# Manifestation of Luttinger liquid effects in a hybrid metal-semiconductor
double-quantum dot device

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

###### Abstract

We theoretically study the transport properties of a hybrid nanodevice
comprised of two large metallic islands incorporated in a two-dimensional
electron gas. The high-tunability of the conducting channels electrically
connecting two islands to each other and to the leads allows us to treat the
setup as a realization of a multi-channel two-site charge Kondo (2SCK) model.
It is shown that the leading temperature dependence of the conductance in the
2SCK circuit satisfies the conductance scaling of a single-impurity problem in
a Luttinger liquid, whose interaction parameter is fully determined by the
number of conducting channels in the device. We demonstrate that the finite
weak backscattering in all conducting channels features the appearance of the
subleading temperature dependencies in linear conductance. At the special
critical point, we predict an equivalency between the 2SCK nanodevice and a
single-site two-channel charge Kondo problem, where one Kondo channel is
implemented by a non-interacting electron gas and the second Kondo channel is
attributed to the Luttinger liquid.

##  I Introduction

Recent advances in the fabrication and control of hybrid metal-semiconductor
nanodevices have induced enormous interest among the theoretical and
experimental community due to their unprecedented access to the multichannel
Kondo physics and promising perspective as a platform for studying quantum
critical phenomena.

Although the original Kondo model [1] is usually attributed to the interaction
between spins of conduction electrons and impurity spins [2], Kondo physics
may arise whenever degenerate quantum states are coupled to one or few bath
continua [3, 5, 4]. A ’charge’ implementation of the Kondo model, suggested in
seminal papers [6, 7, 8, 9], consists of a large quantum dot (QD) in the weak
Coulomb blockade regime attached to one or a few leads via high-controllable
quantum point contacts (QPCs). Two degenerate macroscopic charge states of the
QD tuned by a gate voltage constitute impurity pseudo-spin, and the electron’s
location (whether it is in the lead or in the QD) plays the role of the
conduction electron’s pseudo-spin. The corresponding mapping of the single-
electron transistor model on the anisotropic Kondo model is justified at $T\ll
E_{C}$ (where $E_{C}$ is the QD’s charging energy) and if the QD has a
continuous density of states. The ’charge’ Kondo (CK) model can be easily
generalized to the multi-channel Kondo model, where the number of Kondo
channels is determined either by the electron’s internal degree of freedom
(spin), the number of electronic channels or the number of electrodes coupled
to the QD.

The first experimental implementation of the CK device was developed in a
hybrid metal-semiconductor single-electron transistor on the base of a
Ga(Al)As heterostructure operating in the integer quantum Hall regime [10,
11]. A semiconductor component of the setup ensures the high tunability of
electric channels, while a metallic island with negligible level spacing
provides the quantization of charge states and avoids any coherent electronic
transport from a reservoir on one side of the island to another. As a
consequence, the universal conductance scalings for the cases of two- and
three-channel CK models were experimentally reported with unprecedented
control [10, 11]. It is essential to mention that hybrid CK circuit [10, 11]
can also be regarded as a simulator for the metal-to-insulator transition
occurred in 1D interacting gas with a single-impurity [12, 13]. Indeed, a
quantum channel with arbitrary transmission (implemented by the QPC) being in
series with linear resistance determined by $N$ ballistic channels corresponds
to the problem of the Luttinger liquid (LL) of interaction parameter
$K=N/(N+1)$ with an isolated impurity [14, 15].

Nevertheless, the interest generated by the experiments [10, 11] has resulted
in a number of exciting experimental [16, 12, 17, 13, 20, 18, 19] and
theoretical works [23, 43, 44, 31, 27, 28, 30, 35, 33, 36, 21, 22, 26, 32, 24,
40, 25, 37, 38, 29, 34, 39, 41, 42]. Especially, we would like to emphasize
Ref. [23], where a so-called two-site CK (2SCK) nanodevice was first proposed
to test competing between the Fermi- and non-Fermi liquid phases arising in
the setup. Authors in Ref. [23] discussed a double-quantum dot (DQD) device
with strong- and weak inter-dot coupling, where each dot is attached to a
fixed number of leads via single-mode QPCs. Therefore, the theoretical
proposal in [23] established a ’charge’ implementation of a quantum two-
impurity model, where the interplay between an individual (multi-channel)
Kondo screening and Ruderman-Kittel-Kasuya-Yosida-like inter-dot interaction
may take place.

Soon, introduced in [23] model was fabricated in a hybrid metal-semiconductor
double-island device [20] as a platform for probing frustrated interaction at
the exotic quantum critical point with fractional excitation [37, 38]. In the
simplest case of strong inter-dot coupling, when each QD is coupled only to
one electrode via nearly ballistic single-mode QPC, the emergence of a
$\mathbb{Z}_{3}$ parafermion featured by the fractional $\log(3)/2$ residual
entropy was declared [20, 38].

Further, it was revealed in Ref. [41] that the 2SCK nanodevice, where
left/right QD connected via $N$($M$) fully ballistic channels to source/drain
reservoirs, can be treated as a LL simulator with the Luttinger interaction
parameter $K$$=$$NM/(N+M+NM)$. Here, the interaction parameter $K$ may acquire
any value in the interval $1/3<K<1$. The study in Ref. [41] didn’t account for
finite backscattering in the QPCs connected QDs with source and drain leads
(except for a special case $N$$=$$M$$=$$1$). Thus, the possible interplay
between interaction effects and quantum CK criticality was neglected therein.
Meanwhile, such interplay may produce exciting phenomena involving various
quantum phase transitions [45, 46, 47]; see also a series of works [28, 30,
33, 35, 36], where the influence of interactions on the transport properties
in a single-site CK circuit was investigated.

This paper partially corrects this omission by considering the 2SCK device
with strong inter-dot coupling and includes the finite backscattering in all
QPCs.

Figure 1: Schematic representation of the two-site charge Kondo (2SCK)
circuit: a hybrid metal-semiconductor device formed in Ga(Al)As
heterostructure consists of two metallic islands or QDs with the continuous
density of states (marked by green and blue colors) strongly coupled via a
single-mode quantum point contact (QPC) to each other. Besides, left (right)
QD is electrically connected to the 2DEG via $N_{1}$ ($N_{3}$) single-mode
QPCs. The system is exposed to a strong magnetic field, which drives it in the
integer quantum Hall regime with filling factor $\nu=1$. Thus, electric
current propagates along the spin-polarized edge channels (marked by red solid
lines). The single-mode QPCs are assumed to be nearly transparent (featured by
small reflection amplitude $|r_{\alpha}|\ll 1$ with $\alpha=1,2,3$), resulting
in weak backscattering between left- and right-moving electrons. Charge degree
of freedom is characterized via bosonic fields $\phi_{\alpha,i}(x)$ (here,
$i=1,...,N_{\alpha}$), which is the ’difference’ between chiral left- and
right- bosonic fields,
$\phi_{\alpha,i}=(\varphi^{L}_{\alpha,i}-\varphi^{R}_{\alpha,i})/2$ [48].

##  II  Model and Effective Hamiltonian

We study a multichannel 2SCK circuit – a hybrid metal-semiconductor device
consisting of two large metallic islands (QDs) with a continuous density of
states incorporated into a high-mobility Ga(Al)As two-dimensional electron gas
(2DEG), see Fig. 1. The system is in a strong magnetic field, thus realizing
the regime of integer quantum Hall effect with a filling factor $\nu=1$. Two
QDs are strongly coupled to each other via single-mode QPC, while the left
(right) QD is strongly connected to source (drain) electrodes via
$N_{1}$($N_{3}$) single-mode QPCs. The left- and right-moving (chiral)
electrons in the 1D quantum Hall edge channels undergo weak backscattering in
each QPCs. To describe the transport properties of the 1D channels, we use
bosonic operators $\phi_{\alpha,i}(x)$ and $\theta_{\alpha,i}(x)$, which
satisfy the following commutation relation
$[\phi_{\alpha,i}(x),\partial_{x}\theta_{\alpha^{\prime},j}(x^{\prime})]=i\pi\delta(x-x^{\prime})\delta_{\alpha\alpha^{\prime}}\delta_{ij}$.
Here, field $\phi_{2,i}(x)$ is responsible for inter-dot connection, while
fields $\phi_{1(3),i}$ characterize other $N_{1}$($N_{3}$) conducting
channels, see Fig. 1. In this representation, the Hamiltonian of edge states
is simply the Hamiltonian of the spinless Luttinger liquid with interaction
parameter $g=1$ [48, 49].

The whole system presented in Fig. 1 can be described by the total Hamiltonian
$H=H_{0}+H_{C}+H_{BS}$, where (in $\hbar=k_{B}=1$ units)

$\displaystyle
H_{0}=\frac{v_{F}}{2\pi}\sum_{\alpha=1}^{3}\sum_{i=1}^{N_{\alpha}}\int
dx\left\\{[\pi\Pi_{\alpha,i}(x)]^{2}+\left[\frac{\partial\phi_{\alpha,i}(x)}{\partial
x}\right]^{2}\right\\}$

is the kinetic energy of conducting channels, $v_{F}$ is the Fermi velocity
and $\Pi_{\alpha,i}(x)$ is the canonically conjugated to the field
$\phi_{\alpha,i}(x)$ momentum,
$[\phi_{\alpha,i}(x),\Pi_{\alpha^{\prime},i^{\prime}}(x^{\prime})]=i\delta(x-x^{\prime})\delta_{\alpha\alpha^{\prime}}\delta_{ii^{\prime}}$.
Since for the considered model $N_{2}=1$, we use the notation
$\phi_{2,1}(x)\equiv\phi_{2}(x)$ in what follows.

Two metallic islands in the weak Coulomb blockade regime can be described via
the following Hamiltonian:

$\displaystyle
H_{C}=\frac{E_{C}}{\pi^{2}}\left(\sum_{i=1}^{N_{1}}\phi_{1,i}(0)-\phi_{2}(0)+\pi\mathcal{N}_{g}\right)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{E_{C}}{\pi^{2}}\left(\phi_{2}(0)-\sum_{i=1}^{N_{3}}\phi_{3,i}(0)+\pi\mathcal{N}_{g}\right)^{2},$
(2)

where $E_{C}=e^{2}/2C$ is the charging energy of islands, $C$ is islands’
capacitance; $\mathcal{N}_{g}$ is a gate voltage (assumed being equal for both
dots).

A weak scattering between right- and left-moving particles in QPCs is
characterized by

$\displaystyle
H_{BS}=\sum_{\alpha=1}^{3}\sum_{i=1}^{N_{\alpha}}\frac{D|r_{\alpha,i}|}{\pi}\cos\left[2\phi_{\alpha,i}(0)\right].$
(3)

Here, $D$ is an ultra-violet energy cut-off associated with the existence of
bandwidth, and $|r_{\alpha,i}|\ll 1$ are the backscattering matrix elements.
We consider the case when all backscattering amplitudes in QPCs with the same
index $\alpha$ are equal to each other, $|r_{\alpha,i}|\equiv|r_{\alpha}|$.

Next, we perform few approximations to simplify introduced model. First, we
present $\phi_{1,i}$ ($\phi_{3,i}$) via new fields
$\phi_{1,c},\phi_{1,s},...,\phi_{1,f}$
($\phi_{3,c},\phi_{3,s},...,\phi_{3,f}$), which characterize the charge, spin
and other flavor modes. With this representation only three fields, namely,
two charge modes
$\phi_{\alpha,c}=\sum_{i=1}^{N_{\alpha}}\phi_{\alpha,i}/\sqrt{N_{\alpha}}$
($\alpha=1,3$) and $\phi_{2}$, appear in the Coulomb blockade Hamiltonian.
Second, we perform a unitary transformation to diagonalize $H_{0}+H_{C}$ part
of the total Hamiltonian $H$, which reduces the fields
$(\phi_{1,c},\phi_{2},\phi_{3,c})$ to $(\phi_{A},\phi_{B},\phi_{C})$ as
follows:

$\displaystyle\left(\begin{array}[]{c}\phi_{1,c}\\\ \phi_{2}\\\
\phi_{3,c}\end{array}\right)=\hat{P}\left(\begin{array}[]{c}\phi_{A}\\\
\phi_{B}\\\
\phi_{C}\end{array}\right)+\pi\left(\begin{array}[]{c}-\frac{\mathcal{N}_{g}}{\sqrt{N_{1}}}\\\
0\\\ \frac{\mathcal{N}_{g}}{\sqrt{N_{3}}}\end{array}\right).$ (13)

An exact expression for the matrix $\hat{P}$ can be found in Appendix A. After
diagonalization, the kinetic energy Hamiltonian
$H^{\prime}_{0}=\hat{P}^{-1}H_{0}\hat{P}$ has the same quadratic form as in
Eq. (II) (just written in terms of new variables), while the Coulomb blockade
Hamiltonian acquires a more simple form

$\displaystyle
H^{\prime}_{C}=\frac{E_{C}}{\pi^{2}}\left\\{\mathcal{M}_{B}\,\phi_{B}^{2}(0)+\mathcal{M}_{C}\,\phi_{C}^{2}(0)\right\\}.$
(14)

Here, $\mathcal{M}_{B}$, $\mathcal{M}_{C}$ are effective mass parameters of
fields $\phi_{B}$, $\phi_{C}$, whose values are determined by the number of
total conducting channels in the system as

$\displaystyle\mathcal{M}_{B(C)}=1+\frac{(N_{1}+N_{3})}{2}\mp\sqrt{1+\frac{(N_{1}-N_{3})^{2}}{4}}.$
(15)

Thus, one obtains an effective Hamiltonian for $N_{1}+N_{3}+1$ modes, two of
which $(\phi_{B},\phi_{C})$ are gapped due to the Coulomb blockade and
$N_{1}+N_{3}-1$ are massless.

In this paper, we calculate the linear conductance through the device in zero-
frequency limit using the Kubo formula and current operator
$\hat{I}=-(e/\pi)\sqrt{N_{1}}\partial_{t}\phi_{1,c}(0,t)$, see, e.g., [9, 13].
In terms of new variables $\phi_{\alpha}$ ($\alpha=A,B,C$), it reads

$\displaystyle G=KG_{0}\frac{2T}{\pi i}\lim_{\omega\rightarrow
0}\omega\lim_{i\omega_{n}\rightarrow\omega+i0^{+}}\langle\phi_{A}(i\omega_{n})\phi_{A}(-i\omega_{n})\rangle,$
(16)

where $G_{0}=e^{2}/2\pi$ is a unitary conductance,
$K=N_{1}N_{3}/(N_{1}+N_{3}+N_{1}N_{3})$ is an effective LL interaction
parameter (see discussion below), and
$\phi_{A}(i\omega_{n})=\int_{0}^{T^{-1}}d\tau\phi_{A}(\tau)\exp(i\omega_{n}\tau)$,
while $\omega_{n}=2\pi Tn$ is the Matsubara frequency and $\tau$ imaginary
time. The thermal average
$\langle\phi_{A}(i\omega_{n})\phi_{A}(-i\omega_{n})\rangle$ can be calculated
by utilizing the functional integration technique and second-order
perturbation theory over $|r_{\alpha}|\ll 1$. It is important that only one
gapless mode $\phi_{A}(x)$ out of $N_{1}+N_{3}-1$ determines the charge
transport through the 2SCK setup.

Further simplification of the model is possible since the charging energy
suppresses the fluctuations of $\phi_{B}$, $\phi_{C}$ at low temperatures
$T\ll E_{C}$. Thus, one can integrate out these modes by replacing the
backscattering Hamiltonian $H^{\prime}_{BS}$ with its value averaged over the
fluctuations of $\phi_{B}$, $\phi_{C}$. As a result, one obtains effective
Hamiltonian, which, since we are interested in $\phi_{A}(x)$ mode, can be
written as

$\displaystyle H_{\rm eff}=H_{\rm
SG}[\phi_{A}]+\sum_{j=1}^{N_{1}+N_{3}-2}H_{\rm 1d}[\phi_{j}]+H_{\rm
int}[\phi_{A},\phi_{j}].$

Here,

$\displaystyle H_{\rm SG}[\phi_{A}]=\frac{v_{F}}{2\pi}\int
dx\left\\{[\pi\Pi_{A}(x)]^{2}+[\partial_{x}\phi_{A}(x)]^{2}\right\\}$
$\displaystyle+\beta\cos\left[2\sqrt{K}\phi_{A}(0)\right]$ (18)

is a, so-called, boundary sine-Gordon Hamiltonian, which describes many
different physical systems including a single-impurity scattering problem in
the LL with interaction $K$ [50, 48, 49], conducting channel coupled to an
Ohmic environment [15, 51], and other [52, 53, 54, 55]. Coefficient $\beta$ is
a function of $N_{1}$, $N_{3}$ as it is shown in Ref. [41], see Eq. (23) in
Supplemental Material therein.

Effective model (II) becomes non-trivial if $N_{1}+N_{3}>2$. In this case,
field $\phi_{A}$ interacts with other massless fields, namely, with $N_{1}-1$
flavor modes in the left part of the device and with $N_{3}-1$ flavor modes in
the right part of the device. The corresponding Hamiltonian of $j$th copy of
1D massless quasiparticles and interaction term can be written as:

$\displaystyle H_{\rm 1d}[\phi_{j}]=\frac{v_{F}}{2\pi}\int
dx\left\\{[\pi\Pi_{j}(x)]^{2}+[\partial_{x}\phi_{j}(x)]^{2}\right\\},$ (19)
$\displaystyle H_{\rm
int}=\left\\{\sum_{i=1}^{N_{1}-1}\lambda_{i}\cos[a_{i}\phi_{A}(0)]\prod_{j=1}^{i}\cos[a_{ij}\phi_{j}(0)]\right.$
$\displaystyle\left.+\sum_{i=1}^{N_{3}-1}\lambda^{\prime}_{i}\cos[a^{\prime}_{i}\phi_{A}(0)]\prod_{j=1}^{i}\cos[a^{\prime}_{ij}\phi_{j}(0)]\right\\},$
(20)

where constants $\lambda_{i}$, $a_{i}$, $a_{ij}$ and $\lambda^{\prime}_{i}$,
$a^{\prime}_{i}$, $a^{\prime}_{ij}$ depend on the number of conducting
channels $N_{1}$, $N_{3}$. In addition, $\lambda_{i}$, $\lambda^{\prime}_{i}$
are determined by the reflection coefficients in QPCs. In general form, Eq.
(II) is too cumbersome. Thus, we concentrate our study on particular cases
when $N_{1},N_{3}=\\{1,2,3\\}$. The corresponding effective Hamiltonians Eq.
(II) for these cases in explicit form are shown in Appendix B.

##  III Results

$N_{1}$ | $N_{3}$ | $K$ | $\mathcal{I}_{K}$ | $g$ | $\mathcal{I}_{g}$ | $g^{\prime}$ | $\mathcal{I}_{g^{\prime}}$
---|---|---|---|---|---|---|---
1 | 1 | 1/3 | $T^{-4/3}$ | - | - | - | -
1 | 2 | 2/5 | $T^{-6/5}$ | 3/5 | $T^{-4/5}$ | - | -
1 | 3 | 3/7 | $T^{-8/7}$ | 5/7 | $T^{-4/7}$ | - | -
2 | 2 | 1/2 | $T^{-1}$ | 5/8 | $T^{-3/4}$ | - | -
3 | 3 | 3/5 | $T^{-4/5}$ | 11/15 | $T^{-8/15}$ | - | -
2 | 3 | 6/11 | $T^{-10/11}$ | 7/11 | $T^{-8/11}$ | 8/11 | $T^{-6/11}$
Table 1:  Effective Luttinger liquid interaction parameters $K$, $g$,
$g^{\prime}$, and corresponding leading, sub- and sub-sub-leading temperature
dependencies of the conductance Eq. (III) for different number of conducting
channels $N_{1}$,$N_{3}=\\{1,2,3\\}$, which couple DQD with source and drain
electrodes. For considered in this paper geometry of the system, the Luttinger
liquid interaction parameter $K$ is determined as
$K=N_{1}N_{3}/(N_{1}+N_{3}+N_{1}N_{3})$, while $g$($g^{\prime}$) can be found
from Eq. (24).

First, let’s shortly revise the simplest while the very non-trivial case of
$N_{1}=N_{3}=1$, which has been already studied in great detail theoretically
in [37, 38] and experimentally in [20, 38]. Since there is only one gapless
field $\phi_{A}$, the 2SCK model with $N_{1}=N_{3}=1$ can be described by Eq.
(II) with $K=1/3$ and $\beta$ being the function of gate voltages on both QDs
and the backscattering coefficients. The system ’simulates’ properties of the
LL with $K=1/3$ with a single weak potential barrier. As it is known, the
charge transport through such a device is suppressed at low temperatures since
$K<1$ [50, 48, 49]. However, cos-term from Eq. (II) could be nullified by
tuning constant $\beta$ with $\mathcal{N}_{g}$ and $r_{\alpha}$ at, so-called,
triple point [37, 38]. As it was shown [37, 38], this quantum critical point
corresponds to the existence of the fractionalized $\mathbb{Z}_{3}$
parafermion, which was also identified as a source of $\log(3)/2$ residual
entropy. The problem in [38] was solved by utilizing the Emery-Kivelson
mapping [56] in the Toulouse limit in full analogy with the two- and four-
channel Kondo problems [49].

The same configuration of the 2SCK setup ($N_{1}=N_{3}=1$) in the case of the
Fractional Quantum Hall regime, when the edge states conduct fractional charge
$e^{\ast}=\nu e$ with $\nu=1/m$ ($m$ is an odd integer) being a fractional
filling factor, has been investigated in Ref. [41].

Next, we examine the original results. For other than mentioned above
particular case $N_{1}$$=$$N_{3}$$=$$1$, the linear conductance Eq. (16) is
obtained in perturbation theory with respect to backscattering coefficients
$|r_{\alpha}|$. We omit here the details of calculations, referring to Refs.
[9, 36, 41] and to Appendix B for the explicit expression of used effective
Hamiltonians. In general form, the linear conductance through the device reads

$\displaystyle
G(T)=KG_{0}\left[1-\mathcal{C}_{l}\,\mathcal{I}_{K}(T)-\mathcal{C}_{sl}\,\mathcal{I}_{g}(T)-\mathcal{C}_{ssl}\,\mathcal{I}_{g^{\prime}}(T)\right],$

where $K=(N_{1}^{-1}+1+N_{3}^{-1})^{-1}$,

$\displaystyle\mathcal{I}_{g}(T)=K\left(\frac{\pi
T}{E_{C}}\right)^{2g-2}\left(\frac{\gamma}{\pi}\right)^{2-2g}\frac{\sqrt{\pi}\Gamma(g)}{2\Gamma\left(g+\frac{1}{2}\right)},$
(22)

and $\Gamma(x)$ is the gamma function, $\gamma=\exp(0.577)$ is the Euler
constant. Explicit values of the interactions constants $K,g,g^{\prime}$ from
Eq. (III) for particular values $N_{1}$,$N_{3}$ are shown in the Table I,
while non-universal constants $\mathcal{C}_{l}$, $\mathcal{C}_{sl}$,
$\mathcal{C}_{ssl}$ dependent on the reflection coefficients are presented in
the Table II. Due to the usage of perturbation theory, the result Eq. (III) is
justified only down to ’crossover’ temperatures $T_{cr}^{(\kappa)}\sim
E_{C}\mathcal{C}_{\kappa}^{1/(2-2\kappa)}$ with
$\mathcal{C}_{\kappa}=\\{\mathcal{C}_{s},\mathcal{C}_{sl},\mathcal{C}_{ssl}\\}$
for $\kappa=\\{K,g,g^{\prime}\\}$. For even lower temperatures $T\ll
T_{cr}^{(\kappa)}$, a consideration in the regime of strong backscattering
(tunnel barriers) is required. An exception is the case of $N_{1}=N_{3}=2$
($K=1/2$) when an exact solution is possible due to refermionization of the
problem and its equivalency to the resonant model, see [58].

$N_{1}$ | $N_{3}$ |  $\mathcal{C}_{l}$, $\mathcal{C}_{sl}$, $\mathcal{C}_{ssl}$
---|---|---
1 | 2 | $\mathcal{C}_{l}=|r_{1}|^{2}1.11^{2}+|r_{2}|^{2}2^{2}+4.44|r_{1}||r_{2}|\cos(2\pi\mathcal{N}_{g})$
|  | $\mathcal{C}_{sl}=(1.37)^{2}|r_{3}|^{2}/8$
1 | 3 | $\mathcal{C}_{l}=|r_{1}|^{2}0.9+|r_{2}|^{2}1.86+2.58|r_{1}||r_{2}|\cos(2\pi\mathcal{N}_{g})$
|  | $\mathcal{C}_{sl}=(1.39)^{2}|r_{3}|^{2}/3$
2 | 2 | $\mathcal{C}_{l}=|r_{2}|^{2}/2$
|  | $\mathcal{C}_{sl}=2^{-1}\left[|r_{1}|^{2}+|r_{3}|^{2}\right]$
3 | 3 | $\mathcal{C}_{l}=1.9^{2}|r_{2}|^{2}$
|  | $\mathcal{C}_{sl}=1.41^{2}(|r_{1}|^{2}+|r_{3}|^{2})/3$
2 | 3 | $\mathcal{C}_{l}=1.94^{2}|r_{2}|^{2}$
|  |  $\mathcal{C}_{sl}=1.42^{2}|r_{1}|^{2}/2$; $\mathcal{C}_{ssl}=1.4^{2}|r_{3}|^{2}/3$
Table 2:  Explicit dependencies of non-universal constants $\mathcal{C}_{l}$,
$\mathcal{C}_{sl}$, $\mathcal{C}_{ssl}$ from Eq. (III) on reflection
coefficients at different QPCs and on different numbers of conductance
channels $N_{1}$,$N_{3}=\\{1,2,3\\}$ in 2SCK device. The origin of numerical
constants can be tracked from the particular effective models presented in
Appendix B.

Equation (III) is the first main result of this paper. The first two terms in
Eq. (III) give the leading temperature dependence via the 2SCK device, which
is the same as for the conductance scaling of the LL with a single weak
potential barrier, whose interaction parameter is $K$ [50, 48, 49]. The
following correspondence in the multi-channel 2SCK device was predicted in
[41] for the case $|r_{1}|$$=$$|r_{3}|$$=$$0$ and could be attributed to a
well-known result by Safi and Saleur [15]. Indeed, a one-channel coherent
conductor coupled in series with the resistance $R$ is equivalent to the
impurity problem in the LL with interaction parameter $K=1/(1+r)$, where
$r=R/R_{0}$ is a dimensionless environmental resistance ($R_{0}=h/e^{2}$). For
the 2SCK device shown in Fig. 1, $\phi_{2}$ field ’scattered’ in $\alpha=2$
QPC plays the role of one-channel coherent conductor, while $N_{1}$($N_{3}$)
channels connected to the first (second) QD can be treated as $N_{1}$($N_{3}$)
resistors coupled in parallel, $R_{\alpha}=R_{0}/(\sum_{i=1}^{N_{\alpha}})$.
As a result, all the channels coupled to DQD implement the total dimensionless
resistance $r=(R_{1}+R_{3})/R_{0}$, and thus, $K=1/[1+N_{1}^{-1}+N_{3}^{-1}]$
[41]. As it is discussed in Ref. [9], an alternative explanation of the power-
law temperature behavior of the conductance in CK circuits is related to the
Anderson orthogonality catastrophe [57].

Nevertheless, a finite backscattering in $\alpha$th QPCs ($\alpha=1,3$)
results in the appearance of the sub-leading temperature dependencies, see
third and fourth terms in Eq. (III). From Table I, we conclude that the
temperature scaling of the sub-leading term is associated with the effective
interaction constant
$g(g^{\prime})=[N_{1}N_{3}+N_{1}(N_{3})-1]/[N_{1}N_{3}+N_{1}+N_{3}]$. If the
numbers of channels in $\alpha=1$ and $\alpha=3$ QPCs are different and not
equaled to one, $N_{1}\neq N_{3}\neq 1$, then the linear conductance has two
sub-leading dependencies, see, e.g., the case of $N_{1}=2$, $N_{3}=3$ in the
Table I.

Figure 2:  (a) Two-site charge Kondo (2SCK) nanodevice from Fig. 1 in a
particular case when $N_{1}$$=$$1$ and $|r_{3}|=0$. (b) An equivalent scheme
of the above device: a single-site two-channel charge Kondo (2CCK) model,
where one channel is attributed with the non-interacting electrons ($K_{1}=1$)
characterized by the bosonic field $\phi_{1,1}(x)\equiv\phi_{1}(x)$, and the
second channel is implemented by the Luttinger liquid (LL) with interaction
parameter $K_{2}=N_{3}/(N_{3}+1)$. The following mapping is possible since a
conducting channel $\phi_{2}$ being in a series with linear resistance
determined by $N_{3}$ ballistic channels, $R=R_{0}/N_{3}$, is equivalent to
the LL with a single impurity. Effective bosonic field $\tilde{\phi}_{2}(x)$
of the second channel appears after integrating out all fields $\phi_{3,i}(x)$
with $i=1,...,N_{3}$. A universal Kondo scaling in the vicinity of the 2CCK
fixed point reads $G(T)/G_{0}=(K_{red}/2)[1-(T/T_{K})^{K_{red}}]$, where
$K_{red}=2K_{1}K_{2}/(K_{1}+K_{2})$ is the reduced LL interaction parameter.

Thus, if we ignore for a while the ’interference terms’
($\propto|r_{\alpha}||r_{\alpha^{\prime}}|$) and the possible gate-voltage
dependence of these terms, one can re-write Eq. (III) in more general form as

$\displaystyle
G(T)=KG_{0}\left[1-\sum_{\alpha=1}^{3}c_{\alpha}|r_{\alpha}|^{2}\mathcal{I}_{g_{\alpha}}(T)\right],$
(23)

where $g_{2}=K$,

$\displaystyle g_{1(3)}=\frac{N_{1}N_{3}+N_{1(3)}-1}{N_{1}+N_{3}+N_{1}N_{3}},$
(24)

and $c_{\alpha}$ are non-universal constants of the order of one. The weak
backscattering in $\alpha$th QPC ($\alpha=1,2,3$) leads to its own power-law
temperature correction to the maximal conductance. It’s also clear that the
sub-leading term does not appear in case $N_{1}=N_{3}=1$ since
$g_{1}=g_{2}=g_{3}$. In the same way, the sub-sub-leading term does not arise
for $N_{1}=1$, $N_{3}=\\{2,3\\}$, because $g_{1}=g_{2}\neq g_{3}$.
Nevertheless, tuning the corresponding reflection coefficient can obtain the
necessary temperature dependence in the device. For instance, in the symmetric
case $N_{1}=N_{3}\neq 1$, the sub-leading terms become leading if the inter-
dot QPC is fully open ($|r_{2}|=0$).

However, as one can see from Table II, the ’interference terms’ and their gate
voltage dependence may also play an important role. The reason is that tuning
the backscattering coefficients $|r_{\alpha}|$ together with $\mathcal{N}_{g}$
may result in the vanishing of the leading temperature correction. It inspires
us to consider a special case when $N_{1}=1$ (but any $N_{3}$) and
$|r_{3}|=0$, which allows us to study a quantum critical point associated with
the least irrelevant perturbation of the 2SCK model.

##  IV Quantum critical point at $N_{1}=1$ and $r_{3}=0$

Let’s consider the 2SCK circuit when there is only one channel that connects
the left QD with 2DEG, $N_{1}=1$, while all $N_{3}$ single-mode QPCs that
couple the right QD with 2DEG are fully open, $|r_{3}|=0$, see Fig. 2(a). The
linear conductance is obtained in a standard way [9], within the second-order
perturbation theory over the weak backscattering amplitudes
$|r_{1}|$,$|r_{2}|$ and also accounting for next-to-leading order correction
in $T/E_{C}$:

$\displaystyle G(T)=KG_{0}\left[1-\mathcal{C}_{+}\left(\frac{\pi
T}{E_{C}}\right)^{2K-2}-\mathcal{C}_{-}\left(\frac{\pi
T}{E_{C}}\right)^{2K}\right].$ (25)

Here, constants $\mathcal{C}_{\pm}$ read as

$\displaystyle\mathcal{C}_{+}=K\frac{\sqrt{\pi}\Gamma(K)}{2\Gamma(K+1/2)}\left(\frac{\gamma}{\pi}\right)^{2K\frac{1+N_{3}}{N_{3}}}r_{+}^{2},$
(26)
$\displaystyle\mathcal{C}_{-}=2K\pi^{2}\frac{\sqrt{\pi}\Gamma(K+1)}{2\Gamma(K+3/2)}\left(\frac{\gamma}{\pi}\right)^{2K\frac{1+N_{3}}{N_{3}}}r_{-}^{2},$
(27)

while the effective reflection coefficients are

$\displaystyle
r_{+}^{2}=|r_{1}|^{2}\mathcal{M}_{B}^{2p_{12}^{2}}\mathcal{M}_{C}^{2p_{13}^{2}}+|r_{2}|^{2}\mathcal{M}_{B}^{2p_{22}^{2}}\mathcal{M}_{C}^{2p_{23}^{2}}+2|r_{1}||r_{2}|\mathcal{M}_{B}^{p_{12}^{2}+p_{22}^{2}}\mathcal{M}_{C}^{p_{13}^{2}+p_{23}^{2}}\cos(2\pi\mathcal{N}_{g}),$
(28) $\displaystyle
r_{-}^{2}=|r_{1}|^{2}\mathcal{M}_{B}^{2p_{12}^{2}}\mathcal{M}_{C}^{2p_{13}^{2}}\left[\left(\frac{p_{12}}{\mathcal{M}_{B}}\right)^{2}+\left(\frac{p_{13}}{\mathcal{M}_{C}}\right)^{2}\right]+|r_{2}|^{2}\mathcal{M}_{B}^{2p_{22}^{2}}\mathcal{M}_{C}^{2p_{23}^{2}}\left[\left(\frac{p_{22}}{\mathcal{M}_{B}}\right)^{2}+\left(\frac{p_{23}}{\mathcal{M}_{C}}\right)^{2}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2|r_{1}||r_{2}|\mathcal{M}_{B}^{p_{12}^{2}+p_{22}^{2}}\mathcal{M}_{C}^{p_{13}^{2}+p_{23}^{2}}\left[\frac{p_{12}p_{22}}{\mathcal{M}_{B}^{2}}-\frac{p_{13}p_{23}}{\mathcal{M}^{2}_{C}}\right]\cos(2\pi\mathcal{N}_{g}).$
(29)

Explicit expressions for the elements $p_{kl}$ of matrix $\hat{P}$ are shown
in Appendix A.

The Eq. (25) is the second main result of this paper. Compared to Eq. (III),
we neglect the sub-leading term by taking $|r_{3}|=0$, which allows us to
account for the correction originating from the weak fluctuations of charge in
the left QD; see a third term in the Eq. (25). At the resonance (quantum
critical point), when $\mathcal{N}_{g}=1/2$ and reflection coefficients
$|r_{1}|$, $|r_{2}|$ tuned in such a way to nullify constant $\mathcal{C}_{+}$
($|r_{+}|=0$), this term is related to the leading irrelevant operator near
the fixed point of the (two-channel) CK problem [9]. In this case, Eq. (25)
can be treated as a universal charge Kondo scaling

$\displaystyle G(T)=KG_{0}\left[1-\left(\frac{T}{T_{K}}\right)^{2K}\right]$
(30)

with the Kondo temperature $T_{K}\sim E_{C}/|r_{-}|^{1/K}$ obtained in the
case of weak backscattering $|r_{1}|,|r_{2}|$$\ll$$1$. At $T\gg T_{K}$, we
would expect restoring of a logarithmic dependence of the conductance,
$G(T)\propto\log^{-2}(T/T_{K})$. However, it cannot be obtained for the
considered in this paper model since $T\gg T_{K}$ at $|r_{1}|,|r_{2}|\ll 1$
requires $T>E_{C}$ condition. In this case, the CK model is not applicable due
to more than just two charge states of the QD involved into consideration. In
experiments, this problem is resolved by studying the regime of tunnel
barriers, $1-|r_{\alpha}|^{2}\ll 1$ [10, 11].

The dependence of Eq. (30) and the Kondo temperature on the LL interaction
parameter $K=N_{3}/(2N_{3}+1)$ indicates the influence of the effective
electron-electron interaction on the charge Kondo correlations. Interestingly,
the Eq. (30) coincides with the temperature behavior of the conductance in the
LL-based single-site two-channel charge Kondo (2CCK) problem first studied in
Refs. [35, 36], see also [41, 44]. The similarity becomes obvious (see, e.g.,
Eq. (1) from [35] or Eq. (12) from [36]) after presenting $K$ as
$K=K_{red}/2$, where $K_{red}=2K_{1}K_{2}/(K_{1}+K_{2})$ is a reduced
Luttinger parameter of two LLs with different interaction constants, $K_{1}=1$
and $K_{2}=N_{3}/(N_{3}+1)$. Indeed, the right part of the device shown in
Fig. 2(a) – blue QD with all entered edge channels, represents by itself the
LL simulator experimentally studied in Refs. [12, 13]. Thus, we conclude that
the 2SCK device with $N_{1}=1$ and $|r_{3}|=0$ [shown in Fig. 2(a)] at the
special resonant point (when $r_{+}=0$) can be treated as the single-site 2CCK
circuit, where one Kondo channel is attributed with non-interacting electron
gas ($K_{1}=1$) and the second Kondo channel is attributed with the LL
($K_{2}=N_{3}/[N_{3}+1]$), see Fig. 2(b).

Similarly, the physics of the 2CCK problem, where each Kondo channel is
implemented by the Luttinger liquid with its own interaction parameters
$K_{1}\neq 1$ and $K_{2}\neq 1$, can be simulated in a hybrid metal-
semiconductor device consisting of three islands. In this geometry, the middle
QD is strongly coupled to the left (right) QD via single-mode QPC
characterized by weak backscattering amplitude $r_{L}$($r_{R}$). This central
island in the weak Coulomb blockade regime constitutes the Kondo impurity
pseudo-spin. Meanwhile, the left (right) QD electrically connected to 2DEG via
$N_{1}$($N_{2}$) fully open single-mode QPCs provides the LL simulator as in
Refs. [12, 13]. In this case, we expect appearance of the conductance scaling
Eq. (30) with $K=N_{1}N_{2}/(2N_{1}N_{2}+N_{1}+N_{2})$ or in terms of the
reduced Luttinger interaction parameter:
$G(T)/G_{0}=(K_{red}/2)[1-(T/T_{K})^{K_{red}}]$, where
$K_{red}=2K_{1}K_{2}/(K_{1}+K_{2})$ and $K_{\alpha}=N_{\alpha}/(N_{\alpha}+1)$
($\alpha=1,2$). Corresponding calculation and the expression analog to Eq.
(25) will be done elsewhere.

Moreover, a proposal to use a hybrid metal-semiconductor two- or three-island
nanodevice for experimental study of the interaction’s effects on the 2CCK
physics looks more promising than the original proposals in Refs. [35, 36].
The main reason is that the 2SCK circuit is already fabricated [20]. At the
same time, a single-site analog of the setup was successfully used to study a
universal conductor-insulator crossover in circuit quantum simulation of the
LL with impurity [12, 13]. Meanwhile, implementation of the model proposed in
[35] encounters the obstacle associated with the fabrication of wide enough
QPCs to observe strong enough electron-electron interaction between conducting
channels. Another proposal from [35] is naturally constrained due to the
problem of achieving a fractional quantum Hall regime with $\nu=1/3$ in an
already fabricated device [10, 11].

Note. While preparing the current manuscript, the author discovered a work
[44] addressing related questions in a similar hybrid metal-semiconductor
double-quantum dot device but with a different configuration. Study in Ref.
[44] concentrates on the conductance behavior near the quantum critical point
in the case when $N$ ballistic channels connect two islands. As the main
result, equation equivalent to Eq. (25) of the current manuscript has been
obtained. For the particular case of $N=2$, the appearance of the sub-leading
temperature dependence with $g=3/5$ has been also reported.

We also urge the readers to investigate works [23, 39, 43, 31], which are
closely related to this article and devoted to the study of the transport
properties through the 2SCK device in the case of weak inter-dot coupling.

##  V Conclusions

In this paper, we have theoretically investigated the low-temperature charge
transport properties in a multi-channel two-site charge Kondo circuit, a
device comprised of two large metallic islands (QDs) embedded into a high-
mobility two-dimensional electron gas [20]. It is revised that the leading
temperature behavior of the linear conductance obeys the conductance scaling
of a single-impurity problem in the Luttinger liquid, $KG_{0}-G(T)\propto
T^{2K-2}$, whose effective interaction parameter $K$ is fully determined by
the number of conducting channels in the system,
$K=N_{1}N_{3}/(N_{1}+N_{3}+N_{1}N_{3})$ [here, $N_{1}$($N_{3}$) is the number
of channels coupling left (right) QD to 2DEG]. We predict the appearance of
extra sub-leading temperature dependencies $\propto T^{2g-2}$ ($\propto
T^{2g^{\prime}-2}$), where
$g(g^{\prime})=[N_{1}(N_{3})-1+N_{1}N_{3}]/[N_{1}+N_{3}+N_{1}N_{3}]$ is
attributed with a finite backscattering in $N_{1}$($N_{3}$) QPCs. In the
particular case, when the left QD is electrically coupled to the source via
single-mode QPC ($N_{1}=1$) and the right QD is electrically connected to 2DEG
via $N_{3}$ fully open conducting channels, the 2SCK device can be treated as
a single-site CK device with two channels associated with the non-interacting
1DEG and with the LL, respectively. The corresponding finite temperature
correction in the vicinity of intermediate coupling 2CCK fixed point reads
$KG_{0}-G(T)\propto(T/T_{K})^{2K}$. Thus, one concludes that the 2SCK circuit
could be used as a simulator to explore the interplay between interaction
effects and multi-channel Kondo physics.

Acknowledgements. The author would like to especially thank V. M. Kovalev and
I. G. Savenko for fruitful discussions. The author also acknowledges support
from the Institute for Basic Science in Korea (IBS-R024-D1).

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##  Appendix A Diagonalization of the Hamiltonian

In this Appendix, we discuss a unitary transformation, which diagonalizes the
Hamiltonian $H_{0}+H_{C}$ and transfers bose fields
$(\phi_{1,c},\phi_{2},\phi_{3,c})^{T}$ to $(\phi_{A},\phi_{B},\phi_{C})^{T}$.
The corresponding transformation reads
$H^{\prime}_{0}+H^{\prime}_{C}=\hat{P}^{-1}(H_{0}+H_{C})\hat{P}$, where the
matrix $\hat{P}$ reads

$\displaystyle\hat{P}=\left[\begin{array}[]{ccc}p_{11}&p_{12}&p_{13}\\\
p_{21}&p_{22}&p_{23}\\\ p_{31}&p_{32}&p_{33}\end{array}\right],$ (34)

with

$\displaystyle p_{11}=\sqrt{\frac{N_{3}}{N_{1}+N_{3}+N_{1}N_{3}}}\quad,\quad
p_{12}=-\sqrt{N_{1}}\frac{-A+\Delta}{\sqrt{N_{3}+(1+A-\Delta)^{2}+N_{1}(-A+\Delta)^{2}}},$
(35) $\displaystyle
p_{13}=-\sqrt{N_{1}}\frac{-A-\Delta}{\sqrt{N_{3}+(1+A+\Delta)^{2}+N_{1}(-A-\Delta)^{2}}}\quad,\quad
p_{21}=\sqrt{\frac{N_{1}N_{3}}{N_{1}+N_{3}+N_{1}N_{3}}},$ $\displaystyle
p_{22}=-\frac{1+A-\Delta}{\sqrt{N_{3}+(1+A-\Delta)^{2}+N_{1}(-A+\Delta)^{2}}}\quad,\quad
p_{23}=-\frac{1+A+\Delta}{\sqrt{N_{3}+(1+A+\Delta)^{2}+N_{1}(-A-\Delta)^{2}}},$
$\displaystyle p_{31}=\sqrt{\frac{N_{1}}{N_{1}+N_{3}+N_{1}N_{3}}}\quad,\quad
p_{32}=\frac{\sqrt{N_{3}}}{\sqrt{N_{3}+(1+A-\Delta)^{2}+N_{1}(-A+\Delta)^{2}}},$
$\displaystyle
p_{33}=\frac{\sqrt{N_{3}}}{\sqrt{N_{3}+(1+A+\Delta)^{2}+N_{1}(-A-\Delta)^{2}}},$

where we denote $A=(N_{1}-N_{3})/2$ and $\Delta=\sqrt{1+A^{2}}$.

##  Appendix B Effective Hamiltonians

In this Appendix, we present explicitly the effective boundary sine-Gordon
models, which appear after integrating out fluctuations of fields $\phi_{B}$,
$\phi_{C}$ for particular cases of $N_{1}$,$N_{3}=\\{1,2,3\\}$.

i) case: $N_{1}=1$, $N_{3}=2$. Corresponding Luttinger interaction parameter
and effective mass parameters of fields $\phi_{B}$ and $\phi_{C}$ read
$K=2/5$, $\mathcal{M}_{B}=(5-\sqrt{5})/2$, $\mathcal{M}_{C}=(5+\sqrt{5})/2$,
respectively. An effective Hamiltonian has the following form

$\displaystyle H_{\rm eff}^{(i)}=\frac{v_{F}}{2\pi}\sum_{i=s,A}\int
dx\left\\{[\pi\Pi_{i}(x)]^{2}+[\partial_{x}\phi_{i}(x)]^{2}\right\\}+\frac{D}{\pi}\left(\frac{\gamma
E_{C}}{\pi
D}\right)^{2/5}2|r_{3}|c_{3}\cos[\sqrt{2}\phi_{s}(0)]\cos\left[\sqrt{\frac{2}{5}}\phi_{A}(0)+\pi\mathcal{N}_{g}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{D}{\pi}\left(\frac{\gamma
E_{C}}{\pi
D}\right)^{3/5}\left\\{|r_{1}|c_{1}\cos\left[2\sqrt{\frac{2}{5}}\phi_{A}(0)-2\pi\mathcal{N}_{g}\right]\right.\left.+|r_{2}|c_{2}\cos\left[2\sqrt{\frac{2}{5}}\phi_{A}(0)\right]\right\\},$
(36)

where $c_{1}\approx 1.11$, $c_{2}\approx 2.$, $c_{3}\approx 1.37$.

ii) case: $N_{1}=N_{3}=2$. Corresponding Luttinger interaction parameter and
effective mass parameters of fields $\phi_{B}$ and $\phi_{C}$ read $K=1/2$,
$\mathcal{M}_{B}=2$, $\mathcal{M}_{C}=4$, respectively. An effective
Hamiltonian has the following form

$\displaystyle H_{\rm eff}^{(ii)}=\frac{v_{F}}{2\pi}\sum_{i=s,A,\tilde{s}}\int
dx\left\\{[\pi\Pi_{i}(x)]^{2}+[\partial_{x}\phi_{i}(x)]^{2}\right\\}+\frac{D}{\pi}\left(\frac{\gamma
E_{C}}{\pi D}\right)^{1/2}|r_{2}|c_{2}\cos[\sqrt{2}\phi_{A}(0)]$ (37)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{2D}{\pi}\left(\frac{\gamma
E_{C}}{\pi
D}\right)^{3/8}c_{1}\left\\{|r_{1}|\cos[\sqrt{2}\phi_{s}(0)]\cos\left[\frac{1}{\sqrt{2}}\phi_{A}(0)-\pi\mathcal{N}_{g}\right]+|r_{3}|\cos[\sqrt{2}\phi_{\tilde{s}}(0)]\cos\left[\frac{1}{\sqrt{2}}\phi_{A}(0)+\pi\mathcal{N}_{g}\right]\right\\},$

where $c_{1}\approx\sqrt{2}$, $c_{2}=2$.

Interestingly, the above model is equivalent to the single-impurity problem in
the Luttinger liquid with interaction parameter $K=1/2$ in case
$|r_{1}|=|r_{3}|=0$. As it is well-known, the following model allows for an
exact solution due to possibility of refermionization [49, 56, 48]. Thus, the
linear conductance can be exactly obtained from Eq. (37) in case
$|r_{1}|=|r_{3}|=0$, see [58].

iii) case: $N_{1}=1$, $N_{3}=3$. Corresponding Luttinger interaction parameter
and effective mass parameters of fields $\phi_{B}$ and $\phi_{C}$ read
$K=3/7$, $\mathcal{M}_{B}=3-\sqrt{2}$, $\mathcal{M}_{C}=3+\sqrt{2}$,
respectively. An effective Hamiltonian has the following form

$\displaystyle H_{\rm eff}^{(iii)}=\frac{v_{F}}{2\pi}\sum_{i=s,f,A}\int
dx\left\\{[\pi\Pi_{i}(x)]^{2}+[\partial_{x}\phi_{i}(x)]^{2}\right\\}$ (38)
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{D|r_{1}|c_{1}}{\pi}\left(\frac{\gamma
E_{C}}{\pi
D}\right)^{4/7}\cos\left[\frac{2\sqrt{3}}{\sqrt{7}}\phi_{A}(0)-2\pi\mathcal{N}_{g}\right]+\frac{D|r_{2}|c_{2}}{\pi}\left(\frac{\gamma
E_{C}}{\pi D}\right)^{4/7}\cos\left[2\sqrt{\frac{3}{7}}\phi_{A}(0)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{D|r_{3}|c_{3}}{\pi}\left(\frac{\gamma
E_{C}}{\pi
D}\right)^{\frac{2}{7}}\left\\{\cos\left[\frac{2\phi_{A}(0)}{\sqrt{21}}-\frac{2\sqrt{2}}{\sqrt{3}}\phi_{f}(0)-\frac{2\pi\mathcal{N}_{g}}{3}\right]\right.\left.+2\cos\left[\frac{2}{\sqrt{21}}\phi_{A}(0)+\frac{\sqrt{2}}{\sqrt{3}}\phi_{f}(0)-\frac{2\pi\mathcal{N}_{g}}{3}\right]\cos\sqrt{2}\phi_{s}(0)\right\\},$

where $c_{1}\approx 1.34$, $c_{2}\approx 1.93$, $c_{3}\approx 1.39$.

iv) case: $N_{1}=2$, $N_{3}=3$. Corresponding Luttinger interaction parameter
and effective mass parameters of fields $\phi_{B}$ and $\phi_{C}$ read
$K=6/11$, $\mathcal{M}_{B}=(7-\sqrt{5})/2$, $\mathcal{M}_{C}=(7+\sqrt{5})/2$,
respectively. An effective Hamiltonian has the following form

$\displaystyle H^{\rm(iv)}_{\rm
eff}=\frac{v_{F}}{2\pi}\sum_{i=s,\tilde{s},\tilde{f},A}\int
dx\left\\{[\pi\Pi_{i}(x)]^{2}+[\partial_{x}\phi_{i}(x)]^{2}\right\\}$ (39)
$\displaystyle+\frac{D|r_{2}|c_{2}}{\pi}\left(\frac{\gamma E_{C}}{\pi
D}\right)^{\frac{5}{11}}\cos\left[2\sqrt{\frac{6}{11}}\phi_{A}(0)\right]+\frac{2D|r_{1}|c_{1}}{\pi}\left(\frac{\gamma
E_{C}}{\pi
D}\right)^{\frac{4}{11}}\cos\sqrt{2}\phi_{s}(0)\cos\left[\sqrt{\frac{6}{11}}\phi_{A}(0)-\pi\mathcal{N}_{g}\right]$
$\displaystyle+\frac{D|r_{3}|c_{3}}{\pi}\left(\frac{\gamma E_{C}}{\pi
D}\right)^{\frac{3}{11}}\left\\{\cos\left[\sqrt{\frac{8}{33}}\phi_{A}(0)-\frac{2\sqrt{2}}{\sqrt{3}}\phi_{\tilde{f}}(0)+\frac{2\pi\mathcal{N}_{g}}{3}\right]+2\cos\left[\sqrt{\frac{8}{33}}\phi_{A}(0)+\sqrt{\frac{2}{3}}\phi_{\tilde{f}}(0)+\frac{2\pi\mathcal{N}_{g}}{3}\right]\cos\sqrt{2}\phi_{\tilde{s}}(0)\right\\},$

where $c_{3}\approx 1.42$, $c_{2}\approx 1.94$, $c_{1}\approx 1.4$.

v) case: $N_{1}=3$, $N_{3}=3$. Corresponding Luttinger interaction parameter
and effective mass parameters of fields $\phi_{B}$ and $\phi_{C}$ read
$K=6/11$, $\mathcal{M}_{B}=3$, $\mathcal{M}_{C}=5$, respectively. An effective
Hamiltonian has the following form

$\displaystyle H^{\rm(v)}_{\rm
eff}=\frac{v_{F}}{2\pi}\sum_{i=s,\tilde{s},A,f,\tilde{f}}\int
dx\left\\{[\pi\Pi_{i}(x)]^{2}+[\partial_{x}\phi_{i}(x)]^{2}\right\\}+\frac{D|r_{2}|c_{2}}{\pi}\left(\frac{\gamma
E_{C}}{\pi D}\right)^{2/5}\cos\left[2\sqrt{\frac{3}{5}}\phi_{A}(0)\right]$
(40) $\displaystyle+\frac{D|r_{1}|c_{1}}{\pi}\left(\frac{\gamma E_{C}}{\pi
D}\right)^{\frac{4}{15}}\left\\{\cos\left[\frac{2\phi_{A}(0)}{\sqrt{15}}-\frac{2\sqrt{2}}{\sqrt{3}}\phi_{f}(0)-\frac{2\pi\mathcal{N}_{g}}{3}\right]+2\cos\left[\frac{2\phi_{A}(0)}{\sqrt{15}}+\frac{\sqrt{2}}{\sqrt{3}}\phi_{f}(0)-\frac{2\pi\mathcal{N}_{g}}{3}\right]\cos\sqrt{2}\phi_{s}(0)\right\\}$
$\displaystyle+\frac{D|r_{3}|c_{3}}{\pi}\left(\frac{\gamma E_{C}}{\pi
D}\right)^{\frac{4}{15}}\left\\{\cos\left[\frac{2\phi_{A}(0)}{\sqrt{15}}-\frac{2\sqrt{2}}{\sqrt{3}}\phi_{\tilde{f}}(0)+\frac{2\pi\mathcal{N}_{g}}{3}\right]+2\cos\left[\frac{2\phi_{A}(0)}{\sqrt{15}}+\frac{\sqrt{2}}{\sqrt{3}}\phi_{\tilde{f}}(0)-\frac{2\pi\mathcal{N}_{g}}{3}\right]\cos\sqrt{2}\phi_{\tilde{s}}(0)\right\\},$

where $c_{2}\approx 1.9$, $c_{1}=c_{3}\approx 1.41$.


# Evidence of thermodynamics and magnetic monopole plasma formation by photon-
magnon interaction in artificial spin ice

D. G. Duarte  Laboratory of Spintronics and Nanomagnetism (LabSpiN),
Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG, Brazil  S.
F. de Souza  Laboratory of Spintronics and Nanomagnetism (LabSpiN),
Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG, Brazil  L.
B. de Oliveira  Laboratory of Spintronics and Nanomagnetism (LabSpiN),
Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG, Brazil  E.
B. M. Junior  Laboratory of Spintronics and Nanomagnetism (LabSpiN),
Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG, Brazil  E.
N. D. de Araujo  Laboratory of Spintronics and Nanomagnetism (LabSpiN),
Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG, Brazil  J.
M. Fonseca  Laboratory of Spintronics and Nanomagnetism (LabSpiN),
Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG, Brazil  C.
I. L. de Araujo<EMAIL_ADDRESS>Laboratory of Spintronics and Nanomagnetism
(LabSpiN), Departamento de Física, Universidade Federal de Viçosa, Viçosa-MG,
Brazil

###### Abstract

Artificial spin ices (ASI), containing magnetic monopole quasi-particles
emerging at room temperature, have been investigated as a promising system to
be applied in alternative low-power information technology devices. However,
restrictions associated with the intrinsic energetic connections between
opposing magnetic monopoles in conventional ASI need to be overcome to achieve
this purpose. Here, photon-magnon scattering in nanomagnets is examined as an
approach to locally activate the collective dynamics of interacting magnetic
systems at the nanoscale. Low-power white and polarized light were employed as
a new tool to manipulate magnetic monopole intensity, leading to tuning on the
particles response to external magnetic field and spontaneous magnetization
flipping without external field (thermodynamics). Our findings showing
evidence of magnetic monopole plasma formation in a regular square ASI system
are explained by an analytical model of photon-magnon conversion acting
directly on the ASI nanomagnet dipole. Micromagnetic simulations based on the
samples parameters and values obtained from the model present a very good
qualitative correspondence between theory and observations for the
investigated ASI system.

##  I Introduction

Artificial Spin Ice ($ASI$) systems, consisting of nanomagnet arrays
nanofabricated in planar geometries in order to present magnetic frustration
[1], have been extensively investigated in the last decade. Such
nanofabricated network have the potentiality to mimic, at room temperature [2,
3], the properties previously observed only at very low temperatures in
natural spin ice Pyrochlore crystals, such as emergency of magnetic monopoles
[4] and thermodynamic phase transitions [5].

In the most common square $ASI$ system, the four nanomagnets at each vertex
can retain magnetic configurations with different energies [1], as shown in
Figure 1a. By describing the magnetic dipole of each nanomagnet as a magnetic
charge dumbbell, it is possible to see that the least energetic configuration
T1 has no residual magnetic charge and no magnetic dipole, whereas
configuration T2 has no residual magnetic charge but a magnetic dipole at the
vertex. Configurations T3 and T4 are the most interesting because they carry
residual magnetic charge resembling Nambu magnetic monopoles. Such sort of
monopoles have the characteristics of always been attached to an opposite
charge by an energetic string composed by T2 configurations to keep the system
charge balance [6] (Figure 1b).

When it comes to the goal of magnetic monopole information transport in a
lower dissipation system, compared to conventional electronics, such an active
string may provide a challenge [7]. To circumvent this constraint, geometries
presenting degeneracy with system vacuum composed by both configurations T1
and T2 have been investigated. These systems allow higher magnetic monopole
mobility [8, 9, 10, 11], because they present a non-energetic string
connecting magnetic monopoles [12, 13, 14, 15, 16], resembling what has been
called Dirac monopoles in natural crystals [17]. Some of the major issues that
still need to be resolved are the difficulty of building a system ground state
manifold with zero magnetic charges at vertices and a controlled method of
selectively regulating magnetic charge mobility.

Figure 1: Magnetic monopole mobility control by photon-magnon conversion a,
vertex configurations possible in square $ASI$, with ground state in T1
holding zero residual charge, followed energetically by T2 with emergent
dipole, T3 configuration presenting both emergent dipole and magnetic charge
$q=\pm 2$ and T4 with magnetic charge $q=\pm 4$. b, example of excitation’s
over the vacuum (ground state), with opposite charges T3 connected by an
energetic string composed by T2 configurations and c cartoon summering our
findings, with a circularly polarized laser generating magnons in the
nanoislands, consequently decreasing its dipole moment and magnetic monopole
strength.

Recent research has looked into the use of nanomagnets thin enough to be
almost superparamagnetic, which allows thermal activation close to room
temperature in the context of ground state manifold accomplishment [18]. But
the time frame reported to reach the lowest energy state up to this moment is
too long to be useful in practical device applications.

Furthermore, effective methods for manipulating magnetic monopole mobility are
currently missing. Recently, we have suggested a method with a reasonable
mobility change that involves monopole screening by free electrons on metals
[19]. That approach is limited by the permanent charge carriers on metal thin
films, resulting in non-reversibility. Regulated methods for quick reversible
monopole mobility adjustment in ASI would represent a significant next step in
the development of magnetronics. A recent study [20] revealed an intriguing
method of plasmonic nanomagnets heating by using both constant and pulsed high
laser power of 60mW. They showed a reasonable heating and consecutive
nanomagnet magnetization change in the timescale of $\mu$s and $ns$ for
constant and pulsed light successively.

In this paper, we will provide a novel approach for tuning magnetic properties
in ASI systems using considerably lower laser power, bypassing the nanomagnet
heating process. Using an analytical model, we propose that low energetic
photon incidence in traditional ASI nanomagnets can produce magnons, which
will alter the nanomagnets dipolar energy and hence decrease the monopole
strength at the vertex, as illustrated in the cartoon of Figure 1c. We
subsequently demonstrate experimentally, using magnetic force microscopy (MFM)
a non-optical magnetic measurement technique, that such an effect can
reasonably change the evolution of vertex configurations and system hysteresis
as a function of the external magnetic field. Micromagnetic simulations based
on materials parameters and estimations from the analytical model were used to
support our findings.

Despite of provide a tool for magnetic monopole mobility modification and
plasma formation, as it will be further demonstrated, such observed behavior
should be considered in general ASI investigations using optical magnetic
characterizations, like photoemission electron microscopy combined with
magnetic dichroism (PEEM-XMCD) [21, 22] and magnetic optic Kerr effect (MOKE)
[23, 24], once we show here that light can affect the ASI properties. That
could be the reason why predicted ground states were not experimentally fully
demonstrated in some ASI systems investigations [25, 14].

##  II Magnon-photon interaction model

Here, we develop our theoretical model to study photon scattering anomalies in
the magnetization of Permalloy nanomagnets. Since we are working with a
magnetized medium, prior photon scattering in magnon models is taken into
consideration when deriving our Hamiltonian. [26, 27, 28]:

$\hat{H}=-g\left(\hat{b}^{\dagger}_{q}+\hat{b}_{q}\right)\hat{a}^{\dagger}_{\omega}\hat{a}_{\omega^{{}^{\prime}}}$
(1)

where $g$ is the optomechanical coupling constant,
$q=\omega-\omega^{{}^{\prime}}$ are magnon and photon frequencies respectively
and $\hat{b}$ and $\hat{a}$ are respectively magnon and photon bosonic
operators.

Assuming that the photon scatterings causes small disturbances
($\delta\hat{S}<<1$) in the magnetization
$\hat{S}=\left(\hat{S}^{x},\hat{S}^{y},\hat{S}^{z}\right)$, we can apply the
Holstein-Primakoff transformation and obtain
$\hat{S}^{x}=\sqrt{S/2}\left(\hat{b}^{\dagger}_{q}+\hat{b}_{q}\right)$, with
$S=|\hat{S}|$. Thus, eq (1) becomes:

$\hat{H}=-G\hat{S}^{x}\hat{a}^{\dagger}_{\omega}\hat{a}_{\omega^{{}^{\prime}}}$
(2)

where the optomagnonic coupling constant $G=g\sqrt{2/S}$ is defined [27].
Thus, photons scattering into magnons can cause perturbations in the spin;
these perturbations are minor deviations in the projection of $\hat{S}$ in the
plane $xy$ in the limit $\delta\hat{S}<<1$. Since $G$ depends on the
magnetization, it is important to note that the coupling shown in eq.(2) only
occurs in a medium whose magnetization is non-zero.

Considering an isotropic, non-dissipative medium with a linear magnetization
response M [29], the permittivity tensor in this case is
$\varepsilon_{i,j}\left(\textbf{M}\right)=\varepsilon_{0}\left(\varepsilon\delta_{i,j}-if\sum_{k}\epsilon_{ijk}M_{k}\right)$,
where $\varepsilon_{0}$ ($\varepsilon$) is the vacuum (relative) permittivity,
$f$ is a material-dependent constant and $\epsilon_{ijk}$ is the Levi-Civita
tensor. Using the complex representation of the electric field
$\textbf{E}=(\textbf{E}^{*}+\textbf{E})/2$, the average energy is [27]:

$\Phi=-\dfrac{if\varepsilon_{0}}{4}\int_{V}\textbf{M({r})}\cdot\left[\textbf{E({r})}^{*}\times\textbf{E({r})}\right]d\textbf{r}.$
(3)

Light rotates in its polarization when it travels through a magnetic medium;
this rotation is related to the permittivity tensor and is determined by the
Faraday angle $\theta_{F}=\omega fM_{S}/2c\sqrt{\varepsilon}$, where $c$
denotes the speed of light, $\omega$ denotes the frequency of light, and
$M_{S}$ denotes the saturation magnetization. Conversely, a local effective
magnetic field can be produced via the permittivity tensor [29, 30]:

$\textbf{B}_{eff}(\textbf{r},t)=-\dfrac{if\varepsilon_{0}}{4}\int_{V}\textbf{E}^{*}(\textbf{r},t)\times\textbf{E}(\textbf{r},t)d\textbf{r}.$
(4)

This field has a similar impact to spin waves, or magnons. As spin waves are
closely related to the light’s electric field, we may proceed further by
quantize the electric field $\hat{E}=E_{\beta}\hat{a}_{\beta}$, where
$E_{\beta}$ denotes the $\beta$ eigenmode of the electric field. A change in
the orientation of local magnetization may be caused by the field $B_{eff}$;
this phenomenon is known as the Inverse Faraday Effect (IEF)[29].

On the other hand, the Holtein-Primakoff representation can be applied if we
take into account tiny spin variations. This allows us to directly derive
eq(1) from eq.(3) and, in turn, eq.(2), where the optomagnetic coupling is
expressed as follows:

$G^{j}_{\beta\gamma}=-i\dfrac{\varepsilon_{0}f}{4\hbar}\sum_{jmn}\int_{V}\dfrac{M_{j}(\textbf{r})}{S_{j}(\textbf{r})}E^{*}_{\beta
m}E_{\gamma n}d\textbf{r}.$ (5)

Specifically, we may describe $G$ in terms of the saturation magnetization and
the macro spin norm $M_{j}/M_{S}=S_{j}/S$ if we assume that the modes are
homogenous (Kittel modes) [27] (this is conceivable if the spins are in
resonance). The macro spin of the nano-island is represented by the operator
$\hat{S}$. $G=c\theta_{F}\lambda/4S\sqrt{\varepsilon}$, which is the result of
decomposing the electric field’s eigenmodes on a circular basis [27]. For flat
waves, lambda is a factor that is close to $1$ [31] and $G\backsim 1Hz$ and
$S\backsim 10^{10}$ are found in materials like Permalloy [31]. The
relationship between the total magnon frequency and photon count is given by
the constant $G$ [27]. Based on this, we can calculate that the maximum magnon
density per volume for low power light is around $10^{17}m^{-3}$. A magnon’s
average wavelength is $1\mu m$. The spin wave magnetic field created by this
low power light has an amplitude of around $B_{eff}\backsim 10mT$, according
to equation 2.

##  III Response to magnetic field in function of white and circularly
polarized light

The studied samples are regular square ASI consisting of Permalloy nanomagnets
developed on top of a silicon substrate (see methods VII.1). We validated the
reproducibility of our results by performing three sets of measurements in
separate areas of $50\mu m\times 50\mu m$ for the two specified samples. We
will concentrate our analysis on the results from the q04 sample. Equivalent
results for sample q08 are presented in Supplementary materials.

Figure 2: Magnetization reversal and monopole evolution charaterized by MFM
measurements a, serie of MFM images obtained during a reversal process
performed by external magnetic field applied in the horizontal direction b,
magnetization reversal in function of white light intensity obtained in
remanent state after successive external magnetic field application c, change
in the magnetic monopoles evolution under different white light power and d,
comparison between histeretic behavior for the sample on dark and under white
and circularly polarized light.

Figure 2 shows the characterizations done by magnetic force microscopy (MFM)
at room temperature, with the sample in the remanence state after each stage
of external magnetic field application. Because the nanomagnet size used here
are appropriate for devices exhibiting athermal behavior, the system will
maintain its magnetic configuration if not subjected to any external
excitation. First, the samples were exposed to an external magnetic field for
magnetization saturation in the diagonal direction (top-right to bottom-left).
The saturation is confirmed in the first image of blow up MFM images with
$20\mu m\times 20\mu m$ area given in Figure 2a.

The stray fields generated by the in plane nanomagnet magnetization results in
brilliant and dark patches coming from positive and negative out-of-plane
magnetic force on the MFM tip. The other four frames were removed from images
obtained during the magnetization reversal process. They were performed with
steps of external magnetic field applied in the x-axis direction, in the
opposite sense to the initial saturated magnetization. That approach was
utilized to study the behavior of individual nanomagnet magnetization flips in
relation to the external magnetic field and system internal magnetization,
which is primarily determined by geometrical frustration. Two main
characteristics are observed in function of external magnetic field: the
system’s hysterical behavior obtained through the sum of nanomagnet
magnetization and magnetic monopoles creation and annihilation obtained by
vertex configuration analysis, following the same procedure utilized in our
recent works [32, 33, 19].

We then executed the above-mentioned characterisation in larger areas of
$50\mu\times 50\mu$ from dark to a sequence of white light with varying power.
The light source was set at a constant distance enough to prevent sample
heating during the operation, with constant temperature been measured by a
commercial thermocouple. In order to validate the ineffectiveness of light
heating in the ASI magnetization we have saturated the nanomagnets
magnetization in the sample diagonal, with vertex in the energetic
configuration T2, and left it under light exposition for 72 hours. After that
period, MFM measurements showed that the imposed saturation state was still
there, meaning that the light power was not enough to impose heat and
consequently thermodynamics to the system.

Figure 2b shows the hysteresis curves obtained by counting each nanomagnet
magnetization in x-axys for each magnetic field step applied in the same
direction. We illustrate the evolution of nanomagnet magnetization in function
of external field under varied white light power. Figure 2c shows the
evolution of magnetic vertex percentage presenting monopole configuration
under same conditions. The presented results show a clear influence of
depolarized white light power on the magnetic properties of the sample, with a
noticeable decrease in the field required to initiate the evolution of
magnetic monopoles and a decrease in vertex population density as the light
intensity increases.

To study the effect of light polarization, we reproduced the previous
experiment, but this time using 100W of white light with circular
polarization. Figure 2d compares the measurements performed in the dark, white
light with 50W of power and circular polarized light with source of 100W,
which is cut in half by the polarizer as predicted by the law of Malus. The
large decrease in coercive field observed in the polarized light is compatible
with the theoretical prediction presented above for scattering enhancement
between photons and magnons. We are going to show latter, by micromagnetic
simulations, the influence of generated magnons in the hysteresis process. A
lower effect was also studied using light under linear polarization, and a
strong direction dependence between polarization and nanomagnet magnetism was
found (not shown).

##  IV Thermodynamics produced by focussed polarized light

As the last stage of the investigation, we will look at possible thermodynamic
effects with nanomagnetic magnetization flipping directed solely by light
excitation and characteristic ASI internal frustration. In this case, we used
a micro Raman spectroscopic setup to excite individual nanomagnets without
applying a magnetic field.

In Figure 3a, we used an argon laser with a wavelength of $632.8nm$ and a beam
diameter of $1.03\mu m$ through a $50\times$ objective lens with a numerical
aperture of $0.75$. The system’s maximum laser power is $2.17mW$, but we are
interested in low power regime, so we have utilized for circular polarization
both $0.5\%$ and $5\%$ of the utmost capacity, namely $10.85\mu W$ and
$0.108mW$, with each nanomagnet in a chosen row, marked in Figure 3a with a
red dotted line, subjected to laser exposure for approximately $1$ second.

Figure 3: ASI Thermodynamics under linear and circular polarized laser
exposition a, Optical image showing the laser spot with average size similar
to the nanomagnet width b, MFM measurements showing vertex configuration after
low power (upper panel) and moderate power (lower panel) for laser exposition
with linear (left pannels) and circular (right pannels) polarization. The
black dotted lines mark the row of nanomagnets exposed to the laser for 1ns
each. c, summary of vertex changing after the processes showing thermodynamics
effect with much more effectiveness for the circular polarization case on the
right panel.

Figure 3b depicts the MFM pictures after laser exposition, as indicated by the
red dotted line. The images on the left were acquired after saturated samples
exposed to a linearly polarized laser with $10.85\mu W$ (top) and $0.108mW$
(bottom), whereas the right panel shows the identical condition with a
circular polarized laser. It is easy to see that some vertices altered
following low-power laser exposure. At $10.85\mu W$, higher energy T3 vertices
(green square) were seen in both linear and circular polarization. At
$0.108mW$, even higher energetic configurations T4 (blue squares) were
achieved in both polarizations.

Figure 3c shows the evolution of vertex configurations based on laser power
for both linear (top) and circular (bottom) polarizations. In Figure 3c, it is
possible to observe not only the formation of high energetic vertex
configurations by moderate laser power, but also the evolution of the vertex
density statistics to the relation $T1/T2\backsim 1/2$, which is expected for
degenerated systems suggesting the appearance of Gauge fields with magnetic
monopole plasma formation [34]. This observation is completely compatible with
the underlying theoretical framework described here, which states that
increasing power by a factor of ten results in a directly corresponding
increase in photon and magnon conversion densities. Magnon creation reduces
nanomagnet dipolar interaction, modifying the relation between $J_{1}$ and
$J_{2}$ in a similar way as it was done before in literature [8] using an
external magnetic field.

The tests given in this chapter were rigorously performed three times in
different sections of the sample, resulting in data with an error margin of
less than 2%. The optical microscope with focused laser setup allowed for the
utilization of light to switch nanoislands due to the laser’s increased photon
density compared to white light, which lacks coherence in earlier research.
The experimental data suggest that at low power non-thermal excitation’s
mediated by photon assisted magnon generation are responsible for the magnetic
switching.

##  V Micromagnetic simulations

To investigate the influence of magnons generated by photon interaction on
nanomagnet magnetization and, as a result, observe the effect of such
modification in a $ASI$ system, we performed micro-magnetic simulations using
our system architecture and materials parameters, as well as the created
magnon intensity recovered from the presented analytical model. The simulation
was conducted using an open-access GPU-based micro-magnetic simulation
software (Mumax³) [35].

We first investigate the resonance frequency of a single Permalloy nanoisland
used in our studies, which has dimensions of $(3000\times 400\times 20)nm$. To
do this, we first saturated the magnetization along the nanomagnet’s x-axis
and then applied a sinusoidal signal in the form $H_{sinc}=\sin{(x)}/x$
perpendicular to the nanomagnet’s (z-axis) for $12ns$. Using the described
technique, we were able to find the nanomagnet magnon resonance frequency of
6.7 Ghz, as presented in Figure 4a. This value is in good agreement with
values previously presented in literature [36, 37] and will be used for the
simulation of magnon generated on the ASI system.

Figure 4: Simulation of magnon influence on nanomagnet magnetization reversal
a, Snapshot of magnetization oscillation, with blue and red colors
representing out-of-plane component, under b ac external magnetic field in z
direction. In c it is possible to follow the behavior of magnetization in the
nanomagnet for x,y and z directions. d The frequency of external field
excitation for magnon generation was obtained by a protocol of sweep in
frequency with magnon modes represented by the peaks. e hysteresis sweep with
static field applied in x-axys with and without ac external excitation,
representing the magnon influence. It is possible to see that the ac
excitation decreases the single nanomagnet coercivity. Same protocol was
extended for f, a portion of an artificial spin ice system under hysteresis
sweep with same ac external excitation where the colors represent the in-plane
magnetization. g simulated hysteresis curve on ASI system presenting similar
behavior to the experiments.

As mentioned in section II, magnons resulting from photon scattering have an
associated effective field on the order of $~{}10mT$. To simulate the effect
of magnons on nanomagnet magnetization reversal, we applied an AC external
field with the analytically predicted intensity of $~{}10mT$ and simulated
frequency of 6.7 Ghz on the z-axis direction, while performing a hysteresis
cycle with a DC field on the x-axys along the nanomagnet length. Figure 4b
demonstrates that nanomagnet coercivity decreases significantly in the
presence of magnons.

We next applied the same procedure to a tiny ASI lattice with a similar
topology as in our experiments. Figure 4c shows declines in coercivity and
magnetization saturation under external field reversal and magnon presence,
which are qualitatively similar to the findings given in Figure 4b.

##  VI Conclusions

This work describes an analysis of the shine light process in an artificial
spin ice system made of normal athermal Permalloy nanomagnets. We observe
changes in the ASI histeretical behavior, as well as the evolution of magnetic
monopoles, by characterizing magnetization reversal in function of external
field under both white and circularly polarized light.

In order to examine possible thermodynamics under light exposure, white light
was not enough to provide nanomagnet magnetization flips after 72 hours even
in an energetic saturated state. We then exposed each nanomagnet to a low
power circularly polarized laser for 1 second, after complete magnetization
saturation in the diagonal direction, and observed numerous nanomagnets flip,
demonstrating ASI thermodynamics led by light. As a best result, at $0.1mW$,
the new vertex configurations obtained were close to the ratio of the
degenerate system, indicating the formation of magnetic monopole plasma under
light exposure.

We explain our findings of nanomagnets magnetization switching under the
photon excitation using an analytical model that describes magnon creation on
nanomagnets via photon conversion. Using micromagnetic simulations, we were
able to determine nanomagnet magnon frequency. Finally, utilizing the
analytical model’s magnon intensity, we simulate the influence of such magnons
in the ASI system. The good qualitative match between experimental and
theoretical hysteresis curves indicates that our model is successful in
explaining the observed shift in monopole mobility due to the action of
moderate power light. The same change in the nanomagnet dipole caused by a
focussed moderate circularly polarized laser beam may be accountable for the
ASI thermodynamics imprinted by light.

###### Acknowledgements.

The authors thank the Brazilian agencies CNPq 432029/2018-1, FAPEMIG and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)- Finance
Code 001 - for the financial support.

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##  VII Methods

###  VII.1 Sample fabrication

To prepare the sample, a trilayer of Ta $3nm$ / Py $20nm$ / Ta $3nm$ was
sputtered on top of a p-type silicon substrate, with Ta acting as both an
adhesion and capping layer. ASI samples in square geometry were manufactured
with a total lattice size of $100\mu m\times 100\mu m$. The samples q04 and
q08 were obtained with different lattice parameters ”$a$,” which were $800nm$
and $1200nm$, respectively. The nanomagnet size of $3000nm\times 400nm\times
20nm$ was carefully created to obtain increased magnetic sensitivity in the
Magnetic Force Microscope ($MFM$) up to the limit while ensuring magnetic
monodomain in each nanomagnet.

###  VII.2 Magnetic force measurements

The experimental procedure involved initially saturating the nanomagnets’
magnetization in the positive $x$-axis direction, followed by stages of
magnetic field up to saturation in the opposite direction (negative $x$-axis),
without the presence of a light source. The method was repeated for several
levels of white light power: $25\%$, $50\%$, $75\%$, and $100\%$. MFM pictures
were consistently captured between each field application in a sample area of
$50\mu m\times 50\mu m$. To improve experimental dependability, each sample’s
saturation process was performed three times in various places, with an error
margin of 3%.

###  VII.3 Micromagnetic simulations

Micromagnetic simulation in the square lattices with nine cells was performed
with the open-source GPU-based software $MUMAX^{3}$ [35]. The Permalloy
numerical parameters utilized in the simulation were magnetic saturation
$M_{\text{S}}=860\times 10^{3}$ A m-1, polarization $P=0.5$, exchange constant
$A_{\text{ex}}=13\times 10^{-12}$ J m-1 and Gilbert damping $\alpha=0.01$.
Finite difference discretization used for the iterations were based on the
Landau–Lifshitz–Gilbert (LLG) equation (Equation (6) with cubic cell of $5$ nm
$\times$ $5$ nm $\times$ $5$ nm).

$\displaystyle\frac{\partial\bm{M}}{\partial
t}=\gamma\bm{H}_{\text{eff}}\times\bm{M}+\frac{\alpha}{M_{\text{S}}}\bm{M}\times\frac{\partial\bm{M}}{\partial
t}-u\frac{\partial\bm{M}}{\partial y}$
$\displaystyle+\frac{\beta}{M_{\text{S}}}\bm{M}\times\frac{\partial\bm{M}}{\partial
y}$ (6)

We investigated magnon resonance in a single nanomagnet using the MUMAXˆ3
catalog technique (vansteenkiste, 2011). We then used our analytical model’s
intensity and the frequency of the major peak to apply an AC magnetic field
out-of-plane in the nanomagnet and ASI system, while completing a hysteretical
cycle with a DC field in plane.

##  VIII Extended Material

Here we show the results obtained in the second sample denominated q08. It is
possible to see that both behavior under white light are similar to the one
presented for sample q04 in Figure 2.

Figure 5: Magnetization reversal and monopole evolution charaterized by MFM
measurements a change in the magnetic monopoles evolution under different
white light power, b slope zoom view of magnetization reversal in function of
white light intensity obtained in remanent state after successive external
magnetic field application and c parameters of the sample q08.