# Probing Luttinger Liquid Properties in Multichannel Two-Site Charge Kondo
Simulator

A. V. Parafilo<EMAIL_ADDRESS>Center for Theoretical Physics of Complex
Systems, Institute for Basic Science, Expo-ro 55, Yuseong-gu, Daejeon 34126,
Republic of Korea  V. M. Kovalev  Novosibirsk State Technical University,
Novosibirsk 630073, Russia  I. G. Savenko  Department of Physics, Guangdong
Technion – Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong,
515063, China  Technion – Israel Institute of Technology, 32000 Haifa, Israel
Guangdong Provincial Key Laboratory of Materials and Technologies for Energy
Conversion, Guangdong Technion–Israel Institute of Technology, Guangdong
515063, China

###### Abstract

We study the influence of many-body interactions on the transport properties
in a two-site charge Kondo circuit recently implemented in a hybrid metal-
semiconductor double-quantum dot device [W. Pouse et al., Nat. Phys. 19, 492
(2023)]. There emerge two principal types of interactions: (i) an intrinsic
one, described by the Luttinger liquid model, and (ii) an induced one, which
appears due to the coupling of the system to an Ohmic environment. Case (i)
could be achieved if the charge Kondo circuit operates in the fractional
quantum Hall regime, while case (ii) can be implemented via a finite number of
open ballistic channels coupled to both the quantum dots. We demonstrate that
the conductance scaling for the case of strong and weak interdot coupling is
fully determined by the effective interaction parameter, which is the
combination of the fractional filling factor $\nu=1/m$ and the number of
transmitting channels. Furthermore, we predict that the fractional filling
factor $\nu$ defines a universal Kondo scaling in the vicinity of a special
triple quantum critical point featured by the emergence of a $\mathbb{Z}_{3}$
parafermion.

The Kondo model introduced in a pioneering work [1] allowed for the
explanation of the minimum in the dependence of resistance on temperature in
dilute magnetic alloys. Soon, it became clear that this model represents a
powerful tool giving access to a wide range of strongly correlated phenomena
[2], providing a testbed for various many-body theoretical methods [3, 4, 5,
6, 7, 8]. The rebirth of the model occurred with the development of mesoscopic
physics, when it became clear that a quantum dot (QD), a principal element of
the single-electron transistor [9, 10, 11], under certain conditions behaves
as a quantum spin-1/2 impurity, thus manifesting the properties of the Kondo
model [12, 13, 14, 15, 16].

It turned out, that the Kondo model has many different implementations: Kondo-
like effects occur in systems where few local degrees of freedom are coupled
to one or more bath continua [17, 18, 19]. One such system, which we focus on
in this Letter, is called the charge Kondo (CK) circuit. It was proposed by
Matveev, Flensberg, and Furusaki in the early 1990s [20, 21, 22, 23]. They
considered a large QD formed in a two-dimensional electron gas (2DEG)
connected to the leads via single-mode quantum point contacts (QPCs). In
contrast to the original Kondo problem, in which the Kondo effect is
attributed to the spin degree of freedom, a quantum pseudo-spin in the charge
Kondo implementation is represented by two degenerate macroscopic charge
states of the QD [20, 22].

The first experimental implementation of the CK device represented a hybrid
metal-semiconductor single-electron transistor on the base of a GaAs/AlGaAs
heterostructure operating in the integer quantum Hall (IQH) regime [24, 25,
26]. Due to remarkable control over the conducting channels entering the QD,
the CK device in [24, 25, 26] provides unprecedented access to multichannel
Kondo physics. A breakthrough series of experiments devoted to various aspects
of a two- and three-channel CK problem [24, 25, 26, 27, 28] induced enormous
interest among the theoretical [29, 34, 35, 37, 42, 40, 41, 33, 39, 30, 45,
32, 31, 38, 43, 36, 44, 46] and experimental [47, 48, 49, 50, 51] scientific
community in the field.

A very recent experiment [51] suggested a two-site (two-QD) generalization of
the CK circuit as a platform for probing frustrated interaction at the exotic
quantum critical point with fractional excitation [38, 43]. The circuit
represents two hybrid metal-semiconductor islands coupled in series via QPCs,
with each island connected to its own lead. In a way, such a setup ‘simulates’
the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction
of two QDs/impurities and an individual Kondo screening of each QD/impurity
[52, 53]. It should be noted that a two-site CK (2SCK) circuit [51] with weak
and strong interdot couplings was initially proposed in Ref. 29 to identify
the competition between Fermi- and non-Fermi liquid behavior featured by
individual multichannel CK simulators.

In this Letter, we address a fundamental question: How does the many-body
interaction in the conducting channels of the circuit [51] affect low-
temperature transport properties in the 2SCK device? (Is it possible to probe
the competition between Fermi- and non-Fermi liquid behavior of a 2SCK device
via conductance scaling?) Note that the influence of many-body interactions on
the properties of the single-site CK circuit was studied in a series of works
[35, 37, 42, 40, 41]. To answer the question for the case of 2SCK circuit, we
examine two types of interaction: (i) an ’intrinsic’ one, which we describe by
the Luttinger liquid (LL) model [54, 55, 56, 57, 58], and (ii) an ‘induced’
one, which may appear due to the coupling of the coherent conductors to an
Ohmic dissipative environment [18, 59, 60, 48, 50]. The first type of
interaction can be experimentally studied in the setup [51] in the fractional
quantum Hall (FQH) regime [61]. The second type of interaction can be
implemented by controlling the number of open ballistic channels connected to
the QDs. We show, that the interaction mechanisms mentioned above uniquely
determine the quantum transport behavior of the system, in particular, the
maximum of the conductance and the temperature scaling.

Figure 1: (a) System schematic: a hybrid metal-semiconductor device consists
of two metallic quantum dots (QDs) strongly coupled via a single-mode quantum
point contact (QPC). The $2N_{1}$ and $2N_{2}$ edge modes ballistically enter
the first and the second QD, respectively. The system is exposed to a strong
magnetic field, which drives it in the fractional quantum Hall regime. In this
regime, 1D chiral edge (Laughlin) excitations with a fractional filling factor
$\nu=1/m$ (with $m$ an odd integer) flow along the top and the bottom edges.
They are characterized by the bosonic fields
$\varphi_{\alpha,i}^{\textrm{t}(\textrm{b})}(x)$. (b) An equivalent electric
circuit of the device: a conducting channel with a backscattering coefficient
$v$ formed in the Luttinger liquid with the interaction constant $\nu$ is
coupled to two resistors $R_{1}=(\nu N_{1}G_{0})^{-1}$ and $R_{2}=(\nu
N_{2}G_{0})^{-1}$ (with $G_{0}=R_{0}^{-1}=e^{2}/2\pi\hbar$). This circuit
corresponds to the Luttinger liquid with the weak potential barrier and an
effective interaction parameter $K=1/(\nu^{-1}+r_{1}+r_{2})$, where
$r_{i}=R_{i}/R_{0}$ [60].

Model.— Consider a multichannel 2SCK circuit – a hybrid metal-semiconductor
device consisting of two large metallic islands or QDs incorporated into the
2DEG hosted in a GaAs/AlGaAs heterostructure [Fig. 1(a)]. The circuit is in
the FQH regime since it is exposed to a strong magnetic field. Furthermore, we
assume that the two QDs are strongly coupled via a single-mode QPC, while
$2N_{1}$ and $2N_{2}$ edge modes freely enter the first and the second QDs,
respectively. Suppose an external magnetic field and the density of 2DEG
satisfy such conditions that the FQH has a filling factor $\nu=1/m$, where $m$
is an odd integer. In this regime, 1D chiral Laughlin [62] excitations conduct
the fractional charge $e^{\ast}=\nu e$ along the top and bottom edge channels.

The low-energy physics of the FQH edge states can be characterized by the
scalar bosonic fields $\varphi^{\rm t(b)}_{\alpha,i}(x)$ [63, 64], which
satisfy the Kac-Moody commutation relation
$[\partial_{x}\varphi_{\alpha,i}^{\rm
t(b)}(x),\varphi_{\alpha^{\prime},i^{\prime}}^{\rm t(b)}(x^{\prime})]=\pm
2i\pi\nu\delta_{\alpha\alpha^{\prime}}\delta_{ii^{\prime}}\delta(x-x^{\prime})$.
Here, $\varphi^{\rm t(b)}_{1,i}$($\varphi^{\rm t(b)}_{2,i}$) denotes the
$N_{1}$($N_{2}$) incoming and $N_{1}$($N_{2}$) outgoing edge states, which
enter the first (second) QD, while $\varphi_{3,i}^{\rm t(b)}$ describe the
modes connecting both QDs. The whole system can be described by the
Hamiltonian $H=H_{0}+H_{C}+H_{BS}$, where

$\displaystyle H_{0}=\frac{\hbar
v_{F}}{4\pi\nu}\sum_{\alpha=1}^{3}\sum_{i=1}^{N_{\alpha}}\int
dx\left\\{\left(\partial_{x}\varphi_{\alpha,i}^{\textrm{t}}\right)^{2}+\left(\partial_{x}\varphi_{\alpha,i}^{\textrm{b}}\right)^{2}\right\\}$
(1)

is the kinetic energy of the top and bottom fractional edge excitation, and
$v_{F}$ is the Fermi velocity. Since $N_{3}=1$, we use the simplified notation
$\varphi_{3,1}^{\rm t(b)}\equiv\varphi_{3}^{\rm t(b)}$ in what follows.

Two large metallic islands (QDs) can be described by the mesoscopic Coulomb
blockade Hamiltonian,

$\displaystyle H_{C}=\frac{1}{2C}\left(Q_{1}^{2}+Q_{2}^{2}\right),$ (2)

where $C$ is the capacitance of metallic islands, and

$\displaystyle Q_{1}=e\sum_{\sigma={\rm
t,b}}\left[\sum_{i=1}^{N_{1}}\int_{0}^{\infty}dx\,\rho_{1,i}^{\sigma}(x)+\int_{-\infty}^{0}dx\,\rho_{3}^{\sigma}(x)\right],$
$\displaystyle Q_{2}=e\sum_{\sigma={\rm
t,b}}\left[\int_{0}^{\infty}dx\,\rho_{3}^{\sigma}(x)+\sum_{i=1}^{N_{2}}\int_{-\infty}^{0}dx\,\rho_{2,i}^{\sigma}(x)\right]$

are the charges accumulated at the first and second QD, respectively, while
$\rho_{\alpha,i}^{\rm t(b)}=\pm(1/2\pi)\partial_{x}\varphi_{\alpha,i}^{\rm
t(b)}$ is the density operator [65].

A weak tunneling of the Laughlin quasiparticles between counter-propagating
(top and bottom) channels at the QPC placed between two QDs is covered by the
Hamiltonian

$\displaystyle
H_{BS}=\frac{D}{2\pi}v\left[e^{i\varphi_{3}^{\textrm{t}}(0)-i\varphi_{3}^{\textrm{b}}(0)}+\textrm{H.c.}\right],$
(3)

where $v\ll 1$ is the backscattering matrix element, and $D$ is the energy
cutoff.

In the spirit of early works [21, 23, 66], it is more common to describe the
system via an Euclidean action
$\mathcal{S}=\mathcal{S}_{0}+\mathcal{S}_{C}+\mathcal{S}_{BS}$. In this case,
combining the top and bottom chiral fields $\varphi^{\rm t(b)}_{\alpha,i}(x)$
is more convenient to represent the action in terms of new variables:
$\phi_{\alpha,i}=(\varphi_{\alpha,i}^{\rm t}-\varphi_{\alpha,i}^{\rm b})/2$
and $\theta_{\alpha,i}=(\varphi_{\alpha,i}^{\rm t}+\varphi_{\alpha,i}^{\rm
b})/2\nu$, which obey the canonical commutation relation
$[\phi_{\alpha,i}(x),\partial_{x}\theta_{\alpha^{\prime},i^{\prime}}(x^{\prime})]=i\pi\delta_{\alpha\alpha^{\prime}}\delta_{ii^{\prime}}\delta(x-x^{\prime})$.
As a result, the free Euclidean action of the Laughlin edge modes
$\mathcal{S}_{0}$ acquires the form of a non-chiral Luttinger liquid action
[58] (in $\hbar=k_{B}=1$ units),

$\displaystyle\mathcal{S}_{0}=\frac{v_{F}}{2\pi
K}\sum_{\alpha=1}^{3}\sum_{i=1}^{N_{\alpha}}\int
dx\int\limits_{0}^{\beta}d\tau\left(\frac{[\partial_{\tau}\phi_{\alpha,i}]^{2}}{v_{F}^{2}}+[\partial_{x}\phi_{\alpha,i}]^{2}\right),$
(4)

where the Luttinger interaction parameter $K$$=$$\nu$ and $\beta=T^{-1}$ is
the inverse temperature. It should be noted that the fields
$\theta_{\alpha,i}(x,\tau)$ were integrated out in the derivation of Eq. (4);
see, e.g., [58] and the Supplemental Material [67].

The action of the double-QD Coulomb blockade corresponding to Eq. (2) has the
form

$\displaystyle\mathcal{S}_{C}=\frac{E_{C}}{\pi^{2}}\int_{0}^{\beta}d\tau\left[\hat{n}^{2}_{1}+\hat{n}_{2}^{2}\right],$
(5)

where $E_{C}=e^{2}/2C$ is the charging energy of two Ohmic contacts (with
zero-energy level spacing) [65], and

$\displaystyle\hat{n}_{1(2)}=\pm\left[\phi_{3}(0,\tau)-\sum_{i=1}^{N_{1(2)}}\phi_{1(2),i}(0,\tau)\right]$

is the total number of quasiparticles that enter the first (second) QD.

Finally, $\mathcal{S}_{BS}$ is the small contribution to the action
$\mathcal{S}$ due to backscattering at the QPC,

$\displaystyle\mathcal{S}_{BS}\\!=\frac{D}{\pi}v\\!\int_{0}^{\beta}\\!\\!\\!d\tau\cos[2\phi_{3}(0,\tau)].$
(6)

Effective model.— Next, let us implement all the reasonable assumptions in
order to derive an effective model (see Supplemental Material for details
[67]). First, we present $\phi_{1,i}(\phi_{2,i})$ in terms of the new fields
$\phi_{1,c},\phi_{1,s},...,\phi_{1,f}$
($\phi_{2,c},\phi_{2,s},...,\phi_{2,f}$), which characterize the charge, spin,
and other flavor modes. Interestingly, only the charge
$\phi_{\alpha,c}=(N_{\alpha})^{-1/2}\sum_{i=1}^{N_{\alpha}}\phi_{\alpha,i}$
($\alpha=1,2$) and $\phi_{3}$ modes enter the Coulomb blockade action, while
all other flavor modes are excluded from further consideration. The
coefficients $\sqrt{N_{\alpha}}$, which appears in Eq. (5) after the
redefinition of variables, effectively renormalize the Luttinger interaction
parameters of the charge fields $\phi_{1,c}$, $\phi_{2,c}$:
$K=\nu\rightarrow\nu N_{\alpha}$ (one can see it by rescaling
$\sqrt{N_{\alpha}}\phi_{\alpha,c}\rightarrow\phi_{\alpha,c}$).

Since the action $\mathcal{S}$ is Gaussian everywhere except for the $x=0$
point, a further simplification of the problem to one that is local in space
but not local in time is possible by integrating out all the degrees of
freedom $\phi_{1,c}(x\neq 0,\tau),\phi_{2,c}(x\neq 0,\tau),\phi_{3}(x\neq
0,\tau)$. After this procedure, the resulting action $\mathcal{S}$ can be
diagonalized by a unitary transformation, such that it reduces
$(\phi_{1,c},\phi_{3},\phi_{2,c})$ to $(\phi_{A},\phi_{B},\phi_{C})$ [67].

After the rescaling $\phi_{\eta}/\sqrt{\nu}\rightarrow\phi_{\eta}$
($\eta=A,B,C$), the effective actions turn into

$\displaystyle\mathcal{S}_{0}=\sum_{\eta=A,B,C}\sum_{\omega_{n}}\frac{|\omega_{n}|T}{\pi}\phi_{\eta}(-i\omega_{n})\phi_{\eta}(i\omega_{n}),$
(7) $\displaystyle\mathcal{S}_{C}=\frac{\nu
E_{C}}{\pi^{2}}\int_{0}^{\beta}d\tau\left\\{A_{-}\phi_{B}^{2}(\tau)+A_{+}\phi_{C}^{2}(\tau)\right\\},$
(8)
$\displaystyle\mathcal{S}_{BS}\\!=\frac{Dv}{\pi}\\!\int_{0}^{\beta}\\!\\!\\!d\tau\cos\left\\{\sqrt{4\nu}\left[A_{A}\phi_{A}-A_{B}\phi_{B}-A_{C}\phi_{C}\right]\right\\},$

where
$\phi_{\eta}(i\omega_{n})=\int_{0}^{\beta}d\tau\phi_{\eta}(\tau)\exp(i\omega_{n}\tau)$
and $\omega_{n}=2\pi Tn$ is the Matsubara frequency;
$A_{\pm}=1+(N_{1}+N_{2})/2\pm\sqrt{1+(N_{1}-N_{2})^{2}/4}$, and coefficients
$A_{\eta}$ are explicitly presented in [68]. Thus, we found the effective
action for three modes, two of which are gapped due to the Coulomb blockade.

Strong interdot coupling.— Let us estimate the linear conductance through the
2SCK circuit in the zero-frequency limit. Using the current conservation
(Kirchoff’s) law, the Kubo formula can be presented

$\displaystyle G=G_{\rm max}\frac{2T}{\pi i}\lim_{\omega\rightarrow
0}\omega\lim_{i\omega_{n}\rightarrow\omega+i0^{+}}\langle\phi_{A}(i\omega)\phi_{A}(-i\omega)\rangle,$
(10)

where $G_{\rm max}=\nu G_{0}N_{1}N_{2}/(N_{1}+N_{2}+N_{1}N_{2})$, and
$G_{0}=R_{0}^{-1}=e^{2}/2\pi$ is a unitary conductance. We treat Eq. (10) by
using the functional integration technique and by utilizing the perturbation
theory over the weak backscattering amplitude, $v\ll 1$. Applying Wick’s
theorem and performing the Gaussian integration of all the fields, in the low-
temperature limit $T\ll E_{C}$ we obtain [67]

$\displaystyle G(T)=G_{0}K-G_{0}K^{2}\tilde{v}^{2}\mathcal{C}_{\rm
sc}\frac{\sqrt{\pi}\Gamma(K)}{2\Gamma\left(K+\frac{1}{2}\right)}\left(\frac{\pi
T}{\nu E_{C}}\right)^{2K-2}.$ (11)

Here, $K=\nu N_{1}N_{2}/(N_{1}+N_{2}+N_{1}N_{2})$ is an effective Luttinger
parameter, $\tilde{v}=v(\nu E_{C}/D)^{\nu-1}$ is the interaction-renormalized
backscattering amplitude (the Kane-Fisher effect [69, 70]), $\Gamma(x)$ is a
gamma-function, and $\mathcal{C}_{\rm
sc}(\nu,N_{1},N_{2})=\left(\gamma/\pi\right)^{\frac{2\nu(N_{1}+N_{2})}{N_{1}+N_{2}+N_{1}N_{2}}}\left(A_{+}^{A_{C}^{2}}A_{-}^{A_{B}^{2}}\right)^{2\nu}$,
where $\gamma=\exp(0.577)$ is the Euler’s constant.

We can compare formula (11) with the existing results by noting, that Eq. (11)
coincides with the expression describing the conductance of a multichannel
single-site CK circuit in the FQH regime [41] when one of the QDs connects to
an infinite number of channels ($N_{1}\rightarrow\infty$). Indeed, in this
case $K\rightarrow\nu N_{2}/(1+N_{2})$ and $A_{-}=A_{B}^{2}=1+N_{2}$, while
$A_{C}^{2}\rightarrow 0$ [68]. In the other limiting case $N_{1}=N_{2}=1$ and
in the IQH regime $\nu=1$, Eq. (11) coincides with the first two terms in Eq.
(D7) derived in [38] and with Eq. (3b) reported in [43].

Moreover, Eq. (11) resembles the conductance in the LL with a weak potential
barrier, when the interaction parameter $K$ is determined by the fractional
filling factor $\nu=1/m$ and by the number of ballistic channels that enter
both QDs, $N_{1}$ and $N_{2}$. This can be attributed to a well-known result
by Safi and Saleur [60]. The Luttinger interaction parameter $K^{\prime}$ will
experience renormalization if the quantum conducting channel formed in the LL
is set in series with the resistance $R_{\rm tot}$ (dissipative environment)
as $K=1/(K^{\prime-1}+R_{\rm tot})$. Indeed, bare interaction in the system is
$K^{\prime}=\nu$, while $N_{1}$ ($N_{2}$) channels attached to the first
(second) QD can be treated as $N_{1}$ ($N_{2}$) resistors, each with the
resistance $r=(\nu G_{0})^{-1}/R_{0}$, coupled in parallel,
$r_{\alpha}=1/(\sum_{i=1}^{N_{\alpha}}r^{-1})$ [see Fig.1(b)]. As a result,
all the open channels implement the total resistance $R_{\rm
tot}=r_{1}+r_{2}$, and thus, $K=1/[\nu^{-1}+(\nu N_{1})^{-1}+(\nu
N_{2})^{-1}]$.

The perturbation theory justifies Eq. (11) down to the ‘boundary’ temperature
$T_{B}\sim\nu E_{C}\tilde{v}^{1/(1-K)}$, while descending down to even lower
temperatures requires the consideration of the model with a tunnel barrier
between two QDs.

Figure 2:  The device presented in Fig.1(a) in the case of tunnel coupling
between two QDs.

Weak interdot coupling.— Let us consider the temperatures $T\ll T_{B}$, when
the coupling between two QDs is due to tunneling, $w=\sqrt{1-v^{2}}\ll 1$. In
this regime, it is more convenient to calculate the conductance using the
Hamiltonian formalism. The total Hamiltonian of the system shown in Fig. 2
reads $\tilde{H}=\tilde{H}_{0}+\tilde{H}_{C}+\tilde{H}_{\rm tun}$, where the
first term is a chiral LL Hamiltonian of the $2(N_{1}+N_{2}+2)$ edge modes,

$\displaystyle\tilde{H}_{0}=\frac{v_{F}}{4\pi\nu}\sum_{\alpha=1,2}\sum_{i=1}^{N_{\alpha}+1}\int
dx\left\\{\left(\partial_{x}\varphi_{\alpha,i}^{\textrm{t}}\right)^{2}+\left(\partial_{x}\varphi_{\alpha,i}^{\textrm{b}}\right)^{2}\right\\},$

while the second term is the charging energy of 2SCK circuit,

$\displaystyle\tilde{H}_{C}=\frac{E_{C}}{4\pi^{2}}\left(\hat{n}_{1}^{2}+\hat{n}_{2}^{2}\right)\,,\,\hat{n}_{\alpha}=\sum_{i=1}^{N_{\alpha}+1}\left[\varphi_{\alpha,i}^{\rm
t}(0)-\varphi_{\alpha,i}^{\rm b}(0)\right].$

Since the QPC between two QDs pinches off the 2SCK circuit into two weakly
coupled subsystems, four bosonic fields $\varphi_{N_{\alpha}+1}^{\rm t(b)}$
describe two ‘looped’ edge states. The term ‘looped’ means that the outgoing
edge mode, say $\varphi_{N_{1}+1}^{\rm t}$, from the first QD after nearly
perfect reflection at the QPC enters back the first QD as an incoming state,
$\varphi_{N_{1}+1}^{\rm b}$. Thus, the fields $\varphi_{N_{\alpha}+1}^{\rm t}$
and $\varphi_{N_{\alpha}+1}^{\rm b}$ can only differ by an insignificant phase
factor [46]. Both these fields merge into one looped field, which we denote as
$\varphi_{+}$ ($\varphi_{-}$ is used for the looped state of the second QD) in
what follows, see Fig. 2. Nevertheless, these two looped states participate in
the tunneling events and ‘contain the information’ about other modes that
enter the QDs. Thus, the single-electron tunneling between two QDs can be
described by the Hamiltonian

$\displaystyle\tilde{H}_{\rm tun}=\frac{D}{2\pi
v_{F}}w\left\\{e^{i\nu^{-1}[\varphi_{+}(0)-\varphi_{-}(0)]}+\textrm{H.c.}\right\\}.$
(14)

The average tunneling current in the lowest-order of perturbation theory reads

$\displaystyle I=\frac{e|w|^{2}D^{2}}{(2\pi v_{F})^{2}}\int
dt\left\\{e^{ieVt}\mathcal{G}_{+}(t)\mathcal{G}^{{\dagger}}_{-}(t)-(t\rightarrow-t)\right\\},$
(15)

where $V$ is a bias voltage between two QDs, and the correlation function
$\mathcal{G}_{\pm}(t)$ is given by (see, e.g., [46, 71, 72])

$\displaystyle\mathcal{G}_{\pm}(t)=\langle
e^{i\varphi_{\pm}(0,t)/\nu}e^{-i\varphi_{\pm}(0,0)/\nu}\rangle=\left(\frac{\pi
T}{D}\right)^{\frac{1}{\nu}}$ (16)
$\displaystyle\times\left(\frac{\pi^{2}T}{\gamma N_{1(2)}\nu
E_{C}}\right)^{\frac{2}{\nu N_{1(2)}}}\left[\frac{1}{i\sinh[\pi
T(t-i0^{+})]}\right]^{\frac{1}{\nu}+\frac{2}{\nu N_{1(2)}}}.$

After substituting Eq. (16) in Eq. (15), a straightforward calculation gives
[67]

$\displaystyle G=G_{\rm tun}\mathcal{C}_{\rm
wc}\frac{\sqrt{\pi}\Gamma\left(\frac{1}{K}\right)}{2\Gamma\left(\frac{1}{2}+\frac{1}{K}\right)}\left(\frac{\pi
T}{\nu E_{C}}\right)^{2/K-2},$ (17)

where $G_{\rm tun}=e^{2}|\tilde{w}|^{2}/(2\pi v_{F}^{2})$, with
$\tilde{w}=w(\nu E_{C}/D)^{1/\nu-1}$ a renormalized tunnel amplitude in
accordance with the Kane-Fisher effect [69, 70], and $\mathcal{C}_{\rm
wc}(\nu,N_{1},N_{2})=\left(\pi/\gamma\right)^{\frac{2}{K}\frac{N_{1}+N_{2}}{N_{1}+N_{2}+N_{1}N_{2}}}N_{1}^{-2/\nu
N_{1}}N_{2}^{-2/\nu N_{2}}$.

Formula (17) is consistent with the previous studies [29, 41, 44]. In
particular, if $N_{1}\rightarrow\infty$, Eq. (17) recalls Eq. (21) in [41]. In
the IQH regime ($\nu=1$), Eq. (17) scales as $G(T)\propto
T^{2(N_{1}+N_{2})/N_{1}N_{2}}$ – the scaling, which was for the first time
predicted in Ref. 29 for the cases $N_{1},N_{2}=1,2$, and in [41, 44] for the
arbitrary number of conducting channels coupled to both QDs. In addition, Eqs.
(11) and (17) at the limiting case $N_{1}$$=$$N_{2}$$=$$1$ ($\nu=1$) are in
agreement with the universal crossover scaling of the conductance calculated
by the Bethe ansatz in [38] [see Eq. (27) there] and [73, 74].

The similarity of Eq. (17) with the backscattering correction in Eq. (11) is
simply the $K$ to $K^{-1}$ correspondence, well-known for the LL with ‘weak’
and ‘strong’ potential barriers [58, 60]. Both the formulas (11) and (17)
cover the temperature dependence of the conductance through the 2SCK circuit
at two temperature regimes, $T\gg T_{B}$ and $T\ll T_{B}$.

It is rather easy to explain the power-law temperature dependence of Eqs. (11)
and (17), and their relation to the LL physics. Indeed, the fluctuations of
two gapped modes $\phi_{B}(0,\tau)$ and $\phi_{C}(0,\tau)$ in the effective
model given by Eqs. (7)–(Probing Luttinger Liquid Properties in Multichannel
Two-Site Charge Kondo Simulator) are suppressed due to the Coulomb blockade.
At low temperatures, $T\ll E_{C}$, these degrees of freedom can be integrated
out. The reduced model (see Eq. (23) in [67]), similar to the impurity problem
in LL with Luttinger parameter $K$ [69], is referred to as the boundary sine-
Gordon (BSG) model in the literature [57]. The BSG model describes ‘similar
physics’ in a broad class of physical systems (see, e.g., [76, 77, 79, 78]),
including a quantum impurity embedded in a 1D nanowire [58] or single
conducting channel coupled to an Ohmic dissipative environment [60, 18, 59].
The comparison between different systems, defined by the BSG model, makes it
possible to simulate [48, 50], interpret [60, 41], and even predict physical
effects.

Interestingly, the 2SCK device can also be treated as a mesoscopic circuit
simulating the tunneling between two different LLs equivalent to the tunneling
between two FQH edges with different filling factors [75]. Each LL is
determined by $N_{\alpha}$ channels connected to $\alpha$th QD [the
corresponding interaction constant reads $K_{\alpha}=\nu
N_{\alpha}/(2+N_{\alpha})$]. In this case, $K$ from Eqs. (11), (17) can be
presented as the reduced Luttinger parameter, $K=2/(K_{1}^{-1}+K^{-1}_{2})$ in
accordance with [75]. Thus, Eqs. (11), (17) can be used to analyze transport
properties of two LLs coupled via weak or strong potential barriers in the
spirit of experiments performed in [48, 50].

Concluding this subsection, let us pose an important question: Is it possible
to account for the effects of interaction at the quantum critical point of the
2SCK problem when the non-Fermi liquid behavior prevails? It can only be done
in the particular case of $N_{1}$$=$$N_{2}$$=$$1$ channels when the system is
tuned to the so-called triple point [38, 43]. As it was shown [38, 43], this
critical point corresponds to the existence of the fractionalized
($\mathbb{Z}_{3}$ parafermion) excitation, while the whole problem can be
exactly solved by the analogy of Emery-Kivelson solution [80] in the Toulouse
limit.

Triple quantum critical point.— Let us consider the situation when the 2SCK
circuit operates in the FQH regime with $\nu=1/m$, while
$N_{1}$$=$$N_{2}$$=$$1$. We account for the gate voltage dependence of the
charging energy by considering the weak backscattering at three QPCs,
$v_{\alpha}\ll 1$ ($\alpha=1,2,3$). For this, one needs to replace
$\hat{n}_{\alpha}\rightarrow\hat{n}_{\alpha}-\pi\mathcal{N}_{\alpha,g}$ in Eq.
(5), where $\mathcal{N}_{\alpha,g}$ is the dimensionless gate voltage
parameter at $\alpha=1,2$ QD. The half-integer value
$\mathcal{N}_{\alpha,g}=1/2+n$ ($n$ is integer) corresponds to the charge
degeneracy between $n$ and $n+1$ charge states in $\alpha$th QD. We also use
$\mathcal{S}^{\prime}_{BS}=(D/\pi)\sum_{\alpha=1}^{3}\int d\tau
v_{\alpha}\cos[2\phi_{\alpha}(0,\tau)]$ instead of Eq. (6) (here,
$\phi_{\alpha,1}\equiv\phi_{\alpha}$ with $\alpha=1,2,3$).

Following the calculations made in Ref. 38, we find

$\displaystyle G=\frac{\nu G_{0}}{3}\left\\{1-\mathcal{C}_{1}\left(\frac{\pi
T}{\nu E_{C}}\right)^{\frac{2\nu}{3}-2}-\mathcal{C}_{2}\left(\frac{\pi T}{\nu
E_{C}}\right)^{\frac{2\nu}{3}}\right\\},$

where
$\mathcal{C}_{1(2)}=(\nu/3)\,3^{\nu/3}(\gamma/\pi)^{4\nu/3}\sqrt{\pi}\Gamma(\nu/3)/[2\Gamma(\nu/3+1/2)]\cdot\tilde{\mathcal{C}}_{1(2)}$
(constants $\tilde{\mathcal{C}}_{1}$ and $\tilde{\mathcal{C}}_{2}$ are
presented in [67] and [81]) are $\nu$ and $\mathcal{N}_{\alpha,g}$-dependent
constants fully consistent with Eqs. (D8) and (D9) in Ref. 38. Thus, Eq.
(Probing Luttinger Liquid Properties in Multichannel Two-Site Charge Kondo
Simulator) generalizes the universal conductance scaling in the vicinity of a
quantum critical (triple) point in the case of the FQH regime. The conductance
scaling of the first two terms in Eq. (Probing Luttinger Liquid Properties in
Multichannel Two-Site Charge Kondo Simulator) coincide with Eq. (11), since
$K\rightarrow\nu/3$ at $N_{1}=N_{2}=1$. At $\nu=1$, the conductance Eq.
(Probing Luttinger Liquid Properties in Multichannel Two-Site Charge Kondo
Simulator) reduces to the scaling Eq. (13) from [43].

At the tri-critical point, $\mathcal{N}_{1,g}=\mathcal{N}_{2,g}=1/2$,
$|v_{1}|=|v_{2}|$ and $v_{3}=2\cdot 3^{-\nu/2}|v_{1}|$, (here, we choose the
simplest set of parameters; the detailed analysis of the triple point for
other sets of parameters can be found elsewhere [38]), the constant
$\mathcal{C}_{1}$ vanishes, while the third term in Eq. (Probing Luttinger
Liquid Properties in Multichannel Two-Site Charge Kondo Simulator) associated
with the least irrelevant perturbation at the quantum critical point survives.
Thus, Eq. (Probing Luttinger Liquid Properties in Multichannel Two-Site Charge
Kondo Simulator) at the triple point can be treated as a universal Kondo
scaling of the 2SCK model in the FQH regime, $G(T)=G_{\rm
max}\\{1-(T/T_{K})^{2\nu/3}\\}$, with the Kondo temperature $T_{K}\propto\nu
E_{C}/|\tilde{v}_{1}|^{3m}$ ($m$ is odd integer, which comes from $\nu=1/m$).
At $T\gg T_{K}$, we would expect a logarithmic correction $\log^{-2}(T/T_{K})$
to the conductance. Far from the tri-critical point, the second term in Eq.
(Probing Luttinger Liquid Properties in Multichannel Two-Site Charge Kondo
Simulator) prevails over the third one. In this case, the Kondo screening is
suppressed either by channel asymmetry ($|v_{1}|\neq|v_{2}|\neq|v_{3}|$) or by
pseudomagnetic field (gate voltages $\mathcal{N}_{1,g}$ and
$\mathcal{N}_{2,g}$ that lift charge degeneracy in the QDs). An exaggerated
channel asymmetry ($|v_{3}|=v$, $|v_{1}|=|v_{2}|=0$) allows us to formulate a
universal conductance scaling, see Eqs. (11) and (17), in the case $N_{1}\neq
1,N_{2}\neq 1$.

Conclusions. We studied the effects of interaction in a 2SCK simulator by
considering a hybrid metal-semiconductor double-dot setup in the FQH regime
with a filling factor $\nu=1/m$ ($m$ is an odd integer). We showed that the
number of open ballistic channels $N_{1},N_{2}$ connected to both the QDs
uniquely determines the transport properties at low temperatures. The
conductance scaling at the strong $KG_{0}-G(T)\propto T^{2K-2}$ and weak
$G(T)\propto T^{2/K-2}$ inter-dot coupling is fully featured by the effective
interaction parameter $K=\nu N_{1}N_{2}/(N_{1}+N_{2}+N_{1}N_{2})$. Meanwhile,
the 2SCK circuit can be treated as a simulator of tunneling between two
different LLs, whose interaction parameters are defined by $N_{1}$ and
$N_{2}$, respectively. Moreover, in a particular case of $N_{1}=N_{2}=1$ and
$\nu=1/m\neq 1$, we predict how the finite temperature correction at the
triple quantum critical point, characterized by the emergence of fractional
$\mathbb{Z}_{3}$ excitation, is determined by the fractional filling factor
$\nu$: $1-G(T)/G_{\rm max}\sim(T/T_{K})^{2\nu/3}$.

Note added. – Ref. [44] addresses a closely related problem of transport
properties in a multichannel two-site charge Kondo circuit in the case of weak
interdot coupling.

Acknowledgements. The authors would like to thank Jaeho Han, A. Andreanov,
Dung X. Nguyen and F. Stäbler for fruitful and inspiring discussions. A.V.P.
acknowledges support by the Institute for Basic Science in Korea
(IBS-R024-D1). V.M.K. is grateful to the Project FSUN-2023-0006 and “BASIS”
Foundation.

## References

  * [1] J. Kondo, Prog. Theor. Phys. 32, 37 (1964).
  * [2] A. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1993).
  * [3] P. Nozieres, J. Low Temp. Phys. 17, 31 (1974).
  * [4] P. W. Anderson, J. Phys. C: Solid State Phys. 3, 2436 (1970).
  * [5] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
  * [6] N. Andrei, Phys. Rev. Lett. 45, 379 (1980).
  * [7] P. B. Vigman, JETP Lett. 31, 364 (1980).
  * [8] P. Nozieres, A. Blandin, Journal de Physique 41, 19 (1980).
  * [9] I. O. Kulik, and R. I. Shekhter, Sov. Phys. JETP 41, 308 (1975).
  * [10] L. I. Glazman, and R. I. Shekhter, J. Phys.: Condens. Matter 1, 5811 (1989).
  * [11] M. A. Kastner, Rev. Mod. Phys. 64, 849 (1992).
  * [12] L. Kouwenhoven, and L. Glazman, Physics World 14, 33 (2001).
  * [13] L. I. Glazman, and M. E. Raikh, JETP Lett. 47, 452 (1988).
  * [14] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav & M. A. Kastner, Nature 391 156 (1998).
  * [15] S.M. Cronenwett, T.H. Oosterkamp, L.P. Kouwenhoven, Science 281, 540 (1998).
  * [16] W. G. van der Wiel, S. De Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha, L. P. Kouwenhoven, Science 289, 2105 (2000).
  * [17] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1, (1987).
  * [18] U. Weiss, Quantum dissipative systems (World Scientific, Singapore, 1999).
  * [19] K. Le Hur, L. Henriet, L. Herviou, K. Plekhanov, A. Petrescu, T. Goren, M. Schiro, C. Mora, P. P. Orth, Comptes Rendus Physique 19, 451 (2018).
  * [20] K. A. Matveev, Sov. Phys. JETP 72, 892 (1991).
  * [21] K. Flensberg, Phys. Rev. B 48, 11156 (1993).
  * [22] K. A. Matveev, Phys. Rev. B 51, 1743 (1995).
  * [23] A. Furusaki, K. A. Matveev, Phys. Rev. B 52, 16676 (1995).
  * [24] Z. Iftikhar, S. Jezouin, A. Anthore, U. Gennser, F. D. Parmentier, A. Cavanna, and F. Pierre, Nature 526, 233 (2015).
  * [25] S. Jezouin, Z. Iftikhar, A. Anthore, F. D. Parmentier, U. Gennser, A. Cavanna, A. Ouerghi, I. P. Levkivskyi, E. Idrisov, E. V. Sukhorukov, L. I. Glazman, and F. Pierre, Nature (London) 536, 60 (2016).
  * [26] Z. Iftikhar, A. Anthore, A. K. Mitchell, F. D. Parmentier, U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Simon, and F. Pierre, Science 360, 1315 (2018).
  * [27] C. Han, Z. Iftikhar, Y. Kleeorin, A. Anthore, F. Pierre, Y. Meir, A. K. Mitchell, E. Sela, Phys. Rev. Lett. 128, 146803 (2022).
  * [28] C. Piquard, P. Glidic, C. Han, A. Aassime, A. Cavanna, U. Gennser, Y. Meir, E. Sela, A. Anthore, F. Pierre, arXiv:2303.12039.
  * [29] T. K. T. Nguyen, and M. N. Kiselev, Phys. Rev. B 97 085403 (2018).
  * [30] L. A. Landau, E. Cornfeld, E. Sela, Phys. Rev. Lett. 120, 186801 (2018).
  * [31] K. Snizhko, F. Buccheri, R. Egger, and Y. Gefen Phys. Rev. B 97, 235139 (2018).
  * [32] G. A. R. van Dalum, A. K. Mitchell, and L. Fritz, Phys. Rev. B 102, 041111(R) (2020).
  * [33] J.-Y. M. Lee, C. Han, and H.-S. Sim, Phys. Rev. Lett. 125, 196802 (2020).
  * [34] T. K. T. Nguyen, M. N. Kiselev, Phys. Rev. Lett. 125, 026801 (2020).
  * [35] T. K. T. Nguyen, M. N. Kiselev, Comm. in Physics 30, 1 (2020).
  * [36] D. B. Karki, Phys. Rev. B 102, 245430 (2020).
  * [37] A. V. Parafilo, T. K. T. Nguyen, and M. N. Kiselev, Phys. Rev. B 105, L121405 (2022).
  * [38] D. B. Karki, E. Boulat, and C. Mora, Phys. Rev. B 105, 245418 (2022).
  * [39] J. Shim, D. Kim, H.-S. Sim, Nature Comm. 14, 3521 (2023).
  * [40] A. V. Parafilo, T. K. T. Nguyen, Comm. in Physics 33, 1 (2023).
  * [41] A. V. Parafilo, Phys. Rev. Research 5, 023019 (2023).
  * [42] T. K. T. Nguyen, A. V. Parafilo, H. Q. Nguyen, M. N. Kiselev, Phys. Rev. B 107, L201402 (2023).
  * [43] D. B. Karki, E. Boulat, W. Pouse, D. Goldhaber-Gordon, A. K. Mitchell, C. Mora, Phys. Rev. Lett. 130, 146201 (2023).
  * [44] M. N. Kiselev, Phys. Rev. B 108, L081108 (2023).
  * [45] E. Sela, D. Goldhaber-Gordon, A. Anthore, F. Pierre, Y. Oreg, Phys. Rev. B 107, L161108 (2023).
  * [46] F. Stäbler, and E. Sukhorukov, arXiv: 2306.1057v2 (2023).
  * [47] E. Sivre, A. Anthore, F. D. Parmentier, A. Cavanna, U. Gennser, A. Ouerghi, Y. Jin, F. Pierre, Nature Physics 14, 145 (2018).
  * [48] A. Anthore, Z. Iftikhar, E. Boulat, F. D. Parmentier, A. Cavanna, A. Ouerghi, U. Gennser, and F. Pierre, Phys. Rev. X 8, 031075 (2018).
  * [49] H. Duprez, E. Sivre, A. Anthore, A. Aassime, A. Cavanna, U. Gennser, F. Pierre, Science 366, 1243 (2019).
  * [50] H. Duprez, F. Pierre, E. Sivre, A. Aassime, F. D. Parmentier, A. Cavanna, A. Ouerghi, U. Gennser, I. Safi, C. Mora, and A. Anthore, Phys. Rev. Res. 3, 023122 (2021).
  * [51] W. Pouse, L. Peeters, C. L. Hsueh, U. Gennser, A. Cavanna, M. A. Kastner, A. K. Mitchell, D. Goldhaber-Gordon, Nature Physics 19, 492 (2023).
  * [52] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125 (1988).
  * [53] I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. 68, 1046 (1992).
  * [54] S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950).
  * [55] J. M. Luttinger, J. Math. Phys. 4, 1154 (1963).
  * [56] F. D. M. Haldane, J. Phys. C: Solid State Phys. 14 2585 (1981).
  * [57] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization Approach to Strongly Correlated Systems (Cambridge University Press, Cambridge, England, 1998).
  * [58] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, UK, 2003).
  * [59] M. H. Devoret and H. Grabert, Single Charge Tunneling: Coulomb Blockade Phenomena in Nanostructures (Plenum Press, New York, 1992).
  * [60] I. Safi, and H. Saleur, Phys. Rev. Lett. 93, 126602 (2004).
  * [61] A. M. Chang, Rev. Mod. Phys. 75, 1449 (2003).
  * [62] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  * [63] X. G. Wen, Phys. Rev. Lett. 64, 2206 (1990).
  * [64] X. G. Wen, Phys. Rev. B 43, 11025 (1991).
  * [65] A metallic island (QD) with a continuous density of states placed on the top of a highly doped region in a semiconductor is often called an Ohmic contact. Note that the charging energy Hamiltonian given by Eq. (2) requires additional regularization associated with the decaying of neutral excitations deep inside the Ohmic contact. For this, we account for the following boundary conditions in expressions for the accumulated charges $Q_{1}$,$Q_{2}$: $\varphi_{1,i}^{\rm t(b)}(+\infty)=\varphi_{2,i}^{\rm t(b)}(-\infty)=\varphi_{3}^{\rm t(b)}(\pm\infty)=0$.
  * [66] K. A. Matveev, A. V. Andreev, Phys. Rev. B 66, 045301 (2002).
  * [67] See Supplemental Material [URL] for the details of derivation.
  * [68] Coefficients $A_{\eta}$ ($\eta=A,B,C$) in $\mathcal{S}_{BS}$ explicitly read as $A_{A}=\sqrt{N_{1}N_{2}/(N_{1}+N_{2}+N_{1}N_{2})}$ and $A_{B(C)}=(1+A\mp\Delta)/[N_{2}\sqrt{N_{2}+(1+A\mp\Delta)^{2}+N_{1}(A\mp\Delta)^{2}}]$, where we denote $A=(N_{1}-N_{2})/2$ and $\Delta=\sqrt{1+A^{2}}$.
  * [69] C. L. Kane, M. P. A. Fisher, Phys. Rev. B 46, 15233 (1992).
  * [70] C. L. Kane, M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992).
  * [71] A. O. Slobodeniuk, I. P. Levkivskyi, and E. V. Sukhorukov, Phys. Rev. B 88, 165307 (2013).
  * [72] E. V. Sukhorukov, Physica E: Low-dimensional Systems and Nanostructures 77, 191 (2016).
  * [73] P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. Lett. 74, 3005 (1995).
  * [74] P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. B 52, 8934 (1995).
  * [75] C. C. Chamon, and E. Fradkin, Phys. Rev. B 56, 2012 (1997).
  * [76] S. Chakravarty, Phys. Rev. Lett. 49, 681 (1982).
  * [77] A. Schmid, Phys. Rev. Lett. 51, 1506 (1983).
  * [78] M. P. A. Fisher, and W. Zwerger, Phys. Rev. B 32, 6190 (1985).
  * [79] H. Grabert, P. Schramm, G.-L. Ingold, Phys. Rev. Rep. 168, 115 (1988).
  * [80] V. J. Emery, S. Kivelson, Phys. Rev. B 46, 10812 (1992).
  * [81] Here, we present explicit expressions for constants $\tilde{\mathcal{C}}_{1(2)}$: $\tilde{\mathcal{C}}_{1}(\nu,\tilde{v}_{\alpha},\mathcal{N}_{i,g})=\,\\{\tilde{v}_{1}^{2}+\tilde{v}_{2}^{2}+3^{\nu}\,\tilde{v}_{3}^{2}+2\,\tilde{v}_{1}\tilde{v}_{2}\cos[2\pi(\mathcal{N}_{1,g}+\mathcal{N}_{2,g})]+2\cdot 3^{\nu/2}\tilde{v}_{3}\,\left(\tilde{v}_{1}\cos[2\pi\mathcal{N}_{1,g}]+\tilde{v}_{2}\cos[2\pi\mathcal{N}_{2,g}]\right)\\}$, and $\tilde{\mathcal{C}}_{2}(\nu,\tilde{v}_{\alpha},\mathcal{N}_{i,g})=(\nu\pi^{2}/27)\,[2\nu/(2\nu+3)]\\{28(\tilde{v}_{1}^{2}+\tilde{v}_{2}^{2})+4\cdot 3^{\nu}\,\tilde{v}_{3}^{2}-52\,\tilde{v}_{1}\tilde{v}_{2}\cos[2\pi(\mathcal{N}_{1,g}+\mathcal{N}_{2,g})]-8\cdot 3^{\nu/2}\tilde{v}_{3}\,\left(\tilde{v}_{1}\cos[2\pi\mathcal{N}_{1,g}]+\tilde{v}_{2}\cos[2\pi\mathcal{N}_{2,g}]\right)\\}$. See also Ref. [67].