# Rotating Lifshitz-like black holes in F(R) gravity

Kh. Jafarzade1,2,3111email address<EMAIL_ADDRESS>E.
Rezaei1,2222email address<EMAIL_ADDRESS>and S. H.
Hendi1,2,4333email address<EMAIL_ADDRESS>1Department of Physics, School
of Science, Shiraz University, Shiraz 71454, Iran
2Biruni Observatory, School of Science, Shiraz University, Shiraz 71454, Iran
3ICRANet-Mazandaran, University of Mazandaran, P. O. Box 47415-416, Babolsar,
Iran
4Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada

###### Abstract

One of the alternative theories of gravitation with a possible UV completion
of general relativity is Horava-Lifshitz gravity. Regarding a particular class
of pure $F(R)$ gravity in three dimensions, we obtain an analytical rotating
Lifshitz-like black hole solution. We first investigate some geometrical
properties of the obtained solution that reduces to a charged rotating BTZ
black hole in a special limit. Then, we study the optical features of such a
black hole like the photon orbit and the energy emission rate and discuss how
electric charge, angular momentum, and exponents affect them. In order to have
an acceptable optical behavior, we should apply some constraints on the
exponents. We continue our investigation with the study of the thermodynamic
behavior of the solutions in the extended phase space and examine the validity
of the first law of thermodynamics besides local thermal stability by using
the heat capacity. Evaluating the existence of van der Waals-like phase
transition, we obtain critical quantities and show how they change under the
variation of black hole parameters. Finally, we construct a holographic heat
engine of such a black hole and obtain its efficiency in a cycle. By comparing
the obtained efficiency with the Carnot one, we examine the second law of
thermodynamics.

##  I Introduction

$F(R)$ theory of gravity is one of the straightforward generalization of
Einstein’s theory of general relativity (GR), where the Ricci scalar of GR
Lagrangian is replaced with an arbitrary function of $R$ Akbar:1a ; deSouza:1a
; Cognola:1a . Unlike Einstein’s gravity, $F(R)$ theory can explain the
accelerated expansion as well as structure formation of the Universe without
considering dark sectors Perlmutter:1a ; Riess:1a ; Riess:1b . Other
motivations to consider $F(R)$ gravity are including i) this theory seems to
be the only one that can avoid the long-known and fatal Ostrogradski
instability Woodard:1b . ii) $F(R)$ theories of gravitation can be compatible
with Newtonian and post-Newtonian approximations Capozziello:1a ;
Capozziello:1b . iii) some viable $F(R)$ models have no ghosts ( $dF/dR>0$),
and the stability condition $d^{2}F/dR^{2}\geq 0$ of essentially amounts to
guarantee that the scalaron is not a tachyon Dolgov:1b ; Faraon:1c . iv)
although $F(R)$ theory is the simplest modification of the gravitational
interaction to higher-order known so far, its action is sufficiently general
to encapsulate some of the basic characteristics of higher-order gravity.

From the geometrical point of view, Einstein’s gravity cannot explain the non-
relativistic scale-invariant theory. To describe a non-relativistic scale-
invariant system that enjoys Galilean symmetry, one can employ Horava-Lifshitz
Horava:1a ; Horava:1b  approach. In this approach, an anisotropic scale
invariant between time and space directions is considered, i.e.
$(t,x)\rightarrow(\lambda^{z}t,\lambda x)$, where the degree of anisotropy is
measured by the dynamical exponent $z$. Theories with $z\neq 1$ are invariant
under non-relativistic transformations Kachru:1ab  while for $z=1$, the theory
reduces to the relativistic isotropic scale invariance model corresponding to
the AdS spacetime. Systems with such a Lifshitz scaling appear frequently in
quantum/statistical field theory of condensed matter physics and ultra-cold
atomic gases (see Cardy:1ab  for more details). Motivated by what was
mentioned above, here, we are going to consider a Lifshitz-like geometry in a
class of three dimensional $F(R)$ gravity model.

The study of three dimensional black holes known as BTZ (Banados-Teitelboim-
Zanelli) solutions Banados:1ab  has opened different aspects of physics in low
dimensional spacetimes. The geometry of $(2+1)-$dimensional manifold has
various interesting properties such as the existence of specific relations
between the BTZ black holes and an effective action in string theory Lee:1ab ;
Larranaga:1ac , developing our understanding of gravitational interaction in
low dimensional manifolds Witten:1abc , improvement in the quantum theory of
gravity and gauge field theory Witten:1abd ; Carlip:1abx , the possibility of
the existence of gravitational Aharonov Bohm effect in the noncommutative
spacetime Anacleto:1mn , and so on. Therefore, the study of three-dimensional
black holes has attracted physicists not only in the context of Einstein’s
gravity, but also in modified theories such as massive gravity Hendi:1az ,
dilaton gravity Chan:1az , gravity’s rainbow Hendi:1ax  and also massive
gravity’s rainbow Hendi:1abx . Besides, the static vacuum solutions of a
Lifshitz model in $(2+1)-$dimensions have been investigated in Shu:1abv . In
addition, three dimensional Lifshitz-like charged black hole solutions in
$F(R)$ gravity have been also studied in Hendi:1abs . In this paper, we
introduce a new Lifshitz-like charged rotating black hole solution in three-
dimensional $F(R)$ gravity and investigate its optical and thermodynamical
properties.

From the theoretical viewpoint, one of the challenging subjects of black hole
physics is the thermodynamic behavior of a typical black hole. The possible
identification of a black hole as a thermodynamic object was first realized by
Bardeen, Carter, and Hawking Bardeen:1973 . They clarified the laws of black
hole mechanics and showed that these laws are corresponding to ordinary
thermodynamics. Thereafter, various thermodynamic properties of black holes
have been widely studied, and we now have a considerable understanding of the
microscopic origin of these properties due to a pioneering work by Strominger
and Vafa Strominger:1996 . The investigation of black hole thermodynamics in
anti-de Sitter (AdS) spacetime provided us with a deep insight into
understanding the quantum nature of gravity which has been one of the open
problems of physical communities Zeng:2020 ; Caldarelli:2000 ; Hertog:2005 ;
Lu:2015 . Besides, the existence of the cosmological constant can change both
the geometry and thermodynamic properties of spacetime. Notably, the study in
the context of black hole thermodynamics showed that the correspondence
between black hole mechanics and ordinary thermodynamic systems is completed
by considering the cosmological constant as a variable parameter Kubiznak:2012
. To complete the thermodynamic properties of a system, it is inevitable to
investigate the existence of phase transition and thermal stability. The
investigation of phase transition has a crucial role in exploring the critical
behavior of a system near its critical point. The black hole phase transition
was first studied by Hawking and Page who demonstrated the existence of a
certain phase transition (so-called Hawking-Page) between thermal AdS and
Schwarzschild-AdS BH which corresponds to the confinement/deconfinement phase
transition in the dual strongly coupled gauge theory Hawking:1983 . The
discovery of the first-order phase transition in the charged AdS black hole
spacetime has gained a lot of attention in recent years Chamblin:1999a ;
Chamblin:1999b . This transition displays a classical critical behavior and is
superficially analogous to a van der Waals liquid-gas phase transition.
Especially, considering the cosmological constant as a thermodynamical
variable and working in the extended phase space led to finding the additional
analogy between the black holes and the behavior of the van der Waals
liquid/gas system Gibbons:1996 ; Breton:2005 ; Kastor:2009 ; Hendi:2013 . In
this regard, some efforts have been made in the context of $P-V$ criticality
of black holes in modified theories of gravitation, such as Horava-Lifshitz
gravity Haldar:2018 ; Mo:2015 ; Jafarzade:2017 , Gauss-Bonnet gravity Cai:2013
; Mo:2014 ; Miao:2018 , Lovelock gravity Mo:2014a ; Hendi:2015 , dilaton
gravity Zhao:2013 ; Dehghani:2014 , $F(R)$ gravity Wu:2016 ; Ovgun:2018 ,
massive gravity Mirza:2014 ; Hendi:2016xz ; Chabab:2019 , gravity’s rainbow
Hendi:2016nm ; Feng:2017 , and massive gravity’s rainbow Hendi:2017fs . In
addition, from the thermodynamics point of view, one may consider a black hole
as a heat engine in the extended phase space. Indeed, the mechanical term PdV
in the first law provides the possibility of extracting the mechanical work
and consequently calculating the efficiency of a typical heat engine. The
concept of the holographic heat engine was first proposed by Johnson in
Ref.CVJohnson . He creatively employed the charged AdS black hole as a heat
engine working substance to construct a holographic heat engine and calculated
the heat engine efficiency. Afterward, holographic heat engines were
investigated in different black hole backgrounds, such as the rotating BHs
RAHennigar ; Jafarzade1 , Horava-Lifshitz BHs Jafarzade2 , Born-Infeld BHs
CVJohnson2 , charged BTZ BHs JXMo , accelerating BHs Zhang1 , black holes in
massive gravity Meng  and gravity’s rainbow Panah .

This paper is organized in the following manner: In Sec. II, we consider
three-dimensional Lifshitz-like background spacetime and obtain charged
rotating black hole solutions in a special class of $F(R)$ gravity. We also
determine the null geodesics equations as well as the radius of the photon
orbit, and explore the conditions to find an acceptable optical behavior. The
energy emission rate for these black holes and the influence of the model’s
parameters on the emission of particles are investigated. In Sec. III, we
study the thermodynamic properties of the corresponding black hole. With
thermodynamic quantities in hand, we study the thermal stability of these
black holes in the context of the canonical ensemble. We also investigate the
critical behavior of the system and discuss how the parameters of the black
holes affect critical quantities. The heat engine efficiency is the other
interesting quantity that we will evaluate in an independent subsection.
Finally, we present our conclusions in the last section.

##  II Geometrical properties

Our line of work in this paper includes investigating three-dimensional
rotating Lifshitz-like black holes in $F(R)$ gravity and studying their
geometrical and thermodynamical properties. In this section, we first
introduce the action of $3-$dimensional $F(R)$ gravity, and then we obtain the
metric function and confirm the existence of black holes. At the end of this
section, we investigate optical features of the black hole including the
photon orbit and energy emission rate, and examine the effects of electric
charge, angular momentum and dynamical exponents on the optical properties.

###  II.1  Constructing the solutions

Here, we intend to construct three-dimensional rotating Lifshitz-like black
holes in $F(R)$ gravity and study their geometrical properties. To do so, we
consider the following action

$S=\int_{\mathcal{M}}d^{2+1}x\sqrt{-g}F(R),$ (1)

in which $\mathcal{M}$ is a $3-$dimensional spacetime and $F(R)$ is an
arbitrary function of Ricci scalar $R$. Variation with respect to metric
tensor, $g_{\mu\nu}$, leads to the following field equation

$G_{\mu\nu}F_{R}-\frac{1}{2}g_{\mu\nu}[F(R)-RF_{R}]-[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square]F_{R}=0,$
(2)

where $G_{\mu\nu}$ is the Einstein tensor and $F_{R}\equiv dF(R)/dR$. Now, we
want to obtain the rotating Lifshitz-like solutions of Eq. (2). For this
purpose, we assume that the metric has the following ansatz

$ds^{2}=-\left(\frac{r}{r_{0}}\right)^{z}f\left(r\right)dt^{2}+\frac{dr^{2}}{f\left(r\right)}+r^{2}\left(d\varphi-\frac{J\left(\frac{r}{r_{0}}\right)^{w}}{2r^{2}}dt\right)^{2},$
(3)

where $z$ and $w$ play the role of dynamical exponents and $r_{0}$ is an
arbitrary (positive) length scale. Here, we study black hole solutions with
constant Ricci scalar with the condition of $F(R_{0})=F_{R}=0$, and therefore,
it is easy to show that the equation of motion reduces to the following
differential equation

$R=R_{0}=\frac{J^{2}}{2r^{4}}\left(\frac{r}{r_{0}}\right)^{2w-z}\left(\frac{w^{2}}{4}-w+1\right)-\frac{z^{2}f}{2r^{2}}-\left(2+\frac{3z}{2}\right)\frac{f^{\prime}}{r}-f^{\prime\prime},$
(4)

with the following metric function as the solution

$f\left(r\right)=-\Lambda
r^{2}-\frac{m}{r^{\gamma}}+\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}}{r^{\delta}}+\frac{\left(w-2\right)^{2}\left(\frac{r}{r_{0}}\right)^{\left(2w-z\right)}J^{2}}{8r^{2}\left[4w^{2}-(z+6)w+2\right]},$
(5)

where $m$ and $q$ are two integration constants related to the total mass and
electric charge of the black hole, respectively. It is worth mentioning that
these two integration constants are set in such a way that for the case of
$w=z=0$, the solution reduces to the rotating BTZ black hole solution in the
presence of a special model of the Power-Maxwell field. Here, $\Lambda$ is a
(positive/negative or zero) constant which depends on the sign/value of
$R_{0}$ as

$\Lambda=\frac{2R_{0}}{z^{2}+6z+12}.$ (6)

Besides, $\gamma$ and $\delta$ are defined as

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(3z+2-\sqrt{z^{2}+12z+4}\right),$ (7)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(3z+2+\sqrt{z^{2}+12z+4}\right).$

(a) $z=w=0$ (b) $z=0$ and $w=3$

(c) $z=1$ and $w=0$ (d) $z=3$ and $w=0$

Figure 1:  The admissible parameter space to have a physical solution for
$m=1$, $r_{0}=1$, $\Lambda=-0.5$ (dotted line), $\Lambda=-1$ (dash-dotted
line) and $\Lambda=-1.5$ (dashed line).

###  II.2 Singularity and Event horizon

With the exact solution in hand, we examine whether the obtained solution
could be interpreted as a black hole. The interpretation of solution as a
(singular) black hole has two criteria: I) Presence of singularity, II)
Existence of an event horizon covering the singularity. In order to look for
the singularity, we use the Kretschmann scalar as

$\displaystyle K$ $\displaystyle=$ $\displaystyle
f^{\prime\prime^{2}}+\left(\frac{3zf^{\prime}}{r}+\frac{z\left(z-2\right)f}{r^{2}}-\frac{3J^{2}\left(\frac{r}{r_{0}}\right)^{2w-z}\left(w-2\right)^{2}}{4r^{2}}\right)f^{\prime\prime}+$
(8)
$\displaystyle\frac{(\frac{9z^{2}}{4}+2)}{r^{2}}f^{\prime^{2}}+\left(\frac{3\left(z^{2}-2z+\frac{4}{3}\right)zf}{2r^{3}}-\frac{J^{2}\left(\frac{r}{r_{0}}\right)^{2w-z}\left(w-2\right)^{2}\left(\frac{9z}{4}-1\right)}{2r^{5}}\right)f^{\prime}+$
$\displaystyle\frac{z^{2}\left(\frac{z^{2}}{4}-z+2\right)}{r^{4}}f^{2}-\frac{J^{2}\left(\frac{r}{r_{0}}\right)^{2w-z}\left(w-2\right)^{2}\left[z^{2}-z\left(w+2\right)+w^{2}\right]}{2r^{6}}f+\frac{11J^{2}\left(\frac{r}{r_{0}}\right)^{4w-2z}\left(w-2\right)^{4}}{64r^{8}}.$

Inserting the metric function $f(r)$ into Eq. (8), one finds

$\displaystyle K$ $\displaystyle=$
$\displaystyle\Upsilon_{1}r^{2(w-z-4)}+(\Upsilon_{2}r^{\gamma}+\Upsilon_{3}r^{\gamma+\delta+2}+\Upsilon_{4}r^{\delta})r^{2w-2.5z-7}$
$\displaystyle+\left[(\Upsilon_{5}r^{\gamma+2}+\Upsilon_{6})r^{\gamma+\delta}+(\Upsilon_{7}r^{2\gamma+4}+\Upsilon_{8}r^{\gamma+2}+\Upsilon_{9})r^{2\delta}+\Upsilon_{10}r^{2\gamma}\right]r^{-3(z+2)},$

where $\Upsilon_{i}$’s are functions of $z$, $w$, $J$, $r_{0}$, $\Lambda$, $m$
and $q$. According to our analysis, the scalar curvature diverges in the limit
of $r\rightarrow 0$ which confirms that there is a curvature singularity at
$r=0$.

(a) $z=1$ and $w=0$ (b) $z=1$ and $w=1$ (c) $z=1$ and $w=3$

(d) $z=2$ and $w=0$ (e) $z=2$ and $w=1$ (f) $z=2$ and $w=3$

Figure 2:  The admissible parameter space to have a physical solution for
$m=1$, $r_{0}=1$ and $\Lambda=-1$.

Now, we require to investigate the second condition (existence of a
horizon(s)). Strictly speaking, roots of the metric function, $g^{rr}=f(r)$,
are where black hole’s horizons are. The absence of a root for the metric
function indicates that the solution is not a black hole but a naked
singularity. Due to the fact that the metric function goes to $+\infty$ for
spatial infinity and also near the origin, one can find that such a function
has a minimum ($r_{min}$). Depending the sign of $f(r_{min})$, one may find a
black hole with two horizons ($f(r_{min})<0$), an extreme black hole
($f(r_{min})=0$) or naked singularity ($f(r_{min})>0$). We examine the
condition of extreme BH by studying the following criteria

$f(r_{min})=0=f^{\prime}(r_{min}).$ (9)

Equation (9) shows that the metric function has one degenerate horizon at
$r_{min}$ which corresponds to the radius of extremal black holes (the
coincidence of the inner and outer BH horizons). Solving these two equations,
simultaneously, leads to

$\displaystyle q$ $\displaystyle=$
$\displaystyle\left(\frac{\left[(2w-z-4)\Lambda
r_{min}^{\gamma+2}+m(\gamma+2w-z-2)\right]r_{min}^{\delta}2^{\frac{1}{4}}}{r_{min}^{\gamma}\left(2w-z-2+\delta\right)}\right)^{\frac{2}{3}},$
(10) $\displaystyle J$ $\displaystyle=$
$\displaystyle\left(\frac{8\left(m(\gamma-\delta)-\Lambda(\delta+2)r_{min}^{\gamma+2}\right)\left(w(z+6)-2-4w^{2}\right)}{r_{min}^{\gamma-2}\left(2w-z-2+\delta\right)\left(w-2\right)^{2}\left(\frac{r_{min}}{r_{0}}\right)^{2w-z}}\right)^{\frac{1}{2}}.$

The resultant curve provides a lower bound for the existence of the black
hole, and is depicted in Fig. 1 see dotted, dash-dotted and dashed lines and
Fig. 2 by the dotted line, denoting the extremal limit. Below this line, black
holes (with two horizons) are present, whereas no black hole exists above it.
As it was observed, the second condition for the existence of roots (horizons)
for the metric function can be satisfied. In other words, the curvature
singularity can be covered by an event horizon. So, the obtained solution can
be interpreted as a black hole solution.

A significant point regarding these figures is that the admissible parameter
space is highly affected by values of the exponent $w$. For case of $z=0$,
there is no physical solution for values of $w=1$ and $w=2$. For case of
$z\neq 0$, a physical solution cannot be observed for small values of the
electric charge for $w=1$ (see the middle panels of Fig. 2). For the same $w$,
the admissible parameter space decreases with increase of $z$ (compare left
panels of these two figures with each other). Also, from Fig. 1, one can find
that the cosmological constant has a decreasing effect on admissible parameter
space.

###  II.3 Optical features

Here, we are going to investigate another geometric property of the black
hole, photon orbit. To do so, we investigate the photon orbit radius of the
black hole and explore the effect of the exponents $z$ and $w$ and parameters
$q$ and $J$ on the radii size. At the first step, we employ the Hamilton-
Jacobi equation for null curves as Carter1a

$\frac{\partial S}{\partial\lambda}=-\frac{1}{2}g^{\mu\nu}\frac{\partial
S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu}},$ (11)

where $S$ and $\lambda$ denote, respectively, the Jacobi action of the photon
and the affine parameter of the null geodesic. Using known constants of the
motion, one can separate the Jacobi function as follows

$S=-Et+L\phi+S_{r}(r),$ (12)

where $E$ and $L$ are, respectively, the energy and angular momentum of the
photon in the direction of rotation axis. By inserting the Jacobi action (12)
into the Hamilton-Jacobi equation (11), and using also the metric components,
we acquire

$r^{2}f(r)\left(\frac{r}{r_{0}}\right)^{z}\left(\frac{dS_{r}}{dr}\right)^{2}-\frac{L^{2}J^{2}}{4r^{2}f(r)}\left(\frac{r}{r_{0}}\right)^{2w}+L^{2}\left(\frac{r}{r_{0}}\right)^{z}-\frac{E^{2}r^{2}}{f(r)}-\frac{ELJ}{f(r)}\left(\frac{r}{r_{0}}\right)^{w}=0.$
(13)

Considering $S_{r}^{\prime}(r)=\frac{\sqrt{\mathcal{R}(r)}}{f(r)}$ and
inserting it into Eq. (13), one finds

$\mathcal{R}(r)=\frac{f(r)}{r^{2}\left(\frac{r}{r_{0}}\right)^{z}}\left(\frac{L^{2}J^{2}}{4r^{2}f(r)}\left(\frac{r}{r_{0}}\right)^{2w}-L^{2}\left(\frac{r}{r_{0}}\right)^{z}+\frac{E^{2}r^{2}}{f(r)}+\frac{ELJ}{f(r)}\left(\frac{r}{r_{0}}\right)^{w}\right).$
(14)

Thus, the photon propagation obeys the following three equations of motion,
obtained from the variation of the Jacobi action with respect to the affine
parameter $\lambda$

$\displaystyle\frac{dt}{d\lambda}=\frac{E}{f(r)\left(\frac{r}{r_{0}}\right)^{z}}-\frac{LJ}{2r^{2}f(r)}\left(\frac{r}{r_{0}}\right)^{w-z}$
(15) $\displaystyle\frac{dr}{d\lambda}=\sqrt{\mathcal{R}(r)},$ (16)
$\displaystyle\frac{d\varphi}{d\lambda}=\frac{L}{r^{2}}-\frac{LJ^{2}}{4r^{4}f(r)}\left(\frac{r}{r_{0}}\right)^{2w-z}-\frac{EJ}{2r^{2}f(r)}\left(\frac{r}{r_{0}}\right)^{w-z}.$
(17)

(a) $z=w=0$ (b) $z=w=0$

(c) $z=0$ and $w=2$ (d) $z=0$ and $w=2$

Figure 3:  The admissible parameter space to have a real horizon (left panels)
and a real photon sphere (right panels) for $m=1$, $\Lambda=-1$, $r_{0}=1$ and
$J=0.4$.

In order to investigate the photon trajectories, one usually expresses the
radial geodesics in terms of the effective potential $V_{\text{eff}}$ as

$\left(\frac{dr}{d\lambda}\right)^{2}+V_{\text{eff}}=0,$

with

$V_{\text{eff}}=\frac{f(r)}{r^{2}\left(\frac{r}{r_{0}}\right)^{z}}\left(L^{2}\left(\frac{r}{r_{0}}\right)^{z}-\frac{E^{2}r^{2}}{f(r)}-\frac{ELJ}{f(r)}\left(\frac{r}{r_{0}}\right)^{w}-\frac{L^{2}J^{2}}{4r^{2}f(r)}\left(\frac{r}{r_{0}}\right)^{2w}\right).$
(18)

Now, we are in a position to obtain the photon critical circular orbit.
Therefore, the following unstable conditions should be satisfied,
simultaneously

$V_{\text{eff}}(r_{ph})=0,\quad~{}~{}~{}\frac{dV_{\text{eff}}(r_{ph})}{dr}=0,~{}~{}~{}~{}\frac{d^{2}V_{\text{eff}}(r_{ph})}{dr^{2}}<0.$
(19)

To have an acceptable optical behavior, we need to examine the condition
$r_{e}<r_{ph}$ where $r_{ph}$ and $r_{e}$ are the radius of photon orbit and
event horizon radius, respectively. Figures (3) and (4) display the admissible
parameter space to have a real horizon and real photon orbit. From these two
figures, one can find that an acceptable optical behavior cannot be observed
for $w-z<3$. In fact, by comparing the left panels with the right panels of
each figure, one can see that the photon orbit radius will be imaginary in the
region where the event horizon is real. In other words, in a given region one
cannot observe the real horizon and real photon orbit, simultaneously.

(a) $z=w=2$ (b) $z=w=2$

(c) $z=2$ and $w=4$ (d) $z=2$ and $w=4$

Figure 4:  The admissible parameter space to have a real horizon (left panels)
and a real photon sphere (right panels) for $m=1$, $\Lambda=-1$, $r_{0}=1$ and
$J=0.5$.

As was already mentioned, in order to have an acceptable optical result, the
condition $w-z\geq 3$ should be satisfied which is quite evident in the tables
1-3. Since relation (19) leads to a complicated equation, it is not possible
to solve the equation analytically. Thus, we employ numerical methods to
obtain the radius of photon orbit. In this regard, several values of the event
horizon and the photon orbit radius are listed in tables 1 ($z=0$), 2 ($z=1$),
and 3 ($z=2$). According to these tables, only for limited regions of the
electric charge and cosmological constant, one can observe acceptable optical
results. As one can see, the increase of $q$ and $|\Lambda|$ leads to an
imaginary event horizon which is not a physical consequence. From these three
tables, it can also be seen that the electric charge, angular momentum and the
absolute value of the cosmological constant have decreasing effects on the
event horizon and the radius of photon orbit. Taking a close look at the
tables, one can notice that the effect of parameter $r_{0}$ is opposite of
that of electric charge and angular momentum. Studying the effect of the
exponent $w$ shows that increasing this parameter leads to increasing
(decreasing) the event horizon (photon orbit). Comparing these three tables to
each other, one can examine the effect of exponent $z$. Our analysis shows
that as the parameter $z$ increases the size of the event horizon and photon
orbit radius increase.

Table 1: The event horizon ($r_{e}$)and photon sphere radius ($r_{ph}$) for the variation of $q$, $J$, $r_{0}$, $w$ and $\Lambda$ for $m=1$ and $z=0$. $q$ |  $0.3$ |  $0.4$ |  $0.5$ |  $0.6$
---|---|---|---|---
$r_{e}$ ($\Lambda=-1$, $J=0.4$, $r_{0}=1$, $w=3$) | $0.9216$ | $0.8684$ | $0.7890$ | $0.57+0.05I$
$r_{ph}$ ($\Lambda=-1$, $J=0.4$, $r_{0}=1$, $w=3$) | $1.7455$ | $1.6989$ | $1.6385$ | $1.5576$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | $\times$
$J$ |  $0.3$ |  $0.4$ |  $0.5$ |  $0.6$
$r_{e}$ ($\Lambda=-1$, $q=0.3$, $r_{0}=1$, $w=3$) | $0.9217$ | $0.9216$ | $0.9213$ | $0.9210$
$r_{ph}$ ($\Lambda=-1$, $q=0.3$, $r_{0}=1$, $w=3$) | $1.9067$ | $1.7455$ | $1.6325$ | $1.5474$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
$r_{0}$ |  $0.7$ |  $0.9$ |  $1$ |  $1.1$
$r_{e}$ ($\Lambda=-1$, $q=0.3$, $J=0.4$, $w=3$) | $0.9204$ | $0.9212$ | $0.9216$ | $0.9217$
$r_{ph}$ ($\Lambda=-1$, $q=0.3$, $J=0.4$, $w=3$) | $1.4354$ | $1.5883$ | $1.7455$ | $1.9057$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
$\Lambda$ |  $-0.5$ |  $-0.8$ |  $-1$ |  $-1.2$
$r_{e}$ ($r_{0}=1$, $q=0.3$, $J=0.4$, $w=3$) | $1.3367$ | $1.0404$ | $0.9216$ | $0.83+0.06I$
$r_{ph}$ ($r_{0}=1$, $q=0.3$, $J=0.4$, $w=3$) | $2.0506$ | $1.8348$ | $1.7455$ | $1.6777$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | $\times$
$w$ |  $4$ |  $5$ |  $6$ |  $7$
$r_{e}$ ($r_{0}=1$, $q=0.2$, $J=0.4$, $\Lambda=-1.5$) | $0.77572$ | $0.77576$ | $0.77581$ | $0.77584$
$r_{ph}$ ($r_{0}=1$, $q=0.2$, $J=0.4$, $\Lambda=-1.5$) | $1.2488$ | $1.1189$ | $1.0566$ | $1.0213$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
Table 2: The event horizon ($r_{e}$) and photon sphere radius ($r_{ph}$) for the variation of $q$, $J$, $r_{0}$, $w$ and $\Lambda$ for $m=1$ and $z=1$. $q$ |  $0.3$ |  $0.35$ |  $0.4$ |  $0.45$
---|---|---|---|---
$r_{e}$ ($\Lambda=-1$, $J=0.5$, $r_{0}=1$, $w=4$) | $0.9216$ | $0.8932$ | $0.8588$ | $0.72+0.06I$
$r_{ph}$ ($\Lambda=-1$, $J=0.5$, $r_{0}=1$, $w=4$) | $1.5012$ | $1.4898$ | $1.4768$ | $1.4552$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | $\times$
$J$ |  $0.2$ |  $0.4$ |  $0.6$ |  $0.8$
$r_{e}$ ($\Lambda=-1$, $q=0.4$, $r_{0}=1$, $w=4$) | $0.8564$ | $0.8540$ | $0.8530$ | $0.8515$
$r_{ph}$ ($\Lambda=-1$, $q=0.4$, $r_{0}=1$, $w=4$) | $2.2107$ | $1.6150$ | $1.3783$ | $1.2453$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
$r_{0}$ |  $0.8$ |  $1$ |  $1.2$ |  $1.3$
$r_{e}$ ($\Lambda=-1$, $q=0.4$, $J=0.5$, $w=4$) | $0.8487$ | $0.8588$ | $0.8597$ | $0.8599$
$r_{ph}$ ($\Lambda=-1$, $q=0.4$, $J=0.5$, $w=4$) | $1.1135$ | $1.4768$ | $1.5482$ | $2.1824$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
$\Lambda$ |  $-0.5$ |  $-0.8$ |  $-1$ |  $-1.5$
$r_{e}$ ($r_{0}=1$, $q=0.4$, $J=0.5$, $w=4$) | $1.2808$ | $0.9879$ | $0.8588$ | $0.61+0.09I$
$r_{ph}$ ($r_{0}=1$, $q=0.4$, $J=0.5$, $w=4$) | $1.6002$ | $1.4973$ | $1.4768$ | $1.4681$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | $\times$
$w$ |  $5$ |  $6$ |  $7$ |  $8$
$r_{e}$ ($r_{0}=1$, $q=0.2$, $J=0.4$, $\Lambda=-1.5$) | $0.78831$ | $0.78844$ | $0.78849$ | $0.78852$
$r_{ph}$ ($r_{0}=1$, $q=0.2$, $J=0.4$, $\Lambda=-1.5$) | $1.2809$ | $1.1480$ | $1.0831$ | $1.0455$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
Table 3: The event horizon ($r_{e}$) and photon sphere radius ($r_{ph}$) for the variation of $q$, $J$, $r_{0}$, $w$ and $\Lambda$ for $m=1$ and $z=2$. $q$ |  $0.2$ |  $0.3$ |  $0.4$ |  $0.5$
---|---|---|---|---
$r_{e}$ ($\Lambda=-1$, $J=0.5$, $r_{0}=1$, $w=5$) | $0.9640$ | $0.9248$ | $0.8281$ | $0.81+0.15I$
$r_{ph}$ ($\Lambda=-1$, $J=0.5$, $r_{0}=1$, $w=5$) | $1.4011$ | $1.3891$ | $1.3733$ | $1.3525$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | $\times$
$J$ |  $0.2$ |  $0.4$ |  $0.6$ |  $0.8$
$r_{e}$ ($\Lambda=-1$, $q=0.4$, $r_{0}=1$, $w=5$) | $0.8302$ | $0.8290$ | $0.8270$ | $0.8257$
$r_{ph}$ ($\Lambda=-1$, $q=0.4$, $r_{0}=1$, $w=5$) | $1.9762$ | $1.4922$ | $1.2868$ | $1.2202$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
$r_{0}$ |  $0.9$ |  $1$ |  $1.1$ |  $1.2$
$r_{e}$ ($\Lambda=-1$, $q=0.4$, $J=0.5$, $w=5$) | $0.8248$ | $0.8281$ | $0.8290$ | $0.8300$
$r_{ph}$ ($\Lambda=-1$, $q=0.4$, $J=0.5$, $w=5$) | $1.1836$ | $1.3733$ | $1.5908$ | $1.8348$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓
$\Lambda$ |  $-0.5$ |  $-0.8$ |  $-1$ |  $-1.1$
$r_{e}$ ($r_{0}=1$, $q=0.4$, $J=0.5$, $w=5$) | $0.9872$ | $0.8417$ | $0.8281$ | $0.76+0.07I$
$r_{ph}$ ($r_{0}=1$, $q=0.4$, $J=0.5$, $w=5$) | $1.4472$ | $1.3738$ | $1.3733$ | $1.3726$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | $\times$
$w$ |  $6$ |  $7$ |  $8$ |  $9$
$r_{e}$ ($r_{0}=1$, $q=0.2$, $J=0.4$, $\Lambda=-1.5$) | $0.81788$ | $0.81794$ | $0.81799$ | $0.81800$
$r_{ph}$ ($r_{0}=1$, $q=0.2$, $J=0.4$, $\Lambda=-1.5$) | $1.2768$ | $1.1616$ | $1.1013$ | $1.0649$
$r_{ph}>r_{e}$ | ✓ | ✓ | ✓ | ✓

###  II.4 Energy emission rate

Now, we are interested in studying the effect of the black hole parameters on
the emission of particles around the black hole. It has been known that at
very high energies, the absorption cross-section for black holes oscillates
around a limiting constant value $\sigma_{lim}$ which is defined in the
following form for an arbitrary spacetime dimension Wei:2013

$\sigma_{lim}=\frac{\pi^{\frac{d-2}{2}}b_{c}^{d-2}}{\Gamma(\frac{d}{2})},$
(20)

where the critical impact parameter $b_{c}$ is given by

$b_{c}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}.$ (21)

The energy emission rate for three-dimensional spacetime is obtained as
Decanini

$\frac{d^{2}E(\omega)}{dtd\omega}=\frac{4\pi^{2}\omega^{2}b_{c}}{e^{\frac{\omega}{T}}-1},$
(22)

where $\omega$ is the emission frequency and $T$ is the Hawking temperature.
For the corresponding black hole, the Hawking temperature is given by

$T=\frac{\kappa}{2\pi}=\frac{f^{\prime}(r)}{4\pi}\left(\frac{r}{r_{0}}\right)^{\frac{z}{2}}\Bigg{|}_{r=r_{e}},$
(23)

in which $\kappa$ is the surface gravity. Using Eqs. (23) and (5), one can
find

$T=\frac{\left(\frac{r_{e}}{r_{0}}\right)^{\frac{z}{2}}}{4\pi}\left(-\frac{A_{1}\left(w-2\right)^{2}J^{2}\left(\frac{r_{e}}{r_{0}}\right)^{2w-z}}{32A_{2}r_{e}^{3}}-\Lambda\left(\gamma+2\right)r_{e}-\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}\left(\delta-\gamma\right)}{r_{e}^{\gamma+1}}\right),$
(24)

where

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle 2\delta-2\gamma-8w+z+6,$
(25) $\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle 2-zw-6w+4w^{2},$

To study the impact of black hole parameters on the energy emission rate, we
have plotted Fig. 5. Figure 5(a) illustrates the influence of the electric
charge on the emission rate of a Lifshitz rotating black hole. As it is clear,
there exists a peak of the energy emission rate for the black hole which
shifts to the low frequency with the increase of $q$. From this figure, one
can also find that this parameter has a decreasing contribution to the energy
emission rate. This reveals the fact that the evaporation process would be
slow for a black hole located in a powerful electric field. The effect of the
angular momentum on the emission rate is depicted in Fig. 5(b), indicating
that the impact of this parameter is opposite of that of the electric charge.
Studying the impact of parameter $r_{0}$ and cosmological constant, we observe
that both parameters have a decreasing effect on this optical quantity similar
to the electric charge (see Figs. 5(c) and 5(d)). This reveals the fact that
as the effect of these two parameters get weak the energy emission rate
becomes significant. To study the effects of two exponents, we plot Figs. 5(e)
and 5(f). From Fig. 5(e), it is clear that the parameter $w$ has an increasing
contribution to the energy emission rate, while the effect of the exponent $z$
is to decrease it (see Fig. 5(f)). From what was expressed, one can find that
the black hole has a longer lifetime when it rotates slowly or when it is
located in a high curvature background or a powerful electric field.

(a) $z=0$, $w=3$, $J=0.4$, $r_{0}=1$, $\Lambda=-1$ (b) $z=0$, $w=3$, $q=0.3$,
$r_{0}=1$, $\Lambda=-1$ (c) $z=0$, $w=3$, $J=0.4$, $q=0.3$, $\Lambda=-1$

(d) $z=0$, $w=3$, $J=0.4$, $q=0.3$, $r_{0}=1$ (e) $z=0$, $q=0.3$, $J=0.4$,
$r_{0}=1$, $\Lambda=-1$ (f)  $w=6$, $J=0.4$, $q=0.2$ ,$r_{0}=1$,
$\Lambda=-1.5$

Figure 5: Energy emission rate for the corresponding black hole with $m=1$ and
different values of black hole parameters.

##  III Thermodynamic properties

In this section, we would like to study the thermodynamic structure of the
system. We first calculate the conserved and thermodynamics quantities of the
black hole solution and examine the first law of thermodynamics. Then, we
investigate phase transition and thermal stability of the black hole in the
context of the canonical ensemble by calculating the heat capacity. We also
examine the effects of the black hole parameters on phase transition and
stability of the system and show that a certain relation between the exponents
$z$ and $w$ should be satisfied in order to have a phase transition. We
investigate the possibility of the existence of van der Waals-like phase
transition and critical behavior for the solutions, and determine critical
values. Finally, we construct a heat engine by taking into account this black
hole as the working substance, and obtain the heat engine efficiency.
Comparing the engine efficiency with Carnot efficiency, we investigate the
criteria of having a consistent thermodynamic second law.

###  III.1 Thermodynamic quantities and the first law

In this subsection, we obtain the thermodynamical quantities of the solutions
and check the validity of the first law of black hole thermodynamics. Before
we go on, we introduce a new notion for our solutions. We consider the
negative branch of cosmological constant to be a thermodynamical quantity
known as pressure. Considering the cosmological constant as a thermodynamical
pressure and its conjugate quantity as a thermodynamical volume leads to a new
insight into thermodynamical structure of the black holes, called extended
phase space thermodynamics. From now on, we replace the cosmological constant
with the pressure using the following relation Kubiznak:2012

$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle-8\pi P$

The finite mass is the first quantity that we would like to calculate. In the
non-extended phase space (the cosmological constant is not allowed to vary),
the total mass of the black holes is depicted as internal energy. Considering
the variable cosmological constant, the role of the mass is changed to
enthalpy. There are several methods for calculating the mass. Here, to
calculate this property, we use ADM (Arnowitt-Deser-Misner) approach which
yields

$M=\frac{m}{16\pi}r_{0}^{-\gamma}.$ (26)

Evaluating the metric function on horizon $(f(r=r_{e})=0)$ and solving it with
respect to geometrical mass result into the following relation for total mass
of the black hole

$M=\left(\frac{\left(w-2\right)^{2}\left(\frac{r_{e}}{r_{0}}\right)^{2w-z}}{128\pi
A_{2}r_{e}^{2}}J^{2}+\frac{r_{e}^{2}P}{2}+\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}}{16\pi
r_{e}^{\delta}}\right)\left(\frac{r_{e}}{r_{0}}\right)^{\gamma}.$ (27)

The temperature of the black hole has already been obtained from Eq. (24).
Replacing cosmological constant with pressure, it can be rewritten as

$T=\frac{\left(\frac{r_{e}}{r_{0}}\right)^{\frac{z}{2}}}{4\pi}\left(-\frac{A_{1}\left(w-2\right)^{2}J^{2}\left(\frac{r_{e}}{r_{0}}\right)^{2w-z}}{32A_{2}r_{e}^{3}}+8\pi
P\left(\gamma+2\right)r_{e}-\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}\left(\delta-\gamma\right)}{r_{e}^{\gamma+1}}\right).$
(28)

As the next step, we calculate the entropy of the black hole. The method for
obtaining the entropy of black holes depends on gravities under consideration
and topological structure of the black holes. In Einsteinian black holes,
without higher curvature terms, the entropy could be obtained by using the
area law. But, since our solutions are obtained in a class of $F(R)$ gravity
with $F_{R}=0$, we suppose the validity of the first law of thermodynamics to
calculate the entropy as

$\delta S=\frac{1}{T}\delta M,$ (29)

yielding

$S=\frac{\left(4+2\delta-\gamma\right)r_{e}\left(\frac{r_{e}}{r_{0}}\right)^{\gamma-\frac{z}{2}}}{24}.$
(30)

As we see, entropy depends only on the exponent $z$ and there is no direct
contributions of matter field. Besides, using the concept of enthalpy, one can
obtain the volume of these black holes as

$V=\left(\frac{\partial H}{\partial
P}\right)\Bigg{|}_{S,Q,J}=\frac{r_{e}^{2}\left(\frac{r_{e}}{r_{0}}\right)^{\gamma}}{2}.$
(31)

Evidently, there is a direct relationship between the total volume of the
black holes and the horizon radius, indicating that one can use the horizon
radius instead of using volume in calculations.

The total electric charge of the black hole can be obtained from the power
Maxwell nonlinear electrodynamics as

$Q=\frac{3\sqrt{q}2^{\frac{3}{4}}}{32\pi},$ (32)

and the electric potential is determined as

$U=r_{e}^{-\delta}q\left(\frac{r_{e}}{r_{0}}\right)^{\gamma}.$ (33)

To obtain the angular velocity, we take advantage of the standard equation as

$\Omega=-\frac{g_{t\varphi}}{g_{\varphi\varphi}}=\frac{J\left(\frac{r_{e}}{r_{0}}\right)^{w}}{2r_{e}^{2}}.$
(34)

Considering the above equation and the first law, the angular momentum can be
obtained as

$\xi=\frac{\left(\frac{r_{e}}{r_{0}}\right)^{w+\gamma-z}\left(w-2\right)^{2}}{32\pi\left(2-zw-6w+4w^{2}\right)}J.$
(35)

It is easy to show that the first law of thermodynamics is as follows

$dM=TdS+UdQ+VdP+\Omega d\xi,$ (36)

Taking into account the scaling argument for our Lifshitz like solutions in
the extended phase space, one can find the following Smarr relation holds

$\gamma
M=\frac{\left(4-\delta+2\gamma\right)}{3}TS+\frac{\delta}{3}QU-2PV-\frac{\left(3w-\delta-\gamma\right)}{3}\Omega\xi.$
(37)

It is notable that for $z=w=0$, Eq. (37) reduces to that of nonlinearly
charged rotating BTZ black holes in which mass term has no scaling.

(a) $z=0.5$, $w=1.5$, $J=0.5$, $r_{0}=0.1$, $\Lambda=-0.01$ (b) $z=0.5$,
$w=1.5$, $q=0.2$, $r_{0}=0.1$, $\Lambda=-0.01$ (c) $z=0.5$, $w=1.5$, $q=0.2$,
$J=0.5$, $\Lambda=-0.01$

(d) $z=0.5$, $J=0.5$, $q=0.2$, $r_{0}=0.1$, $\Lambda=-0.01$ (e) $w=1$,
$J=0.3$, $q=0.2$, $r_{0}=0.1$, $\Lambda=-0.01$ (f) $z=0.5$, $w=1.5$, $J=0.5$,
$q=0.2$, $r_{0}=0.1$

Figure 6: Heat capacity versus $r_{e}$ for different values of parameters.

###  III.2 Thermal stability and phase transition

Heat capacity is one of the interesting thermodynamical quantities which could
be used to extract two important properties of the solutions: I) Phase
transition points. II) Thermal stability of the solutions. The signature of
heat capacity determines the thermal stability/instability of the system. The
positivity of heat capacity represents the black hole being in thermally
stable state, while the opposite corresponds to thermally unstable case. As
was already mentioned, the heat capacity can provide a mechanism to study the
phase transition of the system. In fact, this thermodynamic quantity can be
employed to investigate two distinctive points, bound and phase transition
points. The bound point is where the sign of temperature is changed. In other
words, the root of temperature (or heat capacity) indicates a limitation
point, which separates physical solutions (positive temperature) from non-
physical ones (negative temperature). The phase transition point may be
related to the divergence points of $C$. Indeed, the divergencies of the heat
capacity are where the system goes under phase transition. The heat capacity
is given by

$\displaystyle C_{P,Q,J}$ $\displaystyle=$ $\displaystyle
T\left(\frac{\partial S}{\partial T}\right)_{P,Q,J}.$ (38)

(a) $z=0.5$, $w=1.5$, $J=0.5$, $r_{0}=0.1$, $\Lambda=-0.01$ (b) $z=0.5$,
$w=1.5$, $q=0.2$, $r_{0}=0.1$, $\Lambda=-0.01$ (c) $z=0.5$, $w=1.5$, $q=0.2$,
$J=0.5$, $\Lambda=-0.01$

(d) $z=0.5$, $J=0.5$, $q=0.2$, $r_{0}=0.1$, $\Lambda=-0.01$ (e) $w=1.5$,
$J=0.5$, $q=0.2$, $r_{0}=0.1$, $\Lambda=-0.01$ (f) $z=0.5$, $w=1.5$, $J=0.5$,
$q=0.2$, $r_{0}=0.1$

Figure 7: Thermally stable and/or unstable regions of the black holes.

Employing Eqs. (30), (28) and (38), one can find

$\displaystyle C_{P,Q,J}$ $\displaystyle=$ $\displaystyle\frac{\pi
r_{e}(\frac{r_{e}}{r_{0}})^{\gamma-\frac{z}{2}}}{B_{1}+B_{2}-B_{3}}((1+\gamma-\frac{z}{2})(\gamma-2\delta-4)B_{4}),$
(39)

in which

$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle-3A_{1}r_{e}^{1+\delta}\left(w-2\right)^{2}(4w-z-6)\left(\frac{r_{e}}{r_{0}}\right)^{2w-z}J^{2}$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle 768\pi
PA_{2}(2+z)(\gamma+2)r_{e}^{5+\delta}$ $\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle-48A_{2}(A_{1}+8w)(\delta-\gamma)2^{-\frac{1}{4}}q^{\frac{3}{2}}r_{+}^{3}$
$\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle\frac{A_{1}J^{2}\left(w-2\right)^{2}r_{e}^{1+\delta}}{2}\left(\frac{r_{e}}{r_{0}}\right)^{2w-z}-512r_{e}^{3}\left((\gamma+2)\pi
Pr_{e}^{2+\delta}-\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}A_{2}\left(\gamma+2\right)}{32}\right).$

The behavior of the heat capacity with respect to the horizon radius is
addressed in Fig. 6. According to this figure, there are three possibilities
for the heat capacity:

Case I) One root: there are two phases of small and large black holes. The
small one is not physical due to the negativity of the temperature. Whereas,
the large black hole phase is thermally stable.

Case II) One root and one divergency: in this case, there are three phases
small, medium, and large black holes. The small black holes have negative
temperatures and therefore, are not physical. The medium and large phases are
separated by a divergence point. At this point, two phases of medium and large
black holes are in equilibrium and go from one to the other via a critical
process.

Case III) One root and two divergencies: In this case, four distinguishable
phases can be observed for black holes; very small, small, medium, and large
black holes. For the very small black hole phase, the temperature is negative
and so this phase is not a physical one. For small and large black hole
phases, heat capacity is positive and these two phases are thermally stable.
Medium black hole phases which are located between two divergencies have
negative heat capacity. Therefore, this phase is not physical and accessible
to the black holes.

Figure 6 also displays the effects of different parameters on the heat
capacity. In general, we can highlight the following effects of variation of
different parameters on the heat capacity.

i) According to Fig. 6(a), there is a critical value for the electric charge
where for values smaller than it, two divergencies exist for the heat
capacity. These two divergencies coincide with each other for this critical
value of the electric charge. For the electric charges larger than this
critical value, no divergency appears in the structure of the heat capacity.

ii) Figure 6(b) shows that there is a critical value for the angular momentum
as well. The only difference is that for angular momentums smaller than this
critical value, no divergence point is observed.

iii) The effect of parameter $r_{0}$ is depicted in Fig. 6(c), indicating that
its contribution to the heat capacity is the same as the effect of the
electric charge. In other words, for the values of $r_{0}$ smaller than its
critical value, two divergence points appear. While, no divergency observe for
larger than the critical value.

iv) Figure 6(d) illustrates the effect of the exponent $w$ on the heat
capacity. Taking a closer look at this figure, one can find that its effect is
similar to the angular momentum. The difference is that for fixed parameters
$q$, $J$, $z$, $r_{0}$ and $\Lambda$, there is a specific value of $w$ for
which the heat capacity has only one divergency without any root (see the
dashed curve of Fig. 6(d)). For this specific value, two phases exist small
and large black holes. Small black holes have a negative heat capacity and are
thermally unstable. Whereas, large black holes are in a stable state due to
the positivity of heat capacity. For values of $w$ between this specific value
and the critical value, there are two divergencies for the heat capacity.

v) To study the effect of exponent $z$ and the cosmological constant, we plot
Figs. 6(e) and 6(f), indicating that their effects are similar to the electric
charge.

To have a more precise picture regarding the effects of different parameters
on thermal stability/instability of the solutions, we have plotted Fig. 7. As
we see, by decreasing (increasing) of the electric charge, cosmological
constant and parameter $r_{0}$ (angular momentum), the stability region of the
system decreases. In the case of exponents, the effect of each one on the
stability of the system depends on the value of the other. In fact, the value
of $w$ for which the system is thermally unstable are quite dependent on the
value of $z$ and vice versa.

(a) $z=0.5$, $w=1.5$, $J=0.6$, $r_{0}=0.2$ (b) $z=0.5$, $w=1.5$, $J=0.6$,
$r_{0}=0.2$ (c) $z=0.5$, $w=1.5$, $J=0.6$, $r_{0}=0.2$

(d) $z=1$, $w=0.5$, $J=0.3$, $r_{0}=1$ (e) $z=1$, $w=0.5$, $J=0.3$, $r_{0}=1$
(f) $z=1$, $w=0.5$, $J=0.3$, $r_{0}=1$

Figure 8: van der Waals like phase diagrams for $q=0.1$. Left panels:
$P-r_{e}$ diagram for $T<T_{c}$ (continuous line), $T=T_{c}$ (dash-dotted
line), and $T>T_{c}$ (dotted line). Middle panels: $T-r_{e}$ diagram for
$P<P_{c}$ (continuous line), $P=P_{c}$ (dash-dotted line), and $P>P_{c}$
(dotted line). Right panels: $G-T$ diagram for $P<P_{c}$ (continuous line),
$P=P_{c}$ (dash-dotted line) and $P>P_{c}$ (dotted line).

###  III.3 van der Waals like behavior

Here, we look for the possibility of the existence of van der Waals-like phase
transition for the black holes. We also extract critical thermodynamic
quantities and analyze the effects of black hole parameters on critical
values. To do so, we require to determine the equation of state which is
obtained by writing down the pressure as a function of the temperature and
thermodynamic volume. Since there is a direct relationship between the
thermodynamic volume and the horizon radius, we use the horizon radius instead
of using volume in the equation of state. From Eq. (28), the equation of state
is obtained as

$P=\frac{A_{1}J^{2}\left(w-2\right)^{2}\left(\frac{r_{e}}{r_{0}}\right)^{2w-z}}{256A_{2}\pi
r_{e}^{4}}+\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}\left(\delta-\gamma\right)}{8\pi\left(\gamma+2\right)r_{e}^{\delta+2}}+\frac{T}{2r_{e}\left(\frac{r_{e}}{r_{0}}\right)^{\frac{z}{2}}\left(\gamma+2\right)}.$
(40)

The behaviors of the pressure and temperature under variation of the event
horizon radius are depicted in Fig. 8. Evidently, a van der Waals-like phase
transition can be observed for these black holes by suitable choices of
different parameters. Indeed, the presence of subcritical isobars in $T-r_{e}$
and isothermal diagrams in $P-r_{e}$ confirm the existence of van der Waals-
like phase transition. As we know, the van der Waals fluid goes under a first
order phase transition for temperatures smaller than the critical temperature
$(T<T_{c})$. Whereas, at the critical temperature, its phase transition is a
second-order one Kubiznak:2012 . The formation of the swallow-tail shape in
the $G-T$ diagram is another evidence of the first-order (small-large) phase
transition. Figures 8(c) and 8(f), confirm the first-order phase transition
for our black hole solutions.

Our analysis shows that a first-order phase transition occurs for values of
the exponents that satisfy the condition $x_{1}<z+w<x_{2}$. It is worth
mentioning that the values of $x_{1}$ and $x_{2}$ are highly governed by the
parameter $z$. Some values of $x_{1}$ and $x_{2}$ for different values of $z$
are as follows

$\displaystyle z$ $\displaystyle=$ $\displaystyle
0.5~{}~{}\longrightarrow~{}~{}0.9<z+w<2.1$ (41) $\displaystyle z$
$\displaystyle=$ $\displaystyle
1~{}~{}~{}~{}\longrightarrow~{}~{}~{}1.35<z+w<2.75$ $\displaystyle z$
$\displaystyle=$ $\displaystyle 1.5~{}~{}\longrightarrow~{}~{}1.8<z+w<3.35$
$\displaystyle z$ $\displaystyle=$ $\displaystyle
2~{}~{}~{}~{}\longrightarrow~{}~{}~{}2.3<z+w<3.85,$

A significant point here is that if $w>z$ in the mentioned region (see the
relation (41)), the Gibbs free energy is positive. Since the Gibbs free energy
is obtained as $G=H-ST$, this reveals the fact that the system is energy-
dominated. But for $z>w$, the Gibbs free energy is negative, indicating that
the system is entropy-dominated (see Fig. 9).

(a)  (b)

Figure 9: $G-T$ diagram for $q=0.1$ and $r_{0}=0.2$.

To obtain the critical values of thermodynamic quantities, we use the concept
of the inflection point of isothermal $P-V$ diagram given by

$\left(\frac{\partial P}{\partial r_{e}}\right)_{T}=0,\ \ \
\left(\frac{\partial^{2}P}{\partial r_{e}^{2}}\right)_{T}=0$ (42)

It is a matter of calculation to show that the critical horizon radius
(volume), temperature and pressure are given by

$\displaystyle r_{c}$ $\displaystyle=$
$\displaystyle\frac{32wA_{2}r_{0}^{2w-z}2^{-\frac{1}{4}}q^{\frac{3}{2}}\left(\delta+2\right)\left(-2\delta+z-2\right)\left(\gamma-\delta\right)}{A_{1}J^{2}\left(w-2\right)^{2}(A_{2}-2)(2w-z-4)},$
$\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}\left(\delta+2\right)\left(\gamma-\delta\right)\left(\frac{r_{c}}{r_{0}}\right)^{\frac{z}{2}}r_{c}^{\delta-2}}{2(2+z)\pi
r_{c}^{3}}+\frac{A_{1}\left(2w-z-4\right)\left(w-2\right)^{2}J^{2}\left(\frac{r_{c}}{r_{0}}\right)^{2w-\frac{z}{2}}}{32A_{2}(2+z)\pi
r_{c}^{3}},$ $\displaystyle P_{c}$ $\displaystyle=$
$\displaystyle\frac{A_{1}J^{2}\left(w-2\right)^{2}(A_{2}-2)\left(\frac{r_{c}}{r_{0}}\right)^{2w-z}}{256\pi
wA_{2}r_{c}^{4}\left(\gamma+2\right)(2+z)}-\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}\left(\gamma-\delta\right)(z-2-2\gamma)}{8(\gamma+2)(2+z)\pi
r_{c}^{2+\delta}}.$ (43)

Table 4: Critical values for the variation of $q$, $J$, $r_{0}$, $w$ and $z$. $q$ |  $0.1$ |  $0.12$ |  $0.14$ |  $0.16$
---|---|---|---|---
$P_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $r_{0}=0.2$) | $27\times 10^{-5}$ | $22\times 10^{-5}$ | $19\times 10^{-5}$ | $17\times 10^{-5}$
$T_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $r_{0}=0.2$) | $0.032$ | $0.031$ | $0.03$ | $0.029$
$r_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $r_{0}=0.2$) | $1.77$ | $2.01$ | $2.24$ | $2.45$
$P_{c}r_{c}/T_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $r_{0}=0.2$) | $0.0153$ | $0.0148$ | $0.0144$ | $0.0141$
$J$ |  $0.5$ |  $0.6$ |  $0.7$ |  $0.8$
$P_{c}$ ($z=0.5$, $w=1.5$, $q=0.1$, $r_{0}=0.2$) | $14\times 10^{-5}$ | $27\times 10^{-5}$ | $46\times 10^{-5}$ | $73\times 10^{-5}$
$T_{c}$ ($z=0.5$, $w=1.5$, $q=0.1$, $r_{0}=0.2$) | $0.021$ | $0.032$ | $0.045$ | $0.060$
$r_{c}$ ($z=0.5$, $w=1.5$, $q=0.1$, $r_{0}=0.2$) | $2.10$ | $1.77$ | $1.54$ | $2.45$
$P_{c}r_{c}/T_{c}$ ($z=0.5$, $w=1.5$, $q=0.1$, $r_{0}=0.2$) | $0.0147$ | $0.0153$ | $0.0159$ | $0.0163$
$r_{0}$ |  $0.1$ |  $0.15$ |  $0.2$ |  $0.25$
$P_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $q=0.1$) | $51\times 10^{-4}$ | $93\times 10^{-5}$ | $27\times 10^{-5}$ | $11\times 10^{-5}$
$T_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $q=0.1$) | $0.263$ | $0.076$ | $0.032$ | $0.016$
$r_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $q=0.1$) | $0.801$ | $1.27$ | $1.77$ | $2.29$
$P_{c}r_{c}/T_{c}$ ($z=0.5$, $w=1.5$, $J=0.6$, $q=0.1$) | $0.0157$ | $0.0155$ | $0.0153$ | $0.0152$
$w$ |  $1$ |  $1.25$ |  $1.5$ |  $1.6$
$P_{c}$ ($z=0.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $0.038$ | $31\times 10^{-4}$ | $27\times 10^{-5}$ | $19\times 10^{-6}$
$T_{c}$ ($z=0.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $0.188$ | $0.059$ | $0.032$ | $0.026$
$r_{c}$ ($z=0.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $0.342$ | $0.787$ | $1.77$ | $3.87$
$P_{c}r_{c}/T_{c}$ ($z=0.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $0.0695$ | $0.0423$ | $0.0153$ | $0.0029$
$z$ |  $0.5$ |  $1$ |  $1.5$ |  $2$
$P_{c}$ ($w=1.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $27\times 10^{-5}$ | $91\times 10^{-6}$ | $16\times 10^{-6}$ | $19\times 10^{-7}$
$T_{c}$ ($w=1.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $0.032$ | $0.016$ | $0.007$ | $0.002$
$r_{c}$ ($w=1.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $1.77$ | $2.07$ | $2.64$ | $3.45$
$P_{c}r_{c}/T_{c}$ ($w=1.5$, $J=0.6$, $q=0.1$, $r_{0}=0.2$) | $0.0153$ | $0.0113$ | $0.0057$ | $0.0023$

Table 4 shows how critical quantities and universal critical ratio
$\left(\frac{P_{c}r_{c}}{T_{c}}\right)$ change under variation of black hole
parameters. From this table, one can find that as the electric charge
increases, the critical pressure, temperature, and universal critical ratio
decrease, whereas the critical horizon radius (volume) increases. Regarding
the effect of angular momentum on the critical quantities, one can see that
its effect is opposite of that of the electric charge. Studying the effects of
exponents and parameter $r_{0}$ indicates that their contribution to critical
values is the same as the electric charge. In other words, the critical volume
is an increasing function of these three parameters, whereas the critical
pressure, temperature, and universal critical ratio are decreasing functions
of them.

###  III.4 Heat Engine

As the final step, we would like to consider the Lifshitz rotating black hole
as a heat engine and discuss its efficiency.

A heat engine is a physical system that works between two hot and cold
reservoirs and its main role is transferring heat from the hot reservoir to
the cold one. The total mechanical work done, by the First Law, is $W=Q_{H}$
\- $Q_{C}$. So, the efficiency of the heat engine is

$\eta=\frac{W}{Q_{H}}=1-\frac{Q_{C}}{Q_{H}}.$ (44)

Figure 10: Our engine cycle

In order to calculate the efficiency, one may use the heat capacity. According
to Eqs. (30) and (31), the entropy and thermodynamic volume are related to the
horizon radius. So, these two quantities are dependent to each other for this
kind of solution. This shows that the specific heat at constant volume
vanishes $C_{V}=0$ which is the ”isochore equals adiabat” result CVJohnson .
In this case, the specific heat at constant pressure is not zero. An explicit
expression for $C_{P}$ would suggest that we can consider a rectangular cycle
such as Fig. 10, involving two isobars (paths of $1\rightarrow 2$ and
$3\rightarrow 4$) and two isochores/adiabats (paths of $2\rightarrow 3$ and
$4\rightarrow 1$). Figure 10 shows a schematic of the proposed cycle. We can
calculate the work done along the heat cycle as

$\displaystyle W$ $\displaystyle=$ $\displaystyle\oint PdV=W_{1\longrightarrow
2}+W_{2\longrightarrow 3}+W_{3\longrightarrow 4}+W_{4\longrightarrow 1}$ (45)
$\displaystyle=$ $\displaystyle W_{1\longrightarrow 2}+W_{3\longrightarrow
4}=P_{1}\left(V_{2}-V_{1}\right)+P_{4}\left(V_{4}-V_{3}\right).$

The upper isobar will give the net inflow of heat ($Q_{H}$) as follows

$Q_{H}=\int_{T_{1}}^{T_{2}}C_{p}\left(P_{1},T\right)dT=\int\limits_{r_{+1}}^{r_{+2}}C_{p}\left(P_{1},T\right)\frac{\partial
T}{\partial r}dr=Q_{H2}-Q_{H1}.$ (46)

Taking advantage of Eqs. (28) and (39), the input heat flow to the cycle is

$\displaystyle Q_{H}$ $\displaystyle=$
$\displaystyle\frac{A_{1}(w-2)^{2}J^{2}(1+\gamma-\frac{z}{2})\left(2\delta+4-\gamma\right)(\frac{r_{e}}{r_{0}})^{2w-z+\gamma}}{3072\pi
A_{2}\left(2w-z+\gamma-2\right)r_{e}^{2}}\Bigg{|}_{r_{e1}}^{r_{e2}}$ (47)
$\displaystyle+\frac{\left(1+\gamma-\frac{z}{2}\right)\left(\gamma-4-2\delta\right)(2^{\frac{4}{3}}q^{\frac{3}{2}}+16P_{1}\pi
r_{+}^{\delta+2})(\frac{r_{e}}{r_{0}})^{\gamma}}{192r_{e}^{\delta}}\Bigg{|}_{r_{e1}}^{r_{e2}}.$

Using Eq. (46) and (47), one can obtain the engine efficiency as

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{W}{Q_{H}}=\frac{64\pi
D_{1}A_{2}r_{e2}^{2}r_{e1}^{2}}{-(D_{2}+D_{3})\left((6+z)^{2}-(2\delta-2\gamma)^{2}\right)},$
(48)

in which

$\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle\frac{r_{e1}^{2+\gamma}-r_{e2}^{2+\gamma}}{r_{0}^{\gamma}}$
$\displaystyle D_{2}$ $\displaystyle=$ $\displaystyle\left(4A_{2}P_{1}\pi
r_{e1}^{\delta+2}+\frac{J^{2}(w-2)^{2}r_{e1}^{2w-z+\gamma-1}}{32r_{0}^{2w-z}}+\frac{1}{4}A_{2}2^{-\frac{1}{4}}q^{\frac{3}{2}}\right)r_{e2}^{\delta}\left(\frac{r_{e1}}{r_{0}}\right)^{\gamma}$
$\displaystyle D_{3}$ $\displaystyle=$ $\displaystyle\left(4A_{2}P_{1}\pi
r_{e2}^{\delta+2}+\frac{J^{2}(w-2)^{2}r_{e2}^{2w-z+\gamma-1}}{32r_{0}^{2w-z}}+\frac{1}{4}A_{2}2^{-\frac{1}{4}}q^{\frac{3}{2}}\right)r_{e1}^{\delta}\left(\frac{r_{e2}}{r_{0}}\right)^{\gamma}.$

Among different classical cycles, the Carnot cycle is one of the interesting
simplest cycle that can be considered. The efficiency of this cycle is the
maximum efficiency of the heat engines in such a way that any higher
efficiency would violate the second law of thermodynamics. To calculate the
Carnot efficiency, we consider the $T_{H}$ and $T_{C}$ in our cycle to
correspond to $T_{2}$ and $T_{4}$, respectively. So, this efficiency is

$\displaystyle\eta_{c}$ $\displaystyle=$ $\displaystyle
1-\frac{T_{C}}{T_{H}}=1-\frac{X_{1}(\frac{r_{e1}}{r_{0}})^{\frac{z}{2}}}{X_{2}(\frac{r_{e2}}{r_{0}})^{\frac{z}{2}}}$
(49)

where

$\displaystyle X_{1}$ $\displaystyle=$
$\displaystyle-\frac{A_{1}J^{2}(w-2)^{2}(\frac{r_{e1}}{r_{0}})^{2w-z}}{32A_{2}r_{e1}^{3}}+8\pi
P_{4}r_{e1}\left(\gamma+2\right)-\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}(\delta-\gamma)}{r_{e1}^{\delta+1}}$
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle-\frac{A_{1}J^{2}(w-2)^{2}(\frac{r_{e2}}{r_{0}})^{2w-z}}{32A_{2}r_{e2}^{3}}+8\pi
P_{4}r_{e2}\left(\gamma+2\right)-\frac{2^{-\frac{1}{4}}q^{\frac{3}{2}}(\delta-\gamma)}{r_{e2}^{\delta+1}},$

(a)  (b)

(c)  (d)

Figure 11: Variation of $\eta$ and $\frac{\eta}{\eta_{C}}$ versus $V_{2}$ for
$z=0.5$, $w=1.5$, $r_{0}=0.2$, $P_{1}=0.003$, $P_{4}=0.001$ and $V_{1}=1$. Up
panels: bold lines for $q=0.3$ and thin lines for $q=0$; $J=0.4$ (continues
line), $J=0.5$ (dashed line) and $J=0.65$ (dotted line). Down panels: bold
lines for $J=0.4$ and thin lines for $J=0$; $q=0.1$ (continues line), $q=0.2$
(dashed line) and $q=0.3$ (dotted line).

The behavior of the heat engine efficiency $\eta$ and the ratio
$\frac{\eta}{\eta_{C}}$ under variation of black hole parameters is depicted
in Figs. (11)-(13). In Fig. 11, we examine the influence of electric charge
and angular momentum on $\eta$ and the ratio $\frac{\eta}{\eta_{C}}$ for the
fixed exponents, parameter $r_{0}$ and pressures $P_{1}$, $P_{4}$. As one can
check, from the up panels of Fig. 11, both $\eta$ and the ratio
$\frac{\eta}{\eta_{C}}$ are decreasing functions of the angular momentum. For
large values of $J$, the efficiency monotonically increases as the volume
$V_{2}$ grows (see the bold dotted line of Fig. 11(a)). This means that for
rapidly charged rotating black holes, the increase of volume difference
between the small black hole ($V_{1}$) and larger black hole ($V_{2}$) will
make the heat engine more efficient. For small angular momentum, the
efficiency curve has a local minimum value, indicating that there exists a
specific value of the volume $V_{2}$ at which the black hole heat engine works
at the lowest efficiency (see the bold continuous line of Fig. 11(a)). In the
absence of the electric charge, for all values of the angular momentum the
heat engine efficiency monotonously increases with the growth of $V_{2}$ and
then tends to a constant value (see thin lines of Fig. 11(a)). Taking a close
look at Fig. 11(a), one can find that charged rotating black holes have a
bigger efficiency than their uncharged counterparts (compare bold and thin
lines). Just for large volume difference $\Delta V=V_{2}-V_{1}$, their
efficiency becomes smaller compared to that of a rapidly rotating black hole
(compare bold-dotted and thin-dotted lines in Fig. 11(a)).

(a) $w=1.5$ (b) $w=1.5$

(c) $z=0.5$ (d) $z=0.5$

Figure 12: Variation of $\eta$ and $\frac{\eta}{\eta_{C}}$ versus $V_{2}$ for
$J=0.4$, $q=0.2$, $r_{0}=0.2$, $P_{1}=0.003$, $P_{4}=0.001$ and $V_{1}=1$. Up
panels: for different values of $z$ parameter. Down panels: for different
values of $w$ parameter.

Down panels of Fig. 11 displays the effect of the electric charge on $\eta$
and the ratio $\frac{\eta}{\eta_{C}}$. As we see, although this parameter has
an increasing contribution to the efficiency, its effect is to decrease the
ratio $\frac{\eta}{\eta_{C}}$. For non-rotating black holes, all curves
monotonic reduce rapidly firstly, then the efficiency reaches a constant value
after the certain value of volume $V_{2}$ (see thin lines of Fig. 11(c)). For
slowly charged rotating black holes, the efficiency gradually increases as the
volume $V_{2}$ increases and then tends to a constant value (see the bold
continuous line of Fig. 11(c)). While for rapidly charged rotating black
holes, the efficiency decreases to a minimum value with an increase of $V_{2}$
and then gradually grows as the volume increases more, and finally reaches a
constant value in the limit of that $V_{2}$ goes to the infinity (see the bold
dotted line of Fig. 11(c)). Fig. 11(c) also shows that non-rotating black
holes have a bigger efficiency compared to the charged rotating black holes
(compare bold lines to thin lines). Comparing Fig. 11(a) to Fig. 11(c), one
can find that variation of $J$ has a stronger effect on the efficiency than
the electric charge.

Figure 12 shows how $\eta$ and the ratio $\frac{\eta}{\eta_{C}}$ are affected
by the exponents. According to the up panels of this figure, the exponent $z$
has an increasing (a decreasing) effect on the efficiency (the ratio
$\frac{\eta}{\eta_{C}}$). For small values of $z$, the efficiency
monotonically increases as the volume $V_{2}$ grows and then tends to the
saturation value (see the continuous line of Fig. 12(a)). While for large
values, the opposite behavior will be observed. Figure 12(c) displays the
influence of the exponent $w$ on the efficiency. Our findings indicate that
for $w<0.5$, the efficiency increases with the increase of $w$. While for
$w>0.5$, increasing $w$ leads to the decreasing of the heat engine efficiency.
Regarding the effect of this parameter on the ratio $\frac{\eta}{\eta_{C}}$,
in both regions ($w<0.5$ or $w>0.5$), one finds that increasing the this
parameter makes the increasing of the ratio $\frac{\eta}{\eta_{C}}$ (see Fig.
12(d)). Taking a close look at the right panels of Fig. 12, one can notice
that in the region of volume $V_{2}$ near $V_{1}$ and for small (large) values
of $w$ ($z$), the efficiency becomes larger than Carnot efficiency which
violates the second law of thermodynamics. This shows that small (large)
values of $z$ ($w$) should be considered to observe an acceptable efficiency
of the system.

(a) $\Delta P=0.002$ (b) $\Delta P=0.002$

(c) $r_{0}=0.2$ (d) $r_{0}=0.2$

Figure 13: Variation of $\eta$ and $\frac{\eta}{\eta_{C}}$ versus $V_{2}$ for
$J=0.4$, $q=0.2$, $z=0.5$, $w=1.5$ and $V_{1}=1$. Up panels: for different
values of $r_{0}$ parameter. Down panels: for different values of the pressure
difference.

The effects of the parameter $r_{0}$ and pressure on $\eta$ and the ratio
$\frac{\eta}{\eta_{C}}$ are reflected in Fig. 13. We find that the parameter
$r_{0}$ has an increasing effect on both $\eta$ and $\frac{\eta}{\eta_{C}}$.
For small (large) values of $r_{0}$, the efficiency gradually grows (reduces)
with the increase of $V_{2}$ (see Fig. 13(a)). Regarding the pressure, down
panels of Fig. 13 show that increasing the difference of pressure $\Delta P$
makes the increasing of both $\eta$ and $\frac{\eta}{\eta_{C}}$. Taking a look
at right panels of Fig. 13, one can find that the second law of thermodynamics
is satisfied for all values of these two parameters.

##  IV conclusion

In this paper, we have obtained a new Lifshitz-like rotating black hole
solution in three dimensional $F(R)$ gravity. Investigating geometrical
properties of the solution, we have found that this solution reduces to the
charged rotating BTZ-like black hole in special limits. We also studied the
optical features of the black hole and noticed that some constraints should be
imposed on the exponents to have an acceptable optical behavior. Studying the
impact of parameters of the model on the photon orbit radius illustrated that
as the electric charge, angular momentum and absolute value of the
cosmological constant increase, both event horizon and photon orbit radii
decrease. Regarding the effect of exponent $z$, our analysis showed that this
parameter has an increasing contribution to the horizon radii and photon
orbit.

After that, we continued our analysis by investigating the energy emission
rate and examining the influence of parameters on the radiation process. The
results indicated that the angular momentum and exponent $z$ have an
increasing contribution to the emission rate, namely, the emission of
particles around the BH increases by increasing these two parameters.
Regarding the role of the electric charge, cosmological constant and exponent
$w$, we have found that as the effects of these parameters get stronger, the
evaporation process gets slower. In other words, the lifetime of a black hole
would be longer under such conditions.

As the next step, we have studied the thermodynamic properties of the system
in the extended phase space thermodynamics. We calculated thermodynamic
quantities of black holes and showed that these quantities satisfy the first
law of thermodynamics. We also obtained the modified Smarr relation and found
that regardless of the cosmological constant term, scaling other thermodynamic
quantities is modified. Using heat capacity, we investigated the thermal
stability of the system and showed how the parameters of the model affect the
region of the stability. Moreover, we look for possible phase transitions and
found that three-dimensional Lifshitz-like rotating black hole experiences the
first-order/second-order phase transitions with a suitable choice of
parameters.

Finally, we have considered this kind of black hole as a working substance and
studied the holographic heat engine by taking a rectangle heat cycle in the
$P-V$ plot. Investigating the black hole heat engine efficiency and comparing
obtained results with the Carnot efficiency led to the following interesting
results:

I) The angular momentum (electric charge) has a decreasing (an increasing)
contribution to the efficiency of the system. For all values of these two
parameters, the efficiency is always smaller than the Carnot efficiency which
is consistent with the second law of thermodynamics.

II) The charged rotating black hole has a bigger (smaller) efficiency than its
uncharged (non-rotating) counterpart. It is worth pointing out that for very
large volume difference, the efficiency of rapidly rotating black hole becomes
bigger than them.

III) The heat engine efficiency is an increasing function of the exponent $z$.
For small values of this parameter, the condition $\frac{\eta}{\eta_{C}}<1$ is
satisfied all the time. But for large values of $z$, this condition is
violated in the region of volume $V_{2}$ near $V_{1}$. The contribution of the
exponent $w$ on the efficiency is a little different. For $w<0.5$, the
efficiency increases with the increase of $w$, whereas for $w>0.5$, increasing
this parameter leads to the decrease of the heat engine efficiency. For
$w<0.5$, the second law of thermodynamics violates for a very small volume
difference, whereas for $w>0.5$ it is always preserved.

IV) Increasing the pressure difference makes the increasing the efficiency of
the system. For all values of pressure, the efficiency is always smaller than
the Carnot efficiency which is consistent with the second law of
thermodynamics.

###### Acknowledgements.

The authors thank Shiraz University Research Council. KhJ is grateful to the
Iran Science Elites Federation for the financial support.

## References

  * (1) M. Akbar and R. G. Cai, Phys. Lett. B 635, 7 (2006).
  * (2) J. C. C. de Souza and V. Faraoni, Class. Quant. Grav. 24, 3637 (2007).
  * (3) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, Phys. Rev. D 77, 046009 (2008).
  * (4) S. Perlmutter, et al., Astrophys. 517, 565 (1999).
  * (5) A. G. Riess, et al., Astrophys. 116, 1009 (1998).
  * (6) A. G. Riess, et al., Astrophys. 607,665 (2004).
  * (7) R. Woodard, Lect. Notes Phys. 720, 403 (2007).
  * (8) S. Capozziello, A. Troisi, Phys. Rev. D 72, 044022 (2005).
  * (9) S. Capozziello, A. Stabile, A. Troisi, Phys. Rev. D 76, 104019 (2007).
  * (10) A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1 (2003).
  * (11) V. Faraoni, Phys. Rev. D 74, 104017 (2006).
  * (12) P. Horava, J. High Energy Phys. 09 020 (2009).
  * (13) P. Horava, Phys. Rev. D 79, 084008 (2009).
  * (14) S. Kachru, X. Liu and M. Mulligan, Phys. Rev. D 78, 106005 (2008).
  * (15) J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 2002).
  * (16) M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849
  * (17) H. -W. Lee, Y. -S. Myung, and J. -Y. Kim, Phys. Lett. B 466, 211 (1999).
  * (18) A. Larranaga, Commun. Theor. Phys. 50, 1341 (2008).
  * (19) E. Witten, Three-Dimensional Gravity Revisited, [arXiv:0706.3359].
  * (20) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).
  * (21) S. Carlip, Class. Quantum Gravity 22, 85 (2005).
  * (22) M. A. Anacleto, F. A. Brito, and E. Passos, Phys. Lett. B 743, 184 (2015).
  * (23) S. H. Hendi, B. Eslam Panah and S. Panahiyan, JHEP 05, 029 (2016).
  * (24) K. C. K. Chan and R. B. Mann, Phys. Rev. D 50, 6385 (1994).
  * (25) S. H. Hendi, B. Eslam Panah and S. Panahiyan, PTEP 2016, 103A02 (2016).
  * (26) S. H. Hendi, S. Panahiyan, S. Upadhyay and B. Eslam Panah, Phys. Rev. D 95, 084036 (2017).
  * (27) F. W. Shu, K. Lin, A. Wang, Q. Wu, JHEP 04, 056 (2014).
  * (28) S. H. Hendi, R. Ramezani-Arani, E. Rahimi, Phys. Lett. B 805, 135436 (2020).
  * (29) J. M. Bardeen, B. Carter, S. Hawking, Commun. Math. Phys. 31, 161 (1973).
  * (30) A. Strominger, C. Vafa, Phys. Lett. B 379, 99 (1996).
  * (31) X.-X. Zeng, H.-Q. Zhang, Nucl. Phys. B 959, 115162 (2020).
  * (32) M. M. Caldarelli, G. Cognola, D. Klemm, Class. Quantum Grav. 17, 399 (2000).
  * (33) T. Hertog, K. Maeda, Phys. Rev. D 71, 024001 (2005).
  * (34) H. Lu, C. N. Pope, Q. Wen, JHEP 03, 165 (2015).
  * (35) D. Kubiznak, R. B. Mann, JHEP 07, 033 (2012).
  * (36) S. W. Hawking, D. N. Page, Comm. Math. Phys. 87, 577 (1983).
  * (37) A. Chamblin, R. Emparan, C. Johnson, and R. Myers, Phys. Rev. D 60, 064018 (1999).
  * (38) A. Chamblin, R. Emparan, C. Johnson, and R. Myers, Phys. Rev. D 60, 104026 (1999).
  * (39) A. Haldar, R. Biswas, Gen. Relativ. Gravit. 50, 69 (2018).
  * (40) J. X. Mo, Astrophys. Space Sci. 356, 319 (2015).
  * (41) Kh. Jafarzade, J. Sadeghi, Internat. J. Modern Phys. D 26, 1750138 (2017).
  * (42) R. G. Cai, L. M. Cao, L. Li, R. Q. Yang, JHEP 09, 005 (2013).
  * (43) J. X. Mo, W. B. Liu, Phys. Rev. D 89, 084057 (2014).
  * (44) Y. G. Miao, Z. M. Xu, Phys. Rev. D 98, 084051 (2018).
  * (45) J. X. Mo, W. B. Liu, Eur. Phys. J. C 74, 2836 (2014).
  * (46) S. H. Hendi, S. Panahiyan, B. Eslam Panah, Prog. Theor. Exp. Phys. 2015, 103E01 (2015).
  * (47) R. Zhao, H. Zhao, M. S. Ma, L. C. Zhang, Eur. Phys. J. C 73, 2645 (2013).
  * (48) M. H. Dehghani, S. Kamrani, A. Sheykhi, Phys. Rev. D 90, 104020 (2014).
  * (49) J. X. Mo, G. Q. Li, Y. C. Wu, J. Cosmol. Astropart. Phys. 04, 045 (2016).
  * (50) A. Ovgun, Adv. High Energy Phys. 2018, 8153721 (2018).
  * (51) B. Mirza, Z. Sherkatghanad, Phys. Rev. D 90, 084006 (2014).
  * (52) S. H. Hendi, B. Eslam Panah, S. Panahiyan, Class. Quantum Gravit. 33, 235007 (2016).
  * (53) M. Chabab, H. El Moumni, S. Iraoui, K. Masmar, Eur. Phys. J. C 79, 342 (2019).
  * (54) S. H. Hendi, S. Panahiyan, B. Eslam Panah, M. Faizal, M. Momennia, Phys. Rev. D 94, 024028 (2016).
  * (55) Z. W. Feng, S. Z. Yang, Phys. Lett. B 772, 737 (2017).
  * (56) S. H. Hendi, B. Eslam Panah, S. Panahiyan, Phys. Lett. B 769, 191 (2017).
  * (57) G. Gibbons, R. Kallosh, B. Kol, Phys. Rev. Lett. 77, 4992 (1996).
  * (58) N. Breton, Gen. Relativ. Gravit. 37, 643 (2005).
  * (59) D. Kastor, S. Ray, J. Traschen, Class. Quantum Gravity 26, 195011 (2009).
  * (60) S. H. Hendi, M. H. Vahidinia, Phys. Rev. D 88, 084045 (2013).
  * (61) C. V. Johnson, Class. Quantum Grav. 31, 205002 (2014).
  * (62) R. A. Hennigar, F. McCarthy, A. Ballon, and R. B. Mann, Class. Quantum Grav. 34, 175005 (2017).
  * (63) Kh. Jafarzade, and J. Sadeghi, Int. J. Theor. Phys. 56 , 3387 (2017).
  * (64) Kh. Jafarzade, and J. Sadeghi, Int. J. Mod. Phys. D 26, 1750138 (2017).
  * (65) C. V. Johnson, Class. Quantum Grav. 33, 135001 (2016).
  * (66) J. X. Mo , F. Liang and G. Q. Li, JHEP 03, 010 (2017).
  * (67) J. Zhang, Y. Li, and H. Yu, Eur. Phys. J. C 78, 645 (2018).
  * (68) S. H. Hendi, B. Eslam Panah, S. Panahiyan, H. Liu, and X. H. Meng, Phys. Lett. B 781, 40 (2018).
  * (69) B. Eslam Panah, Phys. Lett. B 787, 45 (2018).
  * (70) B. Carter, Phys. Rev. 174, 1559 (1968).
  * (71) S. W. Wei, and Y. X. Liu, J. Cosmol. Astropart. Phys. 11, 063 (2013).
  * (72) Y. Decanini, G. Esposito-Farèse, A. Folacci, Phys. Rev. D 83, 044032 (2011).


# Deep Transfer Clustering of Radio Signals

Qi Xuan,  _Member, IEEE_ , Xiaohui Li, Zhuangzhi Chen, Dongwei Xu, Shilian
Zheng, and Xiaoniu Yang This work was supported in part by the National
Natural Science Foundation of China under Grants 61973273 and 61903334, and by
the Zhejiang Provincial Natural Science Foundation of China under Grants
LR19F030001 and LY21F030016. _(Corresponding authors: Qi Xuan.)_ Q. Xuan, X.
Li, Z. Chen, and D. Xu are with the Institute of Cyberspace Security, College
of Information Engineering, Zhejiang University of Technology, Hangzhou
310023, China (e-mail: xuanqi@zjut.edu.cn).S. Zheng is with the Science and
Technology on Communication Information Security Control Laboratory, Jiaxing
314033, China.X. Yang is with the Institute of Cyberspace Security, Zhejiang
University of Technology, Hangzhou 310023, China, and also with the Science
and Technology on Communication Information Security Control Laboratory,
Jiaxing 314033, China.

###### Abstract

Modulation recognition is an important task in radio signal processing. Most
of the current researches focus on supervised learning. However, in many real
scenarios, it is difficult and cost to obtain the labels of signals. In this
letter, we turn to the more challenging problem: can we cluster the modulation
types just based on a large number of unlabeled radio signals? If this problem
can be solved, we then can also recognize modulation types by manually
labeling a very small number of samples. To answer this problem, we propose a
deep transfer clustering (DTC) model. DTC naturally integrates feature
learning and deep clustering, and further adopts a transfer learning mechanism
to improve the feature extraction ability of an embedded convolutional neural
network (CNN) model. The experiments validate that our DTC significantly
outperforms a number of baselines, achieving the state-of-the-art performance
in clustering radio signals for modulation recognition.

###### Index Terms:

Signal clustering, deep learning, modulation recognition, transfer learning,
convolutional neural network.

##  I Introduction

With the development of radio communication technology, the electromagnetic
environment has become increasingly complex, and the amount of radio signals
has also exploded. As the basis of radio communication, signal modulation
recognition is of particular importance. Recently, many deep learning models
have been applied to signal modulation classification. Most of these works
focus on supervised learning, which relies on a large number of labeled
signals. However, labeling a large number of signals could be difficult and
costly in reality. In order to make better use of available unlabeled signals,
clustering is a promising direction. As a kind of unsupervised learning
method, clustering can directly captures the correlation between signals, so
as to group them into multiple clusters without the need for signal labels in
advance. However, it is real a challenge to cluster radio signals for
modulation recognition since the signal waves of the same modulation type
could be quite different, while those of different modulation types may be
close to each other, since the difference of signal waves could be largely
determined by the transmitted information. To the best of our knowledge, there
are few studies on the modulation clustering of radio signals, which is used
to analyze the importance of various features in modulation recognition [1],
and to reconstruct the cluster center vector of the constellation diagram [2,
3, 4, 5].

The researches on clustering are much more active in other areas, such as
computer vision [6, 7, 8, 9, 10] and time-series analysis [11, 12]. Quite
recently, a number of deep learning models are proposed for image clustering,
which can be roughly divided into two groups: end-to-end methods and two-step
methods. For the first group, samples are soft-labeled according to the
clustering results to guide the training of deep learning models. For example,
deep embedded clustering (DEC) [13] soft-labeled samples based on the Student
$t$ distribution, joint unsupervised learning (JULE) [14] used $K$-nearest
neighbor (KNN), and deep adaptive clustering (DAC) [15] is based on the
similarity between samples. For the second group, feature learning is
separated from the clustering process. Most of these methods use deep learning
models such as autoencoders to learn features, and then cluster them, e.g.,
deep density-based image clustering (DDC) [16]. In this case, the feature
learning process is not guided by clustering. Since the two processes are
separated, the features learned by the model may not meet the clustering
requirements, which may hurt the performance of the methods.

The above deep clustering methods are largely determined by the training of
deep learning models, and cannot be directly adopted to realize signal
clustering for modulation recognition due to its essential challenge. Transfer
learning [17, 18, 19, 20], on the other hand, is proposed to solve the problem
of insufficient samples. It can largely utilize knowledge or patterns learned
from a different but related fields or problems. Therefore, it is naturally to
believe that the performance of deep clustering methods could be significantly
enhanced if we use transfer learning to pre-train the deep learning models
based on an auxiliary dataset in the related fields, and then use it to
cluster the samples in the target dataset.

Figure 1: The overall framework of DTC for signal clustering, including three
stages: data preprocessing, pre-training, fine-tuning and clustering.

In this letter, we propose a deep transfer learning (DTC) of radio signals for
modulation recognition for the first time, which naturally integrates feature
learning and clustering, and adopts a transfer learning strategy to enhance
the feature extraction ability. In particular, firstly, a convolutional neural
network (CNN) model is pre-trained with labeled signals of an auxiliary
dataset in the same field as the target dataset. In this process, due to the
true labels of the signals, the effective feature learning could be
guaranteed. After that, iterative cluster training is performed on the target
dataset to fine-tune the CNN model. Note that the pre-trained CNN model is
deployed to perform preliminary feature extraction of the signals for
clustering, while those signals with high confidence of clustering results
will be selected and labeled as soft labels to further fine-tune the CNN
model. The process is carried out iteratively until the clustering accuracy no
longer improves. Since each time the signals with high confidence are used for
the model training, the model has a certain degree of anti-interference
ability, and at the same time, the training efficiency could be improved.
Since the clustering process and the feature learning process are jointly
trained, this method can better obtain hidden features that are more suitable
for signal clustering. The main contributions of this letter are summarized as
follows:

  1. 1.

We propose a deep transfer clustering (DTC) model for radio signals, which
naturally integrates feature learning and deep clustering for the first time
in this area.

  2. 2.

We adopt a transfer learning mechanism for the supervised pre-training of the
CNN model, which effectively improves the feature extraction ability of our
DTC model for signal clustering.

  3. 3.

We create a loss function consisting of two parts: positive loss and negative
loss, the balance between which is adjusted by a hyperparameter $\lambda$.

  4. 4.

Experimental results validate that our DTC model significantly outperform the
other traditional or deep learning based clustering methods on multiple radio
signal datasets, achieving the state-of-the-art performance.

The rest of paper is organized as follows. In Section II, we introduce our DTC
model in detail, including data preprocessing, pre-training, fine-tuning and
clustering. In Section III, we give the experimental results on three public
radio signal datasets, to validate the effectiveness of the DTC model.
Finally, the paper is concluded in Section IV.

##  II Deep transfer clustering

In this section, we introduce the detail of our method. The overall framework
of DTC is shown in Fig. 1, which includes three stages: data preprocessing,
pre-training, and clustering.

###  II-A Data preprocessing

The target dataset $O$ needs to be clustered into $k$ classes. The auxiliary
dataset $B$ is a labeled signal dataset that is independent from $O$. We will
use $B$ to pre-train the model before clustering $O$. Since $O$ and $B$ are
independent from each other, the signals from these two sets may be of
different length. Then, the length of signals in $B$ is first adjusted to be
the same as $O$. When the signals in $B$ are shorter than those in $O$, they
are expanded to the target length by just copying. For example, signal ”$abb$”
of length three is expanded to ”$abbabbab$” of length eight. When signals in
$B$ are longer than those in $O$, they are compressed based on equal-interval
sampling to keep the structural characteristics of signals. Of course, the
number of categories in the two datasets may not be the same. When the number
of sample categories in $B$ is more than $k$, we only select the $k$ category
samples to use, here we choose randomly; When the number of categories is not
enough, all samples are used, but the effect of pre-training process will be
slightly reduced.

###  II-B Pre-training

In order to improve the feature extraction ability of the convolutional neural
network (CNN), the signals in $B$, including their labels, are then used to
pre-train the model.

We randomly select $m$ signals from dataset $B$ as the batch input of the CNN,
and the output is an $m\times{k}$ signal feature matrix
$F_{B}=\\{f_{i}\\}^{m}_{i=1}$, where $f_{i}$ is the feature vector of signal
$i$ in $B$, with feature dimension equal to $k$. The cosine similarity between
signals $i$ and $j$ is defined as:

$\displaystyle sim(x_{i},x_{j})=\frac{f_{i}\cdot
f_{j}}{\left\|f_{i}\right\|\cdot\left\|f_{j}\right\|},$ (1)

which is simplified to

$\displaystyle sim(x_{i},x_{j})=f_{i}\cdot f_{j},$ (2)

when we set $\left\|f_{i}\right\|=1$ for $i=1,2,\cdots,m$. So the similarity
matrix of these $m$ signals is

$\displaystyle S_{B}=F_{B}\cdot F_{B}^{\mathrm{T}}.$ (3)

The labels of these signals are converted into one-hot vectors of length $k$,
which are grouped into an $m\times{k}$ label matrix
$Y_{B}=\\{y_{i}\\}^{m}_{i=1}$. Then the true binary judgment matrix of the
signals is defined as:

$\displaystyle P_{B}=Y_{B}\cdot Y_{B}^{\mathrm{T}},$ (4)

which is a Boolean matrix, with its element $P_{B}(i,j)=1$ if signals $i$ and
$j$ belong to the same category, and $P_{B}(i,j)=0$ otherwise. Based on
$P_{B}$, we define a positive matrix $P_{B}^{p}=P_{B}$ and a negative matrix
$P_{B}^{n}=1-P_{B}$. Then, the loss function in the pre-training process is
defined as:

$\displaystyle\mathcal{L}_{pre}=-P_{B}^{p}\cdot\log{S_{B}}-\lambda
P_{B}^{n}\cdot\log{(1-S_{B})},$ (5)

where $\lambda$, as a hyperparameter, is used to adjust the proportion of
positive losses and negative losses.

The pre-training process stops when the loss value on the validation set of
$B$ no longer drops, and we think the CNN has a good ability of feature
extraction.

###  II-C Fine-tuning and clustering

Now, the target dataset $O$ is also divided into batches as the input of the
pre-trained CNN, with the batch size is set to $m$, as shown in the right of
Fig. 1. For each batch input, we have the output feature matrix $F_{O}$, also
we can obtain the similarity matrix

$\displaystyle S_{O}=F_{O}\cdot F_{O}^{\mathrm{T}}.$ (6)

The elements of $S_{O}$ are then compared with the upper threshold $u$ and
lower threshold $l$, respectively, to determine whether the corresponding
signals belong to the same cluster or not.

We construct the positive matrix $P_{O}^{p}$ and the negative matrix
$P_{O}^{n}$ with their elements defined as

$\displaystyle P_{O}^{p}(i,j)=\left\\{\begin{array}[]{lr}1,\quad if\
S_{O}(i,j)\geq u&\\\ 0,\quad if\ S_{O}(i,j)<u&\\\
\end{array}i,j=1,\dots,m\right.$ (9) $\displaystyle
P_{O}^{n}(i,j)=\left\\{\begin{array}[]{lr}1,\quad if\ S_{O}(i,j)\leq l&\\\
0,\quad if\ S_{O}(i,j)>l&\\\ \end{array}i,j=1,\dots,m\right.$ (12)

Then, it is considered that signals $i$ and $j$ are from the same category if
$P_{O}^{p}(i,j)=1$, while they belong to different categories if
$P_{O}^{n}(i,j)=1$. In the process of fine-tuning, $P_{O}^{p}$ and $P_{O}^{n}$
are used as the soft labels to replace the true labels of the signals, and the
loss function in the cluster training stage is defined as:

$\displaystyle\mathcal{L}_{clu}=-P_{O}^{p}\cdot\log{S_{O}}-\lambda
P_{O}^{n}\cdot\log{(1-S_{O})}.$ (13)

During the training process, the feature vectors of signals from different
categories tend to be perpendicular to each other. Note that the dimension of
feature vector is set to $k$, which is the same as the number of categories,
the feature vector is normalized, and each element is limited between $0$ and
$1$. Therefore, as the training progresses, the output features tend to be in
the form of one-hot vectors. The characteristics of the output actually
represent the probability distribution of the signals in each category. In
other words, the index of the maximum value of the feature vector can be
directly used as the label of the signal.

In particular, our CNN consists of four convolutional layers and two dense
fully connected layers. Each layer use rectified linear (ReLU) activation
function. In order to prevent over-fitting, batch normalization (BN) is added
before the ReLU layers. At the same time, BN layers can adjust the
distribution of the data to a normal distribution to ensure the generalization
performance of the model when the input distribution is different at each
time. To remove redundant information, the max-pooling layers are added after
the ReLU layers of the second and third convolutional layers, and the outputs
are also been adjusted by the BN layers. The illustration of the CNN
architecture is shown in Fig. 2. The model contains 32, 128, 128, and 32
filters in layers 1 to 4, respectively. And the last two dense layers contain
64 and $k$ neurons, respectively. At the end of the model is the softmax
function, which acts as a classifier and outputs the probability distribution.

The training of the model uses the Adam optimizer and the loss functions at
different stages are defined by Eq. (5) and Eq. (13), respectively. All
experiments are run on NDIDIA Tesla V100 based on the TensorFlow deep learning
framework.

Figure 2: The structure of CNN. TABLE I: The modulation types of the three datasets Datasets | Modulation Types
---|---
RML2016.10A | WBFM,QPSK,64QAM,16QAM,4PAM,GFSK,CPFSK,BPSK,8PSK,AM-SSB
RML2016.04C | WBFM,QPSK,64QAM,16QAM,4PAM,GFSK,CPFSK,BPSK,8PSK,AM-SSB
RML2018.01A | 32PSK,16APSK,32QAM,FM,GMSK,32APSK,OQPSK,8ASK,16PSK,64APSK
TABLE II: The clustering results of cross pre-training on the three datasets | Datasets | RML2016.10A | RML2016.04C | RML2018.01A
---|---|---|---|---
| Metrics | NMI | ARI | ACC | NMI | ARI | ACC | NMI | ARI | ACC
| No pre-train | 0.3309 | 0.2372 | 0.3699 | 0.4336 | 0.2662 | 0.3905 | 0.3898 | 0.2205 | 0.3082
Auxiliary | RML2016.10A | —— | —— | —— | 0.8259 | 0.6888 | 0.7137 | 0.6576 | 0.4516 | 0.4831
dataset | RML2016.04C | 0.8547 | 0.7566 | 0.7444 | —— | —— | —— | 0.6674 | 0.4587 | 0.4321
| RML2018.01A | 0.5441 | 0.3716 | 0.4980 | 0.6768 | 0.5213 | 0.6229 | —— | —— | ——
TABLE III: The clustering results of various methods on the three datasets Datasets | RML2016.10A | RML2016.04C | RML2018.01A
---|---|---|---
Metrics | NMI | ARI | ACC | NMI | ARI | ACC | NMI | ARI | ACC
K-means | 0.1345 | 0.0585 | 0.1946 | 0.2674 | 0.1494 | 0.3186 | 0.3573 | 0.0933 | 0.1355
DEC | 0.1150 | 0.0626 | 0.2160 | 0.3034 | 0.2022 | 0.3865 | 0.2741 | 0.0887 | 0.1758
DAC | 0.3081 | 0.2345 | 0.3616 | 0.4707 | 0.2880 | 0.3984 | 0.3768 | 0.2174 | 0.3067
DTC | 0.8547 | 0.7566 | 0.7444 | 0.8259 | 0.6888 | 0.7137 | 0.6576 | 0.4516 | 0.4831

##  III Experiments

###  III-A Datasets

The experiments are conducted on three publicly available datasets [21],
including RML2016.10A, RML2016.04C, and RML2018.01A. However, in our
experiments, only 10 categories of signals in each dataset are used. We set
the modulation types in RML2016.10A and RML2016.04C the same, while they are
totally different from those in RML2018.01A, as presented in Table I.

###  III-B Experimental settings

  * •

Baselines: we compare the proposed DTC with several existing clustering
methods, including K-means, DEC, and DAC. The codes of DEC and DAC used in the
experiments are downloaded from GitHub, with the parameters set as suggested
and the signals reshaped as required.

  * •

Evaluation metrics: we use three popular metrics, including adjusted rand
index (ARI), normalized mutual information (NMI), and clustering accuracy
(ACC), with their values all in [0, 1] and higher scores indicating better
clustering performance.

  * •

Hyperparameters: we set $\lambda=0.1$ for pre-training and $\lambda=100$ for
fine-tuning and clustering, and set the upper threshold $u=0.95$, and the
lower threshold $l=0.7$.

###  III-C The experimental results of DTC

The experimental results of DTC are shown in Table II, where we can see that
DTC can achieve reasonable results even without pre-training, i.e., both NMI
and ACC are above 0.3, while ARI is above 0.2 for all the three datasets.
Meanwhile, the performance of DTC is indeed significantly boosted when the CNN
model is pre-trained by auxiliary dataset. All the clustering results on any
evaluation metric are significantly improved, no matter which auxiliary
dataset is used to pre-train the model for which target dataset. Taking the
RML2016.10A dataset as an example, without pre-training, NMI, ARI, and ACC are
0.3309, 0.2372, and 0.3699, respectively. However, after pre-training on the
dataset RML2016.04C, these three metrics greatly increase to 0.8547, 0.7566,
and 0.7444, respectively. Note that the two datasets RML2016.10A and
RML2016.04C are quite similar to each other, i.e., they share exactly the same
types of modulation and the same length of signals, while the dataset
RML2018.01A is relatively different on both types of modulation and length of
signals. As expected, the DTC for RML2016.10A benefits most from the CNN model
pretrained on RML2016.04C, and vice versa. More interestingly, though quite
different, the CNN models pretrained by RML2018.01A can still help to extract
important features of the signals in the other two datasets, so as to improve
the performance of DTC, which indicates the generalization ability of our
method.

###  III-D Comparison with other clustering methods

Now, we compare our DTC with other clustering methods, including K-means, DEC,
and DAC. K-means is a very popular clustering method in many areas, DEC and
DAC are two typical deep clustering methods with outstanding performance in
computer vision. Note that we also try several latest deep clustering methods,
such as semantic pseudo-labeling for image clustering (SPICE) [22], robust
learning for unsupervised clustering (RUC) [23], and semantic clustering by
adopting nearest neighbors (SCAN) [24], but the results are worse than the
three baselines we choose. The comparison results are shown in Table III,
where we can see that DTC significantly outperforms all the other clustering
methods, achieving the state-of-the-art performance. In particular, the
clustering accuracy of DTC is as high as 0.7444 on RML2016.10A, which is
105.9% higher than the second best method DAC. On RML2016.04C, the clustering
accuracy of DTC is 0.7137, which is 79.1% higher than the second best method
DAC. On RML2018.01A, these numbers are 0.4831 and 57.5%. Such incredible
results suggest that DTC could be a feasible method to cluster radio signals
for modulation recognition as a challenging task in wireless communication.

##  IV Conclusion

Automatic modulation recognition is crucial for many applications in
electromagnetic space, especially when the 5G/6G wireless systems emerge.
However, it is always difficult to label a large number of radio signals in
many real scenarios, making it hard to use supervised learning to recognize
modulation types. Therefore, in this letter, we focus on clustering radio
signals for modulation recognition. Since the signal waves could be largely
determined by the transmitted information, the observed signals of the same
modulation type could be quite different, while those of different modulation
types may be close to each other. This makes clustering radio signals real a
challenge in reality.

With the help of the strong feature extraction ability of convolutional neural
network (CNN), in this letter, we propose a novel end-to-end deep transfer
clustering (DTC) model for radio signals, which naturally integrates deep
learning and transfer learning into a single framework to improve the
clustering performance. The experimental results show that, compared with a
number of baselines, our method achieves significantly better performance on
three public radio signal datasets. In the future, we will apply our DTC model
on more various signal datasets, to validate its generalization ability more
comprehensively.

## References

  * [1] N. Daldal, K. Polat, and Y. Guo, “Classification of multi-carrier digital modulation signals using ncm clustering based feature-weighting method,” Computers in Industry, vol. 109, pp. 45–58, 2019.
  * [2] G. Jajoo, Y. Kumar, S. K. Yadav, B. Adhikari, and A. Kumar, “Blind signal modulation recognition through clustering analysis of constellation signature,” Expert Systems with Applications, vol. 90, pp. 13–22, 2017\.
  * [3] F. Yang, L. Yang, D. Wang, P. Qi, and H. Wang, “Method of modulation recognition based on combination algorithm of k-means clustering and grading training svm,” China Communications, vol. 15, no. 12, pp. 55–63, 2018\.
  * [4] J. Tian, Y. Pei, Y.-D. Huang, and Y.-C. Liang, “Modulation-constrained clustering approach to blind modulation classification for mimo systems,” IEEE Transactions on Cognitive Communications and Networking, vol. 4, no. 4, pp. 894–907, 2018.
  * [5] Z. Zhao, A. Yang, P. Guo, and Q. Tan, “A density clustering algorithm for simultaneous modulation format identification and osnr estimation,” Applied Sciences, vol. 10, no. 3, p. 1095, 2020.
  * [6] F. Tian, B. Gao, Q. Cui, E. Chen, and T.-Y. Liu, “Learning deep representations for graph clustering,” Proceedings of the AAAI Conference on Artificial Intelligence, vol. 28, Jun. 2014.
  * [7] K. Tu, P. Cui, X. Wang, P. S. Yu, and W. Zhu, “Deep recursive network embedding with regular equivalence,” in Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’18, (New York, NY, USA), p. 2357–2366, Association for Computing Machinery, 2018.
  * [8] W. Yu, C. Zheng, W. Cheng, C. C. Aggarwal, D. Song, B. Zong, H. Chen, and W. Wang, “Learning deep network representations with adversarially regularized autoencoders,” in Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD ’18, (New York, NY, USA), p. 2663–2671, Association for Computing Machinery, 2018\.
  * [9] Y. Seo, M. Defferrard, P. Vandergheynst, and X. Bresson, “Structured sequence modeling with graph convolutional recurrent networks,” in Neural Information Processing (L. Cheng, A. C. S. Leung, and S. Ozawa, eds.), (Cham), pp. 362–373, Springer International Publishing, 2018.
  * [10] A. Sperduti and A. Starita, “Supervised neural networks for the classification of structures,” IEEE Transactions on Neural Networks, vol. 8, no. 3, pp. 714–735, 1997.
  * [11] R. McConville, R. Santos-Rodriguez, R. J. Piechocki, and I. Craddock, “N2d: (not too) deep clustering via clustering the local manifold of an autoencoded embedding,” 2019.
  * [12] S. M. Mousavi, W. Zhu, W. Ellsworth, and G. Beroza, “Unsupervised clustering of seismic signals using deep convolutional autoencoders,” IEEE Geoscience and Remote Sensing Letters, vol. 16, no. 11, pp. 1693–1697, 2019\.
  * [13] J. Xie, R. Girshick, and A. Farhadi, “Unsupervised deep embedding for clustering analysis,” in Proceedings of The 33rd International Conference on Machine Learning (M. F. Balcan and K. Q. Weinberger, eds.), vol. 48 of Proceedings of Machine Learning Research, (New York, New York, USA), pp. 478–487, PMLR, 20–22 Jun 2016.
  * [14] J. Yang, D. Parikh, and D. Batra, “Joint unsupervised learning of deep representations and image clusters,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016.
  * [15] J. Chang, L. Wang, G. Meng, S. Xiang, and C. Pan, “Deep adaptive image clustering,” in Proceedings of the IEEE International Conference on Computer Vision (ICCV), Oct 2017.
  * [16] Y. Ren, N. Wang, M. Li, and Z. Xu, “Deep density-based image clustering,” Knowledge-Based Systems, vol. 197, p. 105841, 2020.
  * [17] Y. Yu, “Boosting for transfer learning,” in ICML, 2007.
  * [18] R. Raina, A. Battle, H. Lee, B. Packer, and A. Y. Ng, “Self-taught learning: transfer learning from unlabeled data,” in Proceedings of the 24th international conference on Machine learning, pp. 759–766, 2007.
  * [19] T. Evgeniou and M. Pontil, “Regularized multi–task learning,” in Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 109–117, 2004.
  * [20] J. Yosinski, J. Clune, Y. Bengio, and H. Lipson, “How transferable are features in deep neural networks?,” arXiv preprint arXiv:1411.1792, 2014\.
  * [21] Z. Chen, H. Cui, J. Xiang, K. Qiu, L. Huang, S. Zheng, S. Chen, Q. Xuan, and X. Yang, “Signet: An advanced deep learning framework for radio signal classification,” arXiv preprint arXiv:2011.03525, 2020.
  * [22] C. Niu and G. Wang, “Spice: Semantic pseudo-labeling for image clustering,” arXiv preprint arXiv:2103.09382, 2021.
  * [23] S. Park, S. Han, S. Kim, D. Kim, S. Park, S. Hong, and M. Cha, “Improving unsupervised image clustering with robust learning,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 12278–12287, June 2021.
  * [24] W. Van Gansbeke, S. Vandenhende, S. Georgoulis, M. Proesmans, and L. Van Gool, “Scan: Learning to classify images without labels,” in Computer Vision – ECCV 2020 (A. Vedaldi, H. Bischof, T. Brox, and J.-M. Frahm, eds.), (Cham), pp. 268–285, Springer International Publishing, 2020.