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# Phase transition of non-linear charged Anti-de Sitter black holes

Yun-Zhi Du, Huai-Fan Li, Fang Liu, Ren Zhao, Li-Chun Zhang111the corresponding
author Department of Physics, Shan xi Da tong University, Da tong 037009,
China
Institute of Theoretical Physics, Shan xi Da tong University, Da tong 037009,
China

###### Abstract

Understanding the thermodynamic phase transition of black holes can provide a
deep insight into the fundamental properties of black hole gravity to
establish the theory of quantum gravity. We investigate the condition and
latent heat of phase transition for non-linear charged AdS black holes using
the Maxwell’s equal-area law, and analysis the boundary and curve of the two-
phase coexistence area in the expanded phase space. We suggest that the phase
transition of the non-linear charged AdS black hole with the fixed temperature
($T<T_{c}$) is related to the electric potential at the horizon, not only to
the location of horizon. Recently, the molecular number density was introduced
to study the phase transition and microstructure of black holes. On this
basis, we discuss the continuous phase transition of a non-linear charged AdS
black hole to reveal the potential microstructure of a black hole by
introducing the order parameter and using the scalar curvature.

##  I Introduction

Black hole physics, as an intersection of many disciplines between the GR and
quantum mechanics, such as thermodynamic statistical physics and particle
physics, plays a significant role in modern physics. The study of black holes
from the perspective of thermodynamics has become indisputable. Black holes
have the characteristics of an ordinary thermodynamical system and they can
undergo phase transitions under certain conditions. Black holes have thermal
properties; hence, a key concern regarding black hole is that whether they
have a microstructure like ordinary thermodynamic systems. From a mathematical
perspective, string theory and supersymmetry are used to explain the
microstructure of black holes. From a thermodynamic view point, hypotheses of
black hole molecules and spacetime atoms are proposed to explore the
microscopic behavior of black holes. Research on both areas has made progress
Ruppeiner2014 ; Ruppeiner2018 ; Ruppeiner2008 ; Ruppeiner2012 ; Strominger1996
; Emparan2006 ; Horowitz1996 ; Miao2018 ; Miao2019 ; Miao2017 ; Wei2015 ;
Hendi2017

By regarding the cosmological constant in the AdS spacetime as the pressure of
an ordinary thermodynamic system and comparing the state parameters of the AdS
black holes to that of the van der Waals (vdW) system, the thermodynamical
phase transition continue to be one of the increasingly active areas in black
hole physics. Some satisfactory results on the thermodynamical property of AdS
and dS black holes were obtained Hendi2017a ; Hennigar2017a ; Frassin ;
Kubiznak2012 ; Cai2013 ; Ma2017 ; Ma2017a ; Mir2017b ; Banerjee2017 ;
Banerjee2011 ; Hendi2019 ; Simovic2019 ; Hennigar2019 ; Mbarek2019 ;
Kubiznak2016 ; Li2017a ; Li2017 ; zhao2015 ; Ma2018 ; Ma2016 ; Zhang2016 ;
Zhou2019 ; Wei2019 . However, the statistical mechanics background of black
holes as a thermodynamic system is not understood yet. Therefore, obtaining
the relationship between the thermodynamic properties of various AdS and dS
black holes is crucial. This study will be useful for further understanding
the entropy, temperature, heat capacity, etc, of black holes and developing a
self-consistent black hole thermodynamic geometry theory.

For the AdS black holes, discontinuous exist in the P-V or T-S diagram with
$T<T_{c}$ or $P<P_{c}$ zhao2015 ; Ma2018 . The black holes within the two-
phase coexistence zone have the latent heat of phase transition; they have
different thermodynamical properties at different phases. Therefore, a black
hole as a thermodynamical system must have different microstructures with
different phases.

The linear charged black holes in AdS spacetime Chamblin1999 ; Chamblin1999a
within a second order phase transition shows a scaling symmetry: at the
critical point, the state parameters scale with respect to charge q, i.e.,
$S\sim q^{2},~{}P\sim q^{-2},~{}T\sim q^{-1}$ Johnson2018 ; Johnson2018a . One
can infer whether the scaling symmetry exists in the non-linear charged AdS
black holes Hassaine2008 . As a generalization of the charged AdS Einstein-
Maxwell black holes, exploring new non-linear charged systems is interesting.
Due to infinite self-energy of point like charges in Maxwell’s theory Born1934
; Born1934a ; Kats2007 ; Anninos2009 ; Cai2008 ; Seiberg1999 ; Fradkin1985 ;
Mesaev1987 ; Bergshoeff1987 ; Tseytlin1986 ; Gross1987 , Born and Infeld
proposed a generalization when the field is strong, introducing non-
linearities Born1934 ; Born1934a ; Dirac2013 ; Birula1970 . An interesting
non-linear generalization of charged black holes involves a Yang-Mill field
coupled to Einstein gravity, where several features in extended thermodynamics
have recently been studied Zhang2015 ; Moumni2018 .

As a unique perspective, the thermodynamic geometry (scalar curvature) plays
an important role in studying black hole phase transition. The comprehensible
physical picture of Ruppeiner geometry makes it widely used Ruppeiner2014 ;
Ruppeiner2018 ; Ruppeiner2008 ; Ruppeiner2012 ; Miao2019 ; Miao2017 ; Wei2015
; Chamblin1999a ; Johnson2018 ; Johnson2018a ; Born1934 ; Born1934a ; Kats2007
. In the case of fixed charge, considering the fluctuation of mass and
pressure, we will study the Ruppeiner geometric line element and scalar
curvature. We predict that for the non-linear charged AdS black hole with a
certain temperature, the different electric potentials correspond to different
values of the scalar curvature. In contrast, this result indicate different
microstructures of black hole system with different electric potentials.

In this study, we will investigate the phase transition of the non-linear
charged AdS black hole (EPYM black hole) to obtain the condition and the
latent heat of phase transition by using Maxwell’s equal-area law and adopting
different independent dual parameters. We aim to reveal the microstructure of
black holes by studying the phase transition of the EPYM black hole with
different electric potentials. This work is organized as follows. In Sec. II,
we present the thermodynamic parameters of the EPYM black hole. In Sec. III,
we discuss the phase transition of the EPYM black hole for different choices
of the independent dual parameters by the Maxwell¡¯s equal-area law and
provide a universal condition of phase transition. In Sec. IV, we present the
coexistence curve in the P-T diagram and analyze the influence of non-linear
YM charged parameter $\gamma$ on the latent heat of phase change for the EPYM
AdS black hole. In Sec. V, the order parameter $\phi^{2}$ is introduced to
explain the phase transition for the EPYM black hole. Moreover, the
thermodynamic geometry at the critical point is analyzed using the scalar
curvature $R$. We also investigate the influence of the non-linear YM charge
parameter $\gamma$ on $R$. Finally, a brief summary is provided in Sec. VI.

##  II Thermodynamics for Non-Linear Charged AdS Black Holes

Our starting point is the action for four-dimensional Einstein-power-Yang-
Mills (EPYM) gravity with a cosmological constant $\Lambda$, given by
Zhang2015 ; Lorenci2002 ; Corda2011 ; Mazharimousavi2009

$\displaystyle I=\frac{1}{2}\int
d^{4}x\sqrt{g}\left(R-2\Lambda-[Tr(F^{(a)}_{{\mu\nu}}F^{{(a)\mu\nu}})]^{\gamma}\right)$
(1)

with the Yang-Mills (YM) field

$\displaystyle
F_{\mu\nu}^{(a)}=\partial_{\mu}A_{\nu}^{(a)}-\partial_{\nu}A_{\mu}^{(a)}+\frac{1}{2\xi}C_{(b)(c)}^{(a)}A_{\mu}^{(b)}A_{\nu}^{(c)}.$
(2)

Here,
$Tr(F^{(a)}_{\mu\nu}F^{(a)\mu\nu})=\sum^{3}_{a=1}F^{(a)}_{\mu\nu}F^{(a)\mu\nu}$,
$R$ and $\gamma$ are the scalar curvature and a positive real parameter,
respectively; $C_{(b)(c)}^{(a)}$ represents the structure constants of three
parameter Lie group $G$; $\xi$ is the coupling constant; and $A_{\mu}^{(a)}$
represents the $So(3)$ gauge group YM potentials.

For this system, the four-dimensional EPYM black hole solution with the
negative cosmological constant $\Lambda$ is obtained by adopting the metric
Yerra2018

$\displaystyle ds^{2}=-f(r)dt^{2}+f^{-1}dr^{2}+r^{2}d\Omega_{2}^{2},$ (3)
$\displaystyle
f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}+\frac{\left(2q^{2}\right)^{\gamma}}{2(4\gamma-3)r^{4\gamma-2}},$
(4)

where $d\Omega_{2}^{2}$ is the metric on unit 2-sphere with volume $4\pi$ and
$q$ is the YM charge. Note that this solution is valid for the condition of
the non-linear YM charge parameter $\gamma\neq 0.75$, and the power YM term
holds the weak energy condition (WEC) for $\gamma>0$ Corda2011 . The position
of the black hole event horizon is determined as the larger root of
$f(r_{+})=0$. The parameter $M$ represents the ADM mass of the black hole; in
our set up, it is associated with the enthalpy of the system.

Using the “Euclidean trick”, one can identify the black hole temperature and
entropy of the solution given by Zhang2015

$\displaystyle T=\frac{1}{4\pi r_{+}}\left(1+8\pi
Pr_{+}^{2}-\frac{\left(2q^{2}\right)^{\gamma}}{2r_{+}^{(4\gamma-2)}}\right),~{}~{}~{}~{}~{}S=\pi
r_{+}^{2}.$ (5)

The pressure reads $P=-\Lambda/(8\pi)$ from extended thermodynamics; the YM
potential $\Psi$ is given by Anninos2009 ; Dehyadegari2017

$\displaystyle\Psi=\frac{\partial M}{\partial
q^{2\gamma}}=\frac{r_{+}^{3-4\gamma}2^{\gamma-2}}{(4\gamma-3)}.$ (6)

The above thermodynamic state parameters satisfy the first law

$\displaystyle dM=TdS+\Psi dq^{2\gamma}+VdP$ (7)

with the thermodynamic volume

$\displaystyle V=\left(\frac{\partial M}{\partial
P}\right)_{S,q}=\frac{4\pi}{3}r_{+}^{3}.$

Thus, the ADM mass of a black hole corresponds to enthalpy Kastor2009  as
follows Ma2017a :

$\displaystyle H=M(S,q,P)=\frac{1}{6}\left[8\pi
P\left(\frac{S}{\pi}\right)^{3/2}+3\left(\frac{S}{\pi}\right)^{\frac{3-4\gamma}{2}}\frac{\left(2q^{2}\right)^{\gamma}}{8\gamma-6}+3\sqrt{\frac{S}{\pi}}\right].$
(8)

The equation of state $P(V,T)$ for canonical ensemble (fixed YM charge $q$)
can be obtained from the expression of temperature as

$\displaystyle
P=\left(\frac{4\pi}{3V}\right)^{1/3}\left[\frac{T}{2}-\frac{1}{8\pi}\left(\frac{4\pi}{3V}\right)^{1/3}+\frac{\left(2q^{2}\right)^{\gamma}}{16\pi}\left(\frac{4\pi}{3V}\right)^{\frac{1-4\gamma}{3}}\right].$
(9)

##  III The Equal-Area Law

From Eq. (9), we know that the state equation of the EPYM black hole with
fixed YM charge corresponds to an ordinary thermodynamic system; it can be
written as $f(T,P,V)=0$. Furthermore, the number of particles in the system is
unchanged. In the following, through the Maxwell’ equal-area law, we discuss
the phase transition condition of the EPYM black hole in $P-V$, $T-S$, and
$q^{2\gamma}-\Psi$, respectively. Then, we provide a general condition of the
phase transition for the EPYM black hole.

###  III.1 The construction of equal-area law in P-V diagram

For the EPYM black hole with the given YM charge $q$ and temperature
$T_{0}<T_{c}$, the volume at the boundary of the two-phase coexistence area
are $V_{1}$ and $V_{2}$, respectively; the corresponding pressure $P_{0}$ is
less than the critical pressure $P_{c}$; it is determined by the horizon
radius $r_{+}$. Therefore, from the Maxwell’s equal-area law
$P_{0}(V_{2}-V_{1})=\int^{V_{2}}_{V_{1}}PdV$ and Eq. (5), we have

$\displaystyle P_{0}$ $\displaystyle=$
$\displaystyle\frac{T_{0}}{2r_{1}}-\frac{1}{8\pi
r_{1}^{2}}+\frac{\left(2q^{2}\right)^{\gamma}}{16\pi r_{1}^{4\gamma}},$
$\displaystyle\quad P_{0}$ $\displaystyle=$
$\displaystyle\frac{T_{0}}{2r_{2}}-\frac{1}{8\pi
r_{2}^{2}}+\frac{\left(2q^{2}\right)^{\gamma}}{16\pi r_{2}^{4\gamma}},$ (10)
$\displaystyle 2P_{0}$ $\displaystyle=$
$\displaystyle\frac{3T_{0}(1+x)}{2r_{2}\left(1+x+x^{2}\right)}-\frac{3}{4\pi
r_{2}^{2}\left(1+x+x^{2}\right)}$ (11)
$\displaystyle+\frac{3\left(2q^{2}\right)^{\gamma}\left(1-x^{3-4\gamma}\right)}{8\pi(3-4\gamma)r_{2}^{4\gamma}\left(1-x^{3}\right)}$

where $x=\frac{r_{1}}{r_{2}}$. With Eq. (10), we have

$\displaystyle 0$ $\displaystyle=$ $\displaystyle T_{0}-\frac{1}{4\pi
r_{2}x}(1+x)+\frac{\left(2q^{2}\right)^{\gamma}}{8\pi
r_{2}^{4\gamma-1}x^{4\gamma-1}}\frac{\left(1-x^{4\gamma}\right)}{(1-x)},$ (12)
$\displaystyle 2P_{0}$ $\displaystyle=$
$\displaystyle\frac{T_{0}}{2r_{2}x}(1+x)-\frac{1}{8\pi
r_{2}^{2}x^{2}}\left(1+x^{2}\right)$ (13)
$\displaystyle+\frac{\left(2q^{2}\right)^{\gamma}}{16\pi
r_{2}^{4\gamma}x^{4\gamma}}\left(1+x^{4\gamma}\right).$

From the above equation and Eq. (11), we obtain the following expression:

$\displaystyle\frac{1}{4\pi r_{2}x}$ $\displaystyle=$ $\displaystyle
T_{0}\frac{(1+x)}{\left(1+3x+x^{2}\right)}$ (14)
$\displaystyle+\frac{\left(2q^{2}\right)^{\gamma}\left[(3-4\gamma)\left(1-x^{3+4\gamma}\right)+(3+4\gamma)x^{3}\left(1-x^{4\gamma-3}\right)\right]}{8\pi(3-4\gamma)r_{2}^{4\gamma-1}x^{4\gamma-1}(1-x^{3})(1-x)^{2}\left(1+3x+x^{2}\right)}.$

Considering Eqs. (12) and (14), we have

$\displaystyle r_{2}^{4\gamma-2}$ $\displaystyle=$
$\displaystyle\frac{\left(2q^{2}\right)^{\gamma}\left[(3-4\gamma)(1+x)\left(1-x^{4\gamma}\right)+8\gamma
x^{2}\left(1-x^{4\gamma-3}\right)\right]}{2x^{4\gamma-2}(3-4\gamma)(1-x)^{3}}$
(15) $\displaystyle=$ $\displaystyle\left(2q^{2}\right)^{\gamma}f(x,\gamma).$

For the critical point ($x=1$), the state parameters are

$\displaystyle r_{c}^{4\gamma-2}$ $\displaystyle=$
$\displaystyle\left(2q^{2}\right)^{\gamma}f(1,\gamma),\quad
f(1,\gamma)=\gamma(4\gamma-1),$ (16) $\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\left(2q^{2}\right)^{\gamma/(4\gamma-2)}f^{1/(4\gamma-2)}(1,\gamma)}\frac{2\gamma-1}{4\gamma-1},$
(17) $\displaystyle P_{c}$ $\displaystyle=$
$\displaystyle\frac{2\gamma-1}{16\pi\gamma\left(2q^{2}\right)^{\gamma/(2\gamma-1)}f^{1/(2\gamma-1)}(1,\gamma)}.$
(18)

Because the above state parameters must be positive, the non-linear YM charge
parameter satisfies the condition $\frac{1}{2}<\gamma$.

Substituting Eq. (15) into Eq. (12) and setting $T_{0}=\chi T_{c}$
($0<\chi\leq 1$), we have

$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
x\left(2q^{2}\right)^{\gamma/(4\gamma-2)}f^{1/(4\gamma-2)}(x,\gamma)}$ (19)
$\displaystyle\times\left(1+x-\frac{1}{2f(x,\gamma)x^{4\gamma-2}}\frac{1-x^{4\gamma}}{1-x}\right),$
$\displaystyle\chi\frac{2\gamma-1}{\gamma^{\frac{1}{4\gamma-2}}(4\gamma-1)^{\frac{4\gamma-1}{4\gamma-2}}}$
$\displaystyle=\frac{1}{4xf^{1/(4\gamma-2)}(x,\gamma)}\left(1+x-\frac{1}{2f(x,\gamma)x^{4\gamma-2}}\frac{1-x^{4\gamma}}{1-x}\right).$
(20)

For the given parameter $\gamma$ and temperature $T_{0}$ (i.e., $\chi$), we
can obtain the value of $x$ from Eq. (20). Eq. (15) suggests that for the
given temperature $T_{0}$ ($T_{0}<T_{c}$), that is, for the fixed value of
$x$, the phase transition condition reads

$\displaystyle\frac{\left(2q^{2}\right)^{\gamma}}{r_{2}^{4\gamma-2}}=\frac{1}{f(x,\gamma)}.$
(21)

Therefore, the phase transition of the EPYM black hole with a given
temperature $T_{0}$ ($T_{0}<T_{c}$) is determined by the radio between the YM
charge $(2q^{2})^{\gamma}$ and $r_{2}^{4\gamma-2}$ and not the value of the
horizon alone. Note that we call this radio the YM electric potential with the
horizon radius $r_{2}$. The phase transitions in the $P-V$ diagram with
different temperatures are shown in Fig. 1. The effect of the non-linear
parameter $\gamma$ on phase transition is exhibited in Fig. 2

Figure 1: Phase transition in $P-V$ diagram with the parameters
$q=0.85,~{}\gamma=0.8$. The temperature is set to $0.0419$ (red thin line),
$0.04215$ (black thin line), $0.0424$ (green thick line), and $0.0425$ (red
thick line).  Figure 2: Phase transition in $P-V$ diagram with the parameters
$q=0.85,~{}T_{0}=0.0374$. The parameter $\gamma$ is set to 0.78 (black line),
0.9 (green line), 1 (red line), and 1.2 (blue line), respectively.

###  III.2 The construction of equal-area law in T-S diagram

For the EPYM black hole with the given YM charge $q$ and pressure
$P_{0}<P_{c}$, the entropy at the boundary of the two-phase coexistence area
are $S_{1}$ and $S_{2}$, respectively. The corresponding temperature $T_{0}$
is less than the critical temperature $T_{c}$; it is determined by the horizon
radius $r_{+}$. Therefore, from the Maxwell’s equal-area law
$T_{0}(S_{2}-S_{1})=\int^{S_{2}}_{S_{1}}TdS$ and eq. (5), we have

$\displaystyle 2\pi T_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{r_{2}(1+x)}+\frac{8\pi
Pr_{2}}{3(1+x)}\left(1+x+x^{2}\right)$ (22)
$\displaystyle-\frac{\left(2q^{2}\right)^{\gamma}r_{2}^{1-4\gamma}}{2(3-4\gamma)}\frac{\left(1-x^{3-4\gamma}\right)}{\left(1-x^{2}\right)},$
$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r_{2}}\left(1+8\pi
Pr_{2}^{2}-\frac{\left(2q^{2}\right)^{\gamma}}{2r_{2}^{(4\gamma-2)}}\right),$
$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r_{1}}\left(1+8\pi
Pr_{1}^{2}-\frac{\left(2q^{2}\right)^{\gamma}}{2r_{1}^{(4\gamma-2)}}\right).$
(23)

Considering the above equations, we obtain that the phase transition condition
with the independent dual parameters for $T$-$S$ is the same as that for
$P$-$V$. This result indicates that these two choices of independent dual
parameters ($P$-$V$, $T$-$S$) will provide the same phase transition point.
The curves of phase transition for $T$-$S$ are shown in Fig. 3.

Figure 3: Phase transition in $T-S$ diagram with the parameters
$q=0.85,~{}\gamma=0.8$. The temperature is set to $0.000344$ (red thin line),
$0.0035$ (black thin line), $0.0036$ (green thick line), and $0.00365$ (red
thick line).

###  III.3 The construction of equal-area law in $q^{2\gamma}-\Psi$ diagram

For the EPYM black hole with the given temperature $T_{0}<T_{c}$, if the YM
charge changes, we will choose $q^{2\gamma}-\Psi$ to study the phase
transition. The YM electric potential at the boundary of the two-phase
coexistence area are $\Psi_{1}$ and $\Psi_{2}$, respectively. The
corresponding YM charge term is $q^{2\gamma}_{0}$. Therefore, from the
Maxwell’s equal-area law
$q^{2\gamma}_{0}(\Psi_{2}-\Psi_{1})=\int^{\Psi_{2}}_{\Psi_{1}}q^{2\gamma}d\Psi$
and eq. (6), we have

$\displaystyle q_{0}^{2\gamma}$ $\displaystyle=$
$\displaystyle\frac{(4\gamma-3)(1-x)x^{4\gamma-3}}{2^{\gamma-1}\left(1-x^{4\gamma-31}\right)}r_{2}^{4\gamma-2}$
(24) $\displaystyle\times\left[1+\frac{8\pi
P_{0}}{3}r_{2}^{2}\left(1+x+x^{2}\right)-2\pi T_{0}r_{2}(1+x)\right],$
$\displaystyle\frac{q_{0}^{2\gamma}}{2^{1-\gamma}}$ $\displaystyle=$
$\displaystyle r_{2}^{(4\gamma-2)}+8\pi P_{0}r_{2}^{4\gamma}-T_{0}4\pi
r_{2}^{4\gamma-1},$ $\displaystyle\frac{q_{0}^{2\gamma}}{2^{1-\gamma}}$
$\displaystyle=$ $\displaystyle r_{1}^{(4\gamma-2)}+8\pi
P_{0}r_{1}^{4\gamma}-T_{0}4\pi r_{1}^{4\gamma-1}.$ (25)

With the above equations, we have

$\displaystyle
r_{2}^{4\gamma-2}=\left(2q_{0}^{2}\right)^{\gamma}\frac{(4\gamma-3)(1-x)\left[(1+2x)+x^{4\gamma-1}(2+x)\right]-6x^{2}\left(1-x^{4\gamma-3}\right)}{2x^{4\gamma-2}(4\gamma-3)(1-x)^{3}},$
(26)

which is of the same form as Eq. (15).

For the given temperature $T_{0}$ (i.e., $\chi$), the phase transition
condition for the independent dual parameters $q^{2\gamma}-\Psi$ is consistent
with that for $P-V$ and $T-S$. For the EPYM black hole with the fixed YM
charge and temperature, when the horizon radius $r_{+}$ is smaller than
$r_{1}$, the phase corresponds to the liquid of a van der Waals system; in
contrast, it resembles the gas of a van der Waals system for $r_{+}>r_{2}$.
And the phase corresponds to the two-phase coexistent of a van der Waals
system as $r_{1}<r_{+}<r_{2}$. The phase transition curves for
$q^{2\gamma}-\Psi$ with fixed temperature and pressure are shown in Fig. 4,
respectively.

(a) $\gamma=0.8,~{}P_{0}=0.0034$ (b) $\gamma=0.8,~{}T_{0}=0.0419$

Figure 4: Phase transition curves in $q^{2\gamma}-\Psi$ diagram with the
parameter $\gamma=0.8$. In the left panel, $\gamma=0.8,~{}P_{0}=0.0034$, and
the temperatures is set to $T_{0}=0.0419$ (red thin line),
$T_{0}=T_{c}=0.0417$ (green thick line), and $T_{0}=0.0415$ (red thick line).
In the right panel, $\gamma=0.8,~{}T_{0}=0.0419$, and the pressures is set to
$P_{0}=0.0034$ (red thin line), $P_{0}=P_{c}=0.003476$ (green thick line), and
$P_{0}=0.0035$ (red thick line).

##  IV The coexistent curve in P-T diagram

For an ordinary thermodynamic system, when phases ($\alpha$ phase and $\beta$
phase) are in the two-phase coexistence area, the coexistent curve ($P-T$) is
directly determined by experiments. The slope of the $P-T$ curve is given by
the Clapeyron equation as

$\displaystyle\frac{dP}{dT}=\frac{L}{T\left(\nu^{\beta}-\nu^{\alpha}\right)},$
(27)

where $L=T(S^{\beta}-S^{\alpha})$; $\nu^{\alpha},~{}\nu^{\beta}$ are the molar
volumes of $\alpha$ and $\beta$ phases, respectively. Generally, for an
ordinary thermodynamic system, the Clapeyron equation agrees with the
experimental results, which provide a direct experimental verification for the
correctness of thermodynamics.

Figure 5: Coexistent curve in $P-T$ diagram with the parameter $q=1.2$. The
critical points with different values of the non-linear YM charge parameter
$\gamma$ are marked as black dots.

For the EPYM black hole, from Eqs. (12), (13) and (15), we have

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
x\left(2q^{2}\right)^{\gamma/(4\gamma-2)}f^{1/(4\gamma-2)}(x,\gamma)}$ (28)
$\displaystyle\times\left(1+x-\frac{1}{2f(x,\gamma)x^{4\gamma-2}}\frac{1-x^{4\gamma}}{1-x}\right)=y_{2}(x,\gamma),~{}~{}~{}~{}$
$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
r_{2}^{2}x}\left(1-\frac{\left(2q^{2}\right)^{\gamma}\left(1-x^{4\gamma-1}\right)}{2r_{2}^{4\gamma-2}x^{4\gamma-2}(1-x)}\right)$
(29) $\displaystyle=$ $\displaystyle\frac{1}{8\pi
x\left(2q^{2}\right)^{\gamma/(2\gamma-1)}f^{1/(2\gamma-1)}(x,\gamma)}$
$\displaystyle\times\left(1-\frac{1-x^{4\gamma-1}}{2x^{4\gamma-2}(1-x)f(x,\gamma)}\right)=y_{1}(x,\gamma).$

The coexistent curves ($P-T$) with different non-linearity parameters $\gamma$
are shown in Fig. 5. Furthermore, from eqs. (27), (28), and (29), the latent
heat of phase transition for EPYM black hole reads

$\displaystyle L$ $\displaystyle=$ $\displaystyle
T\left(1-x^{3}\right)r_{2}^{3}\frac{4\pi
y_{1}^{\prime}(x,\gamma)}{3y_{2}^{\prime}(x,\gamma)}$ (30) $\displaystyle=$
$\displaystyle\left(1-x^{3}\right)\frac{4\pi
y_{1}^{\prime}(x,\gamma)}{3y_{2}^{\prime}(x,\gamma)}\left(2q^{2}\right)^{3\gamma/(4\gamma-2)}f^{3/(4\gamma-2)}(x,\gamma)y_{2}(x,\gamma).$

The above equation suggests that for the system with a fixed YM charge, the
latent heat of phase transition is related with the temperature $T_{0}$
($T_{0}\leq T_{c}$), that is, related with $x$. The latent heat of phase
transition with $x$ for non-linear parameters $\gamma$ are shown in Fig. 6.
The latent heat of phase transition for fixed $x$ decreases with an increase
in $\gamma$ ($\gamma>\frac{3}{4}$).

Figure 6: $L-x$ curve with the parameter $q=1.2$.

##  V Thermodynamic geometry

Eq. (15) demonstrates that for the EPYM black hole with $T_{0}$ ($T_{0}\leq
T_{c}$), there is a sudden change of the YM electric potential of the black
hole molecules. The YM electric potentials with different phases are

$\displaystyle\phi_{2}^{2}=\frac{\left(2q^{2}\right)^{\gamma}}{r_{2}^{4\gamma-2}}=\frac{1}{f(x,\gamma)},\qquad\phi_{1}^{2}=\frac{\left(2q^{2}\right)^{\gamma}}{r_{1}^{4\gamma-2}}=\frac{1}{x^{4\gamma-2}f(x,\gamma)}.$
(31)

This indicates that the microstructures of the black hole molecules in
different phases, are inconsistent. Recent studies proposed that the phase
transition of a black hole is due to the different number densities of
molecules for large and small black holes Miao2018 ; Miao2019 ; Miao2017 ;
Altamirano2014 . According to our investigation in the last section, the phase
transition of the EPYM black hole is related with the horizon and the YM
charge. That is, the phase transition is determined by the YM electric
potential at the horizon. Based on this issue and Landau’s theory of
continuous phase transition, we explore to explore the physical mechanism of
phase transition for the EPYM black hole.

The continuous phase transition theory is characterized by the change of the
degree of order and the accompanying change of the symmetry property of
matter. So what is the internal reason for the phase transition of the EPYM
black hole? With the above investigation, the phase transition of the EPYM
black hole with $T_{0}$ ($T_{0}<T_{c}$) occurs when the YM electric potential
at the horizon satisfies the relation (21). For the EPYM black hole with
$T_{0}<T_{c}$ and phase $\phi_{1}^{2}$, the YM electric potential is high;
hence, the black hole molecules are affected by this strong YM electric
potential; consequently, they have a certain orientation and displacement
polarization. In contrast, for this system with the same temperature
$T_{0}<T_{c}$ and another phase $\phi_{2}^{2}$, the black hole molecules
become disordered; these molecules have a higher symmetry due to a lower YM
electric potential. The thermal motion of black hole molecules weakes the
order of black hole molecules with increasing temperature. Especially for
$T_{0}>T_{c}$, the thermal motion of black hole molecules leads to the
disorder of black hole molecules. In other words, for the EPYM black hole with
the lower temperature $T_{0}<T_{c}$, the black hole molecules have a lower
symmetry and higher order, and the order parameter $\phi^{2}(T)$ is not equal
to zero. While the black hole molecules have a higher symmetry and lower
order, and the order parameter $\phi^{2}(T)$ is zero for the black hole with
$T_{0}>T_{c}$. Note that the above results are obtained by comparing the EPYM
black hole to an ordinary thermodynamic system.

We define the order parameter of the EPYM black hole as

$\displaystyle\phi^{2}(T)=\frac{\phi_{1}^{2}-\phi_{2}^{2}}{\phi_{c}^{2}}=\frac{\gamma(4\gamma-1)(1-x^{4\gamma-2})}{f(x,\gamma)x^{4\gamma-2}}$
(32)

with $\phi^{2}_{c}=\frac{1}{f(1,\gamma)}=\frac{1}{\gamma(4\gamma-1)}$. The
order-parameter plotted with the non-linear parameter $\gamma$ is shown in
Fig. 7.

Figure 7: $\phi^{2}$-$\frac{T}{T_{c}}$ curves for the EPYM black hole with the
non-linear YM charge parameter $\gamma$.

Becasue the order parameter is small near the critical point, according to
Landau’s theory of continuous phase transition, Gibbs function can be expanded
as the power of $\phi^{2}(T)$ near the critical temperature $T_{c}$ by the
method in Refs. Guo2020 ; Du2019 . The free energy takes the minimum value in
the equilibrium state; based on this theory, we obtained that the critical
exponents of the EPYM black hole are consistent with that of the RN-AdS black
hole and the single axis ferromagnetic. However, the fluctuation of the order
parameter near the critical point is neglected in the above discussion.
Fortunately, the Ruppeiner geometry is derived from the theory of
thermodynamic fluctuation. The singularity of the scalar curvature reveals the
phase transition structure of black holes Ruppeiner2018 ; Ruppeiner1995 . In
the following, we explore the microstructure of the black hole molecules by
investigating the Ruppeiner geometry. We take ($S$, $P$) fluctuation with the
fixed YM charge. The Ruppeiner scalar curvature (Ricci scalar) reads Miao2018

$\displaystyle R=-\frac{N}{D}$ (33)

with

$\displaystyle N$ $\displaystyle=$ $\displaystyle 2\pi
S^{-1}[2^{1+\gamma}\pi^{1+2\gamma}q^{2\gamma}S^{1-2\gamma}(-1+2\gamma)[-1-48PS$
(34) $\displaystyle+(-1+8PS(7+16PS))\gamma+2(1-8PS)^{2}\gamma^{2}]$
$\displaystyle+8\pi^{2}(-1+2\gamma)[(-1+\gamma)^{2}+256P^{3}S^{3}\gamma(-1+2\gamma)$
$\displaystyle+4PS(1+\gamma-2\gamma^{2})-32P^{2}S^{2}(-3+2\gamma(2+\gamma))]$
$\displaystyle+\pi^{4\gamma}\left(2q^{2}\right)^{2\gamma}S^{2-4\gamma}[-3+\gamma(9-6\gamma+8PS(1+2\gamma))]],$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\left[-\pi^{2\gamma}\frac{\left(2q^{2}\right)^{\gamma}}{S^{2\gamma-1}}+2\pi(1+8PS)\right]$
(35)
$\displaystyle\times\left[\pi^{2\gamma}\frac{\left(2q^{2}\right)^{\gamma}}{S^{2\gamma-1}}+2\pi(1-8PS)(1-2\gamma)\right]^{2}.$

Because the forms of the horizon radius with a given temperature $T<T_{c}$ for
two phases are different, the Ricci scalar of two phases also have two forms

$\displaystyle R_{1}$ $\displaystyle=$ (36)
$\displaystyle-\frac{2(2q^{2})^{\frac{1}{1-2\gamma}}f^{\frac{1}{1-2\gamma}}(x,\gamma)}{\pi\left[-x^{2-4\gamma}f^{-1}(x,\gamma)+2(1+8A_{1})\right]\left[x^{2-4\gamma}f^{-1}(x,\gamma)+2(1-2\gamma)(1-8A_{1})\right]^{2}}$
$\displaystyle\times B,~{}~{}~{}~{}~{}~{}$ $\displaystyle R_{2}$
$\displaystyle=$
$\displaystyle-\frac{2(2q^{2})^{\frac{1}{1-2\gamma}}f^{\frac{1}{1-2\gamma}}(x,\gamma)}{\pi\left[-f^{-1}(x,\gamma)+2(1+8A_{2})\right]\left[f^{-1}(x,\gamma)+2(1-2\gamma)(1-8A_{2})\right]^{2}}$
(37) $\displaystyle\times C~{}~{}~{}~{}~{}~{}$

with the following functions

$\displaystyle B$ $\displaystyle=$ $\displaystyle
8x^{-2}(-1+2\gamma)\left[(-1+\gamma)^{2}+256A_{1}^{3}\gamma(-1+2\gamma)+4A_{1}(1+\gamma-2\gamma^{2})-32A_{1}^{2}(-3+2\gamma(2+\gamma))\right]$
$\displaystyle+2x^{-4\gamma}f^{-1}(x,\gamma)(-1+2\gamma)\left[-1-48A_{1}+8A_{1}\gamma(7+16A_{1})-\gamma+2(1-8A_{1})^{2}\gamma^{2}\right]$
$\displaystyle+x^{2-8\gamma}f^{-2}(x,\gamma)\left[-3+\gamma(9-6\gamma+8A_{1}(1+2\gamma))\right],$
$\displaystyle C$ $\displaystyle=$ $\displaystyle
8(-1+2\gamma)\left[(-1+\gamma)^{2}+256A_{2}^{3}\gamma(-1+2\gamma)+4A_{2}(1+\gamma-2\gamma^{2})-32A_{2}^{2}(-3+2\gamma(2+\gamma))\right]$
$\displaystyle+2f^{-1}(x,\gamma)(-1+2\gamma)\left[-1-48A_{2}+8A_{2}\gamma(7+16A_{2})-\gamma+2(1-8A_{2})^{2}\gamma^{2}\right]$
$\displaystyle+f^{-2}(x,\gamma)\left[-3+\gamma(9-6\gamma+8A_{2}(1+2\gamma))\right],$
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{x}{8}\left(1-\frac{x^{-4\gamma+2}-x}{2(1-x)f(x,\gamma)}\right),~{}~{}~{}~{}~{}~{}A_{2}=\frac{1}{8x}\left(1-\frac{x^{-4\gamma+2}-x}{2(1-x)f(x,\gamma)}\right).$

(a) $q=0.85,~{}1\geq\gamma>\frac{3}{4}$ (b) $q=0.85,~{}\gamma\geq 1$

(c) $\gamma=0.8$

Figure 8: R-x curves with the different values of the non-linear parameter
$\gamma$ and YM charge $q$ for two phases of the EPYM black hole. The dashed
lines represent the geometry $R_{2}$, and the continuous lines correspond to
the geometry $R_{1}$. In the upper left plot,
$q=0.85,~{}1\geq\gamma>\frac{3}{4}$, and the non-linear parameter is set to
$\gamma=0.78$ (black lines), $\gamma=0.82$ (red lines), and $\gamma=1$ (green
lines). In the upper right plot, $q=0.85,~{}\gamma\geq 1$, and the non-linear
parameter is set to $\gamma=1$ (black lines), $\gamma=1.2$ (red lines), and
$\gamma=1.5$ (green lines). In the bottom plot, $\gamma=0.8$, and the YM
charge is set to $q=0.75$ (black lines), $q=0.85$ (red lines), $q=0.95$ (green
lines), and $q=1$ (blue lines)

Note that for the EPYM black hole with a given non-linear parameter $\gamma$
under going a phase transition, because of the conditions $P|_{r=r_{1}}\geq 0$
and $\frac{\partial P}{\partial r}|_{r=r_{1}}\geq 0$, the horizon radius
$r_{1}$ has a minimum value, and $x$ must be from $x_{min}$ to one. In other
words, $x_{min}$ must satisfy the following expression:

$\displaystyle\frac{(3-4\gamma)(1+x_{min})(1-x_{min}^{4\gamma})+8\gamma
x_{min}^{2}(1-x_{min}^{4\gamma-3})}{2(3-4\gamma)(1-x_{min})^{3}}=\frac{4\gamma-1}{2}.$
(38)

Especially, the minimum value of $x$ is independent of the YM charge $q$. The
$R-x$ plots with the range $0\leq x\leq 1$ are shown in Fig. 8. For the given
non-linear parameter $\gamma$, $x$ must be considered from some certain value,
for example, $0.515\leq x\leq 1$ with $\gamma=1.5$. From Fig. 8, we obtain
that for the given YM charge $q$ and $x$, the distance ($\triangle
R(x)=R_{2}(x)-R_{1}(x)$) of the Ricci scalars for two phases decreases with
increaseint $\gamma$ and YM charge $q$. The effect of the non-linear parameter
$\gamma$ on the Ricci scalar is consistent with that on the latent heat of
phase transition given in the last section. It is considerably unique that the
scalar curvature $R_{c}$ at the critical point increases with increasing YM
charge $q$ and $\gamma$ in the range $1\geq\gamma>\frac{3}{4}$, while it
decreases with increasing non-linear parameter $\gamma$ in the range
$\gamma\geq 1$.

Based on the conclusion of B. Mirza and H. Mohammadzadeh Mirza2008 ; Mirza2009
on the scalar curvature, the interactions of molecules are repulsive,
attractive, and zero for $R>0$, $R<0$, and $R=0$, respectively. Fig. 8 shows
that for the EPYM black hole, $0>R_{2}>R_{1}$. That is, the average
interaction of the black hole molecules for phase $\phi_{2}^{2}$ is less than
that of phase $\phi_{1}^{2}$. Furthermore, the Ricci scalars for both phases
increase with increasing of the YM charge $q$, until $R_{2}$ becomes zero.
Since the density of black hole molecules $n=\frac{N}{V}=\frac{3}{\gamma
l_{p}^{2}r_{+}}$, the density for phase $\phi_{2}^{2}$ is less than that for
phase $\phi_{1}^{2}$. From Eq. (21), we know that for the EPYM black hole with
the fixed temperature ($T<T_{c}$), the phase $\phi_{2}^{2}$ corresponds to a
lower YM electric potential, while the other phase $\phi_{1}^{2}$ corresponds
to a higher YM electric potential. The horizon radius increases with
increasing YM charge $q$, while the YM electric potential remains uncharged.
Therefore, we think that the non-linear parameter $\gamma$ plays two roles in
the phase transition for the EPYM black hole: change the order degree of the
black hole molecules, which is determined by the YM electric potential, and
change the density of the black hole molecules.

##  VI Discussions and Conclusions

Because a detailed statistical description of the corresponding thermodynamic
states of black holes is still unclear, the investigation of the thermodynamic
properties and critical phenomenon of black holes becomes crucial. In this
study, we focus on the phase transition and thermodynamic geometry of the
four-dimensional Einstein-power-Yang-Mills (EPYM) black hole. We study the
effect of the non-linear YM charge parameter $\gamma$ on the thermodynamical
properties.

First, we reviewed the thermodynamic state parameters
($T,~{}P,~{}S,~{}M,~{}\Psi$) of the AdS black hole in the four-dimensional
EPYM gravity. Then we reconstructed Maxwell’s equal-area law by adopting
different independent dual parameters to explore the phase transition of the
EPYM black hole. The results showed that the phase transition point is the
same for the three choices ($P-V,~{}T-S,~{}q^{2\gamma}-\Psi$). Especially, the
independent dual parameters of the YM charge and the corresponding potential
must be $q^{2\gamma}-\Psi$, not other forms. This result is attributed to the
fact the phase transition must be independent with the concrete physical
process. Furthermore, the phase transition condition of the EPYM black hole
was presented as described in Eq. (21). We obtain that the pressure of the
EPYM black hole with the unchanged temperature $T\leq T_{c}$ decreases with
increasing $\gamma$. The phase transition is related to the non-linear
parameter $\gamma$ and the horizon radius, while it is not the only pure one
between a small and a big black hole.

Next, we presented the coexistent curve in the $P-V$ diagram and analyzed the
latent heat of phase transition for the EPYM black hole. The coexistent curve
in the $P-T$ diagram showed that the critical point of phase transition
increases with increasing non-linear YM charge parameter $\gamma$. In
addition, the latent heat of phase transition for a fixed $x$ decreases with
increasing non-linear parameter $\gamma~{}(\gamma>3/4).$

Finally, by defining a new order parameter $\phi^{2}(T)$ and comparing the
EPYM black hole to an ordinary thermodynamic system at the microscopic level,
we explored the phase transition microstructure of the EPYM black hole by the
scalar curvature $R$. The result suggested that the phase transition of the
EPYM black hole is accompanied by the change of the black hole molecules’
symmetry. For the EPYM black hole with a certain temperature and pressure
($T<T_{c}$, $P<P_{c}$), the Ricci scalars at the boundary of the two-phase
coexistence area are different due to the different values of the order
parameter $\phi^{2}$ at the boundary of the two-phase coexistence area.
Especially, it was significantly unique that the scalar curvature $R$ at the
critical point increases with increasing YM charge $q$ and the non-linear
parameter $\gamma$ in the range $1\geq\gamma>3/4$, while it decreases with
increasing non-linear parameter $\gamma$ in the range $\gamma\geq 1$.

This work elucidated the phase transition microstructure of the EPYM black
hole, which is useful to explore the microstructure of a black hole and
understand its basic property. In particular, the in-depth study of the black
hole microscopic structure will be significantly important to establish
quantum gravity.

## Acknowledgements

We would like to thank Prof. Zong-Hong Zhu and Meng-Sen Ma for their
indispensable discussions and comments. This work was supported by the
National Natural Science Foundation of China (Grant No. 11705106, Grant No.
11475108, Grant No. 12075143), the Natural Science Foundation of Shanxi
Province, China (Grant No.201901D111315), the Natural Science Foundation for
Young Scientists of Shanxi Province, China (Grant No.201901D211441), the
Scientific Innovation Foundation of the Higher Education Institutions of
Shanxi Province (Grant Nos. 2020L0471, Grant Nos. 2020L0472), and the Science
Technology Plan Project of Datong City, China (Grant Nos. 2020153).

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# Generation of Circularly-Polarised High-Harmonics with Identical Helicity in
Two-Dimensional Materials

Navdeep Rana  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India  M. S. Mrudul  Department of Physics and
Astronomy, Uppsala University, P.O. Box 516, SE 75120, Uppsala, Sweden  Gopal
Dixit<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Bombay, Powai, Mumbai 400076, India

###### Abstract

Generation of circularly-polarized high-harmonics with the same helicity to
all orders is indispensable for chiral-sensitive spectroscopy with attosecond
temporal resolution. Solid-state samples have added a valuable asset in
controlling the polarization of emitted harmonics. However, maintaining the
identical helicity of the emitted harmonics to all orders is a daunting task.
In this work, we demonstrate a robust recipe for efficient generation of
circularly-polarized harmonics with the same helicity. For this purpose, a
nontrivial tailored driving field, consisting of two co-rotating laser pulses
with frequencies $\omega$ and $2\omega$, is utilized to generate harmonics
from graphene. The Lissajous figure of the total driving pulse exhibits an
absence of the rotational symmetry, which imposes no constraint on the
helicity of the emitted harmonics. Our approach to generating circularly-
polarized harmonics with the same helicity is robust against various
perturbations in the setup, such as variation in the subcycle phase difference
or the intensity ratio of the $\omega$ and $2\omega$ pulses, as rotational
symmetry of the total driving pulse remains absent. Our approach is expected
to be equally applicable to other two-dimensional materials, among others,
transition-metal dichalcogenides and hexagonal boron nitride as our approach
is based on absence of the rotational symmetry of the driving pulse. Our work
paves the way for establishing compact solid-state chiral-XUV sources, opening
a new realm for chiral light-matter interaction on its intrinsic timescale.

##  I Introduction

In the last few decades, high-harmonic generation (HHG) has become an
essential ingredient in attosecond science. HHG is a strong-field-driven
highly nonlinear frequency up-conversion process. HHG provides a route not
only to generate attosecond pulses in the extreme ultraviolet (XUV) energy
regime [1] but also to interrogate electron dynamics on its natural timescale
[2, 3, 4, 5]. Owing to the recollision mechanism of HHG in gases [6], the
harmonic yield drops markedly as the polarization of the driving laser pulse
changes from linear to circular. To circumvent this problem, a combination of
two counter-rotating circularly-polarized laser pulses with frequencies
$\omega$ and $2\omega$ has been employed to generate circularly-polarized
harmonics. The resultant harmonic spectrum displays doublets of circularly-
polarized harmonics where $(3n+1)$ and $(3n+2)$ harmonics exhibit the
polarization of $\omega$\- and $2\omega$-fields, respectively. However, $3n$
harmonics do not conserve parity and thus they are forbidden [7, 8]. Different
variants of the bichromatic counter-rotating $\omega-2\omega$ schemes, such as
varying the ratio of the ellipticities and/or of the intensities of the two
driving pulses [9, 10], controlling the subcycle phase between the two pulses
[11], introduction of an additional seed pulse with different polarization
[12, 13], adding plasmonic field enhancement [14], and noncollinear geometry
of the $\omega-2\omega$ scheme [15, 16] are employed to control the
polarization of the emitted harmonics. It is important to emphasis that the
recognition of chiral molecules [17, 18], circular dichroism in magnetic
materials [19], spin and magnetisation dynamics in solids at their natural
timescales [20, 21], to name but a few, are examples of chiral-sensitive
light-matter interaction phenomena where the desired control over the
polarization of the emitted harmonics and resultant attosecond XUV pulses is
quintessential.

The extension of HHG from gases to solids offers an attractive option for
compact tabletop sources of coherent XUV attosecond pulses. Because of the
periodic nature of solids with higher electron density in comparison with
gases, HHG in solids seems a better option for higher yield of harmonics [22,
23]. Moreover, solid HHG has added a new dimension in attosecond spectroscopy
to shed new light on various equilibrium and nonequilibrium aspects of solids
[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

In comparison with HHG from gases, HHG from solids exhibits different
sensitivity toward circularly-polarized laser pulses [40, 41]. It has been
shown that a high degree of control over the polarization of emitted harmonics
can be achieved by the dynamic control of crystal symmetries and intertwined
interband and intraband electronic dynamics [42]. Saito et al. have
demonstrated the generation of harmonics with alternate helicity using a
circularly-polarized laser pulse, owing to the selection rules derived from
the symmetry [43]. Recently, bicircular counter-rotating $\omega-2\omega$
laser pulses have been employed to generate circularly-polarized harmonics in
solids, which has enabled one to preform symmetry-resolved chiral spectroscopy
[44] and to probe valley-selective excitations in two-dimensional materials
[45, 46, 47]. In all cases, the harmonic spectra of solids, similar to gases,
are composed of pairs of circularly-polarized harmonics with opposite helicity
owing to the threefold rotational symmetry of the bicircular counter-rotating
pulses. Thus, the generation of circularly-polarized harmonics with the same
helicity to all orders is missing – a major impediment in the realisation of a
compact all-solid-state XUV source for chiral spectroscopy. The main focus of
the present work is to address this major obstacle.

Our work provides a robust recipe to generate circularly-polarized harmonics
with the same helicity to all orders. For that purpose, a nontrivial tailored
driving field without any rotational symmetry is designed for HHG. The total
electric field of the co-rotating bicircular $\omega-2\omega$ pulses does not
exhibit any symmetry, which is in contrast to the counter-rotating
$\omega-2\omega$ pulses with threefold symmetry. Thus, it is expected that the
absence of any symmetry of the total driving field will impose no symmetry
constraint on the helicity of the emitted harmonics. To test our idea,
monolayer graphene is exposed to a combination of bicircular co-rotating
$\omega-2\omega$ laser pulses. It is worth mentioning that HHG from graphene
and the underlying mechanism have been extensively explored in recent years
[48, 49, 50, 51, 52, 53, 54, 55, 56, 57]. Moreover, the anomalous ellipticity
dependence of HHG from graphene has been explored experimentally [58, 54] as
well as discussed theoretically [58, 54, 59, 60, 61, 53].

##  II Computational Details

Semiconductor Bloch equations in the Houston basis are solved to simulate the
interaction of laser with graphene. The nearest-neighbor tight-binding
approach is used to represent graphene as discussed in our previous work [62].
High-harmonic spectrum is simulated by performing the Fourier transform
($\mathcal{FT}$) of the time derivative of the total current as

$\mathcal{I}(\omega)=\left|\mathcal{FT}\left(\frac{d}{dt}\left[\int_{BZ}\textbf{J}(\mathbf{k},t)~{}d{\textbf{k}}\right]\right)\right|^{2}.$
(1)

Here, $\textbf{J}(\mathbf{k},t)$ is the current at any k-point in the
Brillouin zone as a function of time.

##  III Results and Discussion

Before we discuss results obtained for bicircular co-rotating $\omega-2\omega$
laser pulses, let us revisit the results for single-colour and bicircular
counter-rotating $\omega-2\omega$ laser pulses. The polarization-resolved
high-harmonic spectrum of graphene driven by a single-colour left-handed
circularly-polarized pulse is shown in Fig. 1(a). Owing to the six-fold
rotational symmetry of graphene and conservation of spin angular momentum of
the driving laser pulse, selection rules indicate the generation of (6$n$
$\pm$1)-order harmonics with $n=0,1,2,3,\cdots$. Moreover, the helicity of the
6$n$+1 (6$n$-1) harmonics is the same (opposite) helicity as the driving pulse
[63]. Our results in Fig. 1(a) are in prefect agreement with the selection
rules and with an earlier report [55]. Moreover, the low harmonic yield of the
fifth and seventh harmonics with opposite helicity has been observed in an
experiment [58]. It has been anticipated that the yield of harmonics can be
improved by increasing the intensity of the driving field [55].

The total vector potential corresponding to $\omega-2\omega$ circularly-
polarized laser pulses is given as

$\textbf{A}(t)=\frac{A_{0}f(t)}{\sqrt{2}}\left(\left[\cos(\omega
t+\phi)+\frac{\textsf{R}}{2}\cos(2\omega
t)\right]\hat{\textbf{e}}_{x}+\left[\sin(\omega
t+\phi)\pm\frac{\textsf{R}}{2}\sin(2\omega
t)\right]\hat{\textbf{e}}_{y}\right).$ (2)

Here, $A_{0}$ is the amplitude of the vector potential, $f(t)$ is the temporal
envelope of the driving field, $\phi$ is the subcycle phase difference between
$\omega$ and 2$\omega$ pulses, and $\mathsf{R}$ is the ratio of electric field
strength of the two pulses. +(-) represents the co-rotating (counter-rotating)
laser pulse configuration. In this work, fundamental $\omega$ pulse with a
peak intensity of 0.3 TW/cm2 and a wavelength of 2 $\mu$m is used for HHG from
graphene. The fundamental driving pulse has eight cycles with sin-square
envelope. Similar laser parameters have been employed to study coherent
electron dynamics in graphene [64, 65].

Figure 1: Polarization-resolved high-harmonic spectrum of graphene driven by
(a) single-colour circularly-polarized laser, (b) bicircular counter-rotating
$\omega$-2$\omega$, and (c) bicircular co-rotating $\omega$-2$\omega$ laser
pulses. In all cases, the fundamental $\omega$ pulse is left-handed polarized.
Lissajous figures of the total driving fields are shown in the respective
insets. In all cases, the peak intensity of the $\omega$ pulse is 0.3 TW/cm2
and the wavelength is 2 $\mu$m. The subcycle phase difference between the two
pulses $\phi$ is zero, and the relative strength of the electric fields
between $\omega$ and 2$\omega$ pulses $\mathsf{R}$ is 2 in (b) and (c).

Figure 1(b) presents the polarization-resolved harmonic spectrum of graphene
driven by bicircular counter-rotating $\omega$-2$\omega$ laser pulses. The
resultant vector potential exhibits a trefoil symmetry as shown in the inset.
Following the threefold symmetry and conservation of spin angular momentum,
$n\omega=p\omega+2q\omega$ with $p=q\pm 1$ harmonics are allowed, whereas $3n$
harmonics are symmetry forbidden [8]. $p(q)$ is the number of photons of the
$\omega(2\omega)$ pulse. Moreover, the allowed $(3n+1)$\- and $(3n+2)$-order
harmonics follow the helicity of the $\omega$ and $2\omega$ pulses,
respectively [19, 44, 45]. From the figure, our results are consistent with
the selection rules and earlier report [46]. Note that the polarization of one
of the driving pulses individually impacts the helicity of the emitted
harmonics significantly.

Analysis of results in Figs 1(a) and 1(b) establishes that the generation of
harmonics with circular polarization, using either a single colour circular
pulse or bicircular counter-rotating $\omega$-2$\omega$ pulses, is possible.
In both cases, the adjacent harmonics are of opposite helicity, which has a
serious consequence for the helicity of the resultant attosecond pulses. If
the intensities of adjacent harmonics with opposite helicity are the same then
the generated attosecond pulse exhibits linearly polarization with a rotating
axis of polarization. In such circumstances, one needs to induce an imbalance
in the intensity of the adjacent harmonics to have a control over the helicity
of the resultant attosecond pulses [10, 11, 12].

The total harmonic spectrum in polarization-resolved fashion for the
bicircular co-rotating $\omega$-2$\omega$ laser pulses is presented in Fig.
1(c). As evident from the Lissajous figure in the inset, the total field does
not exhibit any rotational symmetry. Thus, the conservation of spin angular
momentum does not impose any constraint in the helicity of the generated
harmonics and none of the harmonics are symmetry forbidden. As a result, all-
order circularly-polarized harmonics are generated with the same helicity as
that of the driving pulse. Analysis of the relative yield of the spectrum
revels that the left-circularly-polarized harmonics are the leading harmonics.

Figure 2: Time profiles of the emitted harmonics, shown in Fig. 1(c),
generated by bicircular co-rotating $\omega$-2$\omega$ laser pulses (red
color). The $x$ and $y$ components of the harmonics are shown in blue and
green colours, respectively. The Lissajous figures of the emitted harmonics
are shown in cyan colour. All harmonics are left-handed circularly polarized.
The ellipticities of the 3${}^{\textrm{rd}}$, 4${}^{\textrm{th}}$,
10${}^{\textrm{th}}$ and 11${}^{\textrm{th}}$ harmonics are 0.8, 0.8, 0.8 and
0.9, respectively.

After demonstrating the capability of the co-rotating $\omega$-2$\omega$
scheme to generate circularly-polarized harmonics with the same helicity, let
us analysis the temporal evolution of the emitted harmonics in the time
domain. For this purpose, Fig. 2 presents the time profiles of a few selected
harmonics corresponding to Fig. 1(c). All the harmonics are left-handed
circularly polarized as evident from the Lissajous figure and the $x$ and $y$
components of the harmonics in the time domain.

Figure 3: High-harmonic spectra of graphene driven by bicircular co-rotating
$\omega$-2$\omega$ laser pulses for different values of (a)-(b) the subcycle
phases $\phi$, and (c)-(d) the relative electric field strengths $\mathsf{R}$.
Left- and right-hand panels show the contributions of left- and right-handed
polarization to the harmonics, respectively. Lissajous figures of the
resultant vector potential, associated with bicircular co-rotating
$\omega$-2$\omega$ laser pulses, are shown in above and below the panels for
different values of $\phi$ and $\mathsf{R}$, respectively. The $x$ and $y$
components of the total vector potential as given in Eq. (2) are plotted
omitting the envelope function along the X and Y axes for different values of
$\phi$ and R as mentioned in the inset. These inset plots are aimed to display
the symmetry of the total field corresponding to different phase and intensity
relations between $\omega$ and $2\omega$ fields. The black dot in each
Lissajous figure corresponds to the start of the vector potential at $t=0$.

At this juncture it is natural to wonder how robust are the features of the
spectrum shown in Fig. 1(c) with respect to the variations in the subcycle
phase $\phi$ and the relative ratio $\mathsf{R}$. To avoid overlapping and
have a better representation, we present the harmonics with left- and right-
handed circular polarization in the left- and right-hand panels of Fig. 3,
respectively. Figures 3(a) and 3(b) display the harmonic spectrum for
different values of $\phi$. From the figure, the polarization state of the
emitted harmonics is insensitive with respect to variations in $\phi$, and all
the harmonics are left-handed circularly polarized in nature. To understand
the insensitivity, let us look at the total electric field of the co-rotating
$\omega$-2$\omega$ scheme for different values of $\phi$ as shown above the
panels. As evident from the Lissajous figures of the total field, the
variation in $\phi$ results in the rotation of the resultant field. However,
the fields have no rotation symmetry. Thus, there are no significantly changes
in the polarization and the intensity of the emitted harmonics [see Figs. 3(a)
and 3(b)].

Figure 4: High-harmonic spectrum of graphene generated by bicircular co-
rotating $\omega$-2$\omega$ laser pulses for different values of dephasing
time. Helicity of the emitted harmonics is the same as that of the $\omega$
driving pulse, i.e., left-handed circularly polarized. The laser parameters
are the same as those used in Fig. 1(c).

Now let us explore how the variation in the $\mathsf{R}$ value affects the
harmonic spectra. The peak intensity of the resultant field is maintained at
0.3 TW/cm2 while varying the $\mathsf{R}$ value. As expected, an increase in
the $\mathsf{R}$ value results in an increase of the harmonic yield [see Figs.
3(c) and 3(d)]. However, the overall nature of the spectra does not change
with respect to $\mathsf{R}$, i.e., harmonics are still circularly polarized
in the direction of the driving field. The reason for that can be attributed
to the similar Lissajous figures of the resultant field for different
$\mathsf{R}$ values as evident from the bottom panel. Thus, the investigation
of Fig. 3 leads us to conclude that the generation of circularly-polarized
harmonics with same helicity via the co-rotating $\omega$-2$\omega$ scheme
does not require fine tuning of subcycle phase $\phi$ or intensity ratio
$\mathsf{R}$ between the pulses. Therefore, the co-rotating scheme is robust
in generating circularly-polarized harmonics with the same helicity to all
ordera.

Before making a summary, let us examine how the harmonic spectrum is sensitive
to the dephasing time – a phenomenological term accounting for the decoherence
between electrons and holes within the semiconductor-Bloch equations
framework. Figure 4 presents how the different values of the dephasing time
affect the harmonic spectrum corresponding to co-rotating $\omega$-2$\omega$
laser pulses. As evident from the figure, lower-order harmonics are
insensitive to the dephasing time as they are dominated by the intraband
current [62, 54, 66]. On the other hand, higher-order harmonics are dominated
by the interband current as their intensities are boosted with an increase in
the dephasing time. The interband channel is prone to the coherence between
the excited electrons and their corresponding holes [54]. Also, longer
electron-hole trajectories are sensitive to the decoherence time as the
corresponding excursion time allows for more scattering events during HHG
[66]. Thus, relatively smaller dephasing time means more scattering events for
the same excursion time. Hence, the yield of higher harmonics will be
suppressed for smaller dephasing time [66]. Note that the increase in the
harmonic yield with increase of dephasing time does not alter the polarization
properties of the emitted harmonics as evident from Fig. 4.

##  IV Conclusion

In summary, we explore the possibility of generating circularly-polarized
high-harmonics with the same helicity in solids. To achieve this ambitious
goal, we harness the no rotational symmetry of the tailored driving field
consisting of two co-rotating circular pulses with frequencies $\omega$ and
$2\omega$. The absence of rotational symmetry of the driving field enforces no
constraint on the helicity of the emitted harmonics. It is observed that the
spectrum of monolayer graphene, driven by the bicircular co-rotating pulses,
consists of circularly-polarized harmonics with the same helicity to all
orders as that of the fundamental $\omega$ pulse. The Lissajous figure of the
total driving pulse exhibits no rotational symmetry for different subcycle
phase differences between $\omega$ and $2\omega$ pulses and different
intensity ratios between the pulses. Moreover, dephasing time, associated with
the decoherence between electrons and holes, does not alter the nature of the
harmonic spectrum. Thus, our approach for generating circularly-polarized
high-harmonics with the same helicity is robust against any imperfection in
the driving pulses. Furthermore, we anticipate that our approach will be
applicable to other hexagonal two-dimensional materials, such as hexagonal
boron nitride and molybdenum disulfide, as our approach is based on the
absence of rotational symmetry of the total system consisting of the total
driving field and the two-dimensional material. However, unlike gapless
graphene, the interplay of the interband and intraband currents is different
in gapped graphene [67], and therefore the value of the dephasing time will
play a crucial role in determining the helicity of the emitted harmonics in
gapped graphene. Thus, laser parameters need to be optimised to obtain a
desired control over the helicity of the emitted harmonics from gapped
graphene. Also, based on the results presented in Ref. [44], our approach
could be extended to bulk materials also. Present work offers an avenue for
detailed chiral-sensitive light-matter interactions on natural timescales.

## Acknowledgements

G. D. acknowledges support from Science and Engineering Research Board (SERB)
India (Project No. MTR/2021/000138).

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