††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Chiral excitation flows of multinode network based on synthetic gauge fields

Fo-Hong Wang  School of Physics, Sun Yat-sen University, Guangzhou 510275,
China  School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai, 200240, China  Xian-Liang Lu  School of Physics, Sun Yat-sen
University, Guangzhou 510275, China  Jia-Jin Zou  School of Physics, Sun Yat-
sen University, Guangzhou 510275, China  Ze-Liang Xiang
<EMAIL_ADDRESS>School of Physics, Sun Yat-sen University,
Guangzhou 510275, China

###### Abstract

Chiral excitation flows have drawn a lot of attention for their unique
unidirectionality. Such flows have been studied in three-node networks with
synthetic gauge fields (SGFs), while they are barely realized as the number of
nodes increases. In this work, we propose a scheme to achieve chiral flows in
$n$-node networks, where an auxiliary node is introduced to govern the system.
This auxiliary node is coupled to all the network nodes, forming sub-triangle
structures with interference paths in these networks. We find the implicit
chiral symmetry behind the perfect chiral flow and propose the universal
criteria that incorporate previous models, facilitating the implementation of
chiral transmission in various networks. By investigating the symmetries
within these models, we present different features of the chiral flow in
bosonic and spin networks. Furthermore, we extend the four-node model into a
ladder network, which is promising for remote state transfer in practical
systems with less complexity. Our scheme can be realized in state-of-the-art
experimental systems, such as superconducting circuits and magnetic photonic
lattices, thereby opening up new possibilities for future quantum networks.

##  I Introduction

The chiral excitation flow is a distinguished form of population transfer
observed in quantum systems, exhibiting a unidirectional circulation
characterized by chirality. A notable example of continuous chiral flows is
dissipation-free edge currents in the quantum Hall effect [1, 2, 3], which was
first observed under a perpendicular magnetic field [4, 5] and later related
to non-trivial topological effects [6]. With the advancement of quantum
technologies, edge currents can also be generated with synthetic gauge fields
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] in photonic systems [18, 19] or
other artificial systems [20, 21]. On the other hand, a discrete version of
chiral excitation flows can also occur in artificial few-body systems
characterized by broken time-reversal symmetry, parity symmetry, or both [22,
23, 24]. In addition, they can serve as fundamental building blocks for
creating larger artificial lattice models to study quantum Hall effects, such
as the integer [18] and fractional quantum Hall states [22].

The quantum network connects various quantum systems via proven-secure
channels and supports quantum state transfer between separated nodes, which is
potentially one of the most wide-ranging applications of current quantum
technologies [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. In
practice, the nodes designed for storing and processing quantum information in
quantum networks can be realized with superconducting qubits [38, 39, 40, 41,
42], atomic systems [43, 26, 44, 45], quantum dots [46, 47], or spin vacancies
[48, 49, 50, 51]. Between these nodes, quantum channels play the role of
bridges to coherently transfer information. In this framework, a chiral flow
can provide site-by-site chiral transmission and enable unidirectional
transfer of quantum states or entanglement in large-scale quantum networks.
The chiral flow in a few-node network has been theoretically studied in
various quantum systems, such as superconducting circuits, photonic lattices
[52], Rydberg-atom arrays [53], giant atoms [54], and cavity-magnonic systems
[55], and even experimentally achieved in superconducting circuits [22, 23,
24]. However, these works demonstrate perfect chiral flows only in a
triangular three-node network and fail to work in larger networks. Whether
perfect chiral excitation flows exist in multinode networks is still an open
question, which is crucial for further applications in lattices with non-
triangular geometries.

In this work, we develop a scheme to generate perfect chiral excitation flows
within $n$-node networks based on synthetic gauge fields (SGFs), where we
introduce an auxiliary node and propose a new mode for realizing the chiral
dynamics. Beyond previous studies, we find the universal criteria for
conducting chiral flows in different systems. For a three-node boson-SGF
network, the broken time-reversal symmetry plays a critical role in the
system’s chiral dynamics, leading to complex hoppings between nodes, where the
hopping phases cannot be simultaneously eliminated by a gauge transformation
[10]. By introducing an auxiliary node, we study the node-assisted SGF (aSGF)
model with broken time-reversal (TR) symmetry in each sub-structure of the
system, in which the inherent relation between the $\pi/2$ hopping phase and a
perfect chiral flow is unveiled. More generally, our aSGF model enables the
emergence of these chiral flows within the $n$-node network by appropriately
designing additional hoppings. In contrast to the boson system, when we
generalize the SGF model to the spin or two-level systems, where TR symmetry
is preserved, chiral flows can still occur but exhibit opposite chiral
dynamics for up and down states. As a result, we study the fundamental
elements for different models that support perfect chiral flows in the single
or multi-excitation subspace and propose the universal criteria that
incorporate previous models: (i) the symmetric and equally spaced energy
spectrum in the corresponding excitation subspace and (ii) a complete set of
chiral modes. In addition, we build an extended four-node network to function
as a larger scale network that also features chirality and can potentially be
employed for directional remote transmission. To make these models feasible in
practice, we investigate the influences of different types of imperfections in
the aSGF network and introduce two experimental platforms for implementing our
model, including superconducting circuits and magnetic photonic lattice, which
contribute to the chiral state transfer and directional entanglement
distribution for future quantum network and quantum simulation applications.

The remainder of the paper is organized as follows. First, in Sec. II, we
introduce the boson-SGF model and show the collapse of chiral excitation flow
when generalizing previous studies to the four-node case. Thus, in Sec. III,
we propose the aSGF model to rebuild the chiral flow in the $n$-node network
by introducing an auxiliary node. We then investigate the spin-SGF model that
preserves TR symmetry and supports opposite chiral dynamics, and derive the
general criteria for different models to conduct chiral flows. In Sec. IV, we
horizontally expand the four-node network to a large ladder, establishing a
chiral flow between remote nodes with less complexity. Furthermore, in Sec. V,
we discuss the sensitivity of the chiral flow to the imperfections in a four-
node boson-SGF network, where the parameters are taken from experimental data.
Also, the experimental feasibility of our scheme is proposed in Sec. VI.
Finally, in Sec. VII, we summarize the main conclusion of this work and
outlook for future applications.

##  II The chiral excitation flow

###  II.1 Synthetic gauge field model

To illustrate the concept of the chiral excitation flow, we first discuss the
three-node bosonic network, as shown in Fig. 1(a). In this configuration,
three boson modes with the identical frequency $\omega_{0}$ are positioned at
the three nodes and coupled with their nearest-neighbor (NN) sites, forming a
triangle arrangement in the real space. Each mode can be described by the
annihilation/creation operators $a_{j},a^{\dagger}_{j}$, where $j=1,2,3$
labels each node site, and the neighboring modes are coupled via the complex
hoppings $J_{jk}=J_{0}e^{i\theta_{jk}}$. In the interaction picture, this
three-node network is described by the following tight-binding Hamiltonian
($\hbar=1$)

$H_{\rm SGF}^{3}=J_{0}\sum_{\langle
jk\rangle}^{3}\left(e^{i\theta_{jk}}a_{j}^{\dagger}a_{k}+\mathrm{H.c.}\right),$
(1)

where the hopping phases $\theta_{jk}$ are induced from the SGF [22, 53]. We
refer to this model as a three-node boson-SGF network, which can be easily
extended to the $n$-node case where the nodes are arranged in a regular
polygon.

Figure 1: (a) Illustration of the three-node boson-SGF model, where the
synthetic flux $\theta_{3}$ determines the dynamical chirality. The evolution
of excited-state populations $P_{e}^{j}(t)$ is shown for (b) $\theta_{3}=\pi$
and (c) $\theta_{3}=\pi/2$. The chiral excitation flow emerges when the
synthetic flux of the loop $1\rightarrow 2\rightarrow 3$ is half-odd-integer
multiples of $\pi$, yet the chirality disappears when the system preserves TR
symmetry, i.e., $\theta_{3}=\mathbb{Z}\pi$. (d) Illustration of the four-node
boson-SGF model. Similarly, (e) and (f) show the populations for
$\theta_{4}=\pi$ and $\theta_{4}=\pi/2$, respectively. There is no chirality
in the dynamics regardless of whether the synthetic flux $\theta_{4}$ breaks
TR symmetry or not.

The hopping phase is analogous to the Peierls phase acquired by a charged
particle moving in a magnetic field. For an $n$-node network, the model can be
visualized as a scenario in which a uniform magnetic field vertically
penetrates the plane. In this scenario, we place the network on the $xy$ plane
with its center located at the coordinate origin and apply a uniform synthetic
magnetic field $\mathbf{B}=B(0,0,1)$ along with the $z$ axis, as shown in Fig.
1(a). Thus, the effective vector potential $\mathbf{A}_{\mathrm{eff}}$ in the
$xy$ induces a gauge-dependent hopping phase
$\theta_{jk}=\int_{\mathbf{r}_{k}}^{\mathbf{r}_{j}}\mathbf{A}_{\mathrm{eff}}\cdot
d\mathbf{r}$ between two nodes, while the synthetic flux through the $n$-node
polygonal network $\theta_{n}=\sum_{\langle jk\rangle,j<k}\theta_{jk}$ is
gauge-independent. In the following context, we mainly adopt the symmetric
gauge with $\theta_{j,j+1}=\theta_{n}/n$, which may greatly simplify our
derivation. In this system, the total excitation number, described by
$\hat{n}_{\mathrm{tot}}=\sum_{j}a^{\dagger}_{j}a_{j}$, is a conserved quantity
since it commutes with the tight-binding Hamiltonian, i.e., $[H_{\rm
SGF}^{n},\hat{n}_{\mathrm{tot}}]=0$. Hence, the dynamics of the $n$-node SGF
model can be confined to a specific excitation subspace and readily addressed
using Schrödinger equations even for large networks, as the dimension of the
reduced Hilbert space is $n$ according to a given total excitation number.

###  II.2 Chiral flow in the single-excitation subspace

The three-node chiral excitation flow has been investigated in different
models [22, 23, 24, 52]. Here, we focus on the single-excitation subspace of
the boson-SGF model, where the 1-st node is initially prepared in its excited
state $|1\rangle$ and others are in their ground state $|0\rangle$. The
wavefunction of the system is written as
$|\psi(t)\rangle=\sum_{j=1}^{n}C_{e}^{j}(t)a^{\dagger}_{j}|G\rangle$, with
initial condition $C_{e}^{j}(0)=\delta_{j1}$, where
$|G\rangle\equiv\otimes_{j=1}^{n}|0_{j}\rangle$ is the ground state of the
system. From the Hamiltonian in Eq. (1), we numerically calculate and plot the
time evolution of excited-state populations of the three bosonic modes. As
shown in Fig 1(b), when the synthetic flux through the three-node loop is an
integer multiple of $\pi$, i.e., $\theta_{3}=\mathbb{Z}\pi$, the system has TR
symmetry [10], and the excitation will flow in both directions (clockwise and
anti-clockwise) with equal probability. However, when TR symmetry is broken,
the excitation flow tends to perform chirality. Particularly, Fig. 1(c) shows
that when $\theta_{3}$ is a half-integer multiples of $\pi$, i.e.,
$\theta_{3}=(2\mathbb{Z}+1)\pi/2$, a perfect chiral excitation flow emerges.
In this case, the excited-state populations $P^{j}_{e}(t)=|C_{e}^{j}(t)|^{2}$
can be analytically solved [52]:

$P^{j}_{e}(t)=\left\\{\frac{1}{3}+\frac{2}{3}\cos\left[\sqrt{3}J_{0}t-\frac{2\pi}{3}\left(j-1\right)\right]\right\\}^{2}.$
(2)

It takes the dimensionless time $J_{0}T_{0}=2\pi/3\sqrt{3}\approx 1.21$ to
fully transfer the excited-state population from site $j$ to site
$(j+1)~{}\text{mod}~{}3$. Furthermore, the dynamics of the excited-state
populations for two nearest-neighboring nodes, e.g., $P^{1}_{e}(t)$ and
$P^{2}_{e}(t)$, turns out to have a $2\pi/3$ phase shift. This feature
explicitly interprets Fig. 1(c) and implies the dynamical chirality of this
network. We also note that as $\theta_{3}=(2\mathbb{Z}+1)\pi/2$, the system
exhibits a symmetric spectrum in the single-excitation subspace with energy
levels $E=\\{-\sqrt{3}J_{0},0,\sqrt{3}J_{0}\\}$, which reveals an underlying
chiral symmetry. We will revisit this point frequently when explaining and
designing multinode chiral flows in Sec. III.

Nevertheless, the SGF model fails to achieve such chiral flows in networks
consisting of four or more nodes. In Figs. 1(e) and 1(f), we plot the time
evolution of the populations for a four-node SGF network. When the synthetic
flux through this four-node loop ($\theta_{4}=\pi/2$) breaks TR symmetry, the
excitation dynamics still shows no chirality. In fact, the populations of the
2-nd node and 4-th node are always the same, $P^{2}_{e}(t)=P^{4}_{e}(t)$, with
details and analytical proof given in Appendix A. If the synthetic flux
$\theta_{4}=\pi$, the population of the 3-rd node vanishes, and thus it can be
referred to as a “dark site” [56, 57]. This results from the Aharonov-Bohm
effect due to the destructive quantum interference from path $1\rightarrow
2\rightarrow 3$ and path $1\rightarrow 4\rightarrow 3$. Further, as discussed
in Appendix B, theoretical analysis suggests that the general $n$-node ($n\geq
4$) SGF model cannot support a perfect chiral flow.

##  III Chiral flows in multinode networks

###  III.1 Perfect chiral flow assisted with auxiliary node in a four-node
network

Motivated by the quantum interference of the two paths in the four-node SGF
model, we here introduce an auxiliary node that is positioned at the center of
the network and couples to all network nodes, dividing the network into four
sub-triangles, as depicted in Fig. 2(a). In this case, with appropriate
parameters, we demonstrate that the quantum interference assisted by sub-
triangles from different paths is able to induce a perfect chiral excitation
flow.

The central-located auxiliary node can be described by annihilation and
creation operators $c$ and $c^{\dagger}$. For simplicity, this auxiliary node
is assumed to homogeneously couple to other nodes with a real-valued hopping
rate. The Hamiltonian of the node-assisted SGF (aSGF) model reads

$H_{\mathrm{aSGF}}^{4}/J_{0}=\sum_{\langle
jk\rangle}^{4}e^{i\theta_{jk}}a_{j}^{\dagger}a_{k}+\beta_{c}\sum_{j=1}^{4}a^{\dagger}_{j}c+\mathrm{H.c.},$
(3)

where $\beta_{c}$ characterizes the hopping strength between the auxiliary
node and other network nodes. This aSGF Hamiltonian will return to $H_{\rm
SGF}^{4}$ as $\beta_{c}=0$. Here, we focus on the single-excitation subspace,
and the system’s time-dependent wavefunction can be written as
$|\psi(t)\rangle=\sum_{j=1}^{N}\left[C_{e}^{j}(t)a_{j}^{\dagger}+C_{e}^{c}(t)c^{\dagger}\right]|G\rangle$,
where $|G\rangle\equiv(\otimes_{j=1}^{N}|0_{j}\rangle)\otimes|0_{c}\rangle$ is
the new ground state, and we set the initial state as
$C_{e}^{j}(0)=\delta_{j1}$ and $C_{e}^{c}(0)=0$. The dynamics of the system
can be analytically solved, and a detailed derivation can be found in Appendix
C. Remarkably, a perfect chiral flow occurs when the central coupling strength
is given by $\beta_{c}=2$ as well as the synthetic flux through the four-node
loop being $\theta_{4}=2\pi$. This optimal value can be directly found in the
solution of excited-state populations $P_{e}^{j}=|C_{e}^{j}(t)|^{2}$, where

$\displaystyle C_{e}^{j}\left(t\right)=$
$\displaystyle\frac{1}{2}\cos\left[2J_{0}t+\frac{(j-1)\pi}{2}\right]$ (4)
$\displaystyle\qquad\quad+\frac{1}{4}\cos\left(2\beta_{c}J_{0}t\right)+\frac{(-1)^{j-1}}{4}.$

When $\beta_{c}=2$, the two cosine functions in Eq. (4) can simultaneously
reach the peak value, meaning the complete transmission of the excited-state
population between neighboring nodes. Particularly, this configuration of
hopping phases ($\theta_{j,j+1}=\pi/2$ and the hopping phase to the auxiliary
node is $0$) can still be interpreted as a vertical uniform magnetic field
with a symmetric gauge chosen. Gauge invariance can be verified numerically:
If one works in Landau gauge instead, $\mathbf{A}=(-By,0,0)$, the same
dynamical results will be obtained.

Figure 2: (a) The four-node network assisted by an auxiliary node, referred to
as the four-node boson-aSGF model. Each network node is connected to the
auxiliary node with the coupling $\beta_{c}J_{0}$. (b) Evolution of the
excited-state populations for $\theta_{4}=2\pi$. In each sub-triangle
structure, TR symmetry is broken with the synthetic flux
$\theta_{\vartriangle}=\pi/2$. (c) Evolution of the excited-state populations
for $\theta_{4}=\pi/2$. In contrast to the three-node SGF model, the dynamics
shows no chirality with the $\pi/2$ total flux.

To intuitively understand the underlying cause for the chiral flow that
emerged in this auxiliary-mediated network, it is instructive to focus on sub-
triangles inside the four-node pattern. The pure four-node boson-SGF model in
Fig. 1(d) does not support a chiral flow due to the absence of the triangular
sub-structures. However, as shown in Fig. 2(a), the auxiliary node can provide
four sub-triangles that share the same hopping phase (or flux)
$\theta_{\vartriangle}=\theta_{4}/4$. When we set $\theta_{\vartriangle}$ as
the phase that totally breaks TR symmetry in the three-node boson-SGF model,
e.g., $\theta_{\vartriangle}=\pi/2$, the perfect chiral flow between these
four nodes indeed occurs, which implies that the three-node structure acts as
a fundamental unit dominating the expected four-node chiral flow. In contrast,
in Fig. 2(c), we show a completely different scenario where the chiral flow
collapses as $\theta_{4}=\pi/2$. This demonstrates that achieving a perfect
chiral flow requires broken TR symmetry in each sub-triangle rather than in
the whole network. In fact, the hopping phase $\theta_{j,j+1}=\pi/2$ ensures a
chiral symmetry that facilitates the chiral flow, as discussed in the
following subsection.

Further, the auxiliary node also contributes to the formation of chiral flow
by mediating the energy spectrum and removing degeneracy. The energy levels of
the four-node boson-SGF model are given by $E_{1,\pm}=\pm 2J_{0}$ and
$E_{0,\pm}=0$, which results in the imperfect oscillations at even sites (see
Appendix B for details), as shown in Fig. 1(e). Due to the internal chiral
symmetry of $H^{4}_{\rm SGF}$, the degenerate zero-energy eigenstates are in
pair, given by $|\varphi_{0,+}\rangle=(1,0,1,0)^{T}/\sqrt{2}$,
$|\varphi_{0,-}\rangle=(0,1,0,1)^{T}/\sqrt{2}$. However, in the presence of
the auxiliary node, these two states are lifted, along with the emergence of a
new zero-energy state. The chiral symmetry also persists as the auxiliary node
is introduced, with the single-particle form of chiral symmetry operator
$\mathcal{C}$ written as

$\mathcal{C}^{-1}\mathcal{H}_{\mathrm{aSGF}}^{4}\mathcal{C}=-\mathcal{H}_{\mathrm{aSGF}}^{4},\quad\mathcal{C}=\begin{pmatrix}1&0&0&0&0\\\
0&0&0&1&0\\\ 0&0&1&0&0\\\ 0&1&0&0&0\\\ 0&0&0&0&-1\end{pmatrix},$ (5)

where $\mathcal{H}_{\mathrm{aSGF}}^{4}$ is the single-particle Hamiltonian:

$H_{\mathrm{aSGF}}^{4}=S^{\dagger}\mathcal{H}_{\mathrm{aSGF}}^{4}S,\quad
S^{\dagger}=(a_{1}^{\dagger},a_{2}^{\dagger},\cdots,c^{\dagger}).$ (6)

Thus, the non-zero energy eigenstates emerge in pair
$\\{|\varphi\rangle,\mathcal{C}|\varphi\rangle\\}$ with opposite
eigenenergies, while the sole unpaired state is the new zero-energy eigenstate
[58]. In Eq. (4), the lifted energies $E_{\beta_{c},\pm}=\pm 2\beta_{c}J_{0}$
result in an extra standing wave to this network, and the newly generated
zero-energy state is denoted by
$|\varphi_{0}^{\prime}\rangle=\sum_{j}^{n}\sigma_{j}^{+}|G\rangle/\sqrt{n}$,
contributing the constant $(-1)^{j}/4$ as described in the last term. In order
to ascertain the value of $\beta_{c}$ and validate the feasibility of
conducting the chiral flow from a physical perspective, we rewrite the
probability amplitudes in Eq. (4) as

$\displaystyle C_{e}^{j}(t)=$
$\displaystyle(-1)^{(j-1)}\left\\{\frac{1}{2}\cos\left[2J_{0}t-\frac{(j-1)\pi}{2}\right]\right.$
(7)
$\displaystyle\left.+\frac{1}{4}\cos\left[2\beta_{c}J_{0}t-(j-1)\pi\right]+\frac{1}{4}\right\\}.$

In this sense, the dynamical superposition from lifted eigenstates
$|\varphi_{\beta_{c},\pm}\rangle$ is equivalent to a $\pi$-shift traveling
wave with frequency $2\beta_{c}J_{0}$, and the zero-energy state
$|\varphi_{0}^{\prime}\rangle$ supplies the last component for a perfect
oscillation. From Eq. (7) we verify that the perfect chiral flow is achieved
if and only if $\beta_{c}=2$.

In short, we conclude that the sub-triangle structure’s TR symmetry determines
the dynamical chirality of the bosonic multinode network. When the additional
auxiliary node is introduced to couple to network nodes and the synthetic flux
of sub-triangles $\theta_{\triangle}$ is chosen to be the phase referred to in
the three-node case, i.e., $(2\mathbb{Z}+1)\pi/2$, the perfect chiral flow can
emerge with a specific value of $\beta_{c}$. In particular, the $\pi/2$
hopping phase and the auxiliary node are the two essential elements for
achieving the perfect chiral flow in the four-node boson-aSGF model. But as we
directly extend the boson-aSGF model to $n$-node networks, due to the
anharmonicity of energy spectrum, the chiral flow collapses regardless of the
strength of $\beta_{c}$, as detailed in Appendix B, and explicit expressions
for six-node aSGF model can be found in Appendix C. Nevertheless, the study of
four-node network demonstrates the principle of conducting chiral flows from
the quantum interference of different states. In the following subsection, we
will show that, by introducing the next-nearest-neighbor (NNN) hopping in
five- and six-node networks, perfect chiral flows can revive with the boson-
aSGF model.

###  III.2 Perfect $n$-node chiral flow beyond NN interactions

Since the pure $n$-node boson-SGF model does not support the chiral flow, we
investigate the prerequisites necessary for achieving a perfect chiral flow in
the $n$-node boson-aSGF network. When an auxiliary node is coupled to network
nodes, it only affects the zero-energy state(s). In the following content, we
use the periodic boundary condition with $a_{n+j}=a_{j}$. The system
Hamiltonian is now described by
$H_{\mathrm{aSGF}}^{n}/J_{0}=\sum_{j=1}^{n}(ia_{j}^{\dagger}a_{j+1}+\beta_{c}a_{j}^{\dagger}c)+\text{H.c.}$,
which can be block diagonalized using the eigenstates of $n$-node boson-SGF
Hamiltonian (see Appendix B and details of four-node and six-node cases in
Appendix C). Analogous to the result of the $n$-node boson-SGF model, the
$\pi/2$ hopping phase yields a chiral symmetry for the single-particle
Hamiltonian $\mathcal{H}_{\mathrm{aSGF}}^{n}$:

$\mathcal{C}^{-1}\mathcal{H}_{\mathrm{aSGF}}^{n}\mathcal{C}=-\mathcal{H}_{\mathrm{aSGF}}^{n},\quad\mathcal{C}=\begin{pmatrix}1&0&\cdots&0&0\\\
0&0&\cdots&1&0\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
0&1&\cdots&0&0\\\ 0&0&\cdots&0&-1\end{pmatrix},$ (8)

leading to a symmetric energy spectrum of the system with paired eigenstates.
Owing to the disparity in zero-energy eigenstates, we separately analyze the
scenarios for even and odd values of $n$. With identical couplings between the
auxiliary node and $n$ network nodes, the Hamiltonian can be written as

$\mathcal{H}_{\mathrm{aSGF}}^{n}=\sum_{\begin{subarray}{c}m\neq
0\end{subarray}}E_{m}|\varphi_{m}\rangle\langle\varphi_{m}|+\sqrt{n}\beta_{c}J_{0}|\varphi_{0}\rangle\langle
c|+\text{H.c.},$ (9)

where $|\varphi_{m}\rangle~{}(-n/2<m\leq n/2)$ are the eigenstates of
$\mathcal{H}_{\mathrm{SGF}}^{n}$, as discussed in Appendix B, and
$|c\rangle=c^{\dagger}|G\rangle$ is the excited state of the auxiliary node.
Here, the energy spectrum $E_{m}=-2J_{0}\sin k_{m}$ exhibits the dispersion
relation of one-dimensional tight-binding Hamiltonian, where $k_{m}=2\pi m/n$
are wave numbers in the first Brillouin zone. We refer to these eigenstates
$|\varphi_{m}\rangle$ as chiral modes in this system because they have chiral
phase shifts $2\pi m/n$ between nearest sites. For an odd number of $n$, the
zero-energy state $|\varphi_{0}\rangle$ hybridize with the product state
$|c\rangle$, and they eventually form the new paired chiral modes
$|\varphi_{\beta_{c},\pm}\rangle=(|\varphi_{0}\rangle\pm|c\rangle)/\sqrt{2}$,
which are independent of the value of $\beta_{c}$, while the eigenenergies
$\pm E_{\beta_{c}}=\pm\sqrt{n}\beta_{c}J_{0}$ are tuned by $\beta_{c}$. If $n$
is an even number, the chiral symmetry of $H_{\mathrm{SGF}}^{n}$ ensures the
existence of two degenerate zero-energy states, corresponding to
$|\varphi_{0}\rangle$ and $|\varphi_{n/2}\rangle$. Similarly, the auxiliary
node results in the hybridization between the two states $|\varphi_{0}\rangle$
and $|c\rangle$, and thus the degeneracy is lifted. Instead,
$|\varphi_{n/2}\rangle$ remains as the unperturbed zero-energy state. We
emphasize that the hybridized states together with those unperturbed states
$|\varphi_{m}\rangle$ form a complete set of required chiral modes, which
makes it possible to realize a perfect chiral flow in this $n$-node network.

Figure 3: Dynamics of the modified (a) five-node and (b) six-node aSGF models.
The NNN hopping is introduced in the modified networks and ensures perfect
chiral flows.

We can now deduce the Hamiltonian from the prerequisites to create a perfect
chiral flow in the $n$-node network. Here, we illustrate the methodology for
constructing chiral flow Hamiltonians from the boson-aSGF model by considering
the examples of five- and six-node networks, corresponding to odd and even
values of $n$, respectively, where the initial condition is still denoted by
an excitation localized on the 1-st node. We first consider the five-node
case, a paradigm for the odd $n$. Since the eigenstates are independent of the
coupling strength $\beta_{c}$, the excitation amplitude for $j$-th node is
given by

$C_{e}^{j}(t)=\sum_{m=1}^{2}\frac{2}{5}\cos\left[E_{m}t-\frac{2(j-1)m\pi}{5}\right]+\frac{1}{5}\cos(E_{\beta_{c}}t).$
(10)

Therefore, conducting a perfect chiral flow requires the complete transmission
condition: $\max[|C_{e}^{j}(t)|^{2}]=1$ for all sites and thus leads to the
conditions $E_{2}=2E_{1}$ and

$\cos\frac{2\pi}{5}+\cos\frac{4\pi}{5}+\frac{1}{2}\cos\frac{2\pi
E_{\beta_{c}}}{5E_{1}}=0,$ (11)

which also reduces to $E_{\beta_{c}}=5E_{1}$. According to the eigenstates and
these constraints on eigenenergies, we yield the chiral flow Hamiltonian from
this five-node boson-aSGF model, with the NNN hopping:

$H_{5}/J_{0}=\sum_{j=1}^{5}-i(a_{j}^{\dagger}a_{j+1}-\alpha
a_{j}^{\dagger}a_{j+2}^{-})-\beta_{c}a_{j}^{\dagger}c+\text{H.c.},$ (12)

where $\alpha=\sqrt{(3-\sqrt{5})/2},~{}\beta_{c}=5\sqrt{2/(5+\sqrt{5})}$. As
shown in Fig. 3(a), we plot the dynamical excitation flow with Hamiltonian
$H_{5}$. Here, for a clockwise chiral flow, the NN hopping phase is $-\pi/2$
instead of $\pi/2$. Both the NN and NNN hoppings exhibit a phase of
$(\mathbb{Z}+1\pi)/2$ to preserve the chiral symmetry.

Similarly, from the eigenstates of the six-node network, we make the ansatz of
the probability amplitude without giving energy levels,

$\begin{split}C_{e}^{j}(t)=&(-1)^{(j-1)}\left\\{\sum_{m^{\prime}=1}^{2}\frac{1}{3}\cos\left[E_{m^{\prime}}t-\frac{(j-1)m^{\prime}\pi}{3}\right]\right.\\\
&\qquad+\left.\frac{1}{6}\cos\left[E_{\beta_{c}}t-(j-1)\pi\right]+\frac{1}{6}\right\\}\\\
=&\sum_{m=1}^{2}\frac{1}{3}\cos\left[E_{3-m}t+\frac{(j-1)m\pi}{3}\right]\\\
&\qquad+\frac{1}{6}\cos E_{\beta_{c}}t+\frac{(-1)^{j-1}}{6},\end{split}$ (13)

where the factor $(-1)^{j-1}$ is interpreted as $\cos(j-1)\pi$. Hence, the
eigenstates $|\varphi_{\pm m}\rangle$ correspond to energies $\mp E_{3-m}$.
From Eq. (13) we can derive the energy-level relationships as follows:
$E_{1}=2E_{2},~{}E_{\beta_{c}}=3E_{2}$. With the eigenstates and energy
levels, we obtain the six-node chiral flow Hamiltonian

$H_{6}/J_{0}=\sum_{j=1}^{6}i(a_{j}^{\dagger}a_{j+1}+\frac{1}{3}a^{\dagger}a_{j+2})-\sqrt{2}a_{j}^{\dagger}c+\text{H.c.}.$
(14)

In Fig. 3(b), we show the perfect chiral flow in the six-node network assisted
by the auxiliary node. In this sense, we have found a universal scheme for
implementing the chiral flow within both odd and even categories of $n$, and
when we go to the $n$-node boson-aSGF network, there are more interactions
involved, with Hamiltonian given by the form

$H_{n}/J_{0}=\sum_{j=1}^{n}\sum_{k=1}^{\lceil
n/2\rceil-1}i\alpha_{k}a_{j}^{\dagger}a_{j+k}+\beta_{c}a_{j}^{\dagger}c+\mathrm{H.c.},$
(15)

where $\lceil\cdot\rceil$ is the ceiling function, and $\alpha_{k}$ denotes
the strength of different kinds of couplings between nodes.

However, as the number of nodes $n$ increases, the complexity of the aSGF
network for conducting a perfect chiral flow also escalates. Specifically, the
number of required couplings exhibits a scaling of $\sim n^{2}$, which results
in the difficulty of constructing a chiral flow in practical systems. Thus, we
will turn to find an alternative way to realize chiral transmission between
remote nodes in a large network in Sec. IV.

###  III.3 Spin-aSGF model and multi-excitation

Figure 4: (a) Schematics of the three-node spin-SGF model. TR symmetry
predicts opposite chiral flows for the spin-up state and spin-down state. (b)
The time evolution of Bell states when the hopping phase is $\pi/2$. The Bell
states $|\Psi_{jk}^{+}\rangle$ and $|\Phi_{jk}^{+}\rangle$ corresponding to
different excitation subspace have opposite chiral dynamics, where the
concurrence of two atoms $\mathscr{C}_{jk}$ transfers with the evolution of
population $P(|\Psi_{jk}^{+}\rangle)$ or $P(|\Phi_{jk}^{+}\rangle)$.

In the two-level atomic and spin systems, we can also develop the spin-SGF
model by synthesizing the complex hoppings between atoms [52, 53] or the spin
interaction [23]. However, the spin-SGF model exhibits very different
symmetries compared to the boson-SGF model, resulting in a distinctive
behavior of its chiral dynamics. By replacing the boson annihilation and
creation operators $a_{j},~{}a_{j}^{\dagger}$ with ladder operators
$\sigma_{j}^{-},\sigma_{j}^{+}$, we obtain the spin-SGF Hamiltonian

$H_{\mathrm{SGF}}^{n}=J_{0}\sum_{\langle
jk\rangle}^{n}(e^{i\theta_{jk}}\sigma_{j}^{+}\sigma_{k}^{-}+\mathrm{H.c.}).$
(16)

This tight-binding Hamiltonian can describe both the excitation hopping
between two-level atoms and the spin-exchange interaction. Different from the
description in spinless lattice models [10], the TR operator for the spin
system is defined as $\hat{T}=i\sigma_{y}\mathcal{K}$, where $\mathcal{K}$
denotes the complex conjugation. The operator $\hat{T}$ flips spin vector as

$\hat{T}\bm{\sigma}\hat{T}^{-1}=-\bm{\sigma}.$ (17)

Under the TR operation, the hopping term transforms to its hermitian conjugate
as
$\hat{T}e^{i\theta_{jk}}\sigma_{j}^{+}\sigma_{k}^{-}\hat{T}^{-1}=e^{-i\theta_{jk}}\sigma_{k}^{+}\sigma_{j}^{-}$.
Thus, the spin-SGF Hamiltonian is always TR invariant regardless of the
specific values of the hopping phases $\theta_{jk}$.

In the three-node spin-SGF network, when we set the hopping phases as
$\theta_{jk}=\pi/2$, the system also supports a perfect chiral flow for the
spin-up state $\left|\uparrow\downarrow\downarrow\right\rangle$ like the
boson-SGF case. On the other hand, the system’s TR symmetry guarantees an
opposite chiral evolution of the spin-down state
$\left|\downarrow\uparrow\uparrow\right\rangle$, since it represents the TR
counterpart of the spin-up state, as shown in Fig. 4(a). Analogous to the
quantum spin Hall effect, the opposite chiral spin currents show the presence
of TR symmetry, whereas in boson-SGF networks the chiral flows are permitted
in a single direction.

Furthermore, based on the unique dynamics in the spin-SGF network, we can
establish opposite chiral transmission channels for different entanglement
states. In Fig. 4(b), we illustrate the opposite chiral dynamics of two types
of Bell states
$|\Psi^{+}_{jk}\rangle=(\left|\uparrow_{j}\downarrow_{k}\right\rangle+\left|\downarrow_{j}\uparrow_{k}\right\rangle)/\sqrt{2}$
and
$|\Phi^{+}_{jk}\rangle=(\left|\downarrow_{j}\downarrow_{k}\right\rangle+\left|\uparrow_{j}\uparrow_{k}\right\rangle)/\sqrt{2}$,
which correspond to the spin-up and spin-down chiralities. Here, we set the
initial states as
$|\Psi_{12}^{+}\rangle\otimes\left|\downarrow_{3}\right\rangle$ and
$|\Phi_{12}^{+}\rangle\otimes\left|\downarrow_{3}\right\rangle$ in the left
and right panels of Fig. 4(b), respectively. Both the populations of the Bell
states and the concurrence $\mathscr{C}_{jk}$ can characterize the chirality
of entanglement in this spin network. Because the dynamics of the two Bell
states is constrained in the up and down subspaces, the preserved TR symmetry
leads to opposite flow directions for them and causes distinctive entanglement
flows. However, in the boson network with broken TR symmetry, the chirality
remains consistent for different excitation subspaces. Hence, the transmission
of entanglement is not sensitive to its initial state. Notably, the three-node
case of the spin-SGF Hamiltonian, with the $\pi/2$ hopping phase, is also
equivalent to the chiral antisymmetric spin exchange interaction (ASI)
Hamiltonian [59, 60], which breaks parity symmetry but preserves TR symmetry
[23].

Figure 5: Time evolution of the four-node spin-aSGF model. The spin-up and spin-down states of network nodes show opposite chirality with the assistance of auxiliary spin. Table 1: A review of different models supporting the three-node chiral flow. model | boson-SGF | ASI (spin-SGF) | SCI
---|---|---|---
Hamiltonian | $\sum_{j=1}^{3}J_{0}(e^{i\pi/2}a_{j}^{\dagger}a_{j+1}+\mathrm{H.c.})$ | $\hbar\kappa C_{z}S_{z},~{}S_{z}=\sum_{j=1}^{3}\sigma_{j}^{z}/2$ | $\hbar\kappa C_{z}=\hbar\kappa\bm{\sigma}_{1}\cdot(\bm{\sigma}_{2}\times\bm{\sigma}_{3})$
PT symmetry |  $\hat{T}$-broken |  $\hat{P}$-broken, $\hat{T}$-preserved |  $\hat{T}$-broken, $\hat{P}$-broken
subspace | single-excitation | single-excitation | double-excitation | single/double-excitation
$\begin{array}[]{c}j\text{-excitation}\\\ \text{Hamiltonian }\mathcal{H}^{(j)}\end{array}$ | $\mathcal{H}^{(1)}\sim\left(\begin{array}[]{ccc}0&i&-i\\\ -i&0&i\\\ i&-i&0\end{array}\right)$ | $\mathcal{H}^{(1)}\sim\left(\begin{array}[]{ccc}0&i&-i\\\ -i&0&i\\\ i&-i&0\end{array}\right)$ | $\mathcal{H}^{(2)}\sim\left(\begin{array}[]{ccc}0&-i&i\\\ i&0&-i\\\ -i&i&0\end{array}\right)$ | $\mathcal{H}^{(1/2)}\sim\left(\begin{array}[]{ccc}0&-i&i\\\ i&0&-i\\\ -i&i&0\end{array}\right)$
flow direction | $1\rightarrow 3\rightarrow 2\rightarrow 1$ | $1\rightarrow 3\rightarrow 2\rightarrow 1$ | $1\rightarrow 2\rightarrow 3\rightarrow 1$ | $1\rightarrow 2\rightarrow 3\rightarrow 1$
chiral symmetry | ✓ | ✓ | ✓ | ✓

Similarly, the $n$-node chiral flow can also be created in the spin or atomic
system by employing the aSGF model with the replacing of operators
$\sigma^{-(+)}\rightarrow a^{(\dagger)}$ in Eq. (15). As shown in Fig. 5(a),
we observe chiral dynamics of the spin-up state in the four-node spin-aSGF
network. If one then flips all the spins, an opposite chiral flow will arise
between the network nodes, predicted by TR symmetry, as shown in Fig. 5(b).
However, in contrast to the boson system, the spin-aSGF model does not support
multi-excitation chiral flows and only permits the chiral dynamics for a
single up or down state. In boson-aSGF networks, the unidirectional chiral
dynamics exists in each excitation subspace. Therefore, it can be utilized to
transfer or prepare more complicated entangled states [23], such as the GHZ
state and the NOON state [55].

###  III.4 Discussion of the criteria

From the study of constructing the Hamiltonian (15) for achieving the $n$-node
chiral flow, we conclude that there are two essential ingredients for inducing
perfect chiral flows: (i) the symmetric and equally spaced energy spectrum in
a specific excitation subspace and (ii) a complete set of the paired chiral
modes. The symmetric spectrum is usually obtained from the chiral symmetry of
Hamiltonians in previous models, and it guarantees that each set of paired
chiral modes with opposite energies contributes a propagating wave of the
excitation in the network. Then, the equally spaced energy levels facilitate
the interference of all pairs of chiral modes on each site, leading to a
perfect site-by-site chiral flow. In particular, the chiral symmetry is not a
necessity, as we can only generate the chiral flow in a specific subspace. For
example, when we add on-site potential $U$ to the three-node boson chiral flow
Hamiltonian such that
$H^{\prime}=\sum_{j=1}^{3}U\hat{n}_{j}^{2}+(iJ_{0}a_{j}^{\dagger}a_{j+1}+\mathrm{H.c.})$,
the chiral symmetry is broken. However, there is still a symmetric spectrum in
the single-excitation subspace, allowing for the existence of the chiral flow.
As the potential $U\gg J_{0}$, the energy spectrum in the double-excitation
subspace also becomes nearly symmetric. In this sense, it forms a hard-core
boson system and mimics the three-node spin-SGF network, resulting in opposite
chiral flows in these two subspaces [22].

On the other hand, additional symmetries can also impose constraints on the
chiral flow Hamiltonian and bring novel phenomena, e.g., the TR symmetry of
the three-node spin-SGF model promotes different chirality like the quantum
spin Hall effect. As shown in Table 1, we summarize different three-node
models that support perfect chiral flows, and all of them manifest the chiral
symmetry mentioned above. These models may break other symmetries within their
systems, such as parity symmetry [23] (described by $\hat{P}$ operator) and TR
symmetry, while the spin chirality interaction (SCI) Hamiltonian [61, 62, 24]
breaks both of them. In particular, all the Hamiltonians of the listed models
have the $U(1)$ symmetry that preserves the excitation number. Therefore, we
can block-diagonalize the Hamiltonians as
$H=\mathcal{H}^{(0)}\otimes\mathcal{H}^{(1)}\otimes\cdots\otimes\mathcal{H}^{(n)}$,
where $H^{(j)}$ is the block Hamiltonian in the $j$-excitation subspace (for
spin system the excitation is replaced by spin-up state). The symmetric energy
spectrum of each block Hamiltonian makes the chiral flow present in each
subspace. In our aSGF model, the chiral symmetry is still preserved after
introducing an auxiliary node if the hopping (including long-range hoppings in
the $n$-node case) phase between network nodes is
$\theta_{j,j+k}=(2\mathbb{Z}+1)\pi/2$. The auxiliary node also generates the
last required pair of chiral modes, which facilitates the completeness of
chiral modes and thus re-induces perfect chiral flows in the $n$-node aSGF
networks.

##  IV Extension of four-node networks

In this section, we extend the four-node chiral flow by connecting four-node
networks to form a ladder. As depicted in Fig. 6(a), we arrange $N$ copies of
the four-node boson-aSGF network alongside one another. Every copy shares two
nodes with its left and right neighboring copies, and we have eliminated the
hoppings between these shared nodes, e.g., node 2 and node $2N+1$. The hopping
strengths and phases are set to be identical in all copies, and the
Hamiltonian reads

$\displaystyle H_{\mathrm{ex}}^{N}/J_{0}$
$\displaystyle=\sum_{j=1}^{2N+2}\left(e^{i\theta_{0}}a_{j}^{\dagger}a_{j+1}+{\rm
H.c.}\right)$ (18) $\displaystyle\qquad+\beta_{c}\sum_{i=1}^{N}\sum_{j\in{\rm
copy}i}\left(c_{i}^{\dagger}a_{j}+{\rm H.c.}\right),$

where the relevant parameters are based on the four-node case. All of the
hopping phases between nearest-neighboring nodes are set to be
$\theta_{0}=\pi/2$ to ensure the chiral symmetry of the system, and the
coupling strengths between auxiliary nodes and network nodes are the same as
the coupling in the four-node case, i.e., $\beta_{c}=2$. However, this
configuration cannot be interpreted as a synthetic uniform magnetic field
model because the hopping phases in this structure cannot be the same within a
uniform magnetic field. In practice, it can still be synthesized in
superconducting circuits by accurately setting the Floquet modulation
parameters [63], see Sec. VI.1 for details.

Figure 6: (a) Illustration of the ladder model with auxiliary nodes, where the
corner nodes are labeled as $c_{j},~{}j=1,2,3,4$. The solid and dashed arrows
are hoppings partitioned into $H_{0}$ and $V_{I}$, respectively. (b) and (c)
correspond to the chiral flows with different copies of the four-node
network($N=2$ and $N=6$ respectively). (e) The effective configuration of the
ladder network. In this pattern, the corner nodes are equivalent. (f) The
fidelity of chiral flow in the ladder network as a function of $N$. Here, the
blue solid line represents the fidelity from identical coupling $\beta_{c}=2$,
while the red dashed line denotes the results from optimal parameters with
$\beta_{j}>2$, $j=1,\cdots,m$.

We study the chiral dynamics of this ladder network by initiating an
excitation at the 1-st node. Figs. 6(b) and 6(c) display the evolution of
excited-state populations of all nodes for $N=2$ and $N=6$ cases respectively,
which demonstrate that the primary characteristic of the chiral flow is
retained. The majority of the excitation is chirally transported between the
corner nodes $c_{j}$ [as labeled in Fig. 6(a)], namely, $1\rightarrow
N+1\rightarrow N+2\rightarrow 2N+2$. We emphasize that the imperfection of
this chiral flow originates from the unevenly spaced oscillation frequencies,
as detailed in Appendix D, where we also provide the analytic expression of
$P_{e}^{1}(t)$ by partitioning the Hamiltonian as
$H_{\mathrm{ex}}^{N}=H_{0}+V_{I}$ and implementing the nonperturbative
calculation of system’s propagator. As the number of copies $N$ increases, the
timescale for transferring an excitation from node $1$ to node $N+1$ also
extends (in a sublinear form). To describe the transmission rate of this
chiral flow, we define the fidelity as
$\mathcal{F}=|\langle\psi(t=0)|\psi_{f}\rangle|^{2}$, where $\psi_{f}$ is the
final state after a cycle period. As shown in Fig. 6(f), the fidelity of
chiral flow $\mathcal{F}$ decays linearly when the network becomes larger,
with less remainder excitation at the 1-st node.

Nevertheless, this extended model can also be viewed as a polygonal network
with auxiliary nodes, as shown in Fig. 6(d). Within this pattern, we can still
maintain equal treatment of the corner nodes by replacing $\beta_{c}$ with a
sequence of coupling parameters $\beta_{j},j=0,1,\cdots m$, where $m=\lceil
N/2\rceil-1$. Thus, we can then optimize the network by modulating the
parameters. Intuitively, the coupling $\beta_{j}J_{0}$ should be enhanced to
connect further nodes when it becomes far away from the corner atoms, i.e.,
$2=\beta_{0}<\beta_{1}<\cdots<\beta_{m}$. As illustrated in Fig. 6(f), we plot
the optimized fidelity as a function of $N$ with a set of optimal parameters,
which shows a linear decline but is improved measurably compared to the
identical coupling case. Even if the network expands to ten times the four-
node network’s scale, the fidelity remains at $\mathcal{F}>0.8$. Therefore, we
are convinced that the chirality persists and dominates within a larger
network, rendering this ladder network highly practical for implementing
chiral transmission between remote nodes.

##  V Imperfections in networks

In practical systems, the inevitable imperfections from various aspects of
quantum networks can affect the quality of chiral flows in the $n$-node aSGF
network and ladder networks. To estimate such influence, in this section, we
investigate the sensitivity of chiral flow to the disorder on the node
(bosonic mode or emitter) frequency $\omega_{0}$, coupling strength $J_{0}$,
and hopping phase $\theta_{ij}$, respectively. Here, we take the four-node
aSGF model as an example for illustration.

As shown in Fig. 7(a), these three types of disorder are denoted as
$\delta_{\square}$, $\square=\omega,J$ and $\theta$, where the first acts on
the node frequency while the latter two affect the hoppings between nodes. In
the single-excitation subspace, the disorder on frequency and hopping is
represented as the diagonal and off-diagonal disorder of the single-particle
Hamiltonian $\mathcal{H}_{\mathrm{aSGF}}^{4}$, respectively. Here, we first
consider the disorder on the frequency by replacing $\omega_{0}$ with
$\omega_{0}+\delta_{\omega}$ for each node, where $\delta_{\omega}$ is sampled
from a uniform distribution within the range
[$-\delta_{\omega}^{m},\delta_{\omega}^{m}$]. Since the maximum excitation
population of $j$-th node is no longer 1, i.e.,
$\max\left[P_{e}^{j}(t)\right]<1$ within a typical cycle period, we use the
average fidelity of the corner excitation to characterize the quality of
chiral flow:

$\mathcal{F}_{\mathrm{avg}}=\sum_{j=1}^{4}\max\left[C_{e}^{j}(t)\right]/4.$
(19)

As illustrated in Fig. 7(a), the frequency disorder significantly influences
the chiral flow. Here, the parameters in this boson-SGF network are taken from
the experimental data [23], with $\omega_{0}=5.6~{}\mathrm{GHz}$ and
$J_{0}=2\pi\times 4.29~{}\mathrm{MHz}$. Compared with the hopping, even a
slight deviation of the node frequency can greatly change the energy spectrum,
leading to a considerable decrease of the fidelity
$\mathcal{F}_{\mathrm{avg}}$. Moreover, the chiral symmetry is broken when
such diagonal (i.e., frequency) disorder is introduced, which results in the
asymmetry of the energy spectrum and thus violates the criteria for conducting
a perfect chiral flow. In contrast, the off-diagonal disorder preserves the
chiral symmetry of the system and causes minor impacts on the energy spectrum.
As plotted in Figs. 7(c) and 7(d), the chiral flow shows robustness
($\mathcal{F}_{\mathrm{avg}}>0.9$ with about $30\%$ deviation of the
parameters) when the disorder stems from the hopping strength and phase.

Figure 7: (a) Illustration of the four-node aSGF model with inevitable
imperfections. The imperfections of this network are described by the disorder
of the node frequencies and hoppings. (b)-(d) Effects of different types of
disorder on the average fidelity.

##  VI Feasibility in experiments

In this section, we elucidate the practical implementation of our proposal,
where we take the four-node aSGF model as an example in the following
discussion. Here, we briefly introduce the schemes for superconducting
circuits [22, 23] and magnetic photonic lattice [11, 18, 12, 52] platforms to
achieve controllable hoppings with phases, thus promising the realization of
multinode chiral flows in bosonic and atomic systems, respectively.

###  VI.1 Superconducting circuits

The hopping phases between the nodes of superconducting setups, such as
resonators and qubits [38, 39, 40, 64, 42], can be synthesized in two ways:
(1) by introducing tunable couplers to periodically modulate coupling
strengths [22, 65, 66], or (2) by periodically adjusting the transition
frequencies of nodes, which are coupled to a mutual bus resonator [23, 67].

Following the first way, we can design a superconducting circuit consisting of
five transmon qubits (or resonators) $Q_{1}$, $Q_{2}$, $Q_{3}$, $Q_{4}$,
$Q_{a}$, coupled with each other via tunable couplers, where the couplings can
be dynamically modulated as
$g_{jk}(t)=2g_{jk}\cos\left(\nu_{jk}t+\phi_{jk}\right)$ [66]. Here, the
modulation frequency $\nu_{jk}$ is chosen to be the frequency detuning between
elements $j$ and $k$, i.e., $\nu_{jk}=\omega_{j}-\omega_{k}$, and we assume
that $\nu_{jk}\gg g$. After transforming to the interaction picture under the
rotating-wave approximation (RWA), we obtain the effective Hamiltonian

$\displaystyle H_{{\rm eff}}=$ $\displaystyle\sum_{\begin{subarray}{c}j,k=1\\\
j<k\end{subarray}}^{4}\left(g_{jk}e^{-i\phi_{jk}}a_{j}^{\dagger}a_{k}+{\rm
H.c.}\right)$ (20)
$\displaystyle+\sum_{j=1}^{4}\left(g_{j}^{c}e^{-i\phi_{j}^{c}}a_{j}^{\dagger}c+{\rm
H.c.}\right)+H_{\mathrm{nl}},$

where
$H_{\mathrm{nl}}\approx-U/2\left(\sum_{j}a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}+c^{\dagger}c^{\dagger}cc\right)$
is the Kerr nonlinearity and arises from the cosine potential of transmon
qubit, with the on-site potential $U\gg g$. The nonlinearity of
superconducting qubits naturally promotes the effective Hamiltonian as a spin-
aSGF model since it prohibits the double-photon occupancy if the transmons are
initially singly occupied with double excitation like
$a_{j}^{\dagger}a_{k}^{\dagger}|G\rangle$. For harmonic resonators with $U=0$,
$H_{\mathrm{eff}}$ mimics a boson-SGF Hamiltonian. To satisfy the requirements
of conducting a four-node chiral flow, we set the parameters as follows:
$g_{j}^{c}/2=g_{j,j+1\mathrm{mod}4}=g$ for $j=1$, 2, 3, 4, $g_{13}=g_{24}=0$,
$\phi_{j}^{c}=0$ and $\phi_{j,j+1\mathrm{mod}4}=-\pi/2$. Notably, the
frequency detunings of the elements have to satisfy the RWA conditions:
$|\omega_{j}-\omega_{k}|>g$ for $j\neq k$. Then, the Hamiltonian reduces to
the form in Eq. (3) with $N=4$, $J_{0}=g$, and $\beta_{c}=2$, allowing for
conducting the perfect chiral flow.

In addition, the first line of Hamiltonian (20) can also be synthesized via a
bus resonator [23]. In this circuit quantum electrodynamics (QED) model, four
elements are interconnected by a bus resonator with couplings $g_{j}$. To
construct synthetic gauge fields, we periodically adjust the frequencies of
qubits around the frequency of the bus resonator $\omega_{r}$ as

$\omega_{j}(t)=\omega_{r}+\Delta\cos\left(\nu t-\phi_{j}\right),$ (21)

where $\Delta$ and $\nu$ represent the modulation amplitude and frequency,
with $\nu\approx\Delta/2.40$, and the phases are denoted as $\phi_{j}=j\pi/2$.
Under the condition $\nu\gg g$, we obtain the effective first-order
Hamiltonian with complex hopping amplitudes
$g_{jk}={g_{j}g_{k}}\beta_{jk}e^{i\pi/2}/{\nu}$, by neglecting higher-order
terms. Here, $\beta_{jk}$ is given by the expression
$\beta_{jk}=\sum_{n=1}^{+\infty}2J_{n}^{2}\left[f\right]\sin\left[n\left(\phi_{k}-\phi_{j}\right)\right]/n$,
where $J_{n}[x]$ represents the $n$th-order Bessel function of the first kind
and the dimensionless variable $f=\Delta/\nu\approx 2.40$ [68, 69] such that
the zeroth-order term $J_{0}[f]=0$. The $\pi/2$ hopping phase of $g_{jk}$ thus
can accurately replicate the four-node chiral flow Hamiltonian. By setting
$\phi_{j}=j\pi/2$, we can establish NN hoppings with the same strength while
preventing the occurrence of NNN hoppings between nodes $j$ and $j+2$.

###  VI.2 Emitters in magnetic photonic lattice

The circuit QED provides a convenient way for synthesizing hopping phases
between artificial atoms or resonators. Here, we show that a hybrid system,
i.e., a magnetic photonic lattice (MPL) coupled to atomic emitters [52], can
also induce hopping phases and be utilized to construct our model. Although
the chiral flow between emitters is imperfect, we still see this dominant
feature in such a hybrid system.

As shown in Fig. 8(a), the MPL is a square array composed of identical
photonic resonators with equal spacing $l_{0}$ and frequency $\omega_{r}$,
where the neighboring lattices are coupled via a synthetic magnetic field
$\mathbf{B}$ with complex tunneling amplitudes. We place five atomic emitters
to the resonators on the center and four corners of a square with side length
$d$. These emitters have the same resonant frequency $\omega_{e}$ and are
labeled as $1,2,3,4$, and $a$. The coupling strengths between the five
emitters and resonators are identical, denoted as $g$.

Figure 8: (a) Schematic of an aSGF model in a magnetic photonic lattice. The
blue hollow circles are resonators that form a square lattice, while the
filled circles with five different colors are emitters coupled to the
resonators locally. The dashed lines between emitters indicate the effective
long-range dipole-dipole interactions induced by the magnetic photonic
lattice. (b) Evolution of the excited-state populations of emitters in the
MPL, where $\ell=0$, $g/(\omega_{e}-\omega_{\ell=0})=0.1$ and $d/l_{0}=4$. And
the unit of time is $\Omega^{-1}=(d/l_{0})g^{-1}$.

In the continuum limit, the Hamiltonian of the MPL can be approximately
diagonalized to a series of Landau levels with energy $\omega_{\ell}$, each of
which possesses a large number of degenerate states. Here, we assume that the
nearest Landau level to the frequencies of emitters is $\ell=0$. Under
conditions $\omega_{c}\gg g$ and $|\omega_{e}-\omega_{\ell=0}|\gg g$, a
Schrieffer-Wolff transformation can be applied to obtain the effective atomic
interaction Hamiltonian

$H_{\rm
eff}=\sum_{j,k}\tilde{J}_{jk}\sigma_{+}^{j}\sigma_{-}^{k}+\mathrm{H.c.},$ (22)

which manifests a long-range dipole-dipole interaction with complex hopping
amplitudes $\tilde{J}_{jk}\sim
g^{2}e^{i\theta_{jk}}/(\omega_{e}-\omega_{\ell=0})$ [52]. By setting the gauge
invariant flux
$\theta_{4}\equiv\theta_{12}+\theta_{23}+\theta_{34}+\theta_{41}=2\pi$ to
break the TR symmetry of each triangular sub-structures, this effective
dipole-dipole interaction can to some extent construct the four-node spin-aSGF
model. Fig. 8(b) shows the chiral dynamics of excitation, where two factors
contribute to the imperfection of this chiral flow. Firstly, in this
configuration, the coupling parameter of the auxiliary emitter is
$\beta_{c}=1.48$ instead of $2$, which is unable to generate a perfect four-
node chiral flow. Secondly, the unavoidable NNN hopping between emitters from
diagonal sites also reduces the fidelity of the chiral flow. However, by
introducing external coupling channels [18, 21, 13, 70] to eliminate these
effects, it is still possible to realize a perfect chiral flow between atoms
in this system.

##  VII Conclusion and Outlook

In conclusion, we investigate the boson-SGF model and show the chiral
excitation flow in the three-node case with a broken TR symmetry. However, the
pure SGF model cannot drive the chiral dynamics when it extends to the four-
node case. By introducing the auxiliary emitter to couple to network nodes, we
recreate the chiral flow in the four-node boson-aSGF network. The excitation
dynamics of this four-node boson-aSGF network suggests that it can exhibit
chiral flow only when the synthetic flux of each triangular sub-structure
breaks TR symmetry, i.e., $\theta_{\vartriangle}\neq\mathbb{Z}\pi$. This links
the well-studied three-node chiral flow and the boson-aSGF model for creating
perfect chiral flows in multinode networks, where these triangular sub-
structures are also retained, with $(\mathbb{Z}+1)\pi/2$ synthetic flux. When
we generalize this framework to the spin or two-level atomic system where TR
symmetry is preserved, it demonstrates opposite chiral dynamics for the spin-
up and spin-down states. Further, we introduce a universal scheme for
constructing the $n$-node chiral flow Hamiltonians with the aSGF model and
provide illustrative examples for odd and even $n$, respectively. During this
process, we find the criteria for a Hamiltonian to support the chiral flow in
a specific excitation subspace, which indicates the requirement of a symmetric
and equally spaced energy spectrum and a complete set of chiral modes. Our
framework for finding chiral flows applies to various systems such as bosons,
spins, and two-level atoms. It is also applicable to different types of
interactions, including the NN complex hopping forming a two-body Hamiltonian,
as well as the three-body interactions like SCI and ASI in spin systems. On
the other hand, to realize the chiral flow in a larger network, the number of
required hoppings scales as $n^{2}$, showing a significant complexity growth.
Therefore, we propose a ladder network with the complexity scaling as $n$.
Instead, it can serve as a more practical scheme for unidirectional
transmission between remote emitters. In addition, the imperfections of
networks are also discussed, where the fidelity of chiral flow shows
robustness in the presence of hopping disorder but is sensitive to the
inhomogeneity of node (boson mode or atomic) frequency.

The multinode aSGF model and the extension of the four-node network can be
realized with superconducting circuits by employing the Floquet modulation,
which provides a new application to implement perfect state transfer in
circuit QED systems. Moreover, the magnetic photonic lattice can also mimic
this model to a certain extent with atomic emitters. The experimental
feasibility, as well as the tolerance to imperfections, make our scheme
practical and promising to realize chiral state transfer and unidirectional
entanglement distribution. Also, we expect that our proposal could be utilized
to develop non-reciprocal quantum devices [63] like $n$-node circulators and
benefit future quantum technologies.

##  VIII Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grant No. 12375025 and No. 11874432), the National Key R&D Program of China
(Grant No. 2019YFA0308200).

##  Appendix A The analytical solution of four-node boson-SGF model with
arbitrary gauge phase

The single-excitation dynamics of the four-node boson-SGF model under a
symmetric gauge can be analytically solved for arbitrary synthetic flux
$\theta_{4}$. Here, we denote the nearest-neighboring hopping phase as
$\theta=\theta_{4}/4$, then the four-node boson-SGF Hamiltonian in the single-
excitation subspace reads

$\mathcal{H}_{{\rm
SGF}}^{4}/J_{0}=\left(\begin{matrix}0&e^{i\theta}&0&e^{-i\theta}\\\
e^{-i\theta}&0&e^{i\theta}&0\\\ 0&e^{-i\theta}&0&e^{i\theta}\\\
e^{i\theta}&0&e^{-i\theta}&0\end{matrix}\right).$ (23)

Here the single-excitation basis kets are $\\{|j\rangle\equiv
a^{\dagger}_{j}|G\rangle\\}$, with $j=1,2,3,4$. The eigenvalues $\lambda_{k}$
and corresponding eigenvectors $\mathbf{f}^{k}$ of
$\mathcal{H}^{4}_{\mathrm{SGF}}$ are given by

$\begin{cases}\mathbf{f}^{1}=\frac{1}{2}\left(1,1,1,1\right)^{T},\lambda_{1}=2\cos\theta,\\\
\mathbf{f}^{2}=\frac{1}{2}\left(-1,1,-1,1\right)^{T},\lambda_{2}=-2\cos\theta,\\\
\mathbf{f}^{3}=\frac{1}{2}\left(-i,-1,i,1\right)^{T},\lambda_{3}=2\sin\theta,\\\
\mathbf{f}^{4}=\frac{1}{2}\left(i,-1,-i,1\right)^{T},\lambda_{4}=-2\sin\theta.\end{cases}$
(24)

When $\theta=\pi/2$, $\mathbf{f}^{1}$ and $\mathbf{f}^{2}$ become the
degenerate zero-energy states.

By setting the initial state as
$|\psi(0)\rangle=a_{1}^{\dagger}|G\rangle=(1,0,0,0)^{T}$, the system’s
dynamics is determined by Schrödinger equations, with
$|\psi(t)\rangle=e^{-iH^{4}_{\rm SGF}t}|\psi(0)\rangle$. We denote that
$|\psi(t)\rangle=\sum_{j}C^{j}_{e}(t)|j\rangle$, and the probability
amplitudes can be written as

$C^{j}_{e}\left(t\right)=\sum_{k}\left(f_{1}^{k}\right)^{*}f_{j}^{k}e^{-iE_{k}t},E_{k}=J_{0}\lambda_{k},$
(25)

where $f_{j}^{k}=\langle j|\mathbf{f}^{k}\rangle$. Substituting the
eigenvalues and eigenvectors from Eq. (24) into the above equation, the
population $P^{j}_{e}(t)\equiv|C^{j}_{e}(t)|^{2}$ can be obtained:

$\begin{split}&P_{e}^{1}(t)=\frac{1}{4}\left(\cos\left[2\cos(\theta)t\right]+\cos\left[2\sin(\theta)t\right]\right)^{2},\\\
&P_{e}^{2}(t)=P_{e}^{4}(t)=\frac{1}{4}\left(\sin^{2}\left[2\sin(\theta)t\right]+\sin^{2}\left[2\cos(\theta)t\right]\right),\\\
&P_{e}^{3}(t)=\frac{1}{4}\left(\cos\left[2\cos(\theta)t\right]-\cos\left[2\sin(\theta)t\right]\right)^{2}.\end{split}$
(26)

The populations of the 2-nd and 4-th nodes are always the same, which reveals
the same dynamical process, as shown in Figs. 1(e) and 1(f). In particular,
when $\theta_{4}=\pi$ and $\theta=\pi/4$, the 3-rd node can be considered as a
dark site whose population vanishes all the time, with
$P^{3}_{e}(t,\theta_{4}=\pi)=0$ as shown in Fig. 1(e). Intuitively, the
transition amplitude of $1\rightarrow 3$ process can be written as
$t_{13}\propto(\exp(i\pi/2)+\exp(-i\pi/2))=0$, with destructive interference
from two paths.

##  Appendix B The dynamics of $n$-node networks

In this part, we study the dynamics of $n$-node networks and show the
destruction of chiral flows for the networks whose node number is larger than
four, i.e., $n\geq 4$.

We consider the $n$-node boson-SGF model that each node has a $\pi/2$ hopping
phase from its next node, and the Hamiltonian reads

$H_{\mathrm{SGF}}^{n}=J_{0}\sum_{j=1}^{n}(ia_{j}^{\dagger}a_{j+1}+\mathrm{H.c.}),$
(27)

where the periodic boundary condition $a_{n+1}=a_{1}$ is used. In the single-
excitation subspace with Hamiltonian $\mathcal{H}_{\mathrm{SGF}}^{n}$, the
eigenstates are plane waves,

Figure 9: Evolution of excited-state population $P_{e}^{j}(t)$ of nodes in (a)
five-node and (b) six-node boson-SGF models, respectively. No chiral flow
appears regardless of whether the synthetic flux breaks TR symmetry.

$|\varphi_{m}\rangle=\sum_{j=1}^{n}e^{ijk_{m}}a_{j}^{\dagger}|G\rangle/\sqrt{n},\quad
m\in\mathbb{Z},-\frac{n}{2}<m\leq\frac{n}{2},$ (28)

with energy spectrum given by $E_{m}=-2J_{0}\sin k_{m}$, where $k_{m}=2\pi
m/n$ are the wave numbers within the first Brillouin zone (BZ). For instance,
when $n=4$, the wave numbers with $m=0,2,-1$ and $1$ reproduce the four
eigenstates in Eq. (24) respectively. We refer to these eigenstates as chiral
modes since they have chiral phase shifts $2\pi m/n$ between nearest sites.
Note that the system has an internal chiral symmetry $\mathcal{C}$, with
$\mathcal{C}^{-1}\mathcal{H}_{\mathrm{SGF}}^{n}\mathcal{C}=-\mathcal{H}_{\mathrm{SGF}}^{n}$,
resulting in a symmetric energy spectrum [58]. The chiral operators for even
and odd numbers of nodes are given by

$\mathcal{C}=\bigoplus_{j=1}^{n/2}\sigma_{z}^{j}\quad\text{and}\quad\mathcal{C}=\begin{pmatrix}0&0&\cdots&0&1\\\
0&0&\cdots&1&0\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
0&1&\cdots&0&0\\\ 1&0&\cdots&0&0\end{pmatrix},$ (29)

respectively.

As shown in Figs. 9(a) and 9(b), we plot the dynamics of five-node and six-
node boson-SGF models and find the perfect chiral flow cannot be supported
within these networks even when the hopping phase is $\pi/2$. To understand
the underlying reasons, we categorize the boson-SGF networks into odd and even
classes based on the value of $n$. When $n$ is odd, the presence of chiral
symmetry gives rise to a single zero-energy state,
$|\varphi_{0}\rangle=\sum_{j}^{n}a_{j}^{\dagger}|G\rangle/\sqrt{n},~{}E_{0}=0$,
corresponding to $k_{m}=0$. Here, we assume that the excitation is initially
on the 1-st node. Then using Eq. (25), the evolution of excitation on $j$-th
node $P_{e}^{j}(t)=|C_{e}^{j}(t)|^{2}$ can be obtained, with the probability
amplitude given by

$C_{e}^{j}(t)=\sum_{k_{m}\in\mathrm{BZ}}\frac{2}{n}\cos\left[E_{m}t-k_{m}(j-1)\right]+\frac{1}{n},$
(30)

which implies that realizing the perfect chiral flow requires a harmonic
energy spectrum, i.e., $E_{m}=mE_{1}$, while this condition cannot be
satisfied when $n>4$, as shown in Fig. 10. For the even class, there are two
degenerate zero-energy states, denoted as

$|\varphi_{0,+}\rangle=\sqrt{\frac{2}{n}}\sum_{j=1}^{n/2}a_{2j-1}^{\dagger}|G\rangle,\;|\varphi_{0,-}\rangle=\sqrt{\frac{2}{n}}\sum_{j=1}^{n/2}a_{2j}^{\dagger}|G\rangle,$
(31)

which are the linear combinations of the two zero-energy eigenstates in Eq.
(28), i.e.,
$|\varphi_{0,\pm}\rangle=(|\varphi_{0}\rangle\pm|\varphi_{n/2}\rangle)/\sqrt{2}$.
Therefore, we can deduce the probability amplitude

$C_{e}^{j}(t)=\sum_{k_{m}\in\mathrm{BZ}}\frac{2}{n}\cos\left[E_{m}t-k_{m}(j-1)\right]+\frac{2}{n}\delta_{j,{\rm
odd}},$ (32)

where $\delta_{j,{\rm odd}}=1$ if $j$ is odd and $\delta_{j,{\rm odd}}=0$ if
$j$ is even. In this sense, the wave function of zero-energy eigenstate
$|\varphi_{0,-}\rangle$ has no distribution on all odd sites, including the
first site. This absence of amplitude leads to imperfect oscillations at all
even sites, with maximum populations
$\max[P_{e}^{j=2\mathbb{N}}(t)]\leq(1-2/n)^{2}$. Furthermore, the energy
spectrum in Fig. 10 also shows degeneracies when $n$ is even, which makes it
impossible to fulfill the harmonicity condition $E_{m}=mE_{1}$.

Figure 10: The energy spectrum of $n$-node boson-SGF model for $n$ ranges from
4 to 11. The dashed lines mean the degenerate energies with two degenerate
eigenstates.

##  Appendix C The analytical solution of four-node and six-node boson-aSGF
models

Since the four-node and six-node boson-aSGF models have simple explicit
expressions of eigenstates and energy levels, we here derive the analytic
solutions for the dynamics of the node excited-state populations
$P_{e}^{j}(t)$. In the single-excitation subspace, the Hamiltonian (3) can be
written as the following matrix form,

$\displaystyle\mathcal{H}_{\mathrm{aSGF}}^{n}/J_{0}=$ (33)
$\displaystyle\begin{pmatrix}0&e^{i\theta_{12}}&0&\cdots&0&e^{-i\theta_{n1}}&\beta_{c}\\\
e^{-i\theta_{12}}&0&e^{i\theta_{23}}&\cdots&0&0&\beta_{c}\\\
0&e^{-i\theta_{23}}&0&\cdots&0&0&\beta_{c}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&0&e^{i\theta_{n-1,n}}&\beta_{c}\\\
e^{i\theta_{n1}}&0&0&\cdots&e^{-i\theta_{n-1,n}}&0&\beta_{c}\\\
\beta_{c}&\beta_{c}&\beta_{c}&\cdots&\beta_{c}&\beta_{c}&0\end{pmatrix},$

where the basis kets are $\\{|j\rangle\equiv
a^{\dagger}_{j}|G\rangle,|c\rangle=c^{\dagger}|G\rangle\\}$ with
$j=1,\dots,n$.

To solve the four-node boson-aSGF model with hopping phase
$\theta_{j,j+1}=\pi/2$, we change the basis kets from
$\\{|1\rangle,|2\rangle,|3\rangle,|4\rangle\\}$ to
$\\{|\alpha\rangle,|\beta\rangle,|\gamma\rangle,|\delta\rangle\\}$, where the
latter set consists of the eigenstates of the four-node SGF Hamiltonian. The
unitary transformation is given by

$\begin{pmatrix}\vspace{0.5ex}a_{\alpha}^{\dagger}\\\
\vspace{0.5ex}a_{\beta}^{\dagger}\\\ \vspace{0.5ex}a_{\gamma}^{\dagger}\\\
\vspace{0.5ex}a_{\delta}^{\dagger}\end{pmatrix}=\begin{pmatrix}\vspace{0.5ex}\frac{i}{2}&-\frac{1}{2}&-\frac{i}{2}&\frac{1}{2}\\\
\vspace{0.5ex}-\frac{i}{2}&-\frac{1}{2}&\frac{i}{2}&\frac{1}{2}\\\
\vspace{0.5ex}0&\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\\
\vspace{0.5ex}\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}&0\end{pmatrix}\begin{pmatrix}\vspace{0.5ex}a_{1}^{\dagger}\\\
\vspace{0.5ex}a_{2}^{\dagger}\\\ \vspace{0.5ex}a_{3}^{\dagger}\\\
\vspace{0.5ex}a_{4}^{\dagger}\end{pmatrix},$ (34)

and the Hamiltonian in the new basis is block-diagonal,

$\widetilde{\mathcal{H}}_{\mathrm{aSGF}}^{4}/J_{0}=\begin{pmatrix}-2\\\ &2\\\
&&0&&\sqrt{2}\beta_{c}\\\ &&&0&\sqrt{2}\beta_{c}\\\
&&\sqrt{2}\beta_{c}&\sqrt{2}\beta_{c}&0\end{pmatrix}.$ (35)

Hence, the dynamics can be separated into two independent subspaces, with the
following Schrödinger equations and initial conditions,

$\displaystyle i\partial_{t}\begin{pmatrix}C_{\alpha}\\\
C_{\beta}\end{pmatrix}=J_{0}\begin{pmatrix}-2\\\
&2\end{pmatrix}\begin{pmatrix}C_{\alpha}\\\
C_{\beta}\end{pmatrix},\left\\{\begin{array}[]{l}{C_{\alpha}(0)=-i/2}\\\
{C_{\beta}(0)=i/2}\end{array}\right.,$ (38) $\displaystyle
i\partial_{t}\begin{pmatrix}C_{\gamma}\\\ C_{\delta}\\\
C_{c}\end{pmatrix}=J_{0}\sqrt{2}\beta_{c}\begin{pmatrix}0&&1\\\ &0&1\\\
1&1&0\end{pmatrix}\begin{pmatrix}C_{\gamma}\\\ C_{\delta}\\\
C_{c}\end{pmatrix},C_{\delta}\left(0\right)=\frac{1}{\sqrt{2}},$ (39)

where the explicit form of the state ket is written as
$|\psi\left(t\right)\rangle=C_{\alpha}(t)|\alpha\rangle+C_{\beta}(t)|\beta\rangle+C_{\gamma}(t)|\gamma\rangle+C_{\delta}(t)|\delta\rangle+C_{c}(t)|c\rangle$.
In the decoupled $\alpha\beta$ subspace, the Hamiltonian is diagonal with
energy $\pm 2J_{0}$, leading to the free-evolution of states $|\alpha\rangle$
and $|\beta\rangle$. However, in the $\gamma\delta c$ subspace, the zero-
energy eigenstates of the four-node boson-SGF Hamiltonian $|\gamma\rangle$ and
$|\delta\rangle$ interact with the auxiliary mode $|c\rangle$. After solving
these time-dependent coefficients, we return to basis
$\\{|1\rangle,|2\rangle,|3\rangle,|4\rangle,|c\rangle\\}$ and obtain the
probability amplitudes for each node,

$\displaystyle C_{e}^{1}\left(t\right)$
$\displaystyle=\frac{1}{4}\left[1+\cos\left(2J_{0}\beta_{c}t\right)+2\cos\left(2J_{0}t\right)\right],$
(40) $\displaystyle C_{e}^{2}\left(t\right)$
$\displaystyle=-\frac{1}{4}\left[1-\cos\left(2J_{0}\beta_{c}t\right)+2\sin\left(2J_{0}t\right)\right],$
$\displaystyle C_{e}^{3}\left(t\right)$
$\displaystyle=\frac{1}{4}\left[1+\cos\left(2J_{0}\beta_{c}t\right)-2\cos\left(2J_{0}t\right)\right],$
$\displaystyle C_{e}^{4}\left(t\right)$
$\displaystyle=-\frac{1}{4}\left[1-\cos\left(2J_{0}\beta_{c}t\right)-2\sin\left(2J_{0}t\right)\right].$

More explicitly, they can be written in a compact form, as given in Eq. (4).

Similarly, in the six-node case with $\theta_{j,j+1}=\pi/2$, we use the
following unitary matrix

$U=\left(\begin{array}[]{cccccc}\vspace{0.5ex}\frac{i}{2}&-\frac{1}{\sqrt{3}}&-\frac{i}{2}&\frac{1}{2\sqrt{3}}&0&\frac{1}{2\sqrt{3}}\\\
\vspace{0.5ex}-\frac{1}{2\sqrt{3}}&0&-\frac{1}{2\sqrt{3}}&-\frac{i}{2}&\frac{1}{\sqrt{3}}&\frac{i}{2}\\\
\vspace{0.5ex}-\frac{i}{2}&-\frac{1}{\sqrt{3}}&\frac{i}{2}&\frac{1}{2\sqrt{3}}&0&\frac{1}{2\sqrt{3}}\\\
\vspace{0.5ex}-\frac{1}{2\sqrt{3}}&0&-\frac{1}{2\sqrt{3}}&\frac{i}{2}&\frac{1}{\sqrt{3}}&-\frac{i}{2}\\\
\vspace{0.5ex}0&\frac{1}{\sqrt{3}}&0&\frac{1}{\sqrt{3}}&0&\frac{1}{\sqrt{3}}\\\
\vspace{0.5ex}\frac{1}{\sqrt{3}}&0&\frac{1}{\sqrt{3}}&0&\frac{1}{\sqrt{3}}&0\\\
\end{array}\right)$ (41)

to block-diagonalize $\mathcal{H}_{\mathrm{aSGF}}^{6}$, i.e., change the basis
from $\\{|1\rangle,\dots,|6\rangle,|c\rangle\\}$ to
$\\{|\alpha\rangle,|\beta\rangle,|\gamma\rangle,|\delta\rangle,|\varepsilon\rangle,|\zeta\rangle,|c\rangle\\}$,
which are the eigenstates of six-node SGF Hamiltonian
$\mathcal{H}_{\mathrm{SGF}}^{6}$. After the transformation, we obtain the
following form,

$\widetilde{\mathcal{H}}_{\mathrm{aSGF}}^{6}/J_{0}=\begin{pmatrix}-\sqrt{3}&&&&&&0\\\
&-\sqrt{3}&&&&&0\\\ &&\sqrt{3}&&&&0\\\ &&&\sqrt{3}&&&0\\\
&&&&0&&\sqrt{3}\beta_{c}\\\ &&&&&0&\sqrt{3}\beta_{c}\\\
0&0&0&0&\sqrt{3}\beta_{c}&\sqrt{3}\beta_{c}&0\end{pmatrix}.$ (42)

The dynamics can be likewise separated into two parts. The first part
corresponds to $\alpha\beta\gamma\delta$ subspace in the upper-left $4\times
4$ block, with a diagonal matrix that indicates the simple free-evolution. The
second part is restricted to the $\varepsilon\zeta c$ subspace in the lower-
right $3\times 3$ block, which denotes the coupling between zero-energy states
and the auxiliary mode. After solving the coefficients of the time-dependent
state
$|\psi\left(t\right)\rangle=C_{\alpha}(t)|\alpha\rangle+C_{\beta}(t)|\beta\rangle+C_{\gamma}(t)|\gamma\rangle+C_{\delta}(t)|\delta\rangle+C_{\varepsilon}(t)|\varepsilon\rangle+C_{\zeta}(t)|\zeta\rangle+C_{c}(t)|c\rangle$,
and transforming back to the original basis
$\\{|1\rangle,\dots,|6\rangle,|c\rangle\\}$, we obtain

$\begin{split}&P^{1}_{e}(t)=\frac{1}{36}\left[1+\cos(\sqrt{6}J_{0}\beta_{c}t)+4\cos(\sqrt{3}J_{0}t)\right]^{2},\\\
&P^{2}_{e}(t)=\frac{1}{36}\left[1-\cos(\sqrt{6}J_{0}\beta_{c}t)+2\sqrt{3}\sin(\sqrt{3}J_{0}t)\right]^{2},\\\
&P^{3}_{e}(t)=\frac{1}{36}\left[1+\cos(\sqrt{6}J_{0}\beta_{c}t)-2\cos(\sqrt{3}J_{0}t)\right]^{2},\\\
&P^{4}_{e}(t)=\frac{1}{36}\left[1-\cos(\sqrt{6}J_{0}\beta_{c}t)\right]^{2},\\\
&P^{5}_{e}(t)=\frac{1}{36}\left[1+\cos(\sqrt{6}J_{0}\beta_{c}t)-2\cos(\sqrt{3}J_{0}t)\right]^{2},\\\
&P^{6}_{e}(t)=\frac{1}{36}\left[1-\cos(\sqrt{6}J_{0}\beta_{c}t)-2\sqrt{3}\sin(\sqrt{3}J_{0}t)\right]^{2}.\end{split}$
(43)

This result suggests that the introduction of an auxiliary node is unable to
create a perfect chiral flow in the six-node network, regardless of the
coupling strength $\beta_{c}J_{0}$.

##  Appendix D Dynamics of the extended four-node network

In the extended four-node network, the energy spectrum is not equally spaced,
which induces accumulated errors as the number of network nodes increases. To
evaluate the imperfection of the chiral flow and investigate its origin, we
here adopt a nonperturbative analysis of the system’s propagator,

$K_{+}(t,t^{\prime})=U(t,t^{\prime})\theta(\tau),\quad\tau=t-t^{\prime},$ (44)

where $U(t,t^{\prime})$ is the time-evolution operator, and $\theta(\tau)$
denotes the Heaviside function. The propagator is related to the resolvent of
the Hamiltonian $H_{\mathrm{ex}}^{N}$ by a Fourier transform

$K_{+}(\tau)=-\frac{1}{2\pi
i}\int_{-\infty}^{+\infty}\mathrm{d}\omega\mathrm{e}^{-i\omega\tau/\hbar}G_{+}(\omega),$
(45)

where the resolvent is given by

$G_{+}(\omega)\equiv\lim_{\eta\to 0_{+}}G(\omega+i\eta)=\lim_{\eta\to
0_{+}}\frac{1}{E-H_{\mathrm{ex}}^{N}+i\eta}.$ (46)

Figure 11: (a) Schematics of the contour integral and the population of the
1-st node. Here, the solid line denotes analytical expression for $P_{e}^{1}$,
and the dots are numerical results obtained from solving the Schrödinger
equation. (b) Time evolution of the excitation on each site for $N=4$. The
right region of the red dashed line represents the evolution of auxiliary
emitters. (c) Normalized poles (with an $\omega_{0}$ shift) for different $N$.
The unequally spaced poles result in the imperfection of fidelity.

To proceed, we divide the network hopping Hamiltonian into two parts,
$H_{\mathrm{ex}}^{N}=H_{0}+V_{I}$, as depicted by the solid and dashed lines
in Fig. 6(a). The unperturbed Hamiltonian $H_{0}$ contains the free
Hamiltonian of all nodes as well as the hoppings between the non-corner nodes,
and the interaction Hamiltonian $V_{I}$ describes the hoppings from the four
corner nodes labeled as $c1$, $c2$, $c3$, and $c4$, as shown in Fig. 6(a). We
constrain the dynamics within the subspace $\mathscr{E}_{0}$ formed by the
corner nodes and define the projector as $P=\sum_{j=1}^{4}|c_{j}\rangle\langle
c_{j}|$. Similarly, the projector of the complementary subspace is given by
$Q=1-P$. Thus, the projection of $G(\omega)$ in subspace $\mathscr{E}_{0}$ is
written as

$PG(\omega)P=\frac{P}{\omega-PH_{0}P-P\Sigma(\omega)P},$ (47)

where $\Sigma(\omega)$ denotes the self-energy and is given by

$\Sigma(\omega)=V_{I}+V_{I}\frac{Q}{\omega-QH_{0}Q-QVQ}V_{I}.$ (48)

We can verify that in our system, the self-energy only extends to its first
order, i.e.,

$\Sigma(\omega)=V_{I}+V_{I}\frac{Q}{\omega-H_{0}}V_{I}.$ (49)

As the system is initially prepared at $|\psi(t=0)\rangle=|c_{1}\rangle$, the
probability amplitudes of the edge nodes are given by

$\displaystyle C_{e}^{cj}(t)=$ $\displaystyle K_{+}(t)|\psi(0)\rangle$ (50)
$\displaystyle=$
$\displaystyle\sum\mathrm{Res}\left\\{[PG(\omega)P]_{j1}e^{-i\omega
t}\right\\},$

with $j=1,2,3,4$. Because the system preserves the chiral symmetry, the real
poles of $PG(\omega)P$ are distributed symmetrically. In order to elucidate
the retrieve of excitation on the 1-st node and identify the cause for the
imperfection of fidelity, here we study the case of $N=4$. We find the simple
poles are given by $x_{i,\pm}=\omega_{0}\pm k\sqrt{2}J_{0}$ with $i=1,2,3$ and
$x_{4,\pm}=\omega_{0}\pm 2\sqrt{7}J_{0}$. By substituting the poles and
calculating the sum of residues, we can then obtain the excited-state
population of each node $|C_{e}^{cj}|^{2}$. The population of the 1-st node is
given by

$\displaystyle P_{e}^{1}=$
$\displaystyle\left[\frac{11}{56}+\frac{3}{8}\cos(\sqrt{2}J_{0}t)+\frac{11}{40}\cos(2\sqrt{2}J_{0}t)\right.$
(51)
$\displaystyle\qquad\left.+\frac{1}{8}\cos(3\sqrt{2}J_{0}t)+\frac{1}{35}\cos(2\sqrt{7}J_{0}t)\right]^{2}.$

As shown in Fig. 11(a), the poles of the contour integral are plotted, and the
numerical result (dots) is consistent with our analytical expression (solid
line). We find that each pair of poles $\pm x_{i}$ contributes to a cosine
oscillation on the probability as in Eq. (51), and the imperfection of
fidelity comes from the fact that these poles are non-uniformly distributed,
i.e., the closed propagation path from $c_{1}$ to itself contains the
interference with non-harmonic frequencies. Fig. 11(b) shows that the leakage
mainly falls into the auxiliary emitters. We emphasize that these results are
independent of the relative value of $J_{0}/\omega_{0}$ and there is no
approximation made here. This implies that these hopping rates can also enter
the non-perturbative regime. For larger networks, the error will accumulate as
$n$ increases, along with the decreasing fidelity. This can be seen from the
normalized poles $x_{i}/|x_{\mathrm{min}}|$ in Fig. 11(c), where
$x_{\mathrm{min}}=\mathrm{min}\\{|x_{i}|,|x_{i}|\neq 0\\}$ induces the lowest
oscillation frequency of $P_{e}^{j}(t)$.

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