#  Quantum Rabi hexagonal ring in an artificial magnetic field

Lin-Jun Li  Department of Physics, and Chongqing Key Laboratory for strongly
coupled Physics, Chongqing University, Chongqing 401330, China  Li-Lu Feng
Department of Physics, and Chongqing Key Laboratory for strongly coupled
Physics, Chongqing University, Chongqing 401330, China  Jia-Hao Dai
Department of Physics, and Chongqing Key Laboratory for strongly coupled
Physics, Chongqing University, Chongqing 401330, China  Yu-Yu Zhang
<EMAIL_ADDRESS>Department of Physics, and Chongqing Key Laboratory for
strongly coupled Physics, Chongqing University, Chongqing 401330, China

(August 28, 2024)

###### Abstract

We present exotic quantum phases in a quantum Rabi hexagonal ring, which is
derived by an analytical solution. We find that an artificial magnetic field
applied in the ring induces an effect magnetic flux in the even and odd
subring. It gives rise to two chiral quantum phases besides a ferro-
superradiant and an antiferro-superradiant phases. With analogy to the
magnetic system, two chiral phases are distinguished by the magnetization
orientation in the $xy$ plane in two subrings, which correspond to skyrmion
structures with different vorticity. In such chiral phases, photons in the
subrings triangle flow in the same or opposite directions by comparing to the
current in the hexagonal ring, which depend on the signs of the induced
magnetic flux in the subrings. Interestingly, the critical exponents of the
excitation energy in two chiral phases are the same as that of the subring
triangle, exhibiting subring-size dependent critical exponents. Our analysis
can be straightforwardly extended to a larger lattice size with subrings of a
triangular or hexagonal structure, predicting a novel universality class of
superradiant phase transitions. An implementation of the system considered is
an exciting prospect in quantum many-body simulations of light-matter
interactions in future.

##  I Introduction

Realizing models of exotic phases and exploring quantum phase transitions
(QPT) has been an attractive objective for studying strongly interacting
quantum many-body systems. A superradiant QPT of a large number of two-level
systems coupled to a bosonic field has been attracting a remarkable amount of
interest Dicke (1954); Emary and Brandes (2003); Chen  _et al._ (2008), and
now has a wide application of many recent studies in cavity Baumann  _et al._
(2011); Viehmann  _et al._ (2011); Zhu  _et al._ (2020) and circuit Baksic and
Ciuti (2014); Zou  _et al._ (2014) quantum electrodynamics. The well-known
quantum Rabi model describing the coupling between a two-level atom and a
single photon exhibits interesting integrability and novel phenomenon Braak
(2011); Irish (2007); Chen  _et al._ (2012); Zhang (2016). Such a model
exhibits a superradiant QPT in the infinite frequency limit (analogous to the
thermodynamic limit) Ashhab (2013); Hwang  _et al._ (2015); Liu  _et al._
(2017); Chen  _et al._ (2020), which has been achieved in quantum simulations
Chen  _et al._ (2021); Cai and et al. (2021). Such QPT in few-body systems is
useful in probing a broad range of quantum phenomenon and exotic phases.

To explore intriguing quantum phenomena, much efforts have been devoted to
realizing synthetic magnetic fields for bosonic excitations such as neutral
atomic Bose-Einstein condensate (BEC) or cold quantum gases Bloch  _et al._
(2008, 2012); Cao  _et al._ (2014); Lin  _et al._ (2009); Dalibard  _et al._
(2011) and photons Umucalılar and Carusotto (2012); Wang  _et al._ (2016);
Roushan  _et al._ (2017). Synthetic magnetic fields have brought forth
remarkable phenomena, such as the chiral ground-state currents of interacting
photons in a 3-qubit loop Roushan  _et al._ (2017) and fractional quantum Hall
physics in the Jaynes-Cummings Hubbard lattice Hayward  _et al._ (2012);
Hayward and Martin (2016); Noh and Angelakis (2016). Recently, a chiral phase
in the Rabi triangle has been discovered by applying a synthetic magnetic
field Zhang  _et al._ (2021). Morever, an artificial magnetic field in the
light-atom coupling system can induce effective XY exchange and the
Dzyaloshinskii-Moriya (DM) interactions Fallas Padilla  _et al._ (2022). A
novel superradiant QPT induced by an artificial magnetic field shows size-
dependent critical exponents in the quantum Rabi ring with $3-5$ coupled
cavities Fallas Padilla  _et al._ (2022); Zhao and Hwang (2022a), which goes
beyond the mean-field type phase transition of the conventional superradiant
QPT in the Dicke model Emary and Brandes (2003); Chen  _et al._ (2008).

Inspired by the size-dependent critical behaviors, we are interested here in
the emergence of exotic quantum phases and critical exponents induced by a
synthetic magnetic field in the quantum Rabi hexagonal ring with six coupled
cavities. We employ an analytical solution and a Bogliubove transformation to
obtain the phase diagram and excitation energies accurately. We show that the
next-nearest-neighbour interactions can be induced by the magnetic filed in
the ring, resulting in even and odd subrings with an effect magnetic field.
There appear two chiral superradiant (CSR), a ferro-superradiant (FSR) and an
antiferro-superradiant phases (AFSR) phases by using analogies to quantum
magnetism. The chiral phases have a complex order paramter, and are
distinguished by different magnetization orientations in $xy$ plane in the
subrings triangle. Such chiral phases are six-fold degenerate, breaking the
$\mathbb{Z}_{2}$ and the chiral symmetries. Photons in the subrings are
flowing unidirectionally in one chiral phase, while they flow in the opposite
direction in the other one. It demontrates that there exist effetive magnetic
fluxs with different signs in the subrings in each chiral phase. In contrast,
there is no current in the FSR and AFSR phases, which only breaks the
$\mathbb{Z}_{2}$ symmetry. The first-order QPT between the FSR (AFSR) and CSR
phases is tuned by the artificial magnetic field. The scaling exponents on
both sides of the chiral phase transitions are different, but are the same as
that of the subring triangle. It demonstrate that the critical exponents
become subring-size dependent as the size increases, which is different from
previous ring-size dependent scaling behavior Zhao and Hwang (2022b).

##  II Quantum Rabi hexagonal Ring

The Hamiltonian of the Rabi ring can be given as follows

$H_{RR}=\sum_{i=1}^{N}H_{R,i}+J\sum_{i=1}^{N}(e^{i\theta}a_{i}^{\dagger}a_{i+1}+h.c.),$
(1)

where $J$ is the hopping strength between neighboring resonators. The complex
factor in the hopping stems from the artificial magnetic field with a flux
$N\theta$ Zhang  _et al._ (2021). The periodic boundary of the cavity ring
satisfies $a_{N+1}=a_{1}$ and $\sigma_{k}^{N+1}=\sigma_{k}^{1}$. Each
resonator interacting on-site with a two-level atom is desribed by the quantum
Rabi model as

$H_{R,i}=\frac{\Delta}{2}\sigma_{z}^{i}+\omega
a^{\dagger}_{i}a_{i}+g\left(a^{\dagger}_{i}+a_{i}\right)\sigma_{x}^{i},$ (2)

where $\Delta$ is qubit energy difference, $a^{\dagger}_{i}$
$\left(a_{i}\right)$ is the photonic creation (annihilation) operator of the
$i$-th single-mode cavity with frequency $\omega$, $g\ $is coupling constants
respectively, and $\sigma_{k}^{i}(k=x,y,z)$ are the Pauli matrices. The scaled
coupling strength is $g_{1}=g/\sqrt{\Delta\omega}$. There exists a
superradiant QPT in the quantum Rabi model in the infinite frequency limit
$\Delta/\omega\rightarrow\infty$ Ashhab (2013); Hwang  _et al._ (2015); Liu
_et al._ (2017); Chen  _et al._ (2020). In this limit, each cavity undergoes
such QPT.

Using the unitrary transformation
$U=\Pi_{n=1}^{N}\mathtt{exp}[-ig\sigma_{y}\left(a_{n}^{\dagger}+a_{n}\right)/\Delta]$,
we obtain the effective Hamiltonian by projecting to the ground state of atom
$|\downarrow\rangle$

$\displaystyle H_{\text{RR}}^{\downarrow}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\omega
a_{n}^{\dagger}a_{n}-\frac{g^{2}}{\Delta}\left(a_{n}^{\dagger}+a_{n}\right)^{2}$
(3)
$\displaystyle+J\sum_{n}^{N}(e^{i\theta}a_{n}^{\dagger}a_{n+1}+h.c.)+E_{0},$

where the constant energy is
$E_{0}=N[-\Delta/2+(\omega+3J)g^{2}/\Delta^{2}-g^{2}/\Delta]$, and high-order
terms are dropped in the limit $\Delta/\omega\rightarrow\infty$.

The Hamiltonian of Eq.(3) can be mapped to an effective magnetic model via a
Holstein-Primakoff transformation Fallas Padilla  _et al._ (2022). After
application of $S_{n}^{z}=a_{n}^{\dagger}a_{n}-S$ and $\quad
S_{n}^{+}=a_{n}^{\dagger}\sqrt{2S-a_{n}^{\dagger}a_{n}}$ with the total spin
angular momentum $S=\Delta/2\omega$, it easily gives approximately
$S_{n}^{+}\approx\sqrt{2S}a_{n}^{\dagger}$ and
$S_{n}^{-}\approx\sqrt{2S}a_{n}$ in the spin limit $S\rightarrow\infty$. We
obtain the effective magnetic Hamiltonian

$\displaystyle H_{\mathtt{MG}}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\omega
S_{n}^{z}-\frac{2g^{2}}{S\Delta}(S_{n}^{x})^{2}+H_{XY}+H_{DM},$ (4)
$\displaystyle H_{XY}$ $\displaystyle=$
$\displaystyle\frac{J_{1}\cos\theta}{S}\sum_{n}(S_{n}^{x}S_{n+1}^{x}+S_{n}^{y}S_{n+1}^{y}),$
(5) $\displaystyle H_{DM}$ $\displaystyle=$
$\displaystyle\frac{J_{1}\sin\theta}{S}\sum_{n}\vec{e_{z}}\cdot(\vec{S}_{n}\times\vec{S}_{n+1}).$
(6)

The XY Heisenberg interaction $H_{XY}$ which can be regarded as ferromagnetic
or antiferromagnetic depending on the sign of $J\cos\theta$. Additionally, the
Dzyaloshinskii-Moriya (DM) interaction strength in $H_{DM}$ depends on
$J\sin\theta$ Dzyaloshinsky (1958); Moriya (1960), which vanishes when
$\theta=0$ or $\pi$. The DM interactions favor noncollinear spin textures such
as magnetic vortices and magnetic skymions in chiral magnates, which plays a
crucial role in topological properties in magnetic systems Nagaosa and Tokura
(2013); Bogdanov and Hubert (1994); Roessler  _et al._ (2006); Mühlbauer  _et
al._ (2009).

It demonstrate that having a complex amplitude photon hopping in the optical
system is fundamental to obtain the DM interaction term in the magnetic
equivalent. Around $\theta=\pm\pi$ the dominant term is the XY Heisenberg
interaction, which is ferromagnetic. On the other hand, around $\theta=0$, the
leading XY Heisenberg interaction is favoring anti-ferromagnetic order. The DM
interaction becomes dominated when $\theta$ approaches $\pm\pi/2$, which cause
magnetization of skyrmions. There exist rich quantum phases in the effect
magnetic model Fallas Padilla  _et al._ (2022). In particular, each magnetic
phase has an equivalent phase in the quantum Rabi ring. Hence, we study the
exotic quantum phases in the quantum Rabi hexagonal ring using analogies to
quantum magnetism.

Table 1: Quamtun phases in the quantum Rabi ring for the even and odd $N$ for $\theta\in[0,\pi)$. Number of different kinds of quantum phases is listed.  Even or odd $N$ |  CSR ($q\neq 0,\pi$) |  FSR ($q=0$) |  AFSR ($q=\pi$)
---|---|---|---
$3$ |  $1$ |  $1$ |  $0$
$4$ |  $1$ |  $1$ |  $1$
$5$ |  $2$ |  $1$ |  $0$
$6$ |  $2$ |  $1$ |  $1$
$7$ |  $3$ |  $1$ |  $0$
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$2n$ |  $n-1$ |  $1$ |  $1$
$2n+1$ |  $n$ |  $1$ |  $0$

##  III Normal phase

For a weak coupling $g_{1}<g_{1c}$, the ground state is a vacuum state with no
excitation. It is so-called normal phase (NP). It is analogy to the
paramagnetic phase where the spin is polarized by external field term along
the z axis.

The quantum Rabi Hamiltonian $H_{\text{RR}}^{\downarrow}$ of Eq.(3) is
diagonalized exactly in the following. By introducing a Fourier transformation
$a_{k}=\sum_{n=1}^{N}a_{n}e^{-ink}$, the Hamiltonian becomes
$H_{\text{RR}}^{\downarrow}=\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k}-\omega
g_{1}^{2}(a_{k}a_{-k}+a_{k}^{\dagger}a_{-k}^{\dagger})+E_{0}$, where the
dispersion relation is $\omega_{k}=\omega(1-2g_{1}^{2})+2J\cos(\theta-k)$ with
$k=0,2\pi/N,\cdots,2(N-1)\pi/N$. One can give the diagonal Hamiltonian as
$H_{\text{RR}}^{\downarrow}=\sum_{k}\varepsilon_{k}a_{k}^{\dagger}a_{k}+E_{g}$,
where the excitation energy is

$\varepsilon_{k}=\frac{1}{2}[\sqrt{(\omega_{k}+\omega_{-k})^{2}-16\omega^{2}g_{1}^{4}}+\omega_{k}-\omega_{-k}].$
(7)

The vanishing of the excitation energy $\varepsilon_{k}=0$ gives the critical
coupling strength as

$g_{1c}(k)=\frac{1}{2}\sqrt{\frac{1+4J/\omega\cos\theta\cos
k+4J_{+}J_{-}}{1+2J/\omega\cos\theta\cos k}},$ (8)

where $J_{\pm}=J/\omega\cos(\theta\pm k)$.

The critical coupling strength $g_{1c}$ depends on the momentum $k$ according
to the value of $\theta$. It predicts different quantum phases emerging by
tuning the flux $\theta$. For example, a CSR and a FSR phases have been found
for the Rabi ring of $N=3$ and $N=4$, and an additional AFSR phase appears for
the case $N=4$ Fallas Padilla  _et al._ (2022). Especially, in the CSR phase,
the critical exponents of the excitation energy for $N=3$, $4$ and $5$ were
different Fallas Padilla  _et al._ (2022); Zhao and Hwang (2022a). It leads to
a distinc universality class of the QPT with the ring-size dependent critical
exponents. As $N$ increase, Table 1 displays the emerging quantum phases.
Since quantum phases for $\theta\in[0,-\pi]$ is a mirror image of the one for
$\theta\in[0,\pi]$. We consider quantum phases for $\theta\geq 0$. For a even
$N$, there exist a FSR phase for $k=0$, an AFSR phase with $k=\pi$, and
$N/2-1$ kinds of CSR phases with $k\neq 0$, respectively. While there is no
AFSR phase for an odd $N$, and there exists $(N-1)/2$ kinds of CSR phases. It
is interesting to explore exotic physical properties of the emerging multiple
chiral phases. We consider $N=6$ Rabi ring with a hexagonal structure as
follows.

##  IV Superradiant Phases

When $g_{1}$ exceeds the critical value, the system enters the superradiant
phases, for which the photon population in each cavity becomes macroscopic.
Additionally, the hopping of photons between neighboring cavities induce
exotic superradiant phases according to the value of $\theta$. For the Rabi
hexagonal ring, the critical coupling strength $g_{1c}(k)$ in Eq. (8) depends
on the momentum $k=0,\pm\pi/3,\pm 2\pi/3,\pi$, which predicts four
superradiant phases. The phase boudary between superradiant phases are
obtained by $\theta_{c}^{\pm}=\pm\pi/2$ and

$\displaystyle\theta_{c}^{\pm}=\pm
cos^{-1}[\mp\frac{1}{4J}\left(\omega-\sqrt{\omega^{2}+8J^{2}}\right)].$ (9)

For $J/\omega=0.05$ the critical values are given explicitly as
$\theta_{c1}^{\pm}=\pm 0.484\pi,\theta_{c2}^{\pm}=\pm
0.5\pi,\theta_{c3}^{\pm}=\pm 0.516\pi$.

In order to describe different superradiant phases, we incorporate that the
bosonic field acquire macroscopic occupations. We start with the Hamiltonian
in Eq.(1) and displace the bosonic mode as $\tilde{a}_{n}=a_{n}+\alpha_{n}$
with $\alpha_{n}=A_{n}+iB_{n}$ being complex. Making the displacements, the
effective low-energy Hamiltonian is obtained by projecting to the spin
subspace $\tilde{|\downarrow\rangle}$, giving

$\displaystyle H_{\text{eff}}^{\downarrow}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\big{[}\omega\tilde{a}_{n}^{\dagger}\tilde{a}_{n}-\frac{\lambda_{n}^{2}}{\Delta_{n}}\left(\tilde{a}_{n}^{\dagger}+\tilde{a}_{n}\right)^{2}$
(10)
$\displaystyle+J\tilde{a}_{n}^{\dagger}(e^{i\theta}\tilde{a}_{n+1}+e^{-i\theta}\tilde{a}_{n-1})\big{]}+E_{g},$

where $\Delta_{n}\equiv\sqrt{\Delta^{2}+16g^{2}A_{n}^{2}}$, $\lambda_{n}\equiv
g\Delta/\Delta_{n}$. The energy is obtained as
$E_{g}=\sum_{n}\omega\alpha_{n}\alpha_{n}^{\ast}-\Delta_{n}/2+J\alpha_{n}^{\ast}(e^{i\theta}\alpha_{n+1}+e^{-i\theta}\alpha_{n-1})$.

Figure 1: Phase diagram in the $\theta-g_{1}$ plane for the quantum Rabi ring
with $N=6$ using $A_{4}$ as an order parameter.The solid black line represents
the second-order phase boundary, while vertical dashed lines represent the
predicted first-order lines. (b) The imagirary of the order parameter $B_{n}$
in different quantum phases. In all our calculations, we set
$\Delta/\omega=50$ and $J/\omega=0.05$, as well as choosing $\omega=1$ as the
units for frequency.

The order parameter $\alpha_{n}$ being site dependent characterizes different
superradiant phases. The solutions of $\\{A_{n},B_{n}\\}$ are obtained by
minimizing the energy $E_{g}$ with respect to all $A_{n}$’s and $B_{n}$’s,
which gives

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\omega
B_{n}+J\sin\theta(A_{n+1}-A_{n-1})$ (11)
$\displaystyle+J\cos\theta(B_{n+1}+B_{n-1}),$ $\displaystyle 0$
$\displaystyle=$ $\displaystyle\omega
A_{n}-\frac{4g^{2}A_{n}}{\sqrt{16g^{2}A_{n}^{2}+\Delta^{2}}}+J\cos\theta(A_{n+1}+A_{n-1})$
(12) $\displaystyle+J\sin\theta(B_{n-1}-B_{n+1}).$

One easily gets $\sum_{n}B_{n}=0$ in the hexagonal ring. Moreover, we derive
the equation
$\omega(B_{n}+B_{n+2}+B_{n-2})+2J\cos\theta(B_{n+1}+B_{n-1}+B_{n+3})=0$. It
leads to the odd and even subrings with a triangle structure, which satisfies

$\displaystyle B_{1}+B_{3}+B_{5}=0,B_{2}+B_{4}+B_{6}=0.$ (13)

It indicate that the nearest-neighborhood complex photon hopping can
effectively induce next-nearest-neighborhood interactions with an effective
magnetic field in the subrings.

Using the conditions, we give the expression as

$\displaystyle
B_{n}=-J\sin\theta\frac{\omega(A_{n+1}-A_{n-1})+J\cos\theta(A_{n-2}-A_{n+2})}{\omega^{2}-J^{2}\cos^{2}\theta}.$

By substituting $B_{n}$ into the Eq.(12), one obtains

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
A_{n}(\omega-\frac{4g^{2}}{\sqrt{16g^{2}A_{n}^{2}+\Delta^{2}}})+J\cos\theta(A_{n+1}+A_{n-1})$
(15)
$\displaystyle-\frac{J^{2}\sin^{2}\theta}{\omega^{2}-J^{2}\cos^{2}\theta}[\omega(2A_{n}-A_{n-2}-A_{n+2})$
$\displaystyle+J\cos\theta(2A_{n+3}-A_{n+1}-A_{n-1})].$

Therefore, we will look for solutions $\\{A_{n},B_{n}\\}$ by solving the above
equation accurately.

It is interesting to understand $\\{A_{n},B_{n}\\}$ using the analogies to
quantum magnetism. Using
$S_{n}^{+}=(a_{n}^{\dagger}+\alpha_{n}^{\ast})\sqrt{2S-(a_{n}^{\dagger}+\alpha^{\ast})(a_{n}+\alpha)}$,
one obtains $\langle
S_{n}^{+}\rangle\approx\sqrt{2S-|\alpha|^{2}}\alpha^{\ast}$. Thus the magnetic
observables can be given as a magnetization vector in the xy-plane as

$\vec{S_{n}}=(\langle\tilde{S}_{n}^{x}\rangle,\langle\tilde{S}_{n}^{y}\rangle)=(A_{n},-B_{n}),$
(16)

where the scaled parameters are $\langle\tilde{S}_{n}^{x}\rangle=\langle
S_{n}^{x}\rangle/\sqrt{2S-|\alpha|^{2}}$ and
$\langle\tilde{S}_{n}^{y}\rangle=\langle
S_{n}^{y}\rangle/\sqrt{2S-|\alpha|^{2}}$.

Fig. 1 shows the phase diagram of the quantum Rabi ring with the order
parameter $\\{A_{n},B_{n}\\}$. In the weak coupling regime $g<g_{1c}$, the
order parameter $A_{n}$ equals to zero in the NP. As the coupling increases
$g>g_{1c}$, $A_{n}$ grows from zero. It indicates that the system undergoes a
second-order phase transition from the NP to different superradiant phases
depending on $\theta$. The nonzero value of the imagary value $B_{n}$ shows
two nontrivial superradiant phases. Four superradiant phases are characterized
by $\theta_{c}$ as follows, which are understood better using analogies to
quantum magnetism.

($1$)Ferro-superradiant phase (FSR) – In the FSR phase for
$|\theta_{c3}^{\pm}|\leq|\theta|\leq\pi$, the order parameter $\alpha_{n}$ is
real. $A_{n}$ are the same for all sites, and $B_{n}=0$. We obtain the
analytical solution

$A_{n}=\pm\frac{1}{4g_{1}}\sqrt{\frac{\Delta}{\omega}}\sqrt{\frac{16g_{1}^{4}}{(1+2J/\omega\cos\theta)^{2}}-1},$
(17)

It demonstrate that the ground state is doubly degenerate due to the breaking
of the $Z_{2}$ symmetry. It leads to the critical strength as
$g_{1c}=\sqrt{1+2J/\omega\cos\theta}/2$.

It is interesting to understand the degenerate configurations of the order
parameters with analogy to the spin magnetizations. In the FSR phase, the spin
vector polarizes along either the $x$ or the $-x$ axis with $\langle
S_{n}^{y}\rangle=0$ in Fig. 2, giving the spin vector $\vec{S_{n}}=(A_{n},0)$.
It is so-called ferromagnetic superadiance.

($2$)Antiferro-superradiant phase (AFSR) – The AFSR phase emerges for
$0\leq|\theta|\leq|\theta_{c1}^{\pm}|$. The corresponding solution is
$A_{n}=-A_{n+1}=a$ and $B_{n}=0$, for which

$a=\pm\frac{1}{4g_{1}}\sqrt{\frac{\Delta}{\omega}}\sqrt{\frac{16g_{1}^{4}}{(1-2J/\omega\cos\theta)^{2}}-1^{2}},$
(18)

It implies that the ground state is doubly degenerate. The critical coupling
strength is obtained as $g_{1c}=\sqrt{1-2J/\omega\cos\theta}/2$.

With analogy to magnetic system, the neighboring spins are antialigned along
the $x$ or the $-x$ axis in Fig. 2, giving the spin vector
$\vec{S_{n}}=((-1)^{n}a,0)$. It is so-called anti-ferromagnetic superadiance.

Figure 2: Magnetization configurations $S_{n}^{x}=A_{n}$ and
$S_{n}^{y}=-B_{n}$ in the quantum Rabi ring in the CSR I and CSR II phases for
$N=6$ for $\theta\in[0,\pi)$ in the $xy$ plane. The arrows indicate the
polarized direction of the spins for the even (blue color) and odd (red color)
subrings. The positive directions of $x$ and $y$ axises is chosen towards left
and up. We choose $g_{1}=0.7>g_{1c}$ in the superradiant phases.

($3$)Chiral superradiant phase (CSR) I – In the chiral phase with
$|\theta_{c1}^{\pm}|\leq|\theta|\leq\frac{\pi}{2}$, the order parameter
$\alpha_{n}$ is complex and depends on $n$. By solving Eq.(15), one of
solutions is given by

$\displaystyle A_{1}=A_{4},A_{2}=A_{3}=A_{5}=A_{6},$ (19)

and

$\displaystyle B_{1}=B_{4}=0,B_{2}=-B_{3}=B_{5}=-B_{6},$ (20)

where $B_{2}=-J\sin\theta(A_{2}-A_{1})/(\omega-J\cos\theta)$. There exist two
degenerate solutions ( $A_{1}>0,A_{2}<0$) and ( $A_{1}<0,A_{2}>0$), which
corresponds to $Z_{2}$ symmetry breaking. Due to the $C_{3}$ symmetry in the
subrings, the ground state is six-fold degenerate.

Fig. 2 shows the corresponding magnetization configurations of spins in the
CSR I phase, which is described by the spin vector
$\vec{S_{n}}=(A_{n},-B_{n})$. One observes in-plane magnetizations of
skyrmions with different helicities. Since the DM interaction in the magnetic
Hamiltonian in Eq.(6) is stronger than the XY coupling, which favors
noncollinear spin structures. It yield in-plane magnetization orientation in
the $xy$ plane, which is so-called chiral superradiant phase. It is observed
that an pair of spins $\vec{S}_{n}$ in the even subring in blue color has the
same orientation as $\vec{S}_{n+3}$ in the odd subring in red color in the CSR
I phase. To character the skymion structure, the velocity of the skyrmion $Q$
can be expressed as
$Q=\frac{1}{4\pi}\int\vec{S_{n}}\cdot(\partial_{x}\vec{S_{n}}\times\partial_{y}\vec{S_{n}})dxdy$,
which counts how many times $\vec{S_{n}}$ wraps the unit sphere Zhang (2017);
Nagaosa (2013); Kuchkin  _et al._ (2020). One can classify the skymion
structures as $Q=2$ in the CSR I phase.

($4$)Chiral superradiant phase (CSR) II –

For $\frac{\pi}{2}\leq|\theta|\leq|\theta_{c3}^{\pm}|$, the system enters the
CSR II phases. Different with the CSR I, one solution of the solutions is

$\displaystyle A_{1}=-A_{4},A_{2}=-A_{3}=-A_{5}=A_{6},$ (21)

and

$\displaystyle B_{1}=B_{4}=0,B_{2}=B_{3}=-B_{5}=-B_{6},$ (22)

where $B_{2}=J\sin\theta(A_{2}+A_{1})/(\omega+J\cos\theta)$. There exist two
degenerate solutions ( $A_{1}<0,A_{2}<0$) and ( $A_{1}>0,A_{2}>0$). The chiral
phase is also six-fold degenerate due to the break of both the $Z_{2}$ and the
chiral symmetries Zhang  _et al._ (2021).

Fig. 2 shows the magnetization configurations with the spin vector
$\vec{S_{n}}=(A_{n},-B_{n})$ in the CSR II phase. An pair of spins,
$\vec{S}_{n}$ and $\vec{S}_{n+3}$, in the even and odd subrings has the
opposite magnetization orientation, which is distinguished from that in the
CSR I phase. The structures of the skymions has different vorticity $Q=1$ by
comparing to the CSR I phase.

##  V Ground-state photon current

To gain further insight into two chiral phases, we consider the ground-state
current of photons in the closed loop of the Rabi ring. Analogous to the
continuity equation in classical systems, the photon current operator is
explicitly defined as

$I=i\Sigma_{n=1}^{N}\left[(a_{n}^{\dagger}a_{n+1}-h.c.\right].$ (23)

Similarily, the photon current in the even and odd subring is given as
$I_{\mathrm{135}}=i\left[(a_{1}^{\dagger}a_{3}+a_{3}^{\dagger}a_{5}+a_{5}^{\dagger}a_{1})-h.c.\right]$
and
$I_{\mathrm{246}}=i\left[(a_{2}^{\dagger}a_{4}+a_{4}^{\dagger}a_{6}+a_{6}^{\dagger}a_{2})-h.c.\right]$.

Figure 3: (a)Photon current $I$ in the ring, $I_{135}$ and $I_{246}$ of the
even and odd subring in the SR regions as a function of $\theta$ for
$g_{1}>g_{1c}$. (b)Arrows represents the direction of photon current in ring
(black color) and in two subrings (blue/red color) the CSR I and CSR II
phases. Figure 4: Lowest excitation energy as a function of $g1/g1c$ across
the critical value for the NP-AFSR (a), NP-CSR I (b), NP-CSR II (c), and NP-
FSR (d) phase transitions. Open markers signal the numerical results while the
fitting functions are denoted by solid lines.

Using the solutions of $\\{A_{n},B_{n}\\}$ of Eqs.(19)-(20) in the CSR I
phase, we obtain the analytical expression of the photons current as

$I=8(A_{2}-A_{1})B_{2},I_{\mathrm{135}}=I_{\mathrm{246}}=-I/2.$ (24)

The current value in the subring is half over the value in the total ring.
Moreover, the direction of current in the subring is opposite to the total
current. Similarily, the current in the CSR II phase can be expressed
according to Eqs.(21)-(22)

$I=-8(A_{2}+A_{1})B_{2},I_{\mathrm{135}}=I_{\mathrm{246}}=I/2.$ (25)

Different from the CSR I phase, we can clearly see that photons in the even
and odd subring moves towards the same direction as well as that in the Rabi
ring. It demonstrates that the effective magnetic flux in the subrings has the
opposite sign in two chiral phases, and is half over the total flux $6\theta$.
We note the positive or negative next-nearest-neighbour interactions have been
found in the $N=5$ Rabi ring Zhao and Hwang (2022a), which can be understood
as a consequence of the effective magnetic field with signs in the subrings.

Fig. 3 shows the photon current in the Rabi ring as well as the even and odd
subring. There is no current in the AFSR and FSR phase. However, $I$ increase
sharply in two chiral phases, and behaves discontinously at
$\theta_{c2}=\pi/2$, indicating a first-order transition from the CSR I to CSR
II phases. Photons in the hexagonal ring moves towards clockwise direction in
both chiral phases. However, in the CSR I phase the photons in the subrings
flow towards anticlockwise direction, which is opposite to that in the CSR II
phase. Thus, the chiral current of photons in the subrings is adjusted by the
effective magnetic field with flux $\pm 3\theta$, which is associated with the
next-nearest-neighbour interactions induced by the artificial magnetic field
in the ring.

##  VI Excitation energy scaling

We consider the excitation energy around the critical value to character the
scaling exponents of the second-order phase transitons. Since the Hamiltonian
in Eq. (10) is bilinear in the creation and annihilation operators
$a_{n}^{\dagger}$ and $a_{n}$. It can be diagonalized as
$H_{\text{eff}}^{\downarrow}=\sum_{n}^{N}\varepsilon_{n}b_{n}^{\dagger}b_{n}$
by the bosonic Bogoliubov transformation Zhang  _et al._ (2021), where
$\varepsilon_{n}$ is the excitation energy, the bosonic operators
$\\{b_{n},b_{n}^{\dagger}\\}$ are a linear combination of
$\\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{1}^{\dagger},a_{2}^{\dagger},a_{3}^{\dagger},a_{4}^{\dagger},a_{5}^{\dagger},a_{6}^{\dagger}\\}$.

The Lowest excitation energy $\varepsilon_{1}$ vanishes at the critical
coupling strength of the second-order phase transition, and behaves as
$\varepsilon_{1}\propto|g_{1}-g_{1c}|^{\gamma}$ across the phase boudary. Fig.
4 shows the a power-law behavior of $\varepsilon_{1}$ when $g_{1}$ approaches
to the critical value from above and below sides. The scaling exponent
$\gamma$ is $1/2$ for the NP-AFSR and NP-FSR phase transitions in Fig. 4(a)
and (d). Both phase transitions belong to the same universality of the
conventional superradiant phase transition in the Dicke and quantum Rabi model
Hwang  _et al._ (2015); Emary and Brandes (2003).

By contrast, the scaling exponents before and after the critical point are
different for the NP-CSR I and NP-CSR II transitions in Fig. 4(b) and (c).
$\gamma$ equals to $1$ and $3/2$ for $g_{1}$ approaching $g_{1c}$ from below
and above, respectively. The unusual critical exponents are as a consequence
of the frustrated configurations in the triangle subring, which is the same as
that in Rabi triangle Zhang  _et al._ (2021). Therefore, scaling exponents in
$N=6$ Rabi ring confirms the phase transitions to be in the universality class
of its subring triangle structure. Our study can be straightforwardly extended
to a larger ring size, which is associated with the subring-size dependnent
critical exponents.

##  VII Conclusion

We present the analytical solution to the Rabi ring with a hexagonal structure
and the phase diagram. We find an effective magnetic flux in the even and odd
subrings, which is associated with the next-nearest-neighbour interactions
induced by the artificial magnetic field in the ring. As the coupling strength
increases, two chiral superradiant phases of the CSR I and CSR II emerge as
well as a ferro-superradiant and an antiferro-superradiant phase. In such
chiral phases, one observes in-plane magnetization of skyrmions, exhibiting
different velocity of the skyrmion in two chiral phases. In the CSR I phase,
the current of photons in the subring has the opposite chirality by compared
to the CSR II phase, which corresponds to the effective magnetic flux with
different signs in the subrings. However, the scaling exponents of the
excitation energy in two chiral phases are the same as that in the subring
triangle case, exhibiting different critical exponents before and after the
critical value. The system exhibits the subring-size dependent critical
exponent, which predicts a nontrivial universality class of the Rabi ring with
a subring of triangle or hexagonal structure. Studying exotic quantum phases
in this few-body system would open intriguing avenues for exploring their
connection to more complicated configurations.

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