# Spectroscopy and topological properties of a Haldane light system

Julian Legendre  Department of Physics and Materials Science, University of
Luxembourg, L-1511, Luxembourg  CPHT, CNRS, Institut Polytechnique de Paris,
Route de Saclay, 91128 Palaiseau, France  Karyn Le Hur  CPHT, CNRS, Institut
Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France

###### Abstract

We present a method to probe the topological properties of a circuit quantum
electrodynamics (cQED) array described through a Haldane model on the
honeycomb lattice. We develop the theory of microwave light propagating in a
local probe or a microscope (a one-dimensional transmission line) capacitively
coupled to the topological cQED lattice model. Interestingly, we show that
even if the microwave light has no transverse polarization, the measured
reflection coefficient, resolved in frequency through the resonance, allows us
to reveal the geometrical properties and topological phase transition
associated to the model. This spectroscopy tool developed for cQED lattice
models reveals the same topological information as circularly polarized light,
locally within the Brillouin zone of the honeycomb lattice. Furthermore, our
findings hold significance for topological magnon systems and are a priori
applicable to all Chern insulators, presenting an intriguing opportunity for
their adaptation to other systems with different particle statistics.

††preprint: APS/123-QED

#### Introduction.—

Topological systems find various interesting applications in physics, in
particular related to the protected mesoscopic transport at the edges. In two
dimensions, the quantum Hall effect, induced by a perpendicular uniform
magnetic field, has been generalized to situations with no net flux in a unit
cell, referring to the Haldane honeycomb lattice model [1], and then generally
to the quantum anomalous Hall effect and Chern insulators. The Haldane’s
seminal model is realized in solid-state systems, in cold atom gases and in
photonic systems (coupled waveguides) [2, 3, 4]. One elegant way to realize
this model for artificial systems is through Floquet engineering [5, 3, 4, 6,
7, 8].

The most common way to probe the topological properties in condensed matter
systems is to determine the Hall conductance [9, 10]. The topological
responses of artificial systems are accessible in several ways [11, 12, 13,
14]. In cold atom gases, topological properties are revealed through transport
or Hall drift [4, 15], interferometry [16, 17, 18], the physics of chiral edge
states [19, 20] or via a measurement of the Berry curvature [21]. For
condensed matter systems and cold atom gases, a circular drive on the system
also enables to probe the topological information [22, 23, 24, 25, 26], even
with a local resolution within the Brillouin zone [27, 28]. Topological
properties of light systems have also attracted attention such as gyromagnetic
photonic crystals [29, 30, 31, 32], arrays of coupled waveguides [3, 33, 34],
optomechanical systems [35, 36], cavity and circuit quantum electrodynamics
(cQED) [37, 38, 39, 40, 41, 42].

In Ref. [43], a protocol to probe the topological properties of a one-
dimensional LC circuit system is proposed. This system is closely connected to
the SSH model which has been implemented recently [44, 45, 46, 47]. In Ref.
[43], the authors considered a transmission line (capacitively) coupled to a
single cell within the chain. From the reflection of an input triggered in the
probe, they reconstructed the Zak phase, which is the topological invariant
characterizing the studied one-dimensional system. The extension of such a
local capacitive probe to two-dimensional systems is not readily apparent.
Previous proposals for light-matter topological probes in two-dimensional
systems have used the transverse polarization of light to detect the chirality
associated with the system’s topological nature [22, 24]. In striking contrast
to these approaches, our study focuses on a local probe, specifically a long
transmission line capacitively coupled to a Haldane bosonic model in circuit
quantum electrodynamics (cQED). Remarkably, we demonstrate how the Chern
number can be measured by analyzing the reflection coefficient, which relates
the input and output voltage signals.

#### Bosonic Haldane model.—

We introduce a cQED system made of an array of resonators coupled together in
such a way [8] that the system is described by a usual Haldane Hamiltonian
$H=\sum_{{\bf k}}\Psi_{{\bf k}}^{\dagger}h_{{\bf k}}\Psi_{{\bf k}}$ [1, 2, 3,
4, 5, 6, 7], with

$h_{{\bf k}}=h_{0}({\bf k})+\textrm{Re}\left[h_{1}({\bf
k})\right]\sigma_{x}-\textrm{Im}\left[h_{1}({\bf
k})\right]\sigma_{y}+h_{2}({\bf k})\sigma_{z},$ (1)

and with $h_{0}({\bf k})=\hbar\Omega_{0}+2t_{2}\cos\phi\sum_{i=1}^{3}\cos({\bf
k}\cdot{\bf b}_{i})$, $h_{1}({\bf k})=t_{1}\sum_{i=1}^{3}\exp(-i{\bf
k}\cdot{\bf a}_{i})$, and $h_{2}({\bf
k})=M-2t_{2}\sin\phi\sum_{i=1}^{3}\sin({\bf k}\cdot{\bf b}_{i})$. We define
$\Psi_{{\bf k}}^{\dagger}=\left(a_{1,{\bf k}}^{\dagger},a_{2,{\bf
k}}^{\dagger}\right)$, with $a^{\dagger}_{j,{\bf k}}$ the creation operator
for a color-$j$ (say $j=1(2)$ correspond to the lattice site A(B) appearing in
Fig. 1(a)) boson with momentum ${\bf k}$, ${\bf a}_{i},\,i\in\\{1,2,3\\}$ and
${\bf b}_{i},\,i\in\\{1,2,3\\}$ are defined in Fig. 1(a), hopping amplitudes
$t_{1}\in\mathbb{R}$ and $t_{2}\in\mathbb{R}$, Semenoff mass $M\in\mathbb{R}$
[48] and $\sigma^{x},\sigma^{y},\sigma^{z}$ the Pauli matrices in sub-lattice
space. Hereafter, we study the case where $t_{2}$ is small compared to
$t_{1}$, as it is often the situation in physical systems. In Ref. [8], a
Haldane effective Hamiltonian is derived from Floquet engineering with a high-
frequency approximation. Such a photonic system is permanently driven to
compensate for the photon decay processes that necessarily happen [14]. In
typical photonic systems the on-site energy $\hbar\Omega_{0}$ is large
(usually $\sim$ GHz order of magnitude) compared to the effective hopping
amplitudes on the lattice (e.g. can be $\sim 10$ MHz to $\sim 100$ MHz) [37,
49, 50, 42].

The Haldane model is characterized by two energy bands in momentum space
$E_{i,{\bf k}}=h_{0}({\bf k})\pm\epsilon({\bf k})$ where the minus (plus) sign
is associated to the band $i=1$ ($i=2$) and with $\epsilon({\bf
k})=\sqrt{|h_{1}({\bf k})|^{2}+h_{2}({\bf k})^{2}}$. Band crossing appears at
$h_{1}({\bf k})=h_{2}({\bf k})=0$. $h_{1}({\bf k})=0$ is reached at both
nonequivalent Dirac points ${\bf K}=\left({\bf g}_{3}-{\bf g}_{2}\right)/3$
and ${\bf K}^{\prime}=\left({\bf g}_{2}-{\bf g}_{3}\right)/3$ (see Fig. 1(c)).
Moreover, we have $h_{2}({\bf K})=0$ if $M=+3\sqrt{3}t_{2}\sin\phi$ and
$h_{2}({\bf K}^{\prime})=0$ if $M=-3\sqrt{3}t_{2}\sin\phi$. When the bands
cross, the dispersion relation around ${\bf K}$ and ${\bf K}^{\prime}$ is
linear. The energies for a chosen point in parameter space are represented in
Fig. 1(b).

(a)

(b)

(c)

(d)

Figure 1: (a) Definition of the sublattices, real space vectors and hopping
amplitudes for the Haldane model on the honeycomb lattice. (b) Haldane model
energies, in units of $t_{1}$, as a function of the momentum $q_{3}=a{\bf
k}\cdot{\bf g}_{3}$ ($a$ is the lattice spacing), with parameters
$\phi=\pi/2$, $t_{2}=0.15t_{1}$ and $M=3\sqrt{3}t_{2}/2$. Both lowest band
energies at the Dirac points, $E_{1,{\bf K}}$ and $E_{1,{\bf K}^{\prime}}$,
are shown. (c) Brillouin zone for the honeycomb lattice and the
$\mathcal{D}_{I}$ and $\mathcal{D}_{II}$ domains. (d) The quantity $1-\int
d\omega S^{\textrm{out}}(\omega)$, computed from Eq. (9); the color scale of
the figure is logarithmic. We considered
$\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)=1$, $j_{0}=1$, and the
energies $\Omega_{0}=10$ GHz, $t_{1}/\hbar=100$ MHz, $t_{2}=0.15t_{1}$,
$\Delta_{\textrm{CI}}=\Delta_{\textrm{P}}=10$MHz and $g_{q_{0}}/\hbar=1$MHz,
as described in the main text. We considered a $30\times 30$ unit cells
Haldane model with periodic boundary conditions.

#### Topological properties.—

Following the approach of Kohmoto [51], we will write the Chern number $C_{i}$
(energy band $i=1$ or $i=2$) in terms of a phase entering in the wavefunction,
which is associated to the definition of the $h_{1}({\bf k})$ kinetic term
only. This will elegantly allow us to show that the light probe can detect the
topological response of the system locally within the Brillouin zone through
the conservation of energy.

Table 1: Sign of $h_{2}$ at the Dirac points, as a function of $M$ and $\textrm{sgn}(\sin\phi)$. | $|M|<3\sqrt{3}t_{2}|\sin\phi|$ | $|M|>3\sqrt{3}t_{2}|\sin\phi|$
---|---|---
$\textrm{sgn}\,h_{2}({\bf K})$ | $-\textrm{sgn}(\sin\phi)$ | $\textrm{sgn}\,M$
$\textrm{sgn}\,h_{2}({\bf K}^{\prime})$ | $\textrm{sgn}(\sin\phi)$ | $\textrm{sgn}\,M$

We write $\ket{u_{i,{\bf k}}}$ the Bloch eigenvectors of the Haldane
Hamiltonian, i.e. $\textrm{e}^{-i{\bf k}\cdot\hat{{\bf r}}}H\textrm{e}^{i{\bf
k}\cdot\hat{{\bf r}}}\ket{u_{i,{\bf k}}}=E_{i,{\bf k}}\ket{u_{i,{\bf k}}}$ and
we define the coefficients $\alpha_{i}^{1}({\bf k})$ and $\alpha_{i}^{2}({\bf
k})$ such that $\ket{u_{i,{\bf k}}}=\left[\alpha_{i}^{1}({\bf k})a_{1,{\bf
k}}^{\dagger}+\alpha_{i}^{2}({\bf k})a_{2,{\bf k}}^{\dagger}\right]\ket{0}$.
These coefficients may vanish only at the Dirac points.

(i) If the sign of $h_{2}$ is opposite at the Dirac points
($|M|<3\sqrt{3}t_{2}|\sin\phi|$, see Table 1), then $\alpha_{i}^{1}({\bf k})$
and $\alpha_{i}^{2}({\bf k})$ vanish at the opposite Dirac points. It follows
that it is impossible to find a unique and smooth phase over all the BZ for
the Bloch state $\ket{u_{i,{\bf k}}}$. Rather, we choose to define two non-
overlapping domains $\mathcal{D}_{I}$ and $\mathcal{D}_{II}$ in the BZ, each
containing a different Dirac point (see Fig. 1(c)), and we use a different
gauge choice for the Bloch states in each domain [51]. We define a closed path
P along the boundary between $\mathcal{D}_{I}$ and $\mathcal{D}_{II}$,
surrounding once the Dirac points, and a smooth phase along P, $\varphi({\bf
k})$, such that $\textrm{e}^{-i\varphi({\bf k})}=\left.\textrm{e}^{i{\bf
k}\cdot{\bf a}_{3}}h_{1}({\bf k})^{*}\right/|h_{1}({\bf k})|$. Then $C_{i}$
reads 111See Supplementary Material S1 for Chern number computation details
using Kohmoto’s approach.

$2\pi C_{i}=(-1)^{i}\,\textrm{sgn}(\sin\phi)\oint_{\textrm{P}}d{\bf
k}\cdot{\bf\nabla}_{{\bf k}}\varphi({\bf k}).$ (2)

From the expressions of $\varphi({\bf k})$ and $h_{1}({\bf k})$, we find that
$\varphi({\bf k})$ changes by $-2\pi$ when moving along the entire closed path
P. This gives $C_{i}=(-1)^{i+1}\,\textrm{sgn}(\sin\phi)$. Let us remark that
the vanishing of Bloch eigenvectors’ components at different points in the BZ,
which prevents the smooth definition of Bloch states, is a characteristic
feature of Chern insulators. As we will see, this fundamental property forms
the basis of the probe proposed in this letter, rendering it a priori relevant
for all Chern insulators.

(ii) If the sign of $h_{2}$ is the same at both Dirac points
($|M|>3\sqrt{3}t_{2}|\sin\phi|$, see Table 1), then $\alpha_{i}^{1}({\bf k})$
or $\alpha_{i}^{2}({\bf k})$ can be chosen non-zero over all the BZ, and it is
possible to find a unique and smooth phase for $\ket{u_{i,{\bf k}}}$, leading
to a unique and smooth Berry gauge field. Because the BZ is a torus, we find
that the Chern numbers $C_{i}$ are vanishing.

From this analysis, for $|M|\neq 3\sqrt{3}t_{2}|\sin\phi|$, we find
$C_{i}=(-1)^{i+1}\,\textrm{sgn}(\sin\phi)\left[1-\textrm{sgn}(h_{2}({\bf
K})h_{2}({\bf K}^{\prime})\right]/2$, i.e.

$C_{i}=\dfrac{(-1)^{i}}{2}\left[\textrm{sgn}\,h_{2}({\bf
K})-\textrm{sgn}\,h_{2}({\bf K}^{\prime})\right].$ (3)

This formula has in fact a simple physical understanding for a Hamiltonian
$h_{\bf k}$ written as a $2\times 2$ matrix. From the Ehrenfest theorem and a
Bloch sphere correspondence the topological number is equivalent to
$C_{i}=(-1)^{i}\left[\langle\sigma_{z}(0)\rangle-\langle\sigma_{z}(\pi)\rangle\right]/2$
with
$\langle\sigma_{z}\rangle=(-1)^{i}\cos\theta=(-1)^{i}\hbox{sgn}\,h_{2}(\theta)$
[53]. In the following, we namely rely on Eq. (3) to show how $C_{i}$ can be
probed from the reflected light in a local probe capacitively coupled to a
Haldane photonic system. The simple idea behind our proposal is that the
topological properties manifest as discernible sublattice weight variations of
the wave function, enabling to reveal the topological transition through the
coupling of a probe to one of the sublattice sites.

#### Spectroscopic probe.—

Let us consider a local light probe with weak capacitive coupling to a Haldane
boson model. The probe is a resonator with a certain number of (relevant)
modes, each mode $q$ being characterized by the frequency $\omega_{q}$. The
probe is coupled to the system at position ${\bf R}_{0}$, on the sublattice
identified by the index $j_{0}$, where $j_{0}=1$ ($j_{0}=2$) for the
sublattice A (B). We write the Hamiltonian associated to the probe
$H_{\textrm{prb}}=\sum_{q}\hbar\omega_{q}b_{q}^{\dagger}b_{q}$, with $b_{q}$
the annihilation operators for the mode $q$ of the probe. The coupling is
described by $H_{\textrm{cpl}}=\left(a_{{\bf R}_{0}}+a_{{\bf
R}_{0}}^{\dagger}\right)\sum_{q}g_{q}\left(b_{q}+b_{q}^{\dagger}\right)$, with
$a_{{\bf R}_{0}}^{\dagger}$ the creation operator for a boson at site ${\bf
R}_{0}$. For simplicity, first of all, we disregard the dissipation effects
induced by the probe in the Chern insulator and we assume that the light modes
$\ket{u_{i,{\bf k}}}$ exhibit an infinitely long lifetime.

Table 2: Coefficients $\alpha_{i}^{j_{0}}$ which, at the Dirac points, are directly related to the Chern number, as a function of $\textrm{sgn}\,M$ and $\textrm{sgn}(\sin\phi)$. | $\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)$ | $\textrm{sgn}\,M=-\textrm{sgn}\left(\sin\phi\right)$
---|---|---
$\textrm{sgn}\left(\sin\phi\right)=(-1)^{i}$ | $\alpha_{i}^{2}(\bm{K})=-C_{i}$ | $\alpha_{i}^{1}(\bm{K}^{\prime})=(-1)^{i+1}C_{i}$
$\textrm{sgn}\left(\sin\phi\right)=(-1)^{i+1}$ | $\alpha_{i}^{1}(\bm{K})=(-1)^{i}C_{i}$ | $\alpha_{i}^{2}(\bm{K}^{\prime})=C_{i}$
Table 3: Choice of the probe’s input frequency $\hbar\omega_{q_{0}}=E_{i,{\bf k}}$, given by the indices $i$ and ${\bf k}$, as a function of $\textrm{sgn}\,M$ and $\textrm{sgn}\left(\sin\phi\right)$, such that the transition rate depends on the Chern number: $\Gamma=J\left[E_{i,{\bf k}}\right]|C_{i}|/\hbar$. | $\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)$ | $\textrm{sgn}\,M=-\textrm{sgn}\left(\sin\phi\right)$
---|---|---
$\textrm{sgn}\left(\sin\phi\right)=1$ |  $i=j_{0}$, ${\bf k}={\bf K}$ |  $i=\overline{j_{0}}$, ${\bf k}={\bf K}^{\prime}$
$\textrm{sgn}\left(\sin\phi\right)=-1$ |  $i=\overline{j_{0}}$, ${\bf k}={\bf K}$ |  $i=j_{0}$, ${\bf k}={\bf K}^{\prime}$

To build intuition, let us show that the transition rate $\Gamma$ from a state
$\ket{\psi(t)}$ which, at initial time $t_{i}$, is a probe’s mode with
frequency $\omega_{q_{0}}$, i.e. $\ket{\psi(t_{i})}=\ket{b_{q_{0}}}$, to the
eigenstates $\ket{u_{i,{\bf k}}}$ of the Haldane Hamiltonian’s bears
information about the topological character of the system. Fermi Golden rule
states that at sufficiently long times $t$,
$\Gamma\left[\hbar\omega_{q_{0}}\right]=\dfrac{2\pi}{\hbar}\sum_{i,{\bf
k}}|\bra{b_{{q_{0}}}}H_{\textrm{cpl}}\ket{u_{i,{\bf
k}}}|^{2}\delta\left(\hbar\omega_{q_{0}}-E_{i,{\bf k}}\right)$.
$\bra{b_{{q_{0}}}}H_{\textrm{cpl}}\ket{u_{i,{\bf k}}}$ involves the components
of the Bloch state in the basis $\left(a_{1,{\bf k}}^{\dagger},a_{2,{\bf
k}}^{\dagger}\right)$ and a factor $\textrm{e}^{i{\bf k}\cdot{\bf R}_{0}}$
(transformation to the real space representation), such that we obtain
$\bra{b_{{q_{0}}}}H_{\textrm{cpl}}\ket{u_{i,{\bf
k}}}=g_{q_{0}}\alpha_{i}^{j_{0}}({\bf k})\textrm{e}^{i{\bf k}\cdot{\bf
R}_{0}}$. As one can see from Table 2, which is constructed using Eq. (3) and
the related analysis of the coefficients $\alpha_{i}^{j_{0}}({\bf k})$,
depending on $\textrm{sgn}\,M$ and $\textrm{sgn}\left(\sin\phi\right)$, it is
possible to express the Chern number as a function of the coefficients
$\alpha_{i}^{j_{0}}({\bf k})$. If
$\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)$
($\textrm{sgn}\,M=-\textrm{sgn}\left(\sin\phi\right)$) we notice that the
Chern number is directly related to the coefficients $\alpha_{i}^{j_{0}}({\bf
k})$ evaluated at $\bm{K}$ ($\bm{K}^{\prime}$) and $E_{i,\bm{K}}$
($E_{i,\bm{K}^{\prime}}$), $i\in\\{1,2\\}$, is non-degenerate. Therefore,
choosing $\hbar\omega_{q_{0}}=E_{i,{\bf k}}$ with $i$ and ${\bf k}$ according
to Table 3, we find a simple relation between $\Gamma$ and the topological
invariant: $\Gamma\left[E_{i,{\bf k}}\right]=J\left[E_{i,{\bf
k}}\right]|C_{i}|^{2}=J\left[E_{i,{\bf k}}\right]|C_{i}|$, where the spectral
function $J$ is:
$J[\omega]=(2\pi/\hbar)\sum_{q}g_{q}^{2}\left[\delta(\omega-\omega_{q})-\delta(\omega+\omega_{q})\right]$.
In other words,

$\Gamma\left[E_{i,{\bf k}}\right]=J\left[E_{i,{\bf k}}\right]|C_{i}|,$ (4)

where the indices $i$ and ${\bf k}$ are functions of $\textrm{sgn}\,M$ and
$\textrm{sgn}\left(\sin\phi\right)$ as indicated in Table 3. The relation
appearing in Eq. (4) has been established from Table 2 and Table 3, which are
constructed from the previous section. Therefore, Eq. (4) relies on a
fundamental property characterizing a Chern insulator: the impossibility of
defining smooth Bloch states over the BZ, which translates here into the
vanishing of the Bloch eigenvectors’ components $\alpha_{i}^{1}$ and
$\alpha_{i}^{2}$ at the opposite Dirac points.

Motivated by this, we now investigate the relation between an input voltage
$\langle V_{{\bf R}_{0}}^{\textrm{in}}[\omega]\rangle$ and the resulting
output voltage $\langle V_{{\bf R}_{0}}^{\textrm{out}}[\omega]\rangle$, both
at frequency $\omega$ in the probe at ${\bf R}_{0}$. For $\omega$ resolved
around one Dirac point, this relation between $\langle V_{{\bf
R}_{0}}^{\textrm{in}}[\omega]\rangle$ and $\langle V_{{\bf
R}_{0}}^{\textrm{out}}[\omega]\rangle$ enables to rebuild the Haldane
topological phase diagram 222For explicit definitions and demonstrations we
refer to the Supplemental material S2.. To fourth order in the coupling
amplitudes $\\{g_{q}\\}$, we indeed find

$\langle V_{{\bf R}_{0}}^{\textrm{out}}[\omega]\rangle=R(\omega)\langle
V_{{\bf R}_{0}}^{\textrm{in}}[\omega]\rangle,$ (5)

with $R(\omega)=1+iJ[\omega]\chi_{{\bf R}_{0},{\bf R}_{0}}$, and

$\chi_{{\bf R}_{0},{\bf R}_{0}}=\dfrac{1}{N}\sum_{i=1}^{2}\sum_{{\bf
k}}\gamma_{j_{0},{\bf k}}^{i}\left[\dfrac{1}{-\hbar\omega-E_{i,{\bf
k}}+i0^{+}}-\dfrac{1}{-\hbar\omega+E_{i,{\bf k}}+i0^{+}}\right],$ (6)

where $N$ is the number of lattice sites and

$\gamma_{j_{0},{\bf k}}^{i}=\dfrac{1}{2}+\dfrac{(-1)^{j_{0}+i+1}h_{2}({\bf
k})}{2\epsilon({\bf k})}\in\,\mathbb{R}.$ (7)

From $\epsilon({\bf k})=\sqrt{|h_{1}({\bf k})|^{2}+h_{2}({\bf k})^{2}}$, we
notice that $\gamma_{j_{0},{\bf k}}^{i}$ evaluated at the Dirac points depends
on the sign of the function $h_{2}$: we have $2\gamma_{j_{0},{\bf
k}}^{i}=1-(-1)^{j_{0}+i}\textrm{sgn}\,h_{2}({\bf k})$ for ${\bf k}=\\{{\bf
K},{\bf K}^{\prime}\\}$. This is related to the topological invariant via the
Eq. (3) and Table 1. As we show in Table 4, depending on the sign of
$\sin\phi$ and on the sign of the Semenoff mass, the $i^{\textrm{th}}$ band
topological invariant is given by the coefficient $\gamma_{j_{0},{\bf
k}}^{i}$, evaluated at $j_{0}=i$ or $j_{0}=\overline{i}$ and at ${\bf k}={\bf
K}$ or ${\bf k}={\bf K}^{\prime}$, with $\overline{i}=2(1)$ if $i=1(2)$.
Again, this outcome is obtained thanks to a fundamental characteristic
associated to the topological phase: the vanishing of the Bloch eigenvectors’
components $\alpha_{i}^{1}$ and $\alpha_{i}^{2}$ at the opposite Dirac points.

Table 4:  Value of $\gamma_{j_{0},{\bf k}}^{i}$, evaluated at $j_{0}=i$ or $j_{0}=\overline{i}$ and at ${\bf k}={\bf K}$ or ${\bf k}={\bf K}^{\prime}$, as a function of $\textrm{sgn}\left(\sin\phi\right)$ and $\textrm{sgn}\,M$. We remind that $\overline{i}=2(1)$ if $i=1(2)$. | $\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)$ | $\textrm{sgn}\,M=-\textrm{sgn}\left(\sin\phi\right)$
---|---|---
$\textrm{sgn}\left(\sin\phi\right)=1$ | $\gamma_{i,\bm{K}}^{i}=(-1)^{\overline{i}}C_{i}$ | $\gamma_{\overline{i},\bm{K}^{\prime}}^{i}=(-1)^{\overline{i}}C_{i}$
$\textrm{sgn}\left(\sin\phi\right)=-1$ | $\gamma_{\overline{i},\bm{K}}^{i}=(-1)^{i}C_{i}$ | $\gamma_{i,\bm{K}^{\prime}}^{i}=(-1)^{i}C_{i}$

Now, we can understand how a simple measurement of the reflection of a light
input in the probe gives access to the topological invariant. We write
$S^{\textrm{in}}(\omega)=\left|\langle V_{{\bf
R}_{0}}^{\textrm{in}}[\omega]\rangle\right|^{2}$ and
$S^{\textrm{out}}(\omega)=\left|\langle V_{{\bf
R}_{0}}^{\textrm{out}}[\omega]\rangle\right|^{2}$ the energy spectral density
respectively associated to the input and output voltages. To leading order in
the coupling amplitudes, we have
$S^{\textrm{out}}(\omega)=|R(\omega)|^{2}S^{\textrm{in}}(\omega)$ and for
$\omega>0$,

$|R(\pm\omega)|^{2}=1\mp\dfrac{2\pi J[\pm\omega]}{N}\sum_{i=1}^{2}\sum_{{\bf
k}}\gamma_{j_{0},{\bf k}}^{i}\delta(\hbar\omega-E_{i,{\bf k}}).$ (8)

For $\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)$, the energies
$E_{i,\bm{K}},\,i\in\\{1,2\\}$ are non-degenerate, therefore, choosing
$\hbar\omega=E_{i,\bm{K}}$ selects only the ${\bf k}={\bf K}$ point in the
integral appearing in the Eq. (8). Moreover, as indicated in Table 4,
$\gamma_{j_{0},{\bf K}}^{i}$ is related to the topological invariant if we
choose a probe at $j_{0}=i$ ($\overline{i}$) for
$\textrm{sgn}\left(\sin\phi\right)=1$
($\textrm{sgn}\left(\sin\phi\right)=-1$). Therefore, for a well-chosen
frequency $\omega$, $|R(\omega)|^{2}$ clearly depends on the topological
invariant. This is also true for
$\textrm{sgn}\,M=-\textrm{sgn}\left(\sin\phi\right)$, if, in the previous
analysis, we replace ${\bf K}$ by ${\bf K}^{\prime}$ and $j_{0}$ by
$\overline{j_{0}}$.

#### Finite lifetimes for the light modes.—

Eventually, we address the more realistic scenario in which we incorporate
finite lifetimes for both the modes in the probe $\ket{b_{q}}$ and the Chern
insulator’s modes $\ket{u_{i,{\bf k}}}$. For simplicity, we consider the same
bandwidth amplitude $\Delta_{\textrm{CI}}$ ($\Delta_{\textrm{P}}$) for all the
modes $\ket{u_{i,{\bf k}}}$ ($\ket{b_{q}}$). We assume the following ordering
of the energies
$\textrm{max}_{q}(g_{q})\ll\\{\Delta_{\textrm{CI}},\Delta_{\textrm{P}}\\}\ll\\{t_{1},\textrm{min}_{q,q^{\prime}}(|\omega_{q}-\omega_{q^{\prime}}|)\\}$.
We replace the Dirac Delta functions appearing in Eq. (8) by normalized
Gaussian spectral distributions denoted
$G\left(\omega;\overline{\omega},\Delta\right)$ with mean value
$\overline{\omega}$ and standard deviation $\Delta$: $\delta(\hbar\omega-
E_{i,{\bf k}})$ is replaced by $G\left(\omega;E_{i,{\bf
k}}/\hbar,\Delta_{\textrm{CI}}\right)$ and $J[\omega]$ is replaced by
$\tilde{J}[\omega]=2\pi\sum_{q}\left(g_{q}/\hbar\right)^{2}\left[G\left(\omega,\omega_{q},\Delta_{\textrm{P}}\right)-G\left(\omega,-\omega_{q},\Delta_{\textrm{P}}\right)\right]$.
We also consider an input energy spectral density with Gaussian distribution:
$S^{\textrm{in}}(\omega)=G\left(\omega;\omega_{q_{0}},\Delta_{\textrm{P}}\right)$.
For a well chosen $\omega_{q_{0}}$ ($E_{i,\bm{K}}$ or $E_{i,\bm{K}}$),
$|R(\omega)|^{2}$ still depends on the topological invariant because
$\gamma_{j_{0},{\bf k}}^{i}$ is directly related to the Chern number. To
illustrate this point, let us consider the case
$\textrm{sgn}\,M=\textrm{sgn}\left(\sin\phi\right)=1$ for which we choose
$\hbar\omega_{q_{0}}=E_{1,\bm{K}}$ and $j_{0}=i=1$ and we expect
$\gamma_{1,{\bf K}}^{1}=C_{1}$. $S^{\textrm{out}}(\omega)$ depends on
$\gamma_{1,{\bf K}}^{1}$, especially around $\omega=\omega_{q_{0}}$, leading
to a decrease of the output peak’s weight $\int d\omega
S^{\textrm{out}}(\omega)$ compared to the normalized weight of the input peak.
This decrease is given by

$1-\int d\omega
S^{\textrm{out}}(\omega)=\dfrac{2\pi}{N}\sum_{i=1}^{2}\sum_{{\bf
k}}\gamma_{j_{0},{\bf k}}^{i}I_{i,{\bf k}},$ (9)

with $I_{i,{\bf k}}=\int d\omega\tilde{J}[\omega]G\left(\omega;E_{i,{\bf
k}}/\hbar,\Delta_{\textrm{CI}}\right)G\left(\omega;E_{1,{\bf
K}}/\hbar,\Delta_{\textrm{P}}\right)$, which is
$\left(g_{q_{0}}/\hbar\right)^{2}/\left(\sqrt{2\pi}\Delta_{\textrm{CI}}\Delta_{\textrm{P}}^{2}\right)$
times the overlap area $\int d\omega\,\textrm{exp}{-\dfrac{\left(\omega-
E_{1,{\bf
K}}/\hbar\right)^{2}}{\Delta_{\textrm{P}}^{2}}}\,\textrm{exp}{-\dfrac{\left(\omega-
E_{i,{\bf k}}/\hbar\right)^{2}}{2\Delta_{\textrm{CI}}^{2}}}$ . $1-\int d\omega
S^{\textrm{out}}(\omega)$ reproduces the topological phase diagram associated
to the Haldane model, as we show in Fig. 1(d), from a numerical evaluation of
Eq. (9), with the energy scales $\Omega_{0}=10$ GHz, $t_{1}/\hbar=100$ MHz,
$\Delta_{\textrm{CI}}=\Delta_{\textrm{P}}=10$MHz and $g_{q_{0}}/\hbar=1$MHz.
These scales corresponds to a relatively low quality factor
$Q=\Omega_{0}/\Delta_{\textrm{CI}}=10^{3}$ (here the same for both the cQED
Chern insulator and the probe) and a low coupling amplitude and should be
reachable in a cQED experiment.

#### Magnon system and material suggestions.—

The results described here are directly relevant for other Chern insulator
systems. For instance, our probe can be used for a topological magnon
insulator, as the one proposed in Ref. [55], which is described by a bosonic
Haldane model. Indeed, consider a magnetic tip described by a polarization
state denoted ${\bf s}=(s^{x},s^{y},s^{z})$, and assume the tip thin enough to
couple to only one magnetic site ${\bf S}_{i}$ of the system. The Hamiltonian
coupling the tip to the magnon system is $H_{\textrm{cpl}}=J\,{\bf
S}_{i}\cdot{\bf s}_{i}$, with $J$ the coupling amplitude. Suppose now the tip
is polarized along $x$ only. Then $H_{\textrm{cpl}}=J\,{\bf S}_{i}\cdot{\bf
s}_{i}=(J/4)(S_{i}^{+}+S_{i}^{-})(s^{+}+s^{-})$, where $S_{i}^{+}$,
$S_{i}^{-}$, $s^{+}$ and $s^{-}$ are bosonic creation/annihilation operators.
$H_{\textrm{cpl}}$ is completely analogue to the coupling Hamiltonian we
considered for the cQED system. Therefore, measuring the response to a
magnetic excitation provides a way to access the topological number. Such a
measurement should be applicable in topological magnon quantum materials, such
as CrI3 [56] or maybe $\beta$-Cu2V2O7 [57]. Even though the magnon can
condense at the lowest energy mode, thermal excitation shall generate magnon
modes at all energies in the system, namely the ones required for the proposed
probe, with energy around the Dirac point.

#### Remarks.—

Several observations are in order.

(i) Topological insulators are robust against weak disorder. Then we expect
the general structure of the wave function over space, which is related to the
bulk invariant, to be robust against disorder. This central feature give
robustness to the probe proposed in this letter. In a magnon system, even
though the disorder could be high compared to cQED systems, a magnetic tip
enables to scan several sites in the sample; averaging the output signal over
these sites would mitigate disorder effects.

(ii) It is interesting to observe that the probe is able to measure the
topological number based on a real space local coupling to the system and with
a resolution in reciprocal space thanks to the energy conservation, similarly
to circularly polarized light [28].

(iii) We checked the applicability of the probe proposed here to a simple
topological bosonic kagome system.

(iv) From Eqs. (7) and (8), we notice that the energy density of states at
$\hbar\omega=E_{i,\bm{k}}$ can be evaluated by summing the local responses
$1-\textrm{Re}\left[R(\omega)\right]$ measured separately in a probe at
$j_{0}=1$ (sublattice A) and in a probe at $j_{0}=2$ (sublattice B).

(v) If the input is triggered at a site with position ${\bf r}$ lying on the
sublattice $j_{0}$ and the output is measured at a different site with
position ${\bf r}^{\prime}$ and sublattice index $j_{0}^{\prime}$ , the
expression in the summation of Eq. (6) is replaced by

$\dfrac{\left[\beta_{j_{0}}^{i}({\bf k})\alpha_{i}^{j_{0}^{\prime}}({\bf
k})\right]^{*}\textrm{e}^{i{\bf k}\left({\bf r}-{\bf
r}^{\prime}\right)}}{-\hbar\omega-E_{i,{\bf
k}}+i0^{+}}-\dfrac{\beta_{j_{0}}^{i}({\bf k})\alpha_{i}^{j_{0}^{\prime}}({\bf
k})\textrm{e}^{-i{\bf k}\left({\bf r}-{\bf
r}^{\prime}\right)}}{-\hbar\omega+E_{i,{\bf k}}+i0^{+}},$ (10)

with $\beta_{j_{0}}^{i}({\bf k})\alpha_{i}^{j_{0}}({\bf k})=\gamma_{j_{0},{\bf
k}}^{i}$ and for $j_{0}^{\prime}=\overline{j_{0}}\neq j_{0}$,
$\beta_{j_{0}}^{i}({\bf k})\alpha_{i}^{\overline{j_{0}}}({\bf k})\propto
h_{1}({\bf k})/2\epsilon({\bf k})$. At the Dirac points, $h_{1}$ is vanishing,
therefore, in the case $j_{0}^{\prime}=\overline{j_{0}}$, the simple protocol
we sketched above does not help to rebuild the topological phase diagram. This
outcome can be anticipated based on the fact that, at one given Dirac point,
one of both Bloch eigenvectors’ components vanish, in the topological regime.
In the scenario $j_{0}^{\prime}=j_{0}$, because the coefficients in the
numerator of Eq. (10) are complex-valued, the Chern number dependency of
$1-\int d\omega S^{\textrm{out}}(\omega)$ is mitigated. Indeed, the latter
contains principal values of integrals over frequency involving
$\left[1/\left(-\hbar\omega\pm E_{i,{\bf k}}\right)\right]$ terms.

#### Conclusion.—

We have considered a local microwave-light probe with capacitive coupling to a
cQED array described by a Haldane bosonic system, in the regime of small
coupling amplitudes. We have explained how this probe is relevant for the
detection of the topological character of Chern insulators. Firstly, using
FGR, we established a connection between the Chern number and the transition
rate from a probe’s eigenstate (with frequency corresponding to one of the
Dirac points energy) to the eigenstates of the Haldane Hamiltonian. Secondly,
we developed the input-output theory for the probe, enabling us to compute the
reflection coefficient which relates an input voltage and an output voltage.
We showed that for an input with frequency resolved at one of the Dirac
points, this reflection coefficient is directly related to the system’s
topological invariant. The fundamental working principle of this probe makes
it inherently relevant for all Chern insulators. Additionally, as a future
prospect, it appears both possible and intriguing to adapt this probe to other
systems that may exhibit different particle statistics, such as cold atoms or
various material platforms.

#### Acknowledgments.—

This work was supported by the french ANR BOCA grant. JL acknowledges support
from the National Research Fund Luxembourg under Grant No.
INTER/QUANTERA21/16447820/MAGMA.

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Supplemental Material

In this Supplemental Material, we show, in Sec. S1, the details of the Chern
number’s computation used in the main text, in relation with the Bloch
eigenvectors’ components properties over the BZ. In Sec. S2, we show the
computation of the coefficient which relates input voltages in a set of probes
capacitively coupled to a bosonic lattice model and the resulting output
voltage in a one of these probes.

##  S1 Chern number, gauge choices and unique and smooth Berry gauge fields

In this section we show a detailed analytical computation of the Chern number,
based on an approach introduced by Kohmoto [51], for the Haldane system with
Hamiltonian defined in the Eq. (1). This computation relies on the definition
of two distinct gauge choices $G_{I}$ and $G_{II}$ for the Bloch eigenvectors
$\ket{u_{i,{\bf k}}}=\left[\alpha_{i}^{1}({\bf k})a_{1,{\bf
k}}^{\dagger}+\alpha_{i}^{2}({\bf k})a_{2,{\bf k}}^{\dagger}\right]\ket{0}\,,$
that we respectively write $\ket{u_{i,{\bf k},I}}$ and $\ket{u_{i,{\bf
k},II}}$.

(i) Gauge choice $G_{I}$: the coefficient $\alpha_{i}^{1}({\bf k})$ is real.
More precisely, we choose

$\alpha_{i}^{1}({\bf k})=\rho_{i}({\bf k})\,,$ (S1)

with

$\rho_{i}({\bf k})=\dfrac{h_{2}({\bf k})+(-1)^{i}\epsilon({\bf
k})}{2\left[\epsilon({\bf k})^{2}+(-1)^{i}h_{2}({\bf k})\epsilon({\bf
k})\right]^{1/2}},$ (S2)

and we have

$\alpha_{i}^{2}({\bf k})=\lambda_{i}({\bf k})\textrm{e}^{-i\varphi({\bf
k})}\,,$ (S3)

with

$\lambda_{i}({\bf k})=\dfrac{|h_{1}({\bf k})|}{2\left[\epsilon({\bf
k})^{2}+(-1)^{i}h_{2}({\bf k})\epsilon({\bf k})\right]^{1/2}},$ (S4)

and

$\textrm{e}^{-i\varphi({\bf k})}=\dfrac{\textrm{e}^{i{\bf k}\cdot{\bf
a}_{3}}h_{1}({\bf k})^{*}}{|h_{1}({\bf k})|}.$ (S5)

(ii) Gauge choice $G_{II}$: the coefficient $\alpha_{i}^{2}({\bf k})$ is real.
We choose $\ket{u_{j,{\bf k},II}}=\textrm{e}^{i\varphi({\bf k})}\ket{u_{j,{\bf
k},I}}$, i.e.

$\alpha_{i}^{2}({\bf k})=\lambda_{i}({\bf k}),$ (S6)

and we have

$\alpha_{i}^{1}({\bf k})=\rho_{i}({\bf k})\textrm{e}^{i\varphi({\bf k})}.$
(S7)

Table SI: Table giving the values of of $\rho_{i}({\bf k})$ and $\lambda_{i}({\bf k})$ at the Dirac points in the case $|M|<3\sqrt{3}t_{2}|\sin\phi|$ , as a function of the sign of the Semenoff mass $M$ and of the index of the energy band $i$. | $\textrm{sgn}(\sin\phi)=(-1)^{i}\,$ | $\textrm{sgn}(\sin\phi)=(-1)^{i+1}\,$
---|---|---
$\rho_{i}({\bf k}\rightarrow{\bf K})$ | $0$ | $(-1)^{i}$
$\rho_{i}({\bf k}\rightarrow{\bf K}^{\prime})$ | $(-1)^{i}$ | $0$
$\lambda_{i}({\bf k}\rightarrow{\bf K})$ | $1$ | $0$
$\lambda_{i}({\bf k}\rightarrow{\bf K}^{\prime})$ | $0$ | $1$
Table SII: Table giving the values of of $\rho_{i}({\bf k})$ and $\lambda_{i}({\bf k})$ at the Dirac points in the case $|M|>3\sqrt{3}t_{2}|\sin\phi|$, as a function of the sign of the Semenoff mass $M$ and of the index of the energy band $i$. | $\textrm{sgn}\,M=(-1)^{i}\,$ | $\textrm{sgn}\,M=(-1)^{i+1}\,$
---|---|---
$\rho_{i}({\bf k}\rightarrow{\bf K})$ | $(-1)^{i}$ | $0$
$\rho_{i}({\bf k}\rightarrow{\bf K}^{\prime})$ | $(-1)^{i}$ | $0$
$\lambda_{i}({\bf k}\rightarrow{\bf K})$ | $0$ | $1$
$\lambda_{i}({\bf k}\rightarrow{\bf K}^{\prime})$ | $0$ | $1$

The phase of the wavefunction is chosen by requiring that either the
coefficient $\alpha_{i}^{1}({\bf k})$ or the coefficient $\alpha_{i}^{2}({\bf
k})$ is real (and non-zero). $\rho_{i}({\bf k})$ and $\lambda_{i}({\bf k})$
may vanish only at the Dirac points and it depends on the sign of $h_{2}$,
given in Table 1 as a function of $M$ and $\phi$. For
$|M|<3\sqrt{3}t_{2}|\sin\phi|$ ($|M|>3\sqrt{3}t_{2}|\sin\phi|$), the values of
$\rho_{i}({\bf k})$ and $\lambda_{i}({\bf k})$ at the Dirac points are given
in Table SI (Table SII).

(i) In the case $|M|<3\sqrt{3}t_{2}|\sin\phi|$, we notice that (see Table SI),
for all the values of $\phi$, none of both gauge choices $G_{I}$ and $G_{II}$
is well-defined in the entire BZ. Indeed, $\rho_{i}({\bf k})$ and
$\lambda_{i}({\bf k})$ each vanish at one of the Dirac points. As explained in
the main text, we then define two non-overlapping domains in the BZ [51] and
use a different gauge choice in each domain. To be more specific, for
$(-1)^{i}\,\textrm{sgn}(\sin\phi)=+1$, we apply $G_{I}$ for the points
contained in $\mathcal{D}_{II}$ and $G_{II}$ for the points contained in
$\mathcal{D}_{I}$ while for $(-1)^{i}\,\textrm{sgn}(\sin\phi)=-1$, we apply
$G_{I}$ for the points contained in $\mathcal{D}_{I}$ and $G_{II}$ for the
points contained in $\mathcal{D}_{II}$. Then the phase of $\ket{u_{i,{\bf
k},I}}$ (the eigenstate in $\mathcal{D}_{I}$) and $\ket{u_{i,{\bf k},II}}$
(the eigenstate in $\mathcal{D}_{II}$) and the Berry gauge fields ${\bf
A}_{i,{\bf k},I}=\bra{u_{i,{\bf k},I}}{\bf\nabla}_{{\bf k}}\ket{u_{i,{\bf
k},I}}$ and ${\bf A}_{i,{\bf k},II}=\bra{u_{i,{\bf k},II}}{\bf\nabla}_{{\bf
k}}\ket{u_{i,{\bf k},II}}$ are uniquely and smoothly defined respectively on
$\mathcal{D}_{I}$ and $\mathcal{D}_{II}$. The Chern number associated to the
$j^{\textrm{th}}$ band (remind that $j=1$ or $j=2$) reads

$C_{j}=\dfrac{1}{2i\pi}\bigg{[}\int_{\mathcal{D}_{\textrm{I}}}d^{2}k\left({\bf\nabla}_{{\bf
k}}\times{\bf A}_{j,{\bf k},\textrm{I}}\right)\cdot{\bf
e}_{z}+\int_{\mathcal{D}_{\textrm{II}}}d^{2}k\left({\bf\nabla}_{{\bf
k}}\times{\bf A}_{j,{\bf k},\textrm{II}}\right)\cdot{\bf e}_{z}\bigg{]}.$ (S8)

Using Stokes’ theorem and considering the specific choice of path P shown in
Fig. 1(c) leads to

$C_{j}=\dfrac{1}{2i\pi}\left(\oint_{\textrm{P}}d{\bf k}\cdot{\bf A}_{j,{\bf
k},\textrm{I}}-\oint_{\textrm{P}}d{\bf k}\cdot{\bf A}_{j,{\bf
k},\textrm{II}}\right).$ (S9)

Along P, we have $\ket{u_{j,{\bf
k},I}}=\textrm{e}^{i(-1)^{j}\,\textrm{sgn}(\sin\phi)\varphi({\bf
k})}\ket{u_{j,{\bf k},II}}$. We choose $\varphi({\bf k})$ so that it is smooth
along the whole P path and we obtain

$C_{j}=\dfrac{(-1)^{j}\,\textrm{sgn}(\sin\phi)}{2\pi}\oint_{\textrm{P}}d{\bf
k}\cdot{\bf\nabla}_{{\bf k}}\varphi({\bf k}).$ (S10)

$C_{i}$ is found by studying how $\varphi({\bf k})$ evolves when moving along
P. Generally speaking, when the P path surround a $\varphi({\bf k})$’s
divergence (here at the ${\bf K}$ or ${\bf K}^{\prime}$ point), the
accumulated phase increases or decreases by $\pm 2\pi z,\,z\in\mathbb{Z}$,
which gives a quantized value of $C$, as expected. Here, we find that
$\varphi({\bf k})$ changes by $-2\pi$ when moving along the entire closed path
P. One can build intuition from small displacement $\delta{\bf k}$ around $\bf
K$, in which case we have $h_{1}({\bf K}+\delta{\bf k})=3t_{1}(-\delta
k_{x}+i\delta k_{y})/2$ where $\delta k_{x}=\delta{\bf k}\cdot{\bf e}_{x}$,
$\delta k_{y}=\delta{\bf k}\cdot{\bf e}_{y}$, ${\bf e}_{x}=-\left({\bf
b}_{2}+{\bf b}_{3}\right)/\sqrt{3}$ and ${\bf e}_{y}=\left({\bf b}_{2}-{\bf
b}_{3}\right)/3$. In this case, we easily see that $\varphi$, which is defined
by $\textrm{e}^{-i\varphi({\bf k})}=\textrm{e}^{i{\bf k}\cdot{\bf
a}_{3}}h_{1}({\bf k})^{*}/|h_{1}({\bf k})|$, varies by $-2\pi$, for one entire
closed path P around $\bf K$ (oriented anticlockwise).

(iii) In the case $|M|>3\sqrt{3}t_{2}|\sin\phi|$, we gathered the values of
$\rho_{i}({\bf k})$ and $\lambda_{i}({\bf k})$ at the Dirac points in Table
SII. At both Dirac points, either $\rho_{i}({\bf k})$ or $\lambda_{i}({\bf
k})$ are non-vanishing, therefore it is possible to find a unique and smooth
phase for the ket $\ket{u_{i,{\bf k}}}$ everywhere in the BZ which leads to a
unique and smooth Berry gauge field ${\bf A}_{{\bf k}}$. Depending on the
value of $(-1)^{i}\,\textrm{sgn}(M)$, we apply gauge choice $G_{I}$ or
$G_{II}$ for all the points of the BZ, and then we can show that the
associated wave function $\ket{u_{i,{\bf k},I}}$ or $\ket{u_{i,{\bf k},II}}$
(and its phase) is uniquely and smoothly defined, as is the Berry gauge field.

##  S2 Local response to capacitively coupled probes in a two-dimensional
lattice bosonic system

In this section, we consider a set of bosonic probes (typically microwave
light resonators) capacitively coupled to a bosonic lattice model. We derive
the relation between the output voltage operator in a probe at a certain
position on the lattice and the input voltage operators associated to the
ensemble of probes on the lattice. We consider a 2-dimensional lattice system
with periodic boundary conditions.

###  S2.1 Hamiltonian

The Hamiltonian describing the lattice model with capacitively coupled probes
reads

$H=H_{\textrm{lat}}+H_{\textrm{prb}}+H_{\textrm{cpl}},$ (S11)

where $H_{\textrm{lat}}$ is the (topological) lattice Hamiltonian,
$H_{\textrm{prb}}$ is the Hamiltonian associated to the probe(s) and
$H_{\textrm{cpl}}$ is the Hamiltonian associated to the coupling between the
lattice and the probe(s).

Let us call $N_{C}$ the number of sites per unit cell in the lattice we
consider and $N$ the total number of unit cells. We label each site within a
unit cell with different colors, and two different sites belonging to the same
Bravais lattice are labeled by the same color. We define the Fourier transform
of the annihilation operator of a (bosonic) particle at position $\bf r$ and
the inverse relation

$a_{j,{\bf k}}=\sum_{{\bf r}\in R_{j}}\textrm{e}^{-i{\bf k}.{\bf r}}a_{{\bf
r}}\quad\textrm{and}\quad a_{{\bf r}}=\dfrac{1}{N}\sum_{{\bf
k}}\textrm{e}^{i{\bf k}.{\bf r}}a_{j({\bf r}),{\bf k}},$ (S12)

with $R_{j}$ the ensemble containing all the lattice positions of the
color-$j$ sites and the function $j({\bf r})$ returns the color index at the
${\bf r}$ site. We formally write

$H_{\textrm{lat}}=\sum_{\bm{k}}\Psi_{\bm{k}}^{\dagger}h_{\bm{k}}\Psi_{\bm{k}},$
(S13)

with $\Psi_{{\bf k}}^{\dagger}=\left(a_{1,{\bf k}}^{\dagger},\dots a_{N,{\bf
k}}^{\dagger}\right)$. We write $h_{{\bf k}}$’s associated eigenvalues
$E_{i,{\bf k}}$, where $i\in\\{1,\dots,N_{C}\\}$, and we write the associated
eigenvectors

$\ket{\Phi_{i,{\bf k}}}=\Phi_{i,{\bf
k}}^{\dagger}\ket{0}=\sum_{j=1}^{N_{C}}\alpha_{i}^{j}({\bf k})a_{j,{\bf
k}}^{\dagger}\ket{0},$ (S14)

with $\alpha_{i}^{j}({\bf k})\in\mathbb{C}$. We have

$H_{\textrm{lat}}=\sum_{{\bf k}}\sum_{i=1}^{N_{C}}E_{i,{\bf k}}\Phi_{i,{\bf
k}}^{\dagger}\Phi_{i,{\bf k}}.$ (S15)

Each probe are resonators with a certain number of relevant modes, each mode
$q$ being characterized by the frequency $\omega_{q}$. Therefore we write

$H_{\textrm{prb}}=\sum_{{\bf r}\in
R_{p}}\sum_{q}\hbar\omega_{q}b_{\bm{r},q}^{\dagger}b_{{\bf r},q},$ (S16)

with $b_{\bm{r},q}$ the annihilation operators for the mode $q$ of the probe
at $\bm{r}\in R_{p}$, $R_{p}$ being the ensemble of the positions of the nodes
coupled to a probe. We assume a capacitive coupling between each probe and a
node of the lattice. The Hamiltonian reads

$H_{\textrm{cpl}}=\sum_{{\bf r}\in R_{p}}\left(a_{\bf r}+a_{{\bf
r}}^{\dagger}\right)\sum_{q}g_{q}\left(b_{{\bf r},q}+b_{{\bf
r},q}^{\dagger}\right).$ (S17)

The coupling amplitude $g_{q}$ is assumed not to depend on the position of the
probe.

###  S2.2 Input-output analysis

Here we use the input-output formalism as reviewed in Ref. [58]. Let us define
the input voltage in the probe at $\bf r$

$V_{\bf
r}^{\textrm{in}}(t)=\sum_{q}g_{q}\left[\textrm{e}^{-i\omega_{q}(t-t_{i})}b_{{\bf
r},q}(t_{i})+h.c.\right],$ (S18)

where $t_{i}<t$ is an initial time in the distant past, and the output voltage
in the probe at $\bf r$

$V_{\bf
r}^{\textrm{out}}(t)=\sum_{q}g_{q}\left[\textrm{e}^{-i\omega_{q}(t-t_{f})}b_{{\bf
r},q}(t_{f})+h.c.\right],$ (S19)

where $t_{f}>t$ is a final time in the distant future. Let us call $x_{\bf
r}=a_{\bf r}+a_{{\bf r}}^{\dagger}$. The Heisenberg equation of motion (EOM)
for the $b_{jq}$ operator reads

$\dot{b}_{{\bf r},q}=\dfrac{i}{\hbar}\left[H,b_{{\bf
r},q}\right]=-i\omega_{q}b_{{\bf r},q}-\dfrac{ig_{q}}{\hbar}x_{\bf r}.$ (S20)

The solution of this equation of motion is

$b_{{\bf r},q}(t)=\textrm{e}^{-i\omega_{q}(t-t_{i})}b_{{\bf
r},q}(t_{i})-\dfrac{ig_{q}}{\hbar}\int_{t_{i}}^{t}d\tau\textrm{e}^{-i\omega_{q}(t-\tau)}x_{\bf
r}(\tau),$ (S21)

or equivalently

$b_{{\bf r},q}(t)=\textrm{e}^{-i\omega_{q}(t-t_{f})}b_{{\bf
r},q}(t_{f})+\dfrac{ig_{q}}{\hbar}\int_{t}^{t_{f}}d\tau\textrm{e}^{-i\omega_{q}(t-\tau)}x_{\bf
r}(\tau).$ (S22)

Combining the previous equations and their complex conjugate counterparts we
get

$V_{\bf
r}^{\textrm{in}}(t)-2\sum_{q}\dfrac{g_{q}^{2}}{\hbar}\int_{t_{i}}^{t_{f}}d\tau\sin\left[\omega_{q}(t-\tau)\right]x_{\bf
r}(\tau)=V_{\bf r}^{\textrm{out}}(t).$ (S23)

We Fourier transform the previous equation with respect to the time variable
$t$, we define
$J(t)=2i\sum_{q}\dfrac{g_{q}^{2}}{\hbar}\sin\left(\omega_{q}t\right)$ [59] and
we get

$V_{\bf r}^{\textrm{out}}[\omega]=V_{\bf
r}^{\textrm{in}}[\omega]+iJ[\omega]x_{\bf r}[\omega].$ (S24)

Notice that $J[\omega]\in\mathbb{R}$. Its explicit expression is

$J[\omega]=2\pi\sum_{q}\dfrac{g_{q}^{2}}{\hbar}\left[\delta(\omega-\omega_{q})-\delta(\omega+\omega_{q})\right],$
(S25)

where $\delta$ is the Dirac delta function.

Now we express $x_{\bf r}[\omega]$ as a function of $V_{\bf
r}^{\textrm{in}}[\omega]$. The Heisenberg EOM for the $\Phi_{i,{\bf k}}$ modes
reads

$\hbar\dot{\Phi}_{i,{\bf k}}=-iE_{i,{\bf k}}\Phi_{i,{\bf k}}-i\sum_{{\bf r}\in
R_{p}}\left[\alpha_{i}^{j}({\bf k})\textrm{e}^{i{\bf k}\cdot{\bf
r}}\right]^{*}\sum_{q}g_{q}\left(b_{{\bf r},q}+b_{{\bf
r},q}^{\dagger}\right),$ (S26)

where $j$ is a function of ${\bf r}$; it returns the color index at the ${\bf
r}$ site. Using equation S21 we have, up to second order in the coupling
amplitudes,

$i\hbar\dot{\Phi}_{i,{\bf k}}(t)=E_{i,{\bf k}}\Phi_{i,{\bf k}}(t)+\sum_{{\bf
r}\in R_{p}}\left[\alpha_{i}^{j}({\bf k})\textrm{e}^{i{\bf k}\cdot{\bf
r}}\right]^{*}V_{\bf r}^{\textrm{in}}(t).$ (S27)

Now we use the Fourier transformation with respect to the time variable to
write, still up to second order in the coupling amplitudes,

$\Phi_{i,{\bf k}}[\omega]=\dfrac{1}{-\hbar\omega-E_{i,{\bf
k}}+i0^{+}}\sum_{{\bf r}\in R_{p}}\left[\alpha_{i}^{j}({\bf
k})\textrm{e}^{i{\bf k}\cdot{\bf r}}\right]^{*}V_{\bf
r}^{\textrm{in}}[\omega],$ (S28)

and

$\Phi_{i,{\bf k}}^{\dagger}[\omega]=-\dfrac{1}{-\hbar\omega+E_{i,{\bf
k}}+i0^{+}}\sum_{{\bf r}\in R_{p}}\alpha_{i}^{j}({\bf k})\textrm{e}^{i{\bf
k}\cdot{\bf r}}V_{\bf r}^{\textrm{in}}[\omega].$ (S29)

Note that $\Phi_{i,{\bf k}}[\omega]=\textrm{T.F.}\left(\Phi_{i,{\bf
k}}\right)[\omega]$, T.F. denoting the Fourier transform and $\Phi_{i,{\bf
k}}^{\dagger}[\omega]=\textrm{T.F.}\left(\Phi_{i,{\bf
k}}^{\dagger}\right)[\omega]$ so $\Phi_{i,{\bf
k}}[\omega]\neq\left(\Phi_{i,{\bf k}}^{\dagger}[\omega]\right)^{\dagger}$.

We introduce the $\beta_{i}^{j}({\bf k})$ coefficients such that

$a_{j,{\bf k}}^{\dagger}\ket{0}=\sum_{i=1}^{N_{C}}\beta_{i}^{j}({\bf
k})\Phi_{i,{\bf k}}^{\dagger}\ket{0}.$ (S30)

Then we have

$x_{\bf r}=\dfrac{1}{N}\sum_{\bf k}\sum_{i=1}^{N_{C}}\textrm{e}^{i{\bf
k}\cdot{\bf r}}\left[\beta_{j({\bf r})}^{i}({\bf k})\right]^{*}\Phi_{i,{\bf
k}}+h.c.,$ (S31)

and we obtain, up to second order in the coupling amplitudes

$x_{\bf r}[\omega]=\sum_{{\bf r}_{0}\in R_{p}}\chi_{{\bf r},{\bf
r}_{0}}V_{{\bf r}_{0}}^{\textrm{in}}[\omega],$ (S32)

with

$\chi_{{\bf r},{\bf r}_{0}}=\dfrac{1}{N}\sum_{\begin{subarray}{c}i=1\\\ {\bf
k}\end{subarray}}^{N_{C}}\left(\dfrac{C_{i,{\bf k},{\bf r},{\bf
r}_{0}}^{*}}{-\hbar\omega-E_{i,{\bf k}}+i0^{+}}-\dfrac{C_{i,{\bf k},{\bf
r},{\bf r}_{0}}}{-\hbar\omega+E_{i,{\bf k}}+i0^{+}}\right),$ (S33)

and

$C_{i,{\bf k},{\bf r},{\bf r}_{0}}=\beta_{j({\bf r})}^{i}({\bf
k})\alpha^{j({\bf r}_{0})}_{i}({\bf k})\textrm{e}^{-i{\bf k}\cdot({\bf r}-{\bf
r}_{0})}.$ (S34)

Let us notice from the last equation that adding a probe with no input does
not influence the response at the other probes (with or without input). This
is because we restricted the response to first order in $g$.

Finally, Eq. (S24) gives, up to fourth order in the coupling amplitudes
$\\{g_{q}\\}$,

$V_{{\bf r}}^{\textrm{out}}[\omega]=V_{{\bf
r}}^{\textrm{in}}[\omega]+iJ[\omega]\sum_{{\bf r}_{0}\in R_{p}}\chi_{{\bf
r},{\bf r}_{0}}V_{{\bf r}_{0}}^{\textrm{in}}[\omega].$ (S35)

In the simple case ${\bf r}={\bf r}_{0}={\bf R}_{0}$ we consider in the main
text, for simplicity, we define $\gamma_{j_{0},{\bf k}}^{i}$ such that
$\gamma_{j_{0},{\bf k}}^{i}=C_{i,{\bf k},{\bf R}_{0},{\bf
R}_{0}}=\beta_{j_{0}}^{i}({\bf k})\alpha_{i}^{j_{0}}({\bf k})$. Moreover, for
the Haldane system we consider, the Bloch eigenvectors

$\ket{u_{i,{\bf k}}}=u_{i,{\bf k}}^{\dagger}\ket{0}=\left[\alpha_{i}^{1}({\bf
k})a_{1,{\bf k}}^{\dagger}+\alpha_{i}^{2}({\bf k})a_{2,{\bf
k}}^{\dagger}\right]\ket{0},$ (S36)

are defined through the components $\alpha_{i}^{1}({\bf k})$ and
$\alpha_{i}^{2}({\bf k})$, which are given by

$\alpha_{i}^{2}({\bf k})=\dfrac{h_{1}({\bf k})^{*}}{h_{2}({\bf k})-h_{0}({\bf
k})+E_{i,{\bf k}}}\alpha_{i}^{1}({\bf k}),$ (S37)

where $E_{i,{\bf k}}=h_{0}({\bf k})+(-1)^{i}\epsilon({\bf k})$, $\epsilon({\bf
k})=\sqrt{|h_{1}({\bf k})|^{2}+h_{2}({\bf k})^{2}}$ and the coefficients
$\beta_{i}^{j}({\bf k})$ are defined by

$a_{j,\bm{k}}^{\dagger}\ket{0}=\sum_{i=1}^{2}\beta_{i}^{j}({\bf
k})\Phi_{i,\bm{k}}^{\dagger}\ket{0},\,j=\\{A,B\\}.$ (S38)

Using Eqs. (S36), (S37) and (S38), we obtain

$\gamma_{j_{0},{\bf k}}^{i}=\dfrac{1}{2}+\dfrac{(-1)^{j_{0}+i+1}h_{2}({\bf
k})}{2\epsilon({\bf k})}.$ (S39)