# Quantum phase transition of light in coupled optical cavity arrays: A
renormalization group study

Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar,
Bangalore 5600 80, India.

###### Abstract

We study the quantum phase transition of light of a system when atom trapped
in microcavities and interacting through the exchange of virtual photons. We
predict the quantum phase transition between the photonic Coulomb blocked
induce insulating phase and anisotropic exchange induced photonic superfluid
phase in the system due to the existence of two Rabi frequency oscillations.
The renormalization group equation shows explicitly that for this system there
is no self-duality. The system also shows two Berezinskii-Kosterlitz-Thouless
(BKT) transitions for the different physical situation of the system. The
presence of single Rabi frequency oscillation in the system leads to the BKT
transition where system shows the quantum phase transition from photonic
metallic state to the Coulomb blocked induced insulating phase. For the other
BKT transition when the z-component of exchange interaction is absent, the
system shows the transition from the photonic metallic state to the photonic
superfluid phase. We also predict the commensurate to incommensurate
transition under the laser field detuning.

Pacs: 42.50.Dv, 42.50.Pq, 03.67.Bg, 75.10.Jm
Keywords: Cavity Quantum Electrodynamics, Renormalization Group Theory,
Quantum Optics and Quantum Spin Model

Introduction:

The recent experimental success in engineering strong interaction between the
photons and atoms in high quality micro-cavities opens up the possibility to
use light matter system as quantum simulators for many body physics [1-18].
The authors of Ref. (hart1 ,hart2 , suj2 ) have shown that effective spin
lattice can be generated with individual atom in the microcavities that are
coupled to each other via exchange of virtual photons. The two states of spin
polarization are represented by the two long lived atomic levels in the
system.
A Many body Hamiltonians can be created and probed in coupled cavity arrays.
In our previous study, we have explained explicitly the basic physics of the
formation of micro-optical cavity suj2 . Atoms in the cavity are used for
detection and also for generation of interaction between photons in the same
cavity. This artificial system can act as a quantum simulator [1-6]. This
micro-cavity system shows the different quantum phases and quantum phase
transition (QPT) of photonic states of the system.
QPT associate with the fundamental changes that occurs in the macroscopic
nature of the matter at zero temperature due to the variation of some external
parameter. Quantum phase transitions are characterized by the drastic change
in the ground state properties of the system driven by the quantum
fluctuations subir ; indrani .
In this paper, we derive the renormalization group equation for the continuum
field theoretical study of the model Hamiltonian. We predict with the
existence of photonic metallic state, the anisotropic exchange induces the
photonic superfluid phase and photon blocked induces the insulating phase. The
transition from the photonic superfluid phase to the photon blocked induced
insulating phase is the second order quantum phase transition. Whereas the
transition from photonic metallic state to the photonic insulating state is
the Berezinski-Kosterlitz-Thouless (BKT) transition subir ; gia ; druf  where
the anisotropic exchange interaction is absent. The other BKT transition
occurs when we consider the transition from photonic metallic state to the
photonic superfluid state when the z-component interaction is absent.
The quantum state of engineering of the cavity QED system is in the state of
art due to the rapid technological development in this field [1]. Therefore
one can achieve the all states through the proper manipulation of laser
frequencies, detuning field, Rabi frequency oscillations. In our study, the
photonic metallic state is the Luttinger liquid phase of the microcavities
array system. The photonic superfluid phase is originated from the anisotropic
exchange interactions which discuss in Ref.jor . To the best of our knowledge,
the explicit study of the different quantum phases through the RG equations
and the proposal of photonic metallic state is the first in the literature for
the Cavity QED system. Now we discuss in detail about the photon blocked
induce insulating phase. The photon blocked is a dressed state of strongly
coupled atom-cavity system. In this phase the inhivitation of resonant
absorption of a second photon if one photon is already resonantly absorbed.
This phenomena was first observed by Birnbaum $et~{}al.$ birn . In this
studies they have predicted the photon blocked in the single atom in the
cavity birn . Here we predict this phenomena for the array of cavities, where
the atoms in different cavities interacting through virtual photon
emission/absorption process.
We will see after the continuum field theoretical study that our model
Hamiltonian contains two strongly relevant and mutually nonlocal perturbation
over the Gaussian (critical) theory. In such a situation the strong coupling
fixed point is usually determined by the most relevant perturbation whose
amplitude grows up according to its Gaussian scaling dimensions and it is not
much affected by the less relevant coupling terms. However, this is not the
general rule if the two operators exclude each other, i.e., if the field
configurations which minimize one perturbation term do not minimize the other.
In this case interplay between the two competing relevant operators can
produce a novel quantum phase transition through a critical point or a
critical line. Therefore, we would like to study the RG equation to interpret
the quantum phases of the system.
Model Setup and Motivation
We would like to solve the quantum phases and phase boundaries of coupled
cavity arrays by establishing the quantum spin chain model of the system hart1
; hart2 ; suj2 . At first we discuss the relation between the spin operators
and the atom-photon system. Our starting point is the Janes-Cummings
Hamiltonian, $H~{}=g({\sigma}a^{\dagger}+h.c)$. Cavity mode represent by the
bosonic operators ($a,{a}^{\dagger}$ ) and atomic mode represent by the
fermionic operator (${\sigma},{\sigma}^{\dagger}$). Where $\sigma$ is the
Pauli matrix which transform one excitation from the radiation field to the
atomic field. Therefore, one can write

${\sigma}^{\dagger}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right)$

, similarly one can write for ${\sigma}^{-}$ with $<1|=(1,0)$ and $<0|=(0,1)$.
The excitation in this system only transfer between atom and photon in the
cavity. For a fixed number ($n$) of total excitation, one can express the
manifold: ${H_{n}}=\\{|0,n>,|1,n-1>\\}$ provided $n\geq 1$. Here $|0,n>$ and
$|1,n-1>$ are represent atom in the ground state with $n$ photon and excited
states of the atom with ($n-1$) photon respectively. We consider the initial
state $|e,n-1>$, we obtain the state $|g,n>$ by the following operation,
$|g,n>={\sigma}a^{+}|e,n-1>$. Therefore, we may write the following relation
based on the conservation of the number of excitation.
${\sigma}^{\dagger}a|e,n-1>=~{}0~{}={\sigma}a^{\dagger}|g,n>$ and
${\sigma}^{\dagger}a|g,n>=|e,n-1>$. These relations are nothing but the
properties of spin operators acting on the spinors in the z basis.
Micro-cavities of a photonic crystal are coupled through the exchange of
photons. Each cavity consists of one atom with three levels in the energy
spectrum, two of them are long lived and represent two spin states of the
system and the other represent excited states (fig. 1 and fig. 2 of Ref. hart1
; hart2 ). Externally applied laser and cavity modes couple to each atom of
the cavity. It may induce the Raman transition between these two long lived
energy levels. Under a suitable detuning between the laser and the cavity
modes, virtual photons are created in the cavity which mediate interactions
with another atom in a neighboring cavity. One can eliminate the excited
states by choosing the appropriate detuning between the applied laser and
cavity modes. Then one can achieve only two states per atom in the long lived
state and the system can be described by a spin-1/2 Hamiltonian hart1 ; hart2
.
The Hamiltonian of the system consists of three parts:

$H~{}=~{}{H_{A}}~{}+~{}{H_{C}}~{}+~{}{H_{AC}}$ (1)

Hamiltonians are the following

${H_{A}}~{}=~{}\sum_{j=1}^{N}{{\omega}_{e}}|e_{j}><e_{j}|~{}+~{}{\omega}_{ab}|b_{j}><b_{j}|$
(2)

where $j$ is the cavity index. ${\omega}_{ab}$ and ${\omega}_{e}$ are the
energies of the state $|b>$ and the excited state respectively. The energy
level of state $|a>$ is set as zero. $|a>$ and $|b>$ are the two stable state
of a atom in the cavity and $|e>$ is the excited state of that atom in the
same cavity. The following Hamiltonian describes photons in the cavity,

${H_{C}}~{}=~{}{{\omega}_{C}}\sum_{j=1}^{N}{{a_{j}}}^{\dagger}{a_{j}}~{}+~{}{J_{C}}\sum_{j=1}^{N}({{a_{j}}}^{\dagger}{a_{j+1}}+h.c),$
(3)

where ${a_{j}}^{\dagger}({a_{j}})$ is the photon creation (annihilation)
operator for the photon field in the $j$’th cavity, ${\omega}_{C}$ is the
energy of photons and $J_{C}$ is the tunneling rate of photons between
neighboring cavities. Interaction between the atoms and photons and also by
the driving lasers are described by

${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+{g_{a}}{a_{j}})|e_{j}><a_{j}|+h.c]+[a\leftrightarrow
b].$ (4)

Here ${g_{a}}$ and ${g_{b}}$ are the couplings of the cavity mode for the
transition from the energy states $|a>$ and $|b>$ to the excited state.
${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the lasers with
frequencies ${\omega}_{a}$ and ${\omega}_{b}$ respectively.
The authors of Ref. hart1 ; hart2  have derived an effective spin model by
considering the following physical processes: A virtual process regarding
emission and absorption of photons between the two stable states of neghbior
cavity yields the resulting effective Hamiltonian as

${H_{xy}}=\sum_{j=1}^{N}B{{\sigma}_{j}}^{z}~{}+~{}\sum_{j=1}^{N}(\frac{J_{1}}{2}{{\sigma}_{j}}^{\dagger}{{\sigma}_{j+1}}^{-}~{}+~{}\frac{J_{2}}{2}{{\sigma}_{j}}^{-}{{\sigma}_{j+1}}^{-}+h.c)$
(5)

When $J_{2}$ is real then this Hamiltonian reduces to the XY model. Where
${{\sigma}_{j}}^{z}=|b_{j}><b_{j}|~{}-~{}|a_{j}><a_{j}|$,
${{\sigma}_{j}}^{+}=|b_{j}><a_{j}|$, ${{\sigma}_{j}}^{-}=|a_{j}><b_{j}|$

$\displaystyle H_{xy}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}(B{{\sigma}_{i}}^{z}~{}+~{}{J_{1}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}+{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y})$
(6)
$\displaystyle+{J_{2}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}-{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}))$
$\displaystyle=\sum_{i=1}^{N}B({{\sigma}_{i}}^{z}~{}+~{}{J_{x}}{{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}~{}+~{}{J_{y}}{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}).$

with ${J_{x}}=(J_{1}+J_{2})$ and ${J_{y}}=(J_{1}-J_{2})$.
Here we present the analytical expression of $B$, $J_{1}$ and $J_{2}$ in terms
of different physical parameters of the system. We follow the references hart1
; james .

$B=\frac{\delta_{1}}{2}-\beta$ (7) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\frac{{|{\Omega_{b}}|}^{2}}{4{\Delta}_{b}}({\Delta}_{b}-\frac{{|{\Omega_{b}}|}^{2}}{4{\Delta}_{b}}-$
(8)
$\displaystyle\frac{{|{\Omega_{b}}|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-{\gamma_{b}}{g_{b}}^{2}-{\gamma_{1}}{g_{a}}^{2}+{\gamma_{1}}^{2}\frac{{g_{a}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow
b)]$
${J_{1}}=\frac{\gamma_{2}}{4}(\frac{{|{\Omega_{a}}|}^{2}{g_{b}}^{2}}{{{\Delta}_{a}}^{2}}+\frac{{|{\Omega_{b}}|}^{2}{g_{a}}^{2}}{{{\Delta}_{b}}^{2}})$
(9)
${J_{2}}=\frac{\gamma_{2}}{2}(\frac{{\Omega_{a}}{\Omega_{b}}g_{a}g_{b}}{{\Delta}_{a}{\Delta_{b}}}).$
(10)

where $\gamma_{a,b}=\frac{1}{N}\sum_{k}\frac{1}{{\omega}_{a,b}-{\omega}_{k}}$
$\gamma_{1}=\frac{1}{N}\sum_{k}\frac{1}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$
and
$\gamma_{2}=\frac{1}{N}\sum_{k}\frac{e^{ik}}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$
${\delta_{1}}={\omega}_{ab}-({\omega}_{a}-{\omega}_{b})/2$,
${\Delta}_{a}={\omega}_{e}-{\omega}_{a}$.
${\Delta}_{b}={\omega}_{e}-{\omega}_{a}-({\omega}_{ab}-{\delta_{1}})$.
${{\delta}_{a}}^{k}={\omega}_{e}-{\omega}_{k}$,
${{\delta}_{b}}^{k}={\omega}_{e}-{\omega}_{k}-({\omega}_{ab}-{\delta_{1}})$,
$g_{a}$ and $g_{b}$ are the couplings of respective transition to the cavity
mode, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequency of laser with
frequency $\omega_{a}$ and $\omega_{b}$.
Here we discuss very briefly about an effective $z$-component of interactions
(${{\sigma}_{i}}^{z}{{\sigma}_{i+1}}^{z}$) in such a system. The authors of
Ref.hart1 ; hart2  have proposed the same atomic level configuration but
having only one laser of frequency ${\omega}$ that mediates the atom-atom
coupling through virtual photons. Another laser field with frequency $\nu$ is
used to tune the effective magnetic field. In this case the Hamiltonian
${H_{AC}}$ changes but the Hamiltonians $H_{A}$ and $H_{C}$ are the same.

$\displaystyle{H_{AC}}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}[(\frac{{\Omega}}{2}e^{-i{{\omega}}t}+\frac{{\Lambda}}{2}e^{-i{{\nu}_{a}}t}{g_{a}}{a_{j}})|e_{j}><a_{j}|+h.c]$
(11) $\displaystyle+[a\leftrightarrow b].$

Here, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the
driving laser with frequency ${\omega}$ on transition $|a>\rightarrow|e>$ ,
$|b>\rightarrow|e>$, whereas ${\Lambda}_{a}$ and ${\Lambda}_{b}$ are the
driving laser with frequency ${\nu}$ on transition $|a>\rightarrow|e>$ ,
$|b>\rightarrow|e>$. One can eliminate adiabatically the excited atomic levels
and photons by considering the interaction picture with respect to
$H_{0}=H_{A}~{}+~{}H_{C}$ [6,7]. They have considered the detuning parameter
in such a way that the Raman transitions between two level are suppressed and
also chosen the parameter in such a way that the dominant two-photon processes
are those that involve one laser photon and one cavity photon but the atom
makes no transition between levels a and b. Whenever two atoms exchange a
virtual photon both of them experience a Stark shift and play the role of an
effective ${{\sigma}^{z}}{{\sigma}^{z}}$ interaction hart1 ; hart2 ; suj2 .
Then the effective Hamiltonian reduces to

${H_{zz}}~{}=~{}\sum_{j=1}^{N}({B_{z}}{{\sigma}_{j}}^{z}~{}+~{}{J_{z}}{{\sigma}_{j}}^{z}{{\sigma}_{j+1}}^{z})$
(12)

These two parameters can be tuned independently by varying the laser
frequencies. Finally, they have obtained an effective model by combining
Hamiltonians $H_{xy}$ and $H_{zz}$ by using Suzuki-Trotter formalism. The
effective Hamiltonian simulated by this procedure is

$H_{spin}~{}=~{}\sum_{j=1}^{N}(B_{tot}{{\sigma}_{j}}^{z}~{}+~{}\sum_{{\alpha}=x,y,z}J_{\alpha}{{\sigma}_{j}}^{\alpha}{{\sigma}_{j+1}}^{\alpha})$
(13)

where $B_{tot}=B+{B_{z}}$. It has been shown in Ref. hart2  that $J_{y}$ is
less than $J_{x}$. From the analytical expressions of $J_{x}$ and $J_{y}$, it
is clear that the magnitudes of ${J_{1}}$ and $J_{2}$ are different. The
result of numerical simulations trigger us also to define a model to study the
quantum phases of this system. In the next section, we present the RG study of
this model Hamiltonian to extract quantum phases and transitions between them.
Here we present the analytical expression of $B_{tot}$, $J_{z}$ in terms of
different physical parameters of the system. We follow the references hart1 ;
james ; suj2 .
${J_{z}}={\gamma}_{2}{|\frac{{{\Omega}_{b}}^{*}g_{b}}{4\Delta_{b}}-\frac{{{\Omega}_{a}}^{*}g_{a}}{4\Delta_{a}}|}^{2}$

$B_{tot}=-\frac{1}{2}[\frac{{|{\Lambda}_{b}|}^{2}}{16{\tilde{\Delta_{b}}}^{2}}(4\tilde{\Delta_{b}}-\frac{{|{\Lambda_{a}}|}^{2}}{4({\tilde{\Delta}_{a}}-{\tilde{\Delta}_{b}})}-\frac{{|{\Lambda_{b}}|}^{2}}{{\tilde{\Delta}_{b}}}-{\beta_{2}})-{\beta_{3}}].$

${\beta_{2}}=\sum_{j=a,b}\frac{{|{\Omega_{j}}|}^{2}}{4({\Delta}_{j}-\tilde{{\Delta}_{b}})}.4\tilde{\gamma_{jb}}{g_{j}}^{2}$

$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle[\frac{{|{\Omega_{b}}|}^{2}}{16{{\Delta}_{b}}^{2}}(4{\Delta}_{b}-\frac{{|{\Omega_{a}}|}^{2}}{4{\Delta}_{b}}-$
(14)
$\displaystyle\frac{{|{\Omega_{b}}|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-\frac{{|{\Omega_{b}}|}^{2}}{\Delta_{b}}-\sum_{j=a,b}\frac{{|{\Lambda_{j}}|}^{2}}{4({\Delta}_{j}-\tilde{{\Delta}_{b}})}.4{\gamma_{jb}}{g_{j}}^{2})+{\gamma_{bb}}^{2}\frac{{g_{b}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow
b)]$

Here ${\gamma}_{1}=\frac{1}{N}\sum_{k}\frac{1}{\omega-{\omega}_{k}}$,
${\gamma}_{2}=\frac{1}{N}\sum_{k}\frac{e^{ik}}{\omega-{\omega}_{k}}$,
${\gamma}_{aa}={\gamma}_{bb}=\frac{1}{N}\sum_{k}\frac{1}{\omega-\omega_{k}}$.

${\gamma}_{ab}={\gamma}_{ba}=\frac{1}{N}\sum_{k}\frac{1}{\omega\pm\omega_{ab}-{\omega}_{k}}$
$\tilde{{\gamma}}_{ab}=\tilde{{\gamma}}_{ba}=\frac{1}{N}\sum_{k}\frac{1}{\nu\pm\omega_{ab}-{\omega}_{k}}$

$\tilde{{\gamma}}_{aa}=\tilde{{\gamma}}_{bb}=\frac{1}{N}\sum_{k}\frac{1}{\nu-\omega_{k}}$.

Analytical Derivation and Analysis of RG Equations:

To study the different quantum phases of the system described by the
Hamiltonian (Eq. 13), we express this Hamiltonian in more explicit way,

$\displaystyle H_{2}~{}=~{}\sum_{n}~{}[$
$\displaystyle(1+a)~{}S_{n}^{x}S_{n+1}^{x}~{}+~{}(1-a)~{}S_{n}^{y}S_{n+1}^{y}$
(15)
$\displaystyle+~{}\Delta~{}S_{n}^{z}S_{n+1}^{z}~{}+~{}h~{}S_{n}^{z}~{}]~{},$

where $S_{n}^{\alpha}$ are the spin-1/2 operators. We assume that the $XY$
anisotropy $a$ and the $zz$ coupling $\Delta$ satisfy the condition
$-1\leq\Delta\leq 1$, and $0<a\leq 1$ and magnetic field strength is $h\geq
0$. The parameters correspondence between the micro cavities and spin chain
are the following, $h\sim B_{tot}$, ${\Delta=J_{z}}$, ${J_{1}=1}$ and
${J_{2}}=a$. The $XY$ anisotropy breaks the in plane rotational symmetry. The
study of the quantum phases from the perspective of quantum spin system and
magnetism is not entirely a new one suj ; ric ; zamo . Here our main aim is to
study the quantum phases of microcavities array through the RG analysis of
this model Hamiltonian.
Spin operators can be recast in terms of spinless fermions through Jordan-
Wigner transformation and then finally one can express the spinless fermions
in terms of bosonic fields gia . We recast the spinless fermions operators in
terms of field operators by this relation.
${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$
, where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized
fields of right- and the left-moving fermions respectively, and $k_{F}$ is the
Fermi wave vector. We express the fermionic fields in terms of bosonic field
by the relation
${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))},$
where $r$ denotes the chirality of the fermionic fields, right (1) or left
movers (-1). The operators $U_{r}$ is the Klein factor to preserve the anti-
commutivity of fermions. $\phi$ field corresponds to the quantum fluctuations
(bosonic) of spin and $\theta$ is the dual field of $\phi$. They are related
by the relations ${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and
${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$. Hamiltonian
$H_{0}=\frac{v}{2}~{}\int~{}dx~{}[~{}(\partial_{x}\theta)^{2}~{}+~{}(\partial_{x}\phi)^{2}~{}]$
is non-interacting part of $H_{XYZ}$. Here $v$ is the velocity of the low-
energy excitations. It is one of the Luttinger liquid parameters and the other
is $K$, which is related to $\Delta$ by gia ; ric

$K~{}=~{}\frac{\pi}{\pi+2\sin^{-1}(\Delta)}~{}.$ (16)

where $K$ takes the values 1 and 1/2 for $\Delta=0$ (free field), and
$\Delta=1$ (isotropic anti-ferromagnet), respectively. The relation between
$K$ and $\Delta$ is not preserved under the renormalization, so this relation
is only correct for the initial Hamiltonian. The analytical form of the spin
operators in terms of the bosonic fields are:
$S_{n}^{x}~{}=~{}[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\cos({\sqrt{\frac{\pi}{K}}}\theta)$;
$S_{n}^{y}~{}=~{}-[~{}c_{2}\cos(2{\sqrt{\pi
K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\sin({\sqrt{\frac{\pi}{K}}}\theta)$, and
$S_{n}^{z}~{}=~{}{\sqrt{\frac{\pi}{K}}}~{}\partial_{x}\phi~{}+~{}(-1)^{n}c_{1}\cos(2{\sqrt{\pi
K}}\phi)~{}$ where $c_{i}$’s are constants as given in Ref. zamo . The
Hamiltonian $H_{2}$ in terms of bosonic fields is the following,

$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle
H_{0}+\frac{a}{2\pi\alpha}\int\cos(2{\sqrt{\frac{\pi}{K}}}\theta(x))dx$ (17)
$\displaystyle+\frac{\Delta}{{2\pi\alpha}^{2}}\int\cos(4{\sqrt{\pi
K}}\phi(x))dx+\frac{h\sqrt{K}}{\pi\alpha}\int{{\partial}_{x}}{\phi(x)}dx$

One can also write the above Hamiltonian in the following form

$\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle
H_{01}+\frac{a}{2\pi\alpha}\int\cos(2{\sqrt{{\pi}}}\theta(x))dx$ (18)
$\displaystyle+\frac{\Delta}{{2\pi\alpha}^{2}}\int\cos(4{\sqrt{\pi}}\phi(x))dx+\frac{h}{\pi\alpha}\int{{\partial}_{x}}{\phi(x)}dx$

Where $H_{01}$,

$\displaystyle H_{01}=\frac{1}{2\pi}\int
dx[uK{({\nabla\theta(x)})}^{2}+(u/K){({\nabla\phi(x)})}^{2}]$ (19)

One can get the $H_{XY}$ Hamiltonian by simply putting $\Delta=0$ in the above
Hamiltonian. In this derivation, different powers of coefficients $c_{i}$ have
been absorbed in the definition of $a,h$ and $\Delta$. The integration of the
oscillatory terms in the Hamiltonian yield negligible small contributions and
the origin of the oscillatory terms occur due the spin operators. So it’s a
reasonably good approximation to keep only the non-oscillatory terms in the
Hamiltonian. The Gaussian scaling dimension of these coupling terms, $a$ and
$\Delta$ are $1/K$ and $4K$ respectively. The third term ($\Delta$) of the
Hamiltonian tends to order the system into density wave phase , whereas the
second term ($a$) of the Hamiltonian favors the staggered order in the $XY$
plane. Two sine-Gordon coupling terms are from two dual fields. Therefore, the
model Hamiltonian consists of two competing interactions. This Hamiltonian
contains two strongly relevant and mutually nonlocal perturbation over the
Gaussian (critical) theory. In such a situation the strong coupling fixed
point is usually determined by the most relevant perturbation whose amplitude
grows up according to its Gaussian scaling dimensions and it is not much
affected by the less relevant coupling terms. However, this is not the general
rule if the two operators exclude each other, i.e., if the field
configurations which minimize one perturbation term do not minimize the other.
In this case interplay between the two competing relevant operators can
produce a novel quantum phase transition through a critical point or a
critical line. Therefore, we would like to study the RG equation to interpret
the quantum phases of the system. In the RG theory, we not only able to
predict the weak coupling limit but also the the intermediate values of the
coupling. The RG theory is a perturbative theory and it ceases to be valid
when the coupling constant $g(l)\sim 1$.
We now study how the parameters $a$, $\Delta$ and $K$ flow under RG. The
operators in Eq. (17) are related to each other through the operator product
expansion. So the RG equations for their coefficients therefore are coupled to
each other. We use operator product expansion to derive these RG equations
which is independent of boundary condition cardy . In our derivation, we
consider two operators, $X_{1}=e^{(ia_{1}\phi+ib_{1}\theta)}$ and
$X_{2}=e^{(ia_{2}\phi+ib_{2}\theta)}$. In the RG procedure, one can write
these two field operators as a sum of fast and slow mode fields. In the fast
field, the momentum range is $\Lambda e^{-dl}<K<\Lambda$ and for the slow
field $K<\Lambda e^{-dl}$, where $\Lambda$ is the momentum cut-off, $dl$ is
the change in the logarithmic scale. The next step is the integration of the
fast field for the operators $X_{1}$ and $X_{2}$, it yields a third operator
at the same space time point,
$X_{3}=e^{i(a_{1}+a_{2})\phi+i(b_{1}+b_{2})\theta)}$. The prefactor of $X_{3}$
can be found by the relation, ${X_{1}}{X_{2}}\sim
e^{-(a_{1}a_{2}+b_{1}b_{2})}\frac{dl}{2\pi}X_{3}$. Our Hamiltonian consists of
two operators, if we consider $l_{1}$ and $l_{2}$ as the coefficient of the
operators $X_{1}$ and $X_{2}$ respectively. Then the RG expressions for
$\frac{dX_{3}}{dl}$ contains the term
$(a_{1}a_{2}+b_{1}b_{2})\frac{l_{1}l_{2}}{2\pi}$. This is the procedure to
derive these RG equations.
In the RG process, one can write RG equations themselves are established in a
perturbative expansion in coupling constant ($g(l)$). They cease to be valid
beyond a certain length scale, where $g(l)\sim 1$ gia . The RG equations for
the coefficients of Hamiltonian $H_{XYZ}$ are

$\displaystyle\frac{da}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}(2-\frac{1}{K})a,$ $\displaystyle\frac{d{\Delta}}{dl}~{}$
$\displaystyle=$ $\displaystyle~{}(2-4K){\Delta}$
$\displaystyle\frac{dK}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}\frac{a^{2}}{4}~{}-~{}K^{2}{\Delta}^{2}~{},$ (20)

We have followed Ref.suj  during the derivation of these RG equations. These
RG equations have trivial (${a^{*}}=0={{\Delta}^{*}}$) fixed points for any
arbitrary $K$. Apart from that these RG equations have also two non-trivial
fixed lines, $a=\Delta$ and $a=-{{\Delta}}$ for $K=1/2$. The above RG
equations show that there is no duality in flow diagram. Here we mean,
duality, that if $\theta$ and $\phi$ interchange $\theta\leftrightarrow\phi$,
$K\leftrightarrow K^{-1}$ and $\Delta\leftrightarrow a$ will not produce the
same set of RG equations.
For $K>1/2$, the sine-Gordon coupling term correspond to the anisotropic
exchange coupling become relevant and the system flows to the photonic
superfluid phase. For $K<1/2$, the sine-Gordon coupling term correspond to the
z-component exchange interaction become relevant and the system flowing off to
the photon blocked induced insulating phase.
Here we explain the physical significance of different quantum phases of the
atom-cavity system what we find in our study.
(1). When both the anisotropic exchange interaction and the z-component of
exchange interaction is absent then there are no sine-Gordon coupling terms in
the Hamiltonian. Then the system is in the mass less Luttinger liquid phase,
i.e., the system is in the photonic metallic state. The other source of
photonic metallic state is that when the system shows the BKT transition which
we will discuss in the next section.
(2). The photonic insulator state of the atom-cavity array system corresponds
to the insulating state of the system where there is no transmission of photon
between the microcavities in the array due to the interaction between photons.
The appearance of this phase has already discussed in the analysis of three RG
equations. This phase will occurs when the system shows the BKT transition,
which we will discuss in the next section.
(3). Photonic superfluid state of the atom-cavity array system corresponds to
the gapless excitations of the system where the photon transmit from one
cavity to the other without any blocking . For this one dimensional cavity QED
system where there is no order parameter, one can only discriminate between
the photonic metallic state and photonic superfluid state by only finding the
difference of fluctuation in photon number in every sites of the array. This
photonic superfluid phase corresponds to dissipitionless flow of photon in the
system. We have already discussed about the appearance of photonic superfluid
phase from the analysis of three RG equations. The other source of the
appearance of photonic superfluid phase is the BKT transition which we will
discuss in the next section.

Berezinskii-Kosterlitz-Transition Physics in Cavity QED System.
The physics of BKT transition has found in different one dimensional and two
dimensional (classical system) and it has discussed extensively in different
context in the Ref. subir ; gia ; druf .
Before we start to discuss the appearance of BKT transition in our system, we
would like to discuss very briefly why it is necessary to study the BKT
transition. Here we study two different situations of our model Hamiltonian.
For the first case the exchange anisotropy is absent ($J_{2}=0$) and for the
second case z-component of exchange interaction is absent ($J_{z}=0$). For
both of these cases only one of the sine-Gordon coupling term is present,
therefore, there is no competition between the two mutually non local
perturbation. Therefore one can think that there is no need to study the RG to
extract the quantum phases and phase boundaries. But we still apply RG method
for the following reason. Each of these Hamiltonians consist of two part, the
first one ($H_{01}$ ) is the non-interacting where the $\phi$ and $\theta$
fields show the quadratic fluctuations and the other part of these
Hamiltonians are the sine-Gordon coupling terms which of either $\theta$ or
$\phi$ fields. The sine-Gordon coupling term lock the field either $\theta$ or
$\phi$ in the minima of the potential well. Therefore the system has a
competition between the quadratic part of the Hamiltonian and the sine-Gordon
coupling term and this competition will govern the low energy physics of these
Hamiltonians in different limit of the system. The RG process (BKT transition)
will predict the quantum phases of these system correctly.
It is very clear from the analytical expression of ${J_{1}},J_{2}$ and $J_{z}$
that one can control these parameters in the laboratory. The quantum state of
engineering of cavity QED system is in the state of art due to the rapid
technological development of this system [1]. In this study we consider the
situations where the $J_{2}$ is absent, i.e., the system with a single Rabi
frequency oscillation. We also consider the situation where the z-component of
exchange interaction is absent. These two situations lead to the two different
set of RG equations which show BKT transition.
For the first case, there is no anisotropic exchange coupling. In this
situation, the three RG equations of the previous section reduce to two RG
equations, which are the following

$\displaystyle\frac{d{\Delta}}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}(2-4K){\Delta}$ $\displaystyle\frac{dK}{dl}~{}$
$\displaystyle=$ $\displaystyle~{}-~{}K^{2}{\Delta}^{2}~{},$ (21)

Now we express these RG equations in the form of BKT transition form. As we
understand from the RG equations that the transition occurs at $K=1/2$. To
study the flow of the RG equation around this transition point, we recast the
RG equation in suitable form.
Here we follow the following transformation, $K=1/2+\frac{y_{||}}{4}$ and
$\Delta\rightarrow\Delta/2$.
The above equation reduce to the standard BKT equation.

$\displaystyle\frac{dy_{||}}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}-{\Delta}^{2}$ $\displaystyle\frac{d\Delta}{dl}~{}$
$\displaystyle=$ $\displaystyle~{}-~{}y_{||}{\Delta}~{},$ (22)

In our case, $\Delta\frac{d\Delta}{dl}=y_{||}\frac{dy_{||}}{dl}$. Therefore
$X^{2}={y_{||}}^{2}-{\Delta}^{2}$ is a constant of motion. Here we discuss the
relevant physics and the quantum phase transition between the photonic
metallic state and photonic Coulomb blocked state based on these equations.
Here we consider the following situations based on these equations:
(1) When $X>0$ and $y_{||}>0$, the sine-Gordon coupling term corresponding to
$\Delta$ term is irrelevant, the fixed point, ${\Delta}^{*}=0$ and
${y_{||}}^{*}=X$, close to this fixed point, we can write the RG equations as
$\frac{d\Delta}{dl}=2(1-2K^{*}){\Delta(l)}$ and $\frac{dy_{||}}{dl}=0$. Using
the flow equation and constant of motion for ${y_{||}}>{\Delta}$. One can
write the solution of coupling terms as

$y_{||}(l)=\frac{X}{tanh(Xl+atanh(\frac{X}{{y_{||}}^{0}})}$ (23)
$\Delta(l)=\frac{X}{sin(Xl+atanh(\frac{X}{{y_{||}}^{0}})}$ (24)

The condition at the line of seperatix is
${y_{||}}(l)={\Delta(l)}=\frac{{\Delta}^{0}}{1+{\Delta}^{0}l}$. There is no
mass gap excitation in the elementary excitation of the system, i.e, there is
no photon blocked induced insulating phase. Hence the system is in the
photonic metallic state.
To the best of our knowledge for the first time in the literature we predict
the existence of photonic metallic state for the array of Cavity QED system.
Here we would like to present the basic origin of photonic-metallic state
explicitly.
If we do the Jordan-Wigner transformation of our model Hamiltonian as we
present in Ref. 22. It is very clear from the Eq. 32 of Ref. 22 that the first
term represent the photon hopping term across the lattice of the cavity QED
array. The second and third term of the Hamiltonian represent the photonic
pair correlation and photonic density wave respectively in the system. When
the second and third term in the Hamiltonian either are absent or irrelevant
in the RG sense at that situation the system shows the photonic metallic
state.

(2) When ${\Delta}>y_{||}$. The RG equation for the coupling $\Delta(l)$
flowing off to the strong coupling regime. A perturbative expansion in
$\Delta$ cease to be valid beyond a certain length scale for which
$\Delta(l)\sim 1$. The analysis of the RG equation is not valid beyond this
length scale.
The analytical relation between the coupling constant is

$arctan(\frac{{{y}_{||}}^{0}}{\sqrt{{{\Delta}_{0}}^{2}-{{y_{||}}^{0}}^{2}}})-arctan(\frac{{{y}_{||}}}{\sqrt{{{\Delta}_{0}}^{2}-{{y_{||}}^{0}}^{2}}})=\sqrt{{{\Delta}_{0}}^{2}-{{y_{||}}^{0}}^{2}}.$
(25)

In this limit, the RG flowing off to the strong coupling phase. Here we
discuss the relevant physics of the phase. The sine-Gordon coupling term is

$\frac{\Delta_{1}u}{2\pi{\alpha}^{2}}\int dxcos(4\phi(x)).$

where $\Delta_{1}=\frac{\Delta}{\pi u}$. The $\phi(x)$ field locks into one of
the minima of the cosine potential. Now we expand the potential for large
$\Delta_{1}$ in the spirit of usual RG method subir ; gia ; druf . We can
write the effective Hamiltonian near to the minima

$H=H_{0}+\frac{4\Delta_{1}u}{\pi{\alpha}^{2}}{\phi}^{2}(x)$

. The total action of the Hamiltonian of the system can be written as

$S=\frac{1}{2\pi
K}\frac{1}{\beta\Omega}\sum_{k,{\omega}_{n}}[\frac{{{\omega}_{n}}^{2}}{u}+uk^{2}+\frac{8K\Delta_{1}u}{{\alpha}^{2}}]{\phi}^{*}(k,\omega_{n}){\phi}(k,\omega_{n})$
(26)

The excitations of the system which cost a finite energy even at $k=0$. In
this situation the field, $\phi(x)$ is massive. The system posses phononic
type mode. This phononic mode is the small oscillation of the field
${\phi}(x)$ around the minima of the cosine potential.
Suppose we consider the RG equation up to the point where ${\Delta_{1}}(l)\sim
1$. This excitations gap in the spectrum has the dimension of an energy and
the renormalization relation is ${\Omega}_{M}(l)=e^{l}{\Omega}_{M}(l=0)$, for
the case, ${\Delta_{1}}(l^{*})\sim 1$. We can write the expression for the gap
following the action as,
${\Omega}_{M}(l^{*})\sim\sqrt{{\Delta_{1}(l^{*})}}u/{\alpha}$. The true gap of
the system is ${\Omega}_{M}(l=0)\sim e^{-l^{*}}{\Omega}_{M}(l^{*})$,
${\Omega}_{M}(l^{*})=\frac{{u}}{\alpha}$. Thus at this phase of this system,
the explicit dependence of the Luttinger liquid parameter is absent. This gap
of the system is of the order of bandwidth of the cavity QED system.

(3) Now we consider the case, when ${\Delta_{1}}<<|y_{||}|$. The system is in
the deep massive phase. In this phase, one can write the RG equation as
$\frac{dK(l)}{dl}=0$ and $\frac{d\Delta_{1}(l)}{dl}=2(1-2K){\Delta_{1}}(l)$.
For this RG equation, we get ${\Delta_{1}}(l)={\Delta_{1}}(0)e^{2(1-2K)l}$
from this equation $e^{-l^{*}}={\Delta(0)}^{1/2(1-2K)}$. The true gap of the
system is ${\Omega}_{2}(l=0)\sim e^{-l^{*}}{\Omega}_{2}(l^{*})$,
$\frac{\Omega_{2}(l=0)}{\Omega_{2}(l^{*})}\sim{\Delta_{1}(0)}^{+1/2(1-2K)}$.
This gap is the power law dependence of the bare $\Delta_{1}$. This gap varies
with $K$, i.e., the gap of the system is now varying with the interaction of
the system. It is very clear from the analytical expression that the gap gets
smaller and tends to zero at $K=1/2$, this prediction is consistent with the
physical scenario that at $K=1/2$ the system shows the phase transition.

(4) Close to the transition point, Eq.(25) reduces to
$\sqrt{{{\Delta_{1}}_{0}}^{2}-{(y_{||})}^{2}}{l^{*}}=\pi$. So the square root
term goes to zero at the transition point and therefore
${y_{||}}^{0}/{\sqrt{{{\Delta_{1}}_{0}}^{2}-{y_{||}}^{2}}}\rightarrow\infty$
and the gap
${\Omega}(l=0)\sim{\Omega}({l^{*}})e^{-\frac{\pi}{\sqrt{{{\Delta_{1}}_{0}}^{2}-{y_{||}}^{2}}}}$.
As one approach to the transition point, such as
${\Delta_{1}}_{0}\rightarrow|{y_{||}}|$. The gap is exponentially small in the
square root of the distance to the transition.
It is therefore clear from our RG analysis that the system is in the massive
phase for the two limits. For the first case when ${\Delta}>>{y_{||}}$, the
excitation gap of the system is of the order of the bandwidth of the system
and the effect of photonic strong correlation is not explicit. The other limit
of gapped state, i.e, when $\Delta<<|y_{||}|$. The excitation gap of the
system has a power law dependence which vary with $K$.
It is very clear from the analytical expression of $J_{z}$ that one can tune
it to zero by adjusting the Rabi frequencies oscillation of the system. In
this situation, there is no z-component of intercavity exchange interaction.
Therefore the three RG equations are reduce to two RG equations.

$\displaystyle\frac{da}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}(2-1/K)a$
$\displaystyle\frac{dlnK}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}~{}{a^{2}}/{4K}~{},$ (27)

If we do the following transformation $\tilde{K}=1/{2K}$ and $\tilde{a}=a/2$
and after that we follow another transformations $\tilde{K}=1+{y_{||}}/2$ the
above equations reduce to

$\displaystyle\frac{dy_{||}}{dl}~{}$ $\displaystyle=$
$\displaystyle~{}-{\tilde{a}}^{2}$ $\displaystyle\frac{d\tilde{a}}{dl}~{}$
$\displaystyle=$ $\displaystyle~{}-~{}y_{||}{\tilde{a}}~{},$ (28)

The mathematical structure of these equations are the same as that of Eq. 22.
Therefore the mathematical analysis for the different limits are the same but
massive phase and deep massive phase for these situations are the photonic
superfluid phase.
The sine-Gordon coupling term is

$\frac{a_{2}u}{2\pi{\alpha}}\int dxcos(2\theta(x)).$

where $a_{2}=\frac{a}{u}$. The $\theta(x)$ field locks into one of the minima
of the cosine potential. Now we expand the potential for large $a_{2}$. We can
write the effective Hamiltonian near to the minima

$H=H_{0}+\frac{a_{2}u}{{\pi\alpha}}{\theta}^{2}(x).$

The total action of the Hamiltonian of the system can be written as

$S=\frac{1}{2\pi
K}\frac{1}{\beta\Omega}\sum_{k,{\omega}_{n}}[\frac{{{\omega}_{n}}^{2}}{u}+uk^{2}+\frac{2Ka_{2}u}{{\alpha}}]{\theta}^{*}(k,\omega_{n}){\theta}(k,\omega_{n})$
(29)

The excitations of the system which cost a finite energy even at $k=0$. In
this situation the field, $\theta(x)$ is massive. The true gap of the system
is ${\Omega}_{C}(l=0)\sim e^{-l^{*}}{\Omega}_{C}(l^{*})$,
${\Omega}_{C}(l^{*})=(\frac{{u}}{\alpha})\sqrt{\alpha}$. Thus at this phase of
this system, the explicit dependence of the Luttinger liquid parameter is
absent. This gap of the system is much less than the excitation gap of the
photon blocked state due to the presence of the extra factor $\sqrt{\alpha}$.
In the deep massive phase,
$\frac{\Omega_{C}(l=0)}{\Omega_{C}(l^{*})}\sim{a_{2}(0)}^{+1/(2-1/K)}$. This
gap is the power law dependence of the bare $a_{2}$. This gap varies with $K$,
i.e., the gap of the system is now varying with the interaction of the system.
It is very clear from the analytical expression that the gap gets smaller and
tends to zero at $K=1/2$ and this prediction is consistent with the physical
scenario that at $K=1/2$ the system shows the phase transition from photonic
metallic state to photonic superfluid phase.
Now we consider the effect of magnetic field ( ${\delta}_{1}$ term) on the
quantum phases and phase boundaries of this cavity QED arrays.
We do the following transformation to eliminate the magnetic field term from
the Hamiltonian. We substitute $4\pi\sqrt{K}\phi\rightarrow
4\pi\sqrt{K}\phi+4\pi\delta x$, where $\delta=\sqrt{K}h$. This introduce the
spatial oscillation in the $\Delta$ term. When the coupling $a$ is absent and
the applied detuning field is larger than the photonic Mott gap, the system
drives to the gapless photonic metallic state. This is nothing but the well
known Porkovosky-Talapov model which shows the commensurate to incommensurate
transition gia ; suj . When $\delta{a_{1}}>>1$ (where $a_{1}$ is the lattice
spacing), the $cos(4\pi\sqrt{K}(\phi+\delta x))$ term is quickly oscillating
and averages out zero. Thus the system reflects the competition between the
$\Delta$ and $h$. As a results of it, the RG flow for $\Delta$ has to be
cutoff when $4\pi\sqrt{K}\delta(l)a_{1}\sim 1$. To the lowest order in
$\Delta,a$ and $\delta$, the RG flow equation is

$\frac{d\delta}{dl}=\delta$ (30)

As a result of this RG equation, the system shows two different response,
either the flow ${\Delta}_{1}$ and $\delta$ or $a$ and $\delta$ flows of to
the strong coupling phase.
When $\Delta_{1}(l)$ reaches strong coupling before $\delta a_{1}$ become
order one. The phase boundary is the same as we predict without magnetic
field. The condition when $\Delta_{1}(l^{*})\sim 1$ define a new length scale
$l^{*}$. It generates a self-consistent scenario that
$4\pi\delta(l^{*})a_{1}<<{{\Delta_{1}}(0)}^{1/(2-4K)}$. In the other limit
when the RG flow of $a$ flows off to the strong coupling phase does not
affected by the magnetic field term and thus the system is in the photonic
superfluid phase.
Conclusions: We have presented three sets of RG equations for the different
physical situations for the Cavity QED system. We have predicted two different
BKT transitions for the different physical situations. We have predicted the
photonic superfluid phase, photonic metallic phase and photon blocked induced
insulating phase. To the best of our knowledge, this explicit quantum phase
analysis and their behavior based on the RG study for the microcavity array
system is absent in the previous literature.

Acknowledgments

The author would like to acknowledge the series of lectures and also
discussions of Prof. S. M. Girvin during the International Workshop/School on
Dirac Materials and Chandrashekar Discussion Meeting, December’2012. The
author would like to thank The Center for Condensed Matter Theory of the
Physics Department of IISc for extended facility. Finally the author would
like to thank Prof. Prabir Mukherjee for reading this manuscript critically.

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