assisted NOMA systems are expected to achieve greater improvements in the network throughput. However, ideal phase control at the RIS is unrealizable in practice because of the imperfections in the channel estimations and the hardware limitations. This imperfection in phase control can have a significant impact on the system performance. Motivated by this, in this paper, we consider an RIS-assisted uplink NOMA system in the presence of imperfect phase compensation. We formulate the criterion for pairing the users that achieves minimum required data rates. We propose adaptive user pairing algorithms that maximize spectral or energy efficiency. We then derive various bounds on power allocation factors  for the paired users. Through extensive simulation results, we show that the proposed algorithms significantly outperform the state-of-the-art algorithms in terms of spectral and energy efficiency.

Energy efficiency, non-orthogonal multiple access, power allocation, reconfigurable intelligent surface, spectral efficiency, uplink, user pairing.

§ INTRODUCTION

In the 5G and beyond 5G communication systems, significant improvements are expected in terms of spectral efficiency, energy conservation, massive connectivity, and latency requirements. NOMA is considered as a key multiple access technique to enhance the spectral efficiency of future cellular networks [1].  Various NOMA transmission schemes have been evaluated for possible consideration of NOMA in 5G uplink scenario [2]. In NOMA, multiple users are allocated with the same time and frequency resources to achieve multi-fold improvement in the network throughputs. However, the users are multiplexed in either power domain (power-domain NOMA) or code domain (code-domain NOMA), which requires the receiver to employ a SIC and decode the data [3]. Similar to NOMA, IRS is considered as a key technology to improve the spectral efficiency of the cellular networks [4]. An IRS is a two-dimensional planar array consisting of low-cost reflecting antenna elements that can modify the amplitude and phase of the signal. Thus, by using IRS, the BS can perform beamforming and transmit the signal in the desired direction.

Unlike the spatial multiplexing in OMA, NOMA is beneficial in situations where the channel vectors of users are in the same direction [4]. This is not always possible in conventional wireless systems, whereas, in the case of IRS-assisted systems, the BS can control the direction of the user channel vectors by tuning the IRS [4, 5]. For these reasons, IRS-assisted NOMA systems have been widely considered to achieve significant improvements in the network performance [4, 6]. However, the ideal phase control is difficult to achieve in practice for various reasons like hardware impairments, channel estimation errors, etc. These imperfections in the phase control degrade the achievable spectral and energy efficiencies in the network. Further, in NOMA systems, the achievable data rates are significantly dependent on the user pairing, and hence, while pairing the users in IRS-assisted NOMA systems, the network operator has to consider the imperfections in the phase compensation. Otherwise, the expected improvements in the network throughputs will not be realized in practice. For the aforementioned reasons, investigating the effect of imperfection in phase at the IRS is crucial to achieve optimum system performance.

In [7, 8], the authors have proposed various channel estimation techniques for IRS-assisted wireless systems.
In [9, 10], the authors have formulated the sum-rate maximization as an optimization problem, and then, derived a near-optimal solution for IRS-assisted uplink NOMA systems.

limited works in the literature have considered the imperfection in phase for the downlink OMA and NOMA systems [6, 11, 12]. To the best of our knowledge, none of the existing works considered the imperfections in the phase compensation while pairing the users in IRS-assisted uplink NOMA systems. Motivated by this, we present the following key contributions in this paper.

* For IRS-assisted uplink NOMA systems, we derive bounds on imperfection in the phase compensation to achieve minimum required data rates.

* We propose user pairing algorithms that maximize spectral efficiency and energy efficiency, respectively.

* We derive bounds and define the power allocation factors for the paired users.

* Through extensive simulation results, we show that the proposed algorithms significantly outperform the state-of-the-art algorithms.

The rest of the paper is organized as follows.
The system model is presented in Section <ref>.  In Section <ref>, we propose the adaptive user pairing algorithms, derive the bounds on the power allocation factors, and formulate the criterion for pairing the users to maximize spectral and energy efficiencies. In Section <ref>, we present the simulation results for various scenarios. Section <ref> presents some concluding remarks and possible future works.

§ SYSTEM MODEL

System model.

We consider an uplink scenario with $M$ antennae at the BS, $N$ reflective antenna elements at the IRS, and a single antenna at each user as shown in Fig. <ref>. The direct link between the BS and user is assumed to be blocked by the obstacles such as buildings, trees, or human body which is a likely scenario in case of mmwave communications [11]. Further,

the channel coefficients from $i^{th}$ user to IRS and IRS to BS are defined as h_i and h_r, respectively and are formulated as follows [11]

\begin{align}
\gls{h_i}&=\gls{beta}\mathbf{a}_N\left(\gls{psiIa},\gls{psiIe}\right),\label{eqn:hi}\\
\gls{h_r}&=\gls{alpha}\mathbf{a}_N\left(\gls{phiIa},\gls{phiIe}\right)\mathbf{a}_M^H\left(\gls{psiBa},\gls{psiBe}\right),\label{eqn:hr}
\end{align}

where beta and alpha denote the channel gains from $i^{th}$ user to IRS and IRS to BS, respectively, phiIa and phiIe represent the aod in azimuth and elevation at the IRS, respectively, psiBa and psiBe represent aoa in azimuth and elevation at the BS, respectively, psiIa and psiIe represent the aoa in azimuth and elevation at the IRS, respectively, and

$\mathbf{a}_X(\ohm^a,\ohm^e)$ represents the array response vector.

A uspa with $X$ antenna elements, has $\sqrt{X}$ elements in both horizontal and vertical directions, and thus, the array response vector is defined as

\begin{align}
\mathbf{a}_X(\ohm^a,\ohm^e)&=\begin{bmatrix}1\\
\vdots\\e^{j\frac{2\pi d}{\gls{lambda}}(x\sin{\ohm^a}\sin{\ohm^a}+y\cos{\ohm^e})}\\ \vdots\\
e^{j\frac{2\pi d}{\gls{lambda}}((\sqrt{X}-1)(\sin{\ohm^a}\sin{\ohm^a})+(\sqrt{X}-1)\cos{\ohm^e})}\end{bmatrix}^T,\nonumber
\end{align}

where, $d$ is the equispaced elemental distance, lambda is the signal wavelength, $0\le x, y\le\sqrt{X}-1$ are the indices of the uspa elements in the horizontal and vertical directions, respectively.

We denote the reflection matrix at the IRS as theta and define it as follows [4].

\begin{align}
\gls{theta}&=\text{diag}(e^{j\theta_1},\hdots,e^{j\theta_k},\hdots,e^{j\theta_N}),\nonumber
\end{align}

where, thetak $(1\le k \le N)$ is the phase shift introduced by $k^{th}$ reflecting element at the IRS. However, note that the ideal phase control is difficult to achieve in practice because of various reasons like hardware limitations, imperfect channel state information, etc. For this reason, we consider imperfections in the phase control and define the practical reflection matrix as follows.

\begin{align}
\gls{Theta}&=\text{diag}(e^{j\tilde{\theta}_1},\hdots,e^{j\tilde{\theta}_k},\hdots,e^{j\tilde{\theta}_N}),\nonumber
\end{align}

where, $\tilde{\theta}_k=\gls{thetak}+\gls{thetakcap}$ with thetakcap being the error in phase control at $k^{th}$ antenna element. Further, we assume  thetakcap to be uniformly distributed over $[-\gls{delta},\gls{delta}]$, $\gls{delta}\in[0,\pi)$.
Thus, in the case of OMA transmission, the signal received from the $i^{th}$ user at the BS is formulated as

\begin{equation}
y_i^{\text{OMA}}=\gls{h_r^H}\gls{Theta}\gls{h_i}\gls{P_t}\gls{s_i}+\gls{n_i}, \nonumber
\end{equation}

where $\{ . \}^\text{H}$ denotes the Hermitian of a matrix, P_t is the available transmit power at each user, s_i is the data symbol transmitted by the $i^{th}$ user, and n_i denotes the thermal noise. The SINR of an $i^{th}$ user in an IRS-assisted OMA system is formulated as

\begin{equation}
\gls{gammaoma}=\frac{\gls{P_t}{\|\gls{h_r^H}\gls{Theta}\gls{h_i}\|}^2}{\gls{I}+\gls{sigma^2}}\label{gammaomaold},
\end{equation}

where I is the interference power and sigma^2 is the noise variance.
In the case of IRS assisted NOMA, we consider two users multiplexed in power domain, with channel coefficients h_1 and h_2. Additionally, we assume the following

\begin{align}
\|\gls{h_r^H}\gls{Theta}\gls{h_1}\|^2 > \|\gls{h_r^H}\gls{Theta}\gls{h_2}\|^2.\nonumber
\end{align}

Thus, the BS receives P_t(alpha_1s_1+alpha_2s_2), where s_1 and s_2 are the data symbols transmitted by the strong and weak user, respectively. Further, we consider a fractional power control scenario, where $0<\gls{alpha_1}, \gls{alpha_2}\leq 1$ represent the fraction of the total available transmit power used by the strong and weak users, respectively. Thus, the signal received from the $i^{th}$ user in an IRS-assisted NOMA system is formulated as follows

\begin{equation}
\label{eqn:NOMASys}
\end{equation}

From (<ref>), we define the SINR of the users in IRS-assisted NOMA as follows

\begin{align}
\gls{gamma_1}&=\frac{\gls{alpha_1}\gls{P_t}{\|\gls{h_r^H}\gls{Theta}\gls{h_1}\|}^2}{\gls{alpha_2}\gls{P_t}{\|\gls{h_r^H}\gls{Theta}\gls{h_2}\|}^2+\gls{I}+\gls{sigma^2}}\label{gamma1old},\\
\gls{gamma_2}&=\frac{{\gls{alpha_2}\gls{P_t}\|\gls{h_r^H}\gls{Theta}\gls{h_2}\|}^2}{\gls{I}+\gls{sigma^2}}\label{gamma2old}.
\end{align}

From (<ref>)-(<ref>), we define the following [6]:

\begin{align}
\gls{h_r^H}\gls{Theta}\gls{h_i}&= \gls{alpha}\gls{beta}\mathbf{a}_M(\gls{psiBa},\gls{psiBe})\sum_{k=1}^N e^{j\gls{thetakcap}},\nonumber\\
\|\mathbf{a}_M(\gls{psiBa},\gls{psiBe}) \|^2&=M,\nonumber\\
\|\gls{h_r^H}\gls{Theta}\gls{h_i} \|^2 & ={|\gls{alpha}\gls{beta}|}^2M\left|\sum_{k=1}^N e^{j\gls{thetakcap}}\right|^2.\label{normsq}
\end{align}

We adopt the following approximation from [11]:

\begin{align}
\left|\frac{1}{N}\sum_{k=1}^N e^{j\gls{thetakcap}}\right|^2\xrightarrow{(a)}\left|\mathbb{E}\left[e^{j\gls{thetakcap}}\right]\right|^2\overset{(b)}{=}\left|\mathbb{E}\left[\cos \gls{thetakcap}\right]\right|^2\overset{(c)}{=}\sinc^2(\gls{delta}),\label{sum}
\end{align}

where $(a)$ follows the strong law of large numbers [11], in $(b)$, the expectation of odd function $\sin{\gls{thetakcap}}$ vanishes over the interval $\gls{thetakcap}\in[-\gls{delta}, \gls{delta}]$, and  $(c)$ is obtained by using the probability density function $f(\gls{thetakcap})=\frac{1}{2\gls{delta}}$, where $\gls{thetakcap}\in[-\gls{delta}, \gls{delta}]$.
Substituting (<ref>) in (<ref>)

\begin{align}
\|\gls{h_r^H}\gls{Theta}\gls{h_i} \|^2&={|\gls{alpha}\gls{beta}|}^2MN^2\sinc^2(\gls{delta}),\nonumber\\
\|\gls{h_r^H}\gls{theta}\gls{h_i} \|^2&={|\gls{alpha}\gls{beta}|}^2MN^2.\nonumber
\end{align}

Using these approximations, we define the CSI of $i^{th}$ user as

\begin{align}
\gls{csi_i}&\triangleq \frac{\gls{P_t}\|\gls{h_r^H}\gls{theta}\gls{h_i} \|^2}{\gls{I}+\gls{sigma^2}}=\frac{\gls{P_t}|\gls{alpha}\gls{beta}|^2N^2M}{\gls{I}+\gls{sigma^2}}.\label{gammacsi}
\end{align}

Using (<ref>) in (<ref>), (<ref>), and  (<ref>), we formulate the received SINRs as follows

\begin{align}
\gls{gammaoma}&= \gls{csi_i}\sinc^2(\gls{delta}),\nonumber\\
\gls{gamma_1}&=\frac{\gls{alpha_1}\gls{csi_1}\sinc^2(\gls{delta})}{1+\gls{alpha_2}\gls{csi_2}\sinc^2(\gls{delta})},\nonumber \\
\gls{gamma_2}&=\gls{alpha_2}\gls{csi_2}\sinc^2(\gls{delta}).\nonumber
\end{align}

Given a logarithmic rate model, the normalized achievable data rates by the users in OMA and NOMA are  formulated as [12]

\begin{align}
\gls{R_1}&=\log_2\left(1+\frac{\gls{alpha_1}\gls{csi_1}\sinc^2(\gls{delta})}{1+\gls{alpha_2}\gls{csi_2}\sinc^2(\gls{delta})}\right),\label{eqn:r1}\\
\end{align}

Next, we propose various adaptive user pairing algorithms.

§ PROPOSED ALGORITHMS

In this section,  we propose MPA and EEPA algorithms to maximize the sum rate and energy efficiency, respectively.

Additionally, in both the algorithms, we derive a criterion for pairing the users in NOMA which ensures the minimum required data rates are achieved for each user. We denote R1bar and R2bar as the minimum required data rates by the strong and weak users, respectively. Thus, both MPA and EEPA algorithms should satisfy the following constraints

\begin{align}
\gls{R_1}\geq\gls{R1bar},\label{eqn:r1constraint}\\
\gls{R_2}\geq\gls{R2bar}.\label{eqn:r2constraint}
\end{align}

§.§ MPA

In MPA, apart from satisfying (<ref>)-(<ref>), we allocate the transmit powers (alpha_1 and alpha_2) that maximize the ASR of the paired users, where, ASR is defined as

\begin{align}
\gls{ASR}=\gls{R_1}+\gls{R_2}\nonumber.
\end{align}

We formulate the desired optimization as follows.

\begin{align}
\max_{\gls{alpha_1},\gls{alpha_2}} \hspace{0.4cm} &\gls{R_1}+\gls{R_2},\label{prob:mpa}\\
\text{s.t.}\hspace{0.3cm} & \eqref{eqn:r1constraint}, \  \eqref{eqn:r2constraint} \nonumber,\\
& \gls{alpha_1}\ge0, \gls{alpha_2}\ge0,\label{eqn:0constr}\\
& \gls{alpha_1}\le1, \gls{alpha_2}\le1.\label{eqn:1constr}
\end{align}

§.§.§ Bounds on $\gls{alpha_1}$ and $\gls{alpha_2}$

From (<ref>)-(<ref>), we get,
\begin{align}
\gls{R_1}+\gls{R_2}=\log_2(1+(\gls{alpha_1}\gls{csi_1}+\gls{alpha_2}\gls{csi_2})\sinc^2(\gls{delta})).\label{eqn:ASR}
\end{align}

Thus, ASR is an increasing function with respect to alpha_1 and alpha_2. However, from (<ref>), an increase in alpha_2 will decrease the achievable data rate for the strong user, whereas, there is no such impact with an increase in alpha_1. Hence, to maximize the ASR, we assign $\gls{alpha_1}=1$. Thus, we get,

\begin{align}
\gls{R_1}&=\log_2\left(1+\frac{\gls{csi_1}\sinc^2(\gls{delta})}{1+\gls{alpha_2}\gls{csi_2}\sinc^2(\gls{delta})}\right).\label{eqn:r11}
\end{align}

From (<ref>) and (<ref>),  we get,

\begin{align}
\gls{alpha_2}\ge\frac{2^{\gls{R2bar}}-1}{\gls{csi_2}\sinc^2(\gls{delta})}&\triangleq \gls{alpha2lb}.\label{eqn:alpha2lb}
\end{align}

From (<ref>) and (<ref>), we get,

\begin{align}
\gls{alpha_2}\leq \frac{\gls{csi_1}\sinc^2(\gls{delta})+1-2^{\gls{R1bar}}}{\gls{csi_2}\sinc^2(\gls{delta})[2^{\gls{R1bar}}-1]}&\triangleq \gls{alpha2ub}.\label{eqn:alpha2ub}
\end{align}

Illustration of adaptive user pairing.

§.§.§ Criterion for pairing the users

Using (<ref>)-(<ref>), and assuming

$\gls{alpha2ub}\geq\gls{alpha2lb}$, we obtain the pairing criterion for the MPA algorithm as follows.

\begin{align}
\sinc^2(\gls{delta})\geq\frac{2^{\gls{R2bar}}\left[2^{\gls{R1bar}}-1\right]}{\gls{csi_1}}\triangleq \sinc^2(\gls{delubmpa}).\label{eqn:delubmpa}
\end{align}

We define R1bar and R2bar as the achievable OMA rates, and then, pair the users in NOMA iff (<ref>) is satisfied. Otherwise, we consider transmitting the information for the users in an OMA scenario as shown in Fig. <ref>. This way, the proposed algorithm ensures that a minimum of OMA rates are achieved in a worst-case scenario and it maximizes the ASR by switching to NOMA whenever feasible.

§.§.§ Power allocation

For the paired users, we define the power allocation factors as follows.

\begin{align}
\gls{alpha1mpa}&=1,\label{eqn:alpha1mpa}\\
\gls{alpha2mpa}&=\min\{\gls{alpha2ub},1\}.\label{eqn:alpha2mpa}
\end{align}

Note that in (<ref>), the strong user transmits at maximum power to maximize the spectral efficiency,
whereas, in (<ref>), the weak user transmits at a maximum possible power that does not degrade the strong user's data rate beyond the OMA rate. Further, to avoid transmit power violations, we limit the maximum value of alpha2mpa to 1 in (<ref>).

The power allocation factors formulated in (<ref>) - (<ref>) are the optimal values that achieve maximum sum rate while ensuring the individual NOMA rates to be better than the OMA counterparts.

For the ease of understanding, we define

\begin{align}
\gls{eta}&\triangleq\frac{2^{\gls{R1bar}}-1}{\gls{csi_1}\sinc^2(\gls{delta})},\label{eqn:eta}\\
\gls{kappa}&\triangleq\frac{(2^{\gls{R1bar}}-1)\gls{csi_2}}{\gls{csi_1}}.\label{eqn:kappa}
\end{align}

Thus, we reformulate (<ref>) as follows

\begin{align}
\max_{\gls{alpha_1},\gls{alpha_2}} \hspace{0.45cm} &\gls{alpha_1}\gls{csi_1}+\gls{alpha_2}\gls{csi_2},\label{prob:finalmpa}\\
\text{s.t.}\hspace{0.35cm}
& -\gls{alpha_2}+\gls{alpha2lb}\le0,\label{eqn:Lemm2}\\
& \hspace{0.4cm} \gls{alpha_1}-1\le0, \gls{alpha_2}-1\le0.\label{eqn:Lemm1}
\end{align}

where, (<ref>) is obtained by using the fact that logarithmic function is a monotonically increasing function and substituting (<ref>) in (<ref>). The constraint (<ref>) is obtained by substituting (<ref>)-(<ref>) in (<ref>) and the constraint (<ref>) is obtained by further solving the (<ref>). Additionally, since R1bar, R2bar are non negative, we get $\gls{alpha2lb},\ \gls{eta},\ \gls{kappa}>0$, and thus, we consider the constraint (<ref>) is already captured in (<ref>)-(<ref>). Next, the Lagrangian for (<ref>) is formulated as

\begin{align}
\end{align}

Solving stationarity conditions $\frac{\partial L}{\partial \gls{alpha_1}}=0\  \text{and}\  \frac{\partial L}{\partial \gls{alpha_2}}=0$, we get

\begin{align}
\gls{csi_1}-\mu_1+\mu_4&=0,\label{eqn:kkt1}\\
\gls{csi_2}-\mu_2+\mu_3-\mu_4\gls{kappa}&=0.\label{eqn:kkt2}
\end{align}

The complementary slackness conditions are formulated as

\begin{align}
\mu_1(\gls{alpha_1}-1)&=0,\label{eqn:kkt3}\\
\mu_2(\gls{alpha_2}-1)&=0,\label{eqn:kkt4}\\
\mu_3(-\gls{alpha_2}+\gls{alpha2lb})&=0,\label{eqn:kkt5}\\
\mu_4(-\gls{alpha_1}+\gls{alpha_2}\gls{kappa}+\gls{eta})&=0.\label{eqn:kkt6}
\end{align}

The dual feasibility conditions are formulated as

\begin{align}
\mu_i\ge0\hspace{0.1cm},\ \forall\hspace{0.1cm}i \in [1,4].\label{eqn:kkt7}
\end{align}

Solving (<ref>)-(<ref>), we find that the possible optimal values of $(\gls{alpha_1},\gls{alpha_2})$ are
$(\gls{alpha2lb}\gls{kappa}+\gls{eta},\ \gls{alpha2lb}), \ (\gls{kappa}+\gls{eta},1),\ (1,1),\ (1,\gls{alpha2lb}),$ and $\ (1,\frac{1-\gls{eta}}{\gls{kappa}})$. From (<ref>), larger the values of alpha_1 and alpha_2, larger will be the ASR. Since, $\gls{alpha_1}=1$ has no impact on any of the desired constraints, we consider only the solutions with $\gls{alpha_1}=1$ which are
$(1,1), (1,\gls{alpha2lb})$, and $(1,\frac{1-\gls{eta}}{\gls{kappa}})$. Note that considering $(\gls{alpha_1},\gls{alpha_2})=(1,1)$ will violate the constraint (<ref>). Hence, by substituting $\gls{alpha2ub}=\frac{1-\gls{eta}}{\gls{kappa}}$, the optimal values of (alpha_1,alpha_2) are either  $(1,\gls{alpha2ub})$ or $(1,\gls{alpha2lb})$. Since, $\gls{alpha2ub}\geq \gls{alpha2lb}$, we conclude $(\gls{alpha_1},\gls{alpha_2})=(1,\gls{alpha2ub})$ as the optimal solution.

This completes the proof of the Lemma 1.

Next, we present the EEPA algorithm.
Proposed algorithms
$\gls{csi_i},\  \forall i \in [1,G]$.
$i$ represents the user pairing index.
Sort the $G$ users in decreasing order of csi_i

Set $i=1$
Consider $i^{th}$ user as strong user and $(G-i+1)^{th}$ user as the weak user

Calculate R1bar, R2bar from (<ref>)
Calculate delubmpa from (<ref>)
$\gls{delta}\le$ delubmpa
Pair the users in NOMA with alpha_1=1, alpha_2=$\min\{\gls{alpha2ub},1\}$ as per (<ref>), (<ref>)
Consider the users in OMA

Calculate $\gls{delubeepa}=\min\{\gls{delub1},\gls{delub2}\}$
from (<ref>)

$\gls{delta}\le\gls{delubeepa}$Pair the users in NOMA with (alpha_1, alpha_2) obtained from the solution of (<ref>)
Consider the users in OMA

csi_1,csi_2=[8,5] dB, delta=$0\degree$.

csi_1,csi_2=[8,5] dB, delta=$11\degree$.

csi_1,csi_2=[8,2] dB, delta=$0\degree$.

csi_1,csi_2=[8,2] dB, delta=$11\degree$.

Comparison of achievable data rates for varying power allocation factor of the weak user.

§.§ EEPA

In EEPA, apart from satisfying (<ref>)-(<ref>), we allocate the powers (alpha_1 and alpha_2) that maximize the EE of the paired users, where, EE is defined as

\begin{align}
\gls{EE}&=\frac{\gls{ASR}}{\gls{alpha_1}+\gls{alpha_2}},\nonumber\\&=\frac{\log_2\left[1+\left(\gls{alpha_1}\gls{csi_1}+\gls{alpha_2}\gls{csi_2}\right)\sinc^2(\gls{delta})\right]}{\gls{alpha_1}+\gls{alpha_2}}.\nonumber
\end{align}

Thus, we formulate the optimisation problem as follows.

\begin{align}
\underset{\gls{alpha_1},\gls{alpha_2}}{\max } \hspace{0.4cm} &  \frac{\log_2\left[1+\left(\gls{alpha_1}\gls{csi_1}+\gls{alpha_2}\gls{csi_2}\right)\sinc^2(\gls{delta})\right]}{\gls{alpha_1}+\gls{alpha_2}}\label{prob:eepa},\\
\text{s.t.}\hspace{0.3cm} & \eqref{eqn:Lemm3}-\eqref{eqn:Lemm1}.\nonumber
\end{align}

§.§.§ Criterion for pairing the users

Solving (<ref>) for a worst case scenario of alpha_2 $=1$,  we get,

\begin{align}
\gls{alpha_1} \ge \gls{kappa}+\gls{eta}&=\frac{2^{\gls{R1bar}} -1}{\gls{csi_1}}\left[\gls{csi_2}+\frac{1}{\sinc^2(\gls{delta})}\right].\label{eqn:alpha1ub}
\end{align}

Using (<ref>) in (<ref>), we get,

\begin{align}
\frac{2^{\gls{R1bar}} -1}{\gls{csi_1}} \left[\gls{csi_2}+\frac{1}{\sinc^2(\gls{delta})}\right]\le&1,\nonumber\\
\sinc^2(\gls{delta})\ge&\frac{1}{\left(\frac{\gls{csi_1}}{2^{\gls{R1bar}} -1}\right)-\gls{csi_2}}\triangleq \sinc^2(\gls{delub1}).
\nonumber
\end{align}

Using (<ref>) in (<ref>), we get,

\begin{align}
\gls{alpha2lb}&\le 1, \nonumber\\
\sinc^2(\gls{delta})&\ge \frac{2^{\gls{R2bar}}-1}{\gls{csi_2}} \triangleq \sinc^2(\gls{delub2}). \nonumber
\end{align}

Thus, we define the upper bound on the phase imperfection in EEPA as

\begin{align}
\gls{delubeepa}=\min\{\gls{delub1},\gls{delub2}\}.\label{eqn:delubeepa}%\nonumber
\end{align}

We define R1bar and R2bar as the achievable OMA  rates, and then, pair the users in NOMA iff the imperfection in the phase compensation is less than delubeepa. Otherwise, we consider transmitting the information for the users in an OMA scenario as shown in Fig. <ref>.

§.§.§ Power allocation

The objective function formulated in (<ref>) is a strictly pseudo-concave function [13] and an efficient way of obtaining a solution is to use the Dinkelbach's algorithm [13, 14].  An outline of implementing MPA and EEPA is presented in detail in Algorithm <ref>.

§ NUMERICAL RESULTS

csi_1,csi_2=[8,5] dB.

csi_1,csi_2=[8,2] dB.

Comparison of achievable data rates for varying imperfection in phase.

For the simulations, we have considered $M=8$, $N=32$, and $\overline{R_i}=R_i^\text{OMA}$. We have then dropped the BSs and users from Poisson point distribution with densities of 25 BS/km$^2$ and 2000 users/km$^2$, respectively. For each user, we have calculated the path loss and the received signal power from each BS by considering the urban cellular path loss model presented in [15].
The users are then associated to the BS from which they receive the maximum signal power, and the rest of the BSs are considered as interfering BSs. Given this simulation set-up, we have calculated various performance metrics with the proposed and the existing state-of-the-art algorithms which are summarised as follows.

In Fig. <ref>, we present the achievable data rates for varying alpha2mpa. For the evaluation, we consider two configurations of user pairs with $[\gls{csi_1},\gls{csi_2}]=[8,5]~\text{dB}$, $[\gls{csi_1},\gls{csi_2}]=[8,2]~\text{dB}$, and two configurations of imperfection in phase compensation, $\gls{delta}=0^\circ, 11^\circ$. As shown in Fig. <ref>, with increasing $\alpha_2$, the data rates for weak user increases. This increase in $\alpha_2$ also increases interference for the strong user, and thus, the achievable data rate for the strong user decreases. Note that for a fixed delta, the achievable data rates and sum rate are better in Fig. <ref> as compared to Fig. <ref> because of the better SINR conditions. Further, with an increase in the delta, the achievable data rates and sum rate are smaller in Fig. <ref> when compared to Fig. <ref>. As shown in Fig. <ref>, the ASR is a non-decreasing function of alpha_2 which aligns with our formulation in the Section <ref>. Further, beyond the proposed $\alpha_2=\gls{alpha2mpa}$, the individual data rates are not better than the OMA counterparts. Thus, we validate the proposed bounds on power allocation factors in the presence of imperfect phase compensation.

In Fig. <ref>, we present  the comparison of the achievable data rates for varying delta.
For the evaluation, we consider two configurations of user pairs with $[\gls{csi_1},\gls{csi_2}]=[8,5]$ dB and $[\gls{csi_1},\gls{csi_2}]=[8,2]$ dB in Fig. <ref> and Fig. <ref>,  respectively.
Further, we consider $\alpha_1=1$ and $\alpha_2=\gls{alpha2mpa}$ while calculating the achievable data rates. With an increase in delta, the achievable rates decrease. Additionally, whenever $\gls{delta}<\gls{delubmpa}$, the achievable data rates and sum-rates are always better than the OMA counterparts. Thus, we validate the proposed bound on the imperfection in the phase compensation. Note that a similar analysis is extendable for the EEPA scenario.

Mean $R_1$ vs delta.

Mean $R_2$ vs delta.

Comparison of mean achievable data rates of strong and weak user with various algorithms.

In Fig. <ref>, we present the performance comparison of mean of  achievable data rates with various algorithms. As shown in Fig. <ref>, for both strong and weak user, the performance of SRM [9] declines gradually with increase in delta and the data rates fall below OMA for larger delta. In case of the proposed algorithms, both MPA and EEPA consider the imperfections in the phase compensation, and hence, the data rates gradually converge to OMA the rates with an increase in delta.
Further, as shown in Fig. <ref>, the SRM algorithm tries to maximize the ASR and in the process significantly decreases the weak user data rates beyond the required OMA rates. However, the MPA algorithm maximizes the strong user rate to achieve maximum ASR and yet ensures both strong and weak user achieve minimum of OMA rates. The EEPA algorithm allocates minimum power to each user to ensure the minimum required OMA rates are achieved, and hence, the data rates with EEPA are significantly lower than MPA rates and slightly higher than the OMA rates.

Comparison of mean of achievable data rates of strong and weak user with various algorithms.

In Fig. <ref>, we present the comparison of the mean ASR with the proposed algorithms against the SRM and OMA in the presence of imperfect phase compensation. As shown in Fig. <ref>, SRM achieves highest ASR as compared to all the algorithms.  However, as shown in Fig. <ref>, the SRM does not achieve minimum required rates for the weak user,
whereas, the proposed MPA maximizes ASR while ensuring a minimum of OMA rates for both strong and weak users. Further, note that with EEPA, the achievable sum rate is slightly higher than the OMA rates.

In Fig. <ref>, we present the comparison of the mean achievable sum rate and energy efficiency with various algorithms.  As shown in Fig. <ref> and <ref>, the SRM has highest mean ASR and EE at the lower delta. However, note that SRM algorithm does not ensure that individual users achieve a minimum of OMA rates.
Further, with increasing delta, the mean ASR and EE of the both the proposed algorithms converge to the OMA rates, whereas, the performance of the  SRM degrades significantly as compared to the OMA rates.  Hence, it is not always beneficial to pair the users in NOMA. Thus, we conclude that, the proposed algorithms outperform the existing algorithms in presence of imperfection in phase compensation. Additionally, they also maximize the data rates or energy efficiency while ensuring minimum required data rates for each user.

Mean ASR vs delta.

Mean EE vs delta.

Comparison of mean achievable sum rate and energy efficiency with various algorithms.

§ CONCLUSION

We have proposed adaptive user pairing algorithms for IRS-assisted uplink NOMA systems that maximize the achievable sum-rate or energy efficiency. We have formulated the criterion for user pairing based on the derived bounds on imperfection in the phase compensation. Through numerical results, we have validated the derived bounds and the proposed pairing criterion. Further, we have proposed novel power allocation procedures for the paired users. We have performed extensive system-level simulations and have shown that the proposed algorithms achieve significant improvement over the state-of-the-art algorithms with increase in phase imperfection. In the future, we plan to validate the proposed algorithms on the hardware test-beds.

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