# Highly Linear Nonmagnetic Circulator Enabled By A
Temporal Nonreciprocal Phase Shifter

Sajjad Taravati  Current address: The Edward S. Rogers Sr. Department of
Electrical and Computer Engineering, University of Toronto, Toronto, Ontario
M5S 3H7, Canada<EMAIL_ADDRESS>George V. Eleftheriades

###### Abstract

Conventional circulators are made of magnetic ferrites and suffer from a
cumbersome architecture, incompatibility with integrated circuit technology
and inability for high frequency applications. To overcome these limitations,
here we propose a lightweight low-profile non-magnetic circulator comprising a
nonreciprocal time-varying phase shifter. This circulator is composing a
nonreciprocal temporal phase shifter and two reciprocal delay-line-based phase
shifters. The proposed nonreciprocal temporal phase shifter is based on the
generation of time-harmonic signals, enforcing destructive interference for
undesired time harmonics and constructive interference for desired time
harmonics at different locations of the structure. Such a unique task is
accomplished through two phase-engineered temporal loops. The phase and
frequency of these two loops are governed by external signals with different
phases, imparting an effective electronic angular momentum to the system. We
observe large isolation level of greater than 32 dB, a P1dB of +31.7 dBm and
IIP3 of +42.4 dBm. Furthermore, this circulator is endowed with a
reconfigurable architecture and can be directly embedded in a conventional
integrated circuit (IC) technology to realize a class of high power handling
and linear IC circulators.

###### keywords:

American Chemical Society, LaTeX

IR,NMR,UV

##  1 Introduction

Circulators are three port nonreciprocal devices and represent key elements
for full-duplex telecommunication transceivers 1, 2, 3. They are used to
separate the transmitter and receiver paths by introducing nonreciprocity
between their ports 4, 3. Conventional circulators are formed by ferrite
magnetic materials, where nonreciprocity is achieved under a magnetic bias
field 2, 5, 6. However, ferrite-based magnetic circulators are bulky, costly,
heavy and are not available at high frequencies. Recently, nonmagnetic
nonreciprocity has spurred a surge of research activity to eliminate the
issues associated with magnetic nonreciprocal devices.

Nonmagnetic nonreciprocity is conventionally achieved through nonreciprocal
properties of transistors at microwave frequencies 7, 8, 9, 10, 11, 12, 13,
and electro-optical modulators at optical frequencies 14. However, these
techniques substitute the absence of cumbersome magnetic bias for other
tangible drawbacks, such as poor noise performance and strong nonlinearities
of transistors, or the complexity and large size of the electro-optical
modulators. Recently, space-time modulation has been shown to be a remarkable
approach to realize nonreciprocal components, especially for integrated
optical networks where it can be fully realized using silicon photonics 15,
16, 17. It is shown that by taking advantage of one-way progressive wave
coupling in a space-time-modulated structure, nonreciprocal devices and
isolators can be created 15, 18, 19, 20, 21, 22, 23. The space-time modulation
technique exhibits appealing capabilities, such as parametric wave
amplification 24, 25, 26, unidirectional beam splitting 27, pure frequency
generation 28, 29, nonreciprocal metasurfaces 30, 31, 32, antennas 33,
nonreciprocal filters 34, and full-transceiver realization 35, 36.

Nevertheless, the space-time modulation technique requires optically long
structures as they rely on the weak and progressive wave coupling, acousto-
optic and electro-optic effects 37, 38, 18, 19, 31. Time-varying circulators
39, 40, 41, 42, 16, 43 have shown to be a good candidate to realize versatile
nonmagnetic wave circulators. They offer outstanding noise performance, and
are compatible with integrated circuit technology and high frequency
applications. Additionally, phase-engineered time modulation has recently been
considered as an alternative approach to realize unique functionalities and
devices 33, 44. Phase-engineered time modulated structure provide a low-
profile and efficient platform for wave-engineering by taking advantage of
asymmetric frequency-phase transitions.

Within this context, this study proposes a low-profile, lightweight and
nonmagnetic circulator based on a nonreciprocal phase shifter at microwaves.
This nonreciprocal phase shifter is constituted of two time-modulated loops,
each of which comprising phase-engineered temporal transmission lines. The
lightweight and low-profile properties originate from the fact that the time
modulation requires only four temporal varactors. Isolation between different
ports is achieved by imposing specified constructive and destructive
interferences at the ports of the circulator. We show that by proper design of
the band structure of the phase-engineered temporal transmission lines, a
desired nonreciprocal phase shift and consequently, strong signal isolation
between the three ports of the circulator can be achieved. The frequency and
phase transition in a time-varying transmission line are provided by varactors
biased by a time-varying voltage. The circulator exhibits strong isolation of
more than 33 dB, highly linear response, that is, 1 dB compression point of
31.7 dBm, broad-band operation where more than 20 dB isolation is introduced
across a 19.2$\%$ fractional bandwidth. This circulator can be realized using
integrated circuit (IC) technology for a much smaller dimension and
programmable mechanism.

##  2 Results

###  2.1 Circulator Design

Figure 1: Schematic illustration of the time-modulated circulator composed of
a nonreciprocal time-varying phase-shifter and two reciprocal phase shifters.

Figure 1 depicts the architecture of the proposed time-modulated circulator.
The circulator is composed of a nonreciprocal time-varying phase shifter
between ports 1 and 2, and two reciprocal phase shifters (e.g., transmission
lines) between ports 1 and 3 and between ports 2 and 3. To compute the
required phase shifts at the three arms of the circulator, we excite a given
port and ensure constructive interference at the next right-hand port, and
destructive interference at the next left-hand port. For instance, we first
excite port 1. Then, a full transmission (constructive interference) from port
1 to port 2 requires

$\phi_{\text{NR}}^{+}=2\phi_{\text{Rec}},$ (1a) whereas a null (destructive
interference) at port 3 may be achieved by satisfying
$\phi_{\text{Rec}}=\phi_{\text{NR}}^{+}+\phi_{\text{Rec}}-\pi,$ (1b) where
$\phi_{\text{NR}}^{+}$ is the forward phase shift of the nonreciprocal phase
shifter, from left to right, and $\phi_{\text{Rec}}$ is the phase shift
provided by reciprocal phase shifters. Next, we excite port 2, where a full
transmission (constructive interference) from port 2 to port 3 requires
$\phi_{\text{Rec}}=\phi_{\text{NR}}^{-}+\phi_{\text{Rec}},$ (1c) where
$\phi_{\text{NR}}^{-}$ denotes the backward phase shift of the nonreciprocal
phase shifter, from right to left. A null (destructive interference) at port 1
can be satisfied by $\phi_{\text{NR}}^{-}=2\phi_{\text{Rec}}-\pi.$ (1d)

Hence, by setting $\phi_{\text{Rec}}=\pi/2$, $\phi_{\text{NR}}^{+}=\pi$ and
$\phi_{\text{NR}}^{-}=0$, the desired constructive and destructive
interferences between three ports of the circulator in Fig. 1 can be achieved.

The two reciprocal phase shifters may be realized by lumped elements at low
frequencies or using transmission lines at high frequencies. The nonreciprocal
phase shifter is realized by a time-varying lightweight and highly efficient
phase shifter, where the effective electric permittivity of a transmission
line is time modulated.

###  2.2 Temporal Nonreciprocal Phase shifter

Figure 2: Schematic representation of the nonreciprocal phase shifter
integrating two time-modulated loops.

Figure 2 depicts the architecture of the temporal nonreciprocal phase shifter,
formed by two time-varying loops, each of which is composed of two phase-
engineered temporal transmission lines. The left temporal loop is composed of
two (upper and lower) temporal transmission lines, where the capacitance of
the transmission lines is modulated in time considering a $\pi$ phase
difference between the upper and lower temporal transmission lines, i.e.,

$C_{1}(t)=C_{\text{av}}+\delta\cos(\Omega t),$ (2a)
$C_{1}^{\prime}(t)=C_{\text{av}}+\delta\cos(\Omega t+\pi),$ (2b) where
$C_{\text{av}}$ is the average capacitance of the loops and $\delta$ is the
modulation amplitude of the loops. In addition, $\Omega$ denotes the temporal
modulation frequency. The right temporal loop is composed of two (upper and
lower) temporal transmission lines, where the capacitance of the transmission
lines is modulated in time considering a $\pi$ phase difference between the
upper and lower temporal transmission lines and $\phi$ phase difference with
the left loop, i.e., $C_{2}(t)=C_{\text{av}}+\delta\cos(\Omega t+\phi),$ (2c)
$C_{2}^{\prime}(t)=C_{\text{av}}+\delta\cos(\Omega t+\phi+\pi),$ (2d)

where $\phi$ represents the modulation phase difference between the first and
second loops. Next, by proper design of the time modulation of the two loops,
a desired nonreciprocal phase shift is achieved. These two temporal loops are
designed to provide appropriate constructive interference for the desired time
harmonics and destructive interference for the undesired time harmonics.

Figure 3 shows a schematic representation of the dispersion diagram for
asymmetric frequency-phase transitions in a temporal transmission line. Here,
a frequency up-conversion from $\omega_{0}$ to $\omega_{0}+\Omega$ is
accompanied by a positive additive phase, i.e., $+\phi_{1}$ in the forward
transmission and $+\phi_{2}$ in the backward transmission. However, a
frequency down-conversion from $\omega_{0}+\Omega$ to $\omega_{0}$ is
accompanied by a negative additive phase, i.e., $-\phi_{2}$ in the forward
transmission and $-\phi_{1}$ in the backward transmission.

Figure 3: Schematic representation of forward and backward frequency-phase
transitions in two cascaded temporal transmission line.

To best understand the effect of the phase and amplitude of the time
modulation on the output voltage of the temporal transmission line, we first
investigate variation of the voltage through a temporal-varactor-loaded
transmission line. It may be shown that, by proper engineering of the
dispersion of the transmission line, the output voltage acquires a frequency
transition (up/down-conversion) accompanied by a phase transition. We assume
the dispersion bands of the structure are engineered so that only the
fundamental and the first higher order time harmonics are excited, whereas all
higher order time harmonics are suppressed. Hence, the voltage is defined
based on the superposition of the $n=0$ and $n=-1$ space-time harmonics
fields, i.e.,

$V_{\text{S}}(z,t)=v_{0}(z)e^{-i\left(k_{z}z-\omega_{0}t\right)}+v_{1}(z)e^{-i\left(k_{z}z-(\omega_{0}+\Omega)t\right)}.$
(3)

Equation (3) considers an identical wave number $k_{0}$ for both the
fundamental and the higher-order harmonics as the transmission line is
modulated only in time, and not in space. It may be seen from Fig. 3 that, as
a result of the temporal sinusoidally periodic capacitance, the dispersion
diagram of the temporal transmission line is periodic with respect to the
$\omega$ axis 45, that is $k(\omega_{0}+\Omega)=k(\omega_{0})=k_{0}$. Such a
temporal periodicity leads to vertical electromagnetic transitions in the
dispersion diagram, as shown in Fig. 3. The unknown spatially variant
amplitudes $v_{0}(z)$ and $v_{1}(z)$ are to be found through satisfying both
the Telegrapher’s equations and initial conditions at $z=0$. We then consider
a lossless transmission line with time-varying capacitance, the Telegrapher’s
equations read

$\frac{\partial V(z,t)}{\partial z}=-L_{0}\frac{\partial I(z,t)}{\partial t},$
(4a) $\frac{\partial I(z,t)}{\partial
z}=-C_{0}\frac{\partial[C_{\text{eq}}(t)V(z,t)]}{\partial t},$ (4b)

where $L_{0}$ and $C_{0}$ are the inductance and capacitance per unit length
of the transmission line, respectively. Equations (4a) and (4b) yield

$\displaystyle\frac{\partial^{2}V(z,t)}{\partial
z^{2}}=\dfrac{1}{v_{\text{p}}^{2}}\frac{\partial^{2}[C_{\text{eq}}(t)V(z,t)]}{\partial
t^{2}},$ (5)

where $v_{\text{p}}=1/\sqrt{L_{0}C_{0}}$.

We then insert the voltage in (3) into (5) and take the space and time
derivatives, while considering a slowly varying envelope approximation, and
apply $\int_{0}^{2\pi/\Omega}dt$ to both sides of the resultant equation,
which yields

$\begin{split}\frac{dv_{0}(z)}{dz}=\frac{ik_{0}\delta}{4C_{\text{av}}}e^{-i\phi}v_{1}(z),\end{split}$
(6)
$\begin{split}\frac{dv_{1}(z)}{dz}=\frac{ik_{1}^{2}\delta}{4k_{z}C_{\text{av}}}e^{i\phi}v_{0}(z),\end{split}$
(7)

Next, we seek independent differential equations for $a_{0}(z)$ and
$v_{1}(z)$, which yields

$\displaystyle\frac{d^{2}v_{0}(z)}{dz^{2}}-\gamma_{0}\gamma_{1}v_{0}(z)=0,$
(8)
$\displaystyle\frac{d^{2}v_{1}(z)}{dz^{2}}-\gamma_{0}\gamma_{1}v_{1}(z)=0.$
(9)

which are second order differential equations, where $\gamma_{0}=ik_{0}\delta
e^{-i\phi}/4C_{\text{av}}$ and $\gamma_{1}=ik_{1}^{2}\delta
e^{i\phi}/4k_{z}C_{\text{av}}$.

The up-conversion assumes the initial conditions of $v_{0}(0)=A_{0}$ and
$v_{1}(0)=0$, which gives

$v_{0}(z)=A_{0}\cos\left(\dfrac{\delta k_{1}}{4C_{\text{av}}}z\right),$ (10a)
$v_{1}(z)=A_{0}\dfrac{k_{1}}{k_{0}}e^{+i\phi}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right).$ (10b)

Equations (10a) and (10b) show that the change of the frequency and phase of
the up-converted signal corresponds to the frequency and phase of the temporal
modulation signal, $\Omega$ and $+\phi$. In addition, the maximum amplitude of
the up-converted signal $a_{1}(z)$ occurs at $\delta
k_{1}/C_{\text{av}}=2\pi$. This reveals that, one can control the amplitude of
the up-converted signal by changing the modulation amplitude $\delta$.
Following the same procedure, we can determine the input and output voltages
for the down-conversion.

The down-conversion assumes the initial conditions of $v_{0}(0)=0$ and
$v_{1}(0)=A_{1}=\text{Max}[v_{1}(z)]=A_{0}k_{1}/k_{0}\exp(+i\phi_{0})$, which
gives

$v_{1}(z)=A_{1}\cos\left(\dfrac{\delta k_{1}}{4C_{\text{av}}}z\right),$ (11a)
and $v_{0}(z)=A_{1}\dfrac{k_{0}}{k_{1}}e^{-i\phi}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right)=A_{0}e^{i(\phi_{0}-\phi)}\sin\left(\dfrac{\delta
k_{1}}{4C_{\text{av}}}z\right).$ (11b)

Equations (11a) and (11b) show that the change of the frequency and phase of
the down-converted signal corresponds to the frequency and phase of the
temporal modulation signal, $\Omega$ and $-\phi$.

Figure 4: Engineered frequency and phase transitions in the proposed temporal
nonreciprocal phase shifter. (a) Forward signal injection. (b) Backward signal
injection.

Figures 4 and 4 illustrate engineered frequency and phase transitions in the
proposed temporal nonreciprocal phase shifter leading to opposite phase shifts
for forward and backward signal transmissions. The structure is designed is a
way to provide desired constructive and instructive interferences of different
harmonics, and for both forward and backward signal transmissions. Two power
splitters/combiners possessing two quarter-wavelength arms are utilized at two
sides of the structure to attain the desired constructive and destructive
interferences. In addition, a half-wavelength transmission line is placed at
the middle of the structure to separate the two loops from each other and
provide desired constructive and destructive interferences at different time
harmonics. We aim to achieve a two-way complete signal transmission, but with
opposite phase shifts for forward and backward transmissions, by proper
engineering of the forward (Figures 4) and backward ( 4) frequency and phase
transitions in the proposed temporal nonreciprocal phase shifter.

Consider the forward signal injection in Fig. 4. The input signal is injected
to the common port of the left power divider and flows in the upper and lower
arms of the left loop. Next, these two signals make a transition to higher
order time harmonic $\omega_{+1}$ but with the $+\pi$ and $0$ phase shifts in
the upper and lower arms, respectively. Since the half-wavelength transmission
middle interconnector makes a constructive interference at $\omega_{+1}$, the
generated $\omega_{+1}$ time harmonic reaches to the right side loop. This
leads to the generation of $\omega_{0}$ harmonic by the right side loop
(second loop for the forward transmission), possessing identical phase shift
of $-\phi$ in the upper and lower arms of the output of the right side loop.
As a consequence, since the power divider is formed by equal quarter-
wavelength long arms, the signals sum up at the common port of the right side
power divider. This yields a transmission at $\omega_{0}$ with the phase shift
of $-\phi$. We shall stress that, since the higher order time harmonics
$\omega_{+1}$ and $\omega_{-1}$ acquire a $\pi$ phase shift difference at the
two arms of the output (right) power divider, they will form a complete null
at the output of the structure.

Now, consider the backward signal injection shown in Fig. 4. The input signal
is injected to the common port of the right side power divider and flows in
the upper and lower arms of the right loop. Then, these two signals make a
transition to higher order time harmonic $\omega_{+1}$ but with the
$+\phi+\pi$ and $+\phi$ phase shifts in the upper and lower arms,
respectively. Similar to the forward signal transmission case, the half-
wavelength transmission middle interconnector makes a constructive
interference at $\omega_{+1}$ and a destructive interference at $\omega_{0}$.
Hence, the generated $\omega_{+1}$ time harmonic reaches to the left side
loop. This leads to the generation of $\omega_{0}$ harmonic by the left side
loop (second loop for the backward transmission), possessing identical phase
shift of $+\phi$ in the upper and lower arms of the output of the right side
loop. As a result, since the power divider is formed by equal quarter-
wavelength long arms, the signals sum up at the common port of the left side
power combiner. This results in a transmission at $\omega_{0}$ with the phase
shift of $+\phi$ which is the opposite of the phase shift of the forward
direction. Since the higher order time harmonics $\omega_{+1}$ and
$\omega_{-1}$ acquire a $\pi$ phase shift difference at the two arms of the
output (left) power divider, they will form a complete null at the output of
the left power combiner. Such a contrast between the phase shift of the
forward and backward transmissions originates from the asymmetric frequency-
phase transitions in phase-engineered time-modulated structures.

##  3 Experimental Design

Figure 5 shows an implementation of the time-modulated circulator by four
phase-shifted time-varying varactors. Here, four varactors are used for the
creation of a phase-shifted time modulation. Two power splitters feed the four
temporal transmission lines, and eight fixed capacitances are considered for
decoupling of the DC bias and low frequency modulation signal from the
incident and transmitted high frequency incident microwave signal.

Figure 5: Implementation of the time-modulated circulator by four phase-
engineered time-varying varactors.

Figures 6 and 6 show the top and bottom views of the fabricated circulator,
respectively. The circulator is designed at the center frequency of 1.56 GHz
with the modulation frequency of $\Omega=101.1$ MHz and modulation power of 15
dBm. Four BB837 varactors manufactured by Infineon Technologies have been used
for realizing the time modulation. We have utilized a RT6010 substrate with
permittivity $\epsilon_{\text{r}}=10.2$, thickness $h=50$ mil and
$\tan\delta=0.0023$. Port 1-3 are the main high frequency ports of the
circulator (support 1.56 GHz), while two other ports at the bottom of the
structure, i.e., Mod. 1 input and Mod. 2 input, support the low frequency
modulation signal injection with different phase shifts. The two 50 Ohm
coplanar waveguide (CPW) transmission lines at the back of the structure,
shown in Fig. 6, possess a width of W=45 mils and a gap of G=20 mils. We have
used four BB837 varactors manufactured by Infineon Technologies for realizing
the phase-shifted time modulation. We utilized two ZFBT-6GQ+ minicircuits
bias-tees.

Figure 6: A photo of the fabricated time-modulated circulator. (a) Top view.
(b) Bottom view.

Figure 7 shows a photo of the experimental set-up for the measurement of the
unidirectional signal transmission through the temporal circulator, where
$\Omega=2\pi\times 101.1$ MHz. Figures 7 and 7 plot the experimental results
for wave circulation between different ports of the circulator. It may be seen
from the experimental results that an isolation of more than 32 dB between the
ports is achieved, whereas the undesired time harmonics are highly suppressed,
they are more than 40 dB lower than the main signals.

##  4 Discussions

Table 1 summarizes the performance of the fabricated circulator and provides a
comparison between the performance of this circulator and that of recently
proposed circulators with different techniques, including space-time (ST)
modulation, magnetic-ferrite-loaded waveguides, nonlinearity, and transistor-
loaded transmission lines. In this table, IL stands for insertion loss, RL
stands for return loss, and FBW is an acronym for the fractional bandwidth
which is equal to the frequency bandwidth over the center frequency.
Additionally, P1dB is the output 1 dB compression point which shows the
prominent linearity of the circulator. The proposed circulator can be realized
in a much more compact fashion, and may be integrated into chip technology for
high frequencies thanks to the availability of variable capacitors at high
frequencies 46. In addition, in contrast to the transistor-based circulator 47
where nonlinear and low power handling transistors are placed in series with
the incident signal, our circulator is endowed by high power rating as the
varactors are placed in parallel to the incident signal. In our experiment, we
applied up to +33 dBm input signal, and the power rating is expected to exceed
$47$ dBm. Having leveraged the linear properties of the time modulation, the
proposed temporal circulator exhibits an outstanding linear response, where
the P1dB is +31.7 dBm and IIP3 is +45.4 dBm.

Figure 7: Experimental results. (a) A photo of the measurement set up. (b) Power transmission from port 1 to port 2, $P_{21}$, and from port 2 to port 1, $P_{12}$. (c) Power transmission from port 1 to port 3, $P_{31}$, and from port 3 to port 1, $P_{13}$. Table 1: Summary of the efficiency of the time-modulated circulator and comparison with previously reported circulators. |  Technology |  Area |  Freq. (GHz) |  FBW (Isolation) |  Isol. (dB) |  IL (dB), IL${}_{\text{21}}$/IL${}_{\text{13}}$ |  RL (dB) |  P1dB (dBm) |  IIP3 (dBm)
---|---|---|---|---|---|---|---|---|---
This work |  Phase-engineered time modulation |  $0.067\lambda^{2}$ $(2475\text{mm}^{2})$ |  1.56 |  19.2$\%$ (20 dB) |  32 |  7.2/9.1 |  16.2 |  31.7 |  45.4
48 |  Ferrite magnets |  $0.8\lambda^{2}$ $(500\text{mm}^{2})$ |  12 |  33.3$\%$ (10 dB) |  10 |  1 |  10 |  N/A |  N/A
41 |  Time modulated resonators |  $0.006\lambda^{2}$ $(575\text{mm}^{2})$ |  1 |  4$\%$ (20 dB) |  55 |  3.3 |  10.8 |  29 |  33.7
49 |  Chip-space-time modulation |  $0.0003\lambda^{2}$ $(11.76\text{mm}^{2})$ |  1.6 |  18$\%$ (25 dB) |  25 |  2 |  $>10$ |  10.5 |  N/A
40 |  Chip-space-time modulation across delay line |  $0.015\lambda^{2}$ $(2.16\text{mm}^{2})$ |  25 |  18$\%$ (18.5 dB) |  18.5 |  3.3/3.2 |  N/A |  21.5 |  19.9
50 |  Chip-hybrid-coupler N-path |  $0.000006\lambda^{2}$ $(1\text{mm}^{2})$ |  0.725 |  27$\%$ (30 dB) |  30 |  3.1/15 |  N/A |  +5.5 |  14
51 |  Chip-switched TL gyrator |  $0.0001\lambda^{2}$ $(9\text{mm}^{2})$ |  1 |  18.4$\%$ (25 dB) |  25 |  2.07/2.49 |  N/A |  17.3 |  36.5
52 |  Chip-transistor (65-nm CMOS), Delta resonators |  $0.023\lambda^{2}$ $(0.21\text{mm}^{2})$ |  100 |  1.5$\%$ (45 dB) |  46 |  5.6 |  $>10$ |  11.4 |  N/A
53 |  Chip-transistor (180-nm CMOS), Feedforward |  $0.021\lambda^{2}$ $(3.22\text{mm}^{2})$ |  24 |  1$\%$ (15 dB) |  20 |  -12.3/ -22.4 (gain) |  $>8$ |  -19.8 |  -11
54 |  Chip-transistor (180-nm CMOS), Current reuse |  $0.0036\lambda^{2}$ $(0.36\text{mm}^{2})$ |  30 |  7.6$\%$ (15 dB) |  12 |  7.5/5 |  $>6$ |  -7 |  N/A
55 |  Chip-transistor (180-nm CMOS), Cancellation |  $0.0022\lambda^{2}$ $(0.35\text{mm}^{2})$ |  24 |  11.2$\%$ (15 dB) |  30 |  8.5/9 |  $>8.5$ |  -12 |  N/A
56 |  Chip-transistor (90-nm CMOS), Amps |  $0.01\lambda^{2}$ $(0.93\text{mm}^{2})$ |  31.75 |  58$\%$ (15 dB) |  12$\sim$22 |  9/3.5 |  $>3$ |  -3 |  N/A
57 |  Chip-transistor (180-nm CMOS), isolators |  $0.004\lambda^{2}$ $(0.715\text{mm}^{2})$ |  24 |  3$\%$ (15 dB) |  20 |  5.7/5.7 |  $>16$ |  9.5 |  N/A

##  5 Conclusion

In summary, we have proposed a lightweight compact non-magnetic time-varying
circulator. The circulator is based on a time-modulated nonreciprocal phase
shifter. The proposed nonreciprocal phase shifter exhibits great performance
including nonreciprocal phase shift range, and can be adopted for broad-band
operation. The experimental results demonstrate a strong isolation level of
more than 32 dB which can be further improved. Furthermore, the unwanted time
harmonics are 40 dB lower than the main signal. More importantly, this
circulator exhibits a highly linear response, with the P1dB of +31.7 dBm and
the IIP3 of 45.4 dBm. Such promising results and compact profile of the
circulator can be further improved and optimized in future work. Furthermore,
the proposed structure can be redesigned for integration into integrated
circuit technology to achieve much smaller profile and versatile operation.

Author contributions:
S.T. carried out the analytical modeling, numerical simulations, sample
fabrication, and measurements. G.V.E. planned, coordinated, and supervised the
work. All authors discussed the theoretical and experimental aspects and
interpreted the results. All authors contributed to the preparation and
writing of the manuscript. Correspondence and requests for materials should be
addressed to Sajjad Taravati (email: sajjad.taravati@utoronto.ca).

Notes
The authors declare no competing interests.

This work is supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC).

## References

  * Ohm 1956 Ohm, E. A. A broad-band microwave circulator. _IRE Transactions on Microwave Theory and Techniques_ 1956, _4_ , 210–217
  * Fay and Comstock 1965 Fay, C.; Comstock, R. Operation of the ferrite junction circulator. _IEEE Trans. Microw. Theory Tech._ 1965, _13_ , 15–27
  * Linkhart 2014 Linkhart, D. K. _Microwave circulator design_ ; Artech house, 2014
  * Ewing and Weiss 1967 Ewing, S.; Weiss, J. A. Ring circulator theory, design, and performance. _IEEE Trans. Microw. Theory Techn._ 1967, _15_ , 623–628
  * O’Neil and Young 2009 O’Neil, B. K.; Young, J. L. Experimental investigation of a self-biased microstrip circulator. _IEEE Trans. Microw. Theory Techn._ 2009, _57_ , 1669–1674
  * Fan et al. 2012 Fan, F.; Chang, S.-J.; Niu, C.; Hou, Y.; Wang, X.-H. Magnetically tunable silicon-ferrite photonic crystals for terahertz circulator. _Opt. Commun._ 2012, _285_ , 3763–3769
  * Tanaka et al. 1965 Tanaka, S.; Shimomura, N.; Ohtake, K. Active circulators: The realization of circulators using transistors. _Proc. IEEE_ 1965, _53_ , 260–267
  * Smith 1988 Smith, M. A. GaAs monolithic implementation of active circulators. Microw. Sympos. Digest, 1988., IEEE MTT-S Int. 1988; pp 1015–1016
  * Ayasli 1989 Ayasli, Y. Field effect transistor circulators. _IEEE Trans. Magn._ 1989, _25_ , 3242–3247
  * Kalialakis et al. 2000 Kalialakis, C.; Cryan, M. J.; Hall, P. S.; Gardner, P. Analysis and design of integrated active circulator antennas. _IEEE Trans. Microw. Theory Tech._ 2000, _48_ , 1017–1023
  * Carchon and Nanwelaers 2000 Carchon, G.; Nanwelaers, B. Power and noise limitations of active circulators. _IEEE Trans. Microw. Theory Tech._ 2000, _48_ , 316–319
  * Taravati and Eleftheriades 2021 Taravati, S.; Eleftheriades, G. V. Programmable nonreciprocal meta-prism. _Sci. Rep._ 2021, _11_ , 1–12
  * Taravati and Eleftheriades 2021 Taravati, S.; Eleftheriades, G. V. Full-duplex Reflective Beamsteering Metasurface Featuring Magnetless Nonreciprocal Amplification. _Accepted in Nat. Commun., arXiv preprint arXiv:2101.10067_ 2021,
  * Bhandare et al. 2005 Bhandare, S.; Ibrahim, S. K.; Sandel, D.; Zhang, H.; Wust, F.; Noé, R. Novel nonmagnetic 30-dB traveling-wave single-sideband optical isolator integrated in III/V material. _IEEE J. Sel. Top. Quantum Electron._ 2005, _11_ , 417–421
  * Yu and Fan 2009 Yu, Z.; Fan, S. Complete optical isolation created by indirect interband photonic transitions. _Nat. Photonics_ 2009, _3_ , 91 – 94
  * Taravati and Kishk 2020 Taravati, S.; Kishk, A. A. Space-Time Modulation: Principles and Applications. _IEEE Microw. Mag._ 2020, _21_ , 30–56
  * Camacho et al. 2020 Camacho, M.; Edwards, B.; Engheta, N. Achieving asymmetry and trapping in diffusion with spatiotemporal metamaterials. _Nat. Commun._ 2020, _11_ , 1–7
  * Taravati et al. 2017 Taravati, S.; Chamanara, N.; Caloz, C. Nonreciprocal Electromagnetic Scattering from a Periodically Space-Time Modulated Slab and Application to a Quasisonic Isolator. _Phys. Rev. B_ 2017, _96_ , 165144
  * Taravati 2017 Taravati, S. Self-biased broadband magnet-free linear isolator based on one-way space-time coherency. _Phys. Rev. B_ 2017, _96_ , 235150
  * Li et al. 2019 Li, J.; Shen, C.; Zhu, X.; Xie, Y.; Cummer, S. A. Nonreciprocal sound propagation in space-time modulated media. _Phys. Rev. B_ 2019, _99_ , 144311
  * Taravati 2018 Taravati, S. Giant Linear Nonreciprocity, Zero Reflection, and Zero Band Gap in Equilibrated Space-Time-Varying Media. _Phys. Rev. Appl._ 2018, _9_ , 064012
  * Jr. 2021 Jr., H. A. A. Active metasurfaces for time modulation and reconfigurable wavefront control. High Contrast Metastructures X. 2021
  * Taravati and Eleftheriades 2019 Taravati, S.; Eleftheriades, G. V. Generalized Space-Time Periodic Diffraction Gratings: Theory and Applications. _Phys. Rev. Appl._ 2019, _12_ , 024026
  * Tien and Suhl 1958 Tien, P.; Suhl, H. A traveling-wave ferromagnetic amplifier. _Proc. IEEE_ 1958, _46_ , 700–706
  * Tien 1958 Tien, P. Parametric amplification and frequency mixing in propagating circuits. _J. Appl. Phys._ 1958, _29_ , 1347–1357
  * Zhu et al. 2020 Zhu, X.; Li, J.; Shen, C.; Zhang, G.; Cummer, S. A.; Li, L. Tunable unidirectional compact acoustic amplifier via space-time modulated membranes. _Phys. Rev. B_ 2020, _102_ , 024309
  * Taravati and Kishk 2019 Taravati, S.; Kishk, A. A. Dynamic modulation yields one-way beam splitting. _Phys. Rev. B_ 2019, _99_ , 075101
  * Taravati 2018 Taravati, S. Aperiodic Space-Time Modulation for Pure Frequency Mixing. _Phys. Rev. B_ 2018, _97_ , 115131
  * Taravati and Eleftheriades 2021 Taravati, S.; Eleftheriades, G. V. Pure and Linear Frequency-Conversion Temporal Metasurface. _Phys. Rev. Appl._ 2021, _15_ , 064011
  * Zang et al. 2019 Zang, J. W.; Correas-Serrano, D.; Do, J. T. S.; Liu, X.; Alvarez-Melcon, A.; Gomez-Diaz, J. S. Nonreciprocal wavefront engineering with time-modulated gradient metasurfaces. _Phys. Rev. Appl._ 2019, _11_ , 054054
  * Taravati and Kishk 2019 Taravati, S.; Kishk, A. A. Advanced Wave Engineering via Obliquely Illuminated Space-Time-Modulated Slab. _IEEE Trans. Antennas Propagat._ 2019, _67_ , 270–281
  * Taravati and Eleftheriades 2020 Taravati, S.; Eleftheriades, G. V. Four-dimensional wave transformations by space-time metasurfaces. _arXiv preprint arXiv:2011.08423_ 2020,
  * Zang et al. 2019 Zang, J.; Alvarez-Melcon, A.; Gomez-Diaz, J. Nonreciprocal phased-array antennas. _Phys. Rev. Appl._ 2019, _12_ , 054008
  * Wu et al. 2020 Wu, X.; Nafe, M.; Melcón, A. Á.; Gómez-Díaz, J. S.; Liu, X. Frequency Tunable Non-Reciprocal Bandpass Filter Using Time-Modulated Microstrip $\lambda$g/2 Resonators. _IEEE Transactions on Circuits and Systems II: Express Briefs_ 2020,
  * Taravati and Caloz 2017 Taravati, S.; Caloz, C. Mixer-Duplexer-Antenna Leaky-Wave System Based on Periodic Space-Time Modulation. _IEEE Trans. Antennas Propagat._ 2017, _65_ , 442 – 452
  * Taravati and Eleftheriades 2020 Taravati, S.; Eleftheriades, G. V. Space-Time Medium Functions as a Perfect Antenna-Mixer-Amplifier Transceiver. _Phys. Rev. Appl._ 2020, _14_ , 054017
  * Huang and Fan 2011 Huang, X.; Fan, S. Complete all-optical silica fiber isolator via stimulated Brillouin scattering. _J. Light. Technol._ 2011, _29_ , 2267–2275
  * Lira et al. 2012 Lira, H.; Yu, Z.; Fan, S.; Lipson, M. Electrically Driven Nonreciprocity Induced by Interband Photonic Transition on a Silicon Chip. _Phys. Rev. Lett._ 2012, _109_ , 033901
  * Estep et al. 2014 Estep, N. A.; Sounas, D. L.; Soric, J.; Alù, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. _Nat. Phys._ 2014, _10_ , 923–927
  * Dinc et al. 2017 Dinc, T.; Nagulu, A.; Krishnaswamy, H. A Millimeter-Wave Non-Magnetic Passive SOI CMOS Circulator Based on Spatio-Temporal Conductivity Modulation. _IEEE J. Solid-State Circuits_ 2017, _52_ , 3276–3292
  * Kord et al. 2017 Kord, A.; Sounas, D. L.; Alù, A. Magnet-less circulators based on spatiotemporal modulation of bandstop filters in a delta topology. _IEEE Trans. Microw. Theory Techn._ 2017, _66_ , 911–926
  * Nagulu et al. 2018 Nagulu, A.; Dinc, T.; Xiao, Z.; Tymchenko, M.; Sounas, D. L.; Alù, A.; Krishnaswamy, H. Nonreciprocal components based on switched transmission lines. _IEEE Transactions on Microwave Theory and Techniques_ 2018, _66_ , 4706–4725
  * Kadry et al. 2020 Kadry, H. M.; Alhawari, M.; Sounas, D. L. Angular-Momentum Biased Circulator With Locally Generated Modulation. _IEEE Trans. Microw. Theory Techn._ 2020,
  * Taravati and Eleftheriades 2020 Taravati, S.; Eleftheriades, G. V. Full-Duplex Nonreciprocal Beam Steering by Time-Modulated Phase-Gradient Metasurfaces. _Phys. Rev. Appl._ 2020, _14_ , 014027
  * Jorge R Zurita-Sánchez and Cervantes-González 2009 Jorge R Zurita-Sánchez, P. H.; Cervantes-González, J. C. Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function $\epsilon(t)$. _Phys. Rev. A_ 2009, _79_ , 053821
  * Lira et al. 2012 Lira, H.; Yu, Z.; Fan, S.; Lipson, M. Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip. _Phys. Rev. Lett._ 2012, _109_ , 033901
  * Tanaka et al. 1965 Tanaka, S.; Shimomura, N.; Ohtake, K. Active circulators–The realization of circulators using transistors. _Proc. IEEE_ 1965, _53_ , 260–267
  * Martinez et al. 2019 Martinez, L.; Laur, V.; Borja, A. L.; Quéffélec, P.; Belenguer, A. Low Loss Ferrite Y-Junction Circulator Based on Empty Substrate Integrated Coaxial Line at Ku-Band. _IEEE Access_ 2019, _7_ , 104789–104796
  * Qin and Wang 2016 Qin, S.; Wang, Y. E. Broadband parametric circulator with balanced monolithic integrated distributedly modulated capacitors (DMC). 2016 IEEE MTT-S International Microwave Symposium (IMS). 2016; pp 1–4
  * Jain et al. 2018 Jain, S.; Agrawal, A.; Johnson, M.; Natarajan, A. A 0.55-to-0.9 GHz 2.7 dB NF full-duplex hybrid-coupler circulator with 56MHz 40dB TX SI suppression. 2018 IEEE International Solid-State Circuits Conference-(ISSCC). 2018; pp 400–402
  * Nagulu et al. 2020 Nagulu, A.; Chen, T.; Zussman, G.; Krishnaswamy, H. Multi-Watt, 1-GHz CMOS Circulator Based on Switched-Capacitor Clock Boosting. _IEEE J. Solid-State Circuits_ 2020, _55_ , 3308–3321
  * Yang and Gui 2018 Yang, C.; Gui, P. 85–110-GHz CMOS magnetic-free nonreciprocal components for full-duplex transceivers. _IEEE J. Solid-State Circuits_ 2018, _54_ , 368–379
  * Wu et al. 2010 Wu, H.-S.; Wang, C.-W.; Tzuang, C.-K. C. CMOS active quasi-circulator with dual transmission gains incorporating feedforward technique at K-band. _IEEE Trans. Microw. Theory Techn._ 2010, _58_ , 2084–2091
  * Chang et al. 2010 Chang, C.-H.; Lo, Y.-T.; Kiang, J.-F. A 30 GHz Active Quasi-Circulator With Current-Reuse Technique in 0.18$-\mu$m CMOS Technology. _IEEE Microw. Wireless Compon. Lett._ 2010, _20_ , 693–695
  * Huang et al. 2012 Huang, D.-J.; Kuo, J.-L.; Wang, H. A 24-GHz low power and high isolation active quasi-circulator. 2012 IEEE/MTT-S International Microwave Symposium Digest. 2012; pp 1–3
  * Hung et al. 2013 Hung, S.-H.; Cheng, K.-W.; Wang, Y.-H. An ultra wideband quasi-circulator with distributed amplifiers using 90 nm CMOS technology. _IEEE Microw. Wireless Compon. Lett._ 2013, _23_ , 656–658
  * Chang et al. 2015 Chang, J.-F.; Kao, J.-C.; Lin, Y.-H.; Wang, H. Design and Analysis of 24-GHz Active Isolator and Quasi-Circulator. _IEEE Trans. Microw. Theory Techn._ 2015, _63_ , 2638–2649


# Hall transport in the topological non-Hermitian checkerboard lattice

Pedro G. de Oliveiraa and Antônio S. T. Piresa Email<EMAIL_ADDRESS>
###### Abstract

The checkerboard lattice is a two-dimensional non-trivial structure usually
seen as a planar version of the pyrochlore lattice. This geometry supports a
two-band insulating electronic system with Chern topology induced by a complex
hoppping parameter. Inspired by the recent advances in the topology of non-
Hermitian systems, in this work we study a non-Hermitian version of the
topological checkerboard lattice. The complex band structure and Berry
curvature are calculated. In the insulating phase, the Chern number is the
same as in the Hermitian version, but the Hall conductivity is no longer
quantized. The dependence of the Hall conductivity with the non-Hermitian
parameters is investigated. The non-Hermiticity can be seen as a result of
dissipation caused by coupling the system to the environment, so this study
casts light on the topology of open systems in condensed matter physics.

aDepartment of Physics, Federal University of Minas Gerais, Belo Horizonte,
MG, Brazil.
<EMAIL_ADDRESS>
##  1 Introduction

The study of condensed matter systems from a topological point of view has
elucidated interesting novel phenomena in the the last decades, the most
famous example being the concept of topological insulators [1, 2]. The
topological approach was fruitful in a variety of models, like magnetic,
photonic, acoustic and superconducting systems [3, 4, 5, 6, 7, 8]. While
topological effects in regular, energy-conserving Hamiltonians is a
established fact with robust theoretical and experimental evidence, that
investigation in non-conservative open systems is still in progress. One way
to model an open system is with a non-Hermitian Hamiltonian. The non-Hermitian
formalism can be applied to a variety of quantum and classical systems, and
the topology of non-Hermitian systems is an on-going field of study [9, 10,
11, 12, 13]. Specifically in fermionic systems, it is known that the non-
Hermiticity spoils the bulk-edge correspondence of Chern topological
insulators, and some efforts have been done to restore it [14]. Another
consequence is that the Hall conductivity is no longer quantized [15, 16, 17].
For that reason, exploring the interplay between non-Hermiticity and
topological effects on different platforms is a valuable way to elucidate the
physics of open systems.

The checkerboard lattice is a nontrivial two-dimensional lattice structure
with square geometry, which can be seen as a planar version of the pyrochlore
lattice [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. The
experimental realization is challenging, but the structure has already been
discovered in some compounds [34, 35, 36]. The checkerboard lattice has some
interesting properties. For instance, it is a member of a set of structures
known as line-graph lattices, which show flat bands and spatially localized
states [26, 27]. It has also been shown that it can hold variety of
topological states [18, 20, 26, 23, 24, 25, 28, 29, 30, 31, 32, 33]. One
interesting context to study these topological states is in a dissipative
environment, which can be represented by a non-Hermitian Hamiltonian.

Motivated by the non-trivial topology of the fermionic checkerboard lattice,
the main purpose of this work is to study a non-Hermitian version of this
lattice, with a focus on the Hall transport and its response to the non-
Hermitian parameters. This paper is organized as follows: in Section 2 we
discuss the description of open quantum systems with non-Hermitian
Hamiltonians. In section 3 we introduce the checkerboard lattice and show
results for the fermionic Hermitian Hamiltonian. In Section 3 we introduce the
non-Hermitian Hamiltonian and calculate its complex band structure. In Section
5 we calculate the system’s Hall conductivity, and show how it varies in
response to the non-Hermitian parameters. In Section 6 we summarize our
findings.

##  2 Non-Hermitian Hamiltonian for open systems

Non-Hermitian Hamiltonians are a well-known way to describe open quantum
systems, which present mechanisms of gain and loss of energy or particles.
These open systems account for many physical processes, for instance, the
interaction of a quantum system with a surrounding environment or an
experimental apparatus of continuous measures [9]. Regarding the interaction
of a two-band quantum system with its surroundings, one way to describe it is
to locally couple the system to a series of baths (which represents the
environment) [37]. That formalism accounts for a lossy system, where
quasiparticles have finite lifetime due to dissipation. Integrating out the
baths’ degrees of freedom we obtain a Green function of the form
$G\left(\mathbf{k},\epsilon\right)=\left(\epsilon-H\left(\mathbf{k}\right)-\Sigma\left(\mathbf{k},\epsilon\right)\right)^{-1}$.
The term $\Sigma\left(\mathbf{k},\epsilon\right)$ is the self-energy and
contains information about the bath modes. The effective Hamiltonian
$H_{eff}\left(\mathbf{k}\right)$ contains a dissipation term
$H_{diss}=i\left(\gamma_{i}\sigma_{i}\right),$ $i=\left(0,x,y,z\right)$, which
makes it non-Hermitian. Non-Hermitian Hamiltonians have complex eigenvalues
and distinct right and left eigenstates, demanding the use of a biorthogonal
algebra [38, 39]. In the formalism of coupled baths above, the physical
requirement that the system’s density of states is non-negative results in a
constraint to the complex eigenvalues: their imaginary part has to be non-
positive[37].

Regarding the transport properties of quasiparticles, it is established that
the Hall conductivity for a two-dimensional fermionic system can be obtained
from the Matsubara Green function $G$ as [40, 41, 17]:

$\sigma_{xy}=\frac{e^{2}}{h}\frac{\varepsilon_{\mu\nu\rho}}{24\pi^{2}}\int
d^{3}p\,Tr\left[G\frac{\partial G}{\partial p_{\mu}}G\frac{\partial
G}{\partial p_{\nu}}G\frac{\partial G}{\partial p_{\rho}}\right]$ (1)

where we assume $T=0\>K$ so the sum over Matsubara frequencies becomes an
integral. The expression above works for either Hermitians and non-Hermitians
Hamiltonians, provided that the Green function is known. For a Hermitian
Hamiltonian, this Hall conductivity is quantized and gives rise to the integer
Hall effect. That is a consequence of the non-trivial topology of the
Hamiltonian’s Bloch bundle [1, 2]. The quantization is indexed by the Chern
number or TKNN integer [42]. The system is a topological insulator due to the
bulk-edge correspondence. However, for a non-Hermitian system, the self-energy
$\Sigma\left(\mathbf{k},\epsilon\right)$ spoils this quantization even if the
system has non-trivial Chern topology [15, 17]. In this work, this is shown
explicitly in the case of the non-Hermitian checkerboard lattice.

##  3 Hermitian checkerboard lattice

Figure 1: The checkerboard lattice. The inequivalent sites are represented in
blue and red. The NN hopping is complex, with a phase of $+\phi$ in the
direction of the arrows and $-\phi$ against it. The NNN hopping is represent
by the solid vertical and horizontal lines. We chose the lattice parameter as
$a=1$.

To investigate the Hall conductivity of the checkerboard fermionic lattice, we
begin by analyzing the Hermitian case. The checkerboard lattice is represented
in Figure 1, with tight-binding Hamiltonian written as:

$\displaystyle H^{H}$ $\displaystyle=-t\sum_{\langle
i,j\rangle}e^{i\phi_{ij}}\left(c_{i}^{\dagger}c_{j}+h.c.\right)-\sum_{\langle\langle
i,j\rangle\rangle}t^{\prime}_{ij}\left(c_{i}^{\dagger}c_{j}+h.c.\right).$ (2)

Here, $c_{i}^{\dagger}\left(c_{i}\right)$ is the fermion creation
(annihilation) operator at site $i$. The pairs $\langle i,j\rangle$ and
$\langle\langle i,j\rangle\rangle$ represent near-neighbor (NN) and next-near-
neighbor (NNN) sites, respectively. There are two inequivalent sites in the
primitive cell. The NN hopping is complex, carrying a phase factor
$\phi_{ij}=\pm\phi$ with the sign determined by the arrows’ directions in
Figure 1. That term breaks the time-reversal symmetry for $\phi\neq n\pi$
($n\in\mathbb{Z}$) and allows non-trivial topology [25]. The NNN hopping
$t^{\prime}_{ij}=t^{\prime}_{1}\left(t^{\prime}_{2}\right)$ is represented by
solid (dashed) lines. As a minimal model, we take $t^{\prime}_{2}=0$ and call
$t^{\prime}_{1}=t^{\prime}$. In the momentum space we have a two-band system,
and the general Hermitian Hamiltonian is:

$H^{H}=\sum_{k}\psi_{k}^{\dagger}H_{k}^{H}\psi_{k}$ (3)

with
$\psi^{\dagger}=\left(\begin{array}[]{cc}a_{k}^{\dagger}&b_{k}^{\dagger}\end{array}\right)$.
The checkerboard Hamiltonian matrix takes the form

$H_{k}^{H}=d_{0}I+d_{x}\sigma_{x}+d_{y}\sigma_{y}+d_{z}\sigma_{z}$ (4)

with

$\displaystyle d_{0}$
$\displaystyle=-t^{\prime}\left(cos\,k_{x}+cos\,k_{y}\right)$ $\displaystyle
d_{x}$
$\displaystyle=-4t\,cos\,\phi\left(cos\,\frac{k_{x}}{2}\,cos\,\frac{k_{y}}{2}\right)$
$\displaystyle d_{y}$
$\displaystyle=-4t\,sin\,\phi\left(sin\,\frac{k_{x}}{2}\,sin\,\frac{k_{y}}{2}\right)$
$\displaystyle d_{z}$
$\displaystyle=-t^{\prime}\left(cos\,k_{x}-cos\,k_{y}\right).$ (5)

The Hamiltonian’s eigenvalues are

$\epsilon_{\pm}=d_{0}\pm d,\qquad d\equiv\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}$
(6)

and the spectrum is gapped for $\phi\neq m\pi$ ($m\in\mathbb{Z}/2$).

Figure 2: Energy spectrum (real and imaginary parts) of the system for
different values of $\gamma_{z}$. The points of high symmetry are
$\Gamma(0,0),\,X(\pi,0),\,X^{\prime}(0,\pi),\,M(\pi,\pi)$. For $\gamma_{z}=2$,
a hybrid exceptional point (in magenta) is generated in the center of the
$\Gamma-M$ line, which splits in two ordinary EPs for $\gamma_{z}>2$. In this
work, we focus in the $\gamma_{z}<2$ regime, where the bands do not touch and
the Berry curvature is well-defined. The other parameters of the theory are
$t=1.0$, $t^{\prime}=0.5$, $\phi=\pi/4$, $\gamma_{0}=-2.5.$

The Berry curvature of the $n\text{-th}$ band is a local function of the
momenta constructed from the eigenstates and defined as
$\Omega_{n,ij}\left(\mathbf{k}\right)=i\varepsilon_{ij}\left\langle\partial_{i}\psi_{n}\left(\mathbf{k}\right)\middle|\partial_{j}\psi_{n}\left(\mathbf{k}\right)\right\rangle$.
For a two-band system with Hamiltonian written as Eq. 4, the Berry curvature
of the valence band can be written as [1, 2, 43]:

$\Omega_{xy}\left(\mathbf{k}\right)=\hat{\mathbf{d}}\cdot\left(\partial_{x}\hat{\mathbf{d}}\times\partial_{y}\hat{\mathbf{d}}\right)$
(7)

with $\hat{\mathbf{d}}=\mathbf{d}/d$,
$\mathbf{d}=\left(d_{x},d_{y},d_{z}\right).$ From the linear response theory,
the Hall conductivity is given by:

$\sigma_{xy}=-\frac{e^{2}}{h}\int\frac{d^{2}k}{4\pi}\Omega_{xy}\left(\mathbf{k}\right)=-\frac{e^{2}}{h}C.$
(8)

Here, $C\equiv\int\frac{d^{2}k}{4\pi}\Omega_{xy}\left(\mathbf{k}\right)$ is
the Chern number. Hence, the Hall conductivity is quantized, realizing the
integer quantum Hall effect. In the case of a two-band insulator, the Chern
number has an natural geometrical interpretation. As $\mathbf{k}$ spreads over
the Brillouin torus, the parameter vector $\mathbf{d}$ describes a surface.
The Chern number represents the number of times this surface wraps around the
origin (the winding number).

In Hermitian systems, a non-trivial Chern number is related to robust
conducting edge states via bulk-boundary correspondence, making the system a
topological insulator. In the gapped checkerboard lattice, the Chern number is
$C=-1$ if $\phi$ lays in the first and third quadrant, and $C=+1$ for $\phi$
in the second and fourth quadrant. For $\phi=n\pi/2$, the system is ungapped.
The bands touch quadratically at the corners of the Brillouin zone for even
$n$, and in the center for odd $n$.

##  4 Non-Hermitian case

One way of obtaining a non-Hermitian Hamiltonian is to include an anti-
Hermitian diagonal term in the momentum-space Hamiltonian:

$\displaystyle H^{AH}$
$\displaystyle=\sum_{k}\psi_{k}^{\dagger}H_{k}^{AH}\psi_{k}$ (9)

where

$\displaystyle H_{k}^{AH}$
$\displaystyle=i\left(\begin{array}[]{cc}\gamma_{a}&0\\\
0&-\gamma_{b}\end{array}\right)=i\left(\begin{array}[]{cc}\gamma_{0}+\gamma_{z}&0\\\
0&\gamma_{0}-\gamma_{z}\end{array}\right)$ (14)
$\displaystyle=i\left(\gamma_{0}I+\gamma_{z}\sigma_{z}\right).$ (15)

Here, we defined
$\gamma_{0}\equiv\frac{\left(\gamma_{a}-\gamma_{b}\right)}{2}$ and
$\gamma_{z}\equiv\frac{\left(\gamma_{a}+\gamma_{b}\right)}{2}$.

The total non-Hermitian Hamiltonian $H_{k}^{NH}=H_{k}^{H}+H_{k}^{AH}$ can be
written with complex parameters $h_{i}\equiv d_{i}+i\gamma_{i}$ as

$H_{k}=h_{0}I+h_{x}\sigma_{x}+h_{y}\sigma_{y}+h_{z}\sigma_{z}.$ (16)

This is a general form. In our system, the only non-real parameters (which
carry the non-Hermiticity) are $h_{0}=d_{0}+i\gamma_{0}$ and
$h_{z}=d_{z}+i\gamma_{z}$. Diagonalizing the Hamiltonian we obtain the
eigenvalues:

$\displaystyle\epsilon_{\pm}$ $\displaystyle=h_{0}\pm h$ (17)

with

$\displaystyle h$
$\displaystyle=\sqrt{h_{x}^{2}+h_{y}^{2}+h_{z}^{2}}=\sqrt{d_{x}^{2}+d_{y}^{2}+\left(d_{z}+i\gamma_{z}\right)^{2}}$
$\displaystyle=h^{R}+ih^{I}.$ (18)

The eigenvalues are complex, with:

$\displaystyle Re\>\epsilon_{\pm}$ $\displaystyle=d_{0}\pm h^{R}$
$\displaystyle Im\>\epsilon_{\pm}$ $\displaystyle=\gamma_{0}\pm h^{I}.$ (19)

Within the interpretation of system-environment coupling derived in Ref. [37],
it is imperative that imaginary part be negative, ensuring the density of
states is non-negative. To fulfill this property it is sufficient to assume
$\gamma_{0}<0$ and $\left|\gamma_{0}\right|>\gamma_{z}$. The spectrum is shown
in Fig. 2 for different values of $\gamma_{z}.$ For high values of
$\gamma_{z}$ the gap closes in exceptional points (EPs). Exceptional points
are a unique and interesting characteristic of non-Hermitian systems. In an
EP, in addition to the gap closing $\epsilon_{+}=\epsilon_{-}$, the
eigenvalues of the system coalesce and the Hamiltonian is said to be defective
[9]. For our system, Fig. 2 shows hybrid exceptional points at
$\mathbf{k}_{HEP}=\left(\pm\pi/2,\pm\pi/2\right)$ when $\gamma_{z}=2$. Those
are called hybrid because they result from the merging of two ordinary
exceptional points with opposite vorticities or winding numbers [44, 45, 46].
In fact, for $\gamma_{z}>2$, each hybrid EP splits in two ordinary EPs
localized in the $\Gamma-M$ line. These coupled EPs are linked by a path in
which the eingenvalues of both bands are complex-conjugate of each other,
which is sometimes called bulk Fermi arcs [47, 10] and are a consequence of
$\mathcal{PT}$ symmetry of the Hamiltonian [11].

Another unique feature of non-Hermitian systems is the topological
classification derived from the vorticity of EPs, which is related to the
energy eigenvalues [44, 48] (rather than the eigenstates like in Chern or
$\mathbb{Z}_{2}$ insulators [1]). This classification is not the focus of the
present work and is left for forthcoming studies. For the investigation of
transverse transport, we restrict ourselves to the gapped regime.

##  5 Hall Conductivity

Figure 3: Berry curvature of the non-Hermitian checkerboard lattice.
Parameters: $t=1.0$, $t^{\prime}=0.5$, $\phi=\pi/4$, $\gamma_{0}=-2.5$.

We can generalize the Berry curvature in Eq. 7 to non-Hermitian systems,
obtaining a non-Hermitian complex Berry curvature [16]:

$\Omega_{xy}\left(\mathbf{k}\right)=\hat{\mathbf{h}}\cdot\left(\partial_{x}\hat{\mathbf{h}}\times\partial_{y}\hat{\mathbf{h}}\right)$
(20)

where $\hat{\mathbf{h}}=\mathbf{h}/h$,
$\mathbf{h}=\left(h_{x},h_{y},h_{z}\right).$ Actually, because of the
difference between right and left eigenstates, we can construct four different
“Berry curvatures”
$\Omega_{xy}^{\alpha\beta}\left(\mathbf{k}\right)=i\varepsilon_{xy}\left\langle\partial_{x}\psi^{\alpha}\left(\mathbf{k}\right)\middle|\partial_{y}\psi^{\beta}\left(\mathbf{k}\right)\right\rangle$,
where $\alpha,\beta=L/R$ [44]. We use here
$\Omega_{xy}\left(\mathbf{k}\right)=\Omega_{xy}^{LR}\left(\mathbf{k}\right)$,
which is well-suited for systems with loss and gain. The non-Hermitian Chern
number is defined as in the Hermitian case:
$C\equiv\int\frac{d^{2}k}{4\pi}\Omega_{xy}^{\alpha\beta}\left(\mathbf{k}\right)$.
It is a real and quantized number and does not depend on $\alpha,\beta$, even
though $\Omega_{xy}^{\alpha\beta}\left(\mathbf{k}\right)$ are locally
different.

In ordinary Hermitian systems, non-trivial Chern numbers are related to robust
edge states through the bulk-edge correspondence, characterizing a topological
insulator. This bulk-edge correspondence breaks down in non-Hermitian systems.
It has been argued that it can be restored defining a non-Bloch Chern number
as the integral of the Berry curvature over an extended complex Brillouin zone
[14]. Nevertheless, the bulk Hall transport is unaffected by that. The Hall
conductivity in Eq. 1 can be written as [17]:

$\displaystyle\sigma_{xy}$
$\displaystyle=-\frac{e^{2}}{h}\int\frac{d^{2}k}{2\pi^{2}}Re\left[\hat{\mathbf{h}}\cdot\left(\hat{\mathbf{h}}_{x}\times\hat{\mathbf{h}}_{y}\right)\times\right.$
$\displaystyle\times\left.\left(\frac{\pi}{2}sgn\left(Re\,h\right)-\frac{ihh_{0}}{h^{2}-h_{0}^{2}}-i\,arctanh\left(\frac{h_{0}}{h}\right)\right)\right]$
$\displaystyle=-\frac{e^{2}}{h}\int\frac{d^{2}k}{4\pi}Re\left[\Omega_{xy}\left(\mathbf{k}\right)f\left(\mathbf{k}\right)\right].$
(21)

The conductivity now bears a factor $f\left(\mathbf{k}\right)$ in the
integrand, and we cannot identify the integral with the Chern number.
Therefore, there is no topological interpretation of the Hall response, and
the Hall conductivity for a non-Hermitian topological Chern insulator is in
general non-quantized. Taking the Hermitian limit
$\mathbf{h}\rightarrow\mathbf{d}$, we can show that the expression above
reduces to Eq. 8, restoring the integer quantum Hall effect.

In the insulating regime, the non-Hermitian Chern number was numerically
calculated and is equal to $C=\pm 1$, like in the Hermitian system. The values
of $\phi$ for which the gap closes are not only $n\pi/2$ (as in the Hermitian
case), but an interval around those values. In that interval, the gap closes
in exceptional points, the Berry curvature diverges and the Chern number is
not defined. As long as we stay in the insulating phase, the Chern topology is
identical to the Hermitian system. The real and imaginary parts of the Berry
curvature are plotted in Figure 3. Note that
$Im\,\Omega_{xy}\left(\mathbf{k}\right)$ is an odd function
($Im\>\Omega_{xy}\left(-\mathbf{k}\right)=-Im\>\Omega_{xy}\left(\mathbf{k}\right)$),
so its integral is null, making the Chern number a real integer.

The Hall conductivity as a function of $\gamma_{z}$ is plotted in Figure 4.
The conductivity increases with the raise of $\gamma_{z}$, but decreases with
the raise of $\left|\gamma_{0}\right|$. This is in accordance with the
interpretation that, for systems coupled to the environment,
$\left|\gamma_{0}\right|>\gamma_{z}$ represent a net loss, i.e., particles
with finite lifetimes [39]. A high value of $\left|\gamma_{0}\right|$ means
higher loss (or shorter quasiparticle lifetime), decreasing the charge
carriers in equilibrium and damping the conductivity. The opposite occurs for
a high $\gamma_{z}$, which means longer quasiparticle lifetime (less loss),
resulting in a higher conductivity.

Figure 4: Hall conductivity of the non-Hermitian checkerboard lattice as a
function of $\gamma_{z}$. Different curves represent different values of
$\gamma_{0}.$ The other parameters are $t=1.0$, $t^{\prime}=0.5$ and
$\phi=\pi/4$. The dotted sections of the plots are not physical in the coupled
baths model ($\gamma_{z}>\left|\gamma_{0}\right|$). The data does not go
further than $\gamma_{z}=2.0$ because this is the value of gap closing, where
the Berry curvature is divergent.

##  6 Summary and concluding remarks

We studied the fermionic non-Hermitian checkerboard lattice with focus on its
Chern topology and Hall conductivity. The non-Hermiticity is introduced by two
parameters: $\gamma_{0},$ related to dissipation, and $\gamma_{z}$, related to
gain. A complex NN hopping induces a Berry curvature, which renders the system
a non-trivial Chern topology in the insulating phase (identically to the
Hermitian case). The gap closes in exceptional points for some choices of
parameters, in specific for high $\gamma_{z}$. Although the system is a Chern
insulator, the Hall conductivity is not quantized. This is a consequence of
dissipation or finite lifetime of states in a context of a system coupled to
the environment. We verified the response of the Hall conductivity to the non-
Hermitian parameters, and concluded it is maximized for high $\gamma_{z}$ and
low $\gamma_{0},$ which represents long quasiparticle lifetimes. Our work
explores the novel field of non-Hermitian topology and corroborates that two-
dimensional exotic lattices, like the checkerboard lattice, can be a good
platform to investigate topological effects in open quantum systems.

## References

  * [1] M. Z. Hasan, C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82 (4) (2010) 3045–3067.  doi:10.1103/revmodphys.82.3045.
URL http://dx.doi.org/10.1103/RevModPhys.82.3045

  * [2] M. Fruchart, D. Carpentier, An introduction to topological insulators, C. R. Phys. 14 (9–10) (2013) 779–815.  doi:10.1016/j.crhy.2013.09.013.
URL http://dx.doi.org/10.1016/j.crhy.2013.09.013

  * [3] S. Ryu, A. P. Schnyder, A. Furusaki, A. W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12 (6) (2010) 065010.  doi:10.1088/1367-2630/12/6/065010.
URL http://dx.doi.org/10.1088/1367-2630/12/6/065010

  * [4] X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83 (4) (2011) 1057–1110.  doi:10.1103/revmodphys.83.1057.
URL http://dx.doi.org/10.1103/RevModPhys.83.1057

  * [5] R. Süsstrunk, S. D. Huber, Observation of phononic helical edge states in a mechanical topological insulator, Science 349 (6243) (2015) 47–50.  doi:10.1126/science.aab0239.
URL http://dx.doi.org/10.1126/science.aab0239

  * [6] N. Goldman, J. C. Budich, P. Zoller, Topological quantum matter with ultracold gases in optical lattices, Nature Phys. 12 (7) (2016) 639–645.  doi:10.1038/nphys3803.
URL http://dx.doi.org/10.1038/nphys3803

  * [7] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, I. Carusotto, Topological photonics, Rev. Mod. Phys. 91 (1) (Mar. 2019).  doi:10.1103/revmodphys.91.015006.
URL http://dx.doi.org/10.1103/RevModPhys.91.015006

  * [8] M. Malki, G. S. Uhrig, Topological magnetic excitations, EPL 132 (2) (2020) 20003.  doi:10.1209/0295-5075/132/20003.
URL http://dx.doi.org/10.1209/0295-5075/132/20003

  * [9] Y. Ashida, Z. Gong, M. Ueda, Non-Hermitian physics, Adv. Phys. 69 (3) (2020) 249–435.  doi:10.1080/00018732.2021.1876991.
URL http://dx.doi.org/10.1080/00018732.2021.1876991

  * [10] T. Tlusty, Exceptional topology in ordinary soft matter, Phys. Rev. E 104 (2) (Aug. 2021).  doi:10.1103/physreve.104.025002.
URL http://dx.doi.org/10.1103/PhysRevE.104.025002

  * [11] H. M. Hurst, B. Flebus, Non-Hermitian physics in magnetic systems, J. Appl. Phys. 132 (22) (Dec. 2022).  doi:10.1063/5.0124841.
URL http://dx.doi.org/10.1063/5.0124841

  * [12] A. Banerjee, R. Sarkar, S. Dey, A. Narayan, Non-Hermitian topological phases: principles and prospects, J. Phys.: Condens. Matter 35 (33) (2023) 333001\.  doi:10.1088/1361-648x/acd1cb.
URL http://dx.doi.org/10.1088/1361-648X/acd1cb

  * [13] T. Yu, J. Zou, B. Zeng, J. Rao, K. Xia, Non-Hermitian topological magnonics, Phys. Rep. 1062 (2024) 1–86.  doi:10.1016/j.physrep.2024.01.006.
URL http://dx.doi.org/10.1016/j.physrep.2024.01.006

  * [14] S. Yao, F. Song, Z. Wang, Non-Hermitian Chern bands, Phy. Rev. Lett. 121 (13) (Sep. 2018).  doi:10.1103/physrevlett.121.136802.
URL http://dx.doi.org/10.1103/PhysRevLett.121.136802

  * [15] M. Hirschberger, R. Chisnell, Y. S. Lee, N. P. Ong, Thermal Hall effect of spin excitations in a kagome magnet, Phys. Rev. Lett. 115 (2015) 106603\.  doi:10.1103/PhysRevLett.115.106603.
URL https://link.aps.org/doi/10.1103/PhysRevLett.115.106603

  * [16] Y. Chen, H. Zhai, Hall conductance of a non-Hermitian Chern insulator, Phys. Rev. B 98 (24) (Dec. 2018).  doi:10.1103/physrevb.98.245130.
URL http://dx.doi.org/10.1103/PhysRevB.98.245130

  * [17] M. R. Hirsbrunner, T. M. Philip, M. J. Gilbert, Topology and observables of the non-Hermitian Chern insulator, Phys. Rev. B 100 (8) (Aug. 2019).  doi:10.1103/physrevb.100.081104.
URL http://dx.doi.org/10.1103/PhysRevB.100.081104

  * [18] B. Canals, From the square lattice to the checkerboard lattice: Spin-wave and large-n limit analysis, Phys. Rev. B 65 (18) (Apr. 2002).  doi:10.1103/physrevb.65.184408.
URL http://dx.doi.org/10.1103/PhysRevB.65.184408

  * [19] S. Fujimoto, Geometrical-frustration-induced (semi)metal-to-insulator transition, Phys. Rev. Lett. 89 (2002) 226402.  doi:10.1103/PhysRevLett.89.226402.
URL https://link.aps.org/doi/10.1103/PhysRevLett.89.226402

  * [20] J.-S. Bernier, C.-H. Chung, Y. B. Kim, S. Sachdev, Planar pyrochlore antiferromagnet: A large-${N}$ analysis, Phys. Rev. B 69 (21) (Jun. 2004).  doi:10.1103/physrevb.69.214427.
URL http://dx.doi.org/10.1103/PhysRevB.69.214427

  * [21] F. Pollmann, J. J. Betouras, K. Shtengel, P. Fulde, Correlated fermions on a checkerboard lattice, Phys. Rev. Lett. 97 (17) (Oct. 2006).  doi:10.1103/physrevlett.97.170407.
URL http://dx.doi.org/10.1103/PhysRevLett.97.170407

  * [22] T. Yoshioka, A. Koga, N. Kawakami, Frustration effects in an anisotropic checkerboard lattice Hubbard model, Phys. Rev. B 78 (2008) 165113.  doi:10.1103/PhysRevB.78.165113.
URL https://link.aps.org/doi/10.1103/PhysRevB.78.165113

  * [23] E. G. Santos, J. R. Iglesias, C. Lacroix, M. A. Gusmão, A two-band model for superconductivity in the checkerboard lattice, J. Phys.: Condens. Matter 22 (21) (2010) 215701.  doi:10.1088/0953-8984/22/21/215701.
URL https://dx.doi.org/10.1088/0953-8984/22/21/215701

  * [24] K. Sun, H. Yao, E. Fradkin, S. A. Kivelson, Topological insulators and nematic phases from spontaneous symmetry breaking in 2d fermi systems with a quadratic band crossing, Phys. Rev. Lett. 103 (2009) 046811.  doi:10.1103/PhysRevLett.103.046811.
URL https://link.aps.org/doi/10.1103/PhysRevLett.103.046811

  * [25] K. Sun, Z. Gu, H. Katsura, S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106 (23) (Jun. 2011).  doi:10.1103/physrevlett.106.236803.
URL http://dx.doi.org/10.1103/PhysRevLett.106.236803

  * [26] H. Katsura, I. Maruyama, A. Tanaka, H. Tasaki, Ferromagnetism in the Hubbard model with topological/non-topological flat bands, Europhys. Lett. 91 (5) (2010) 57007.  doi:10.1209/0295-5075/91/57007.
URL http://dx.doi.org/10.1209/0295-5075/91/57007

  * [27] H. Liu, G. Sethi, S. Meng, F. Liu, Orbital design of flat bands in non-line-graph lattices via line-graph wave functions, Phys. Rev. B 105 (8) (Feb. 2022).  doi:10.1103/physrevb.105.085128.
URL http://dx.doi.org/10.1103/PhysRevB.105.085128

  * [28] X.-P. Liu, Y. Zhou, Y.-F. Wang, C.-D. Gong, Characterizations of topological superconductors: Chern numbers, edge states and Majorana zero modes, New J. Phys. 19 (9) (2017) 093018.  doi:10.1088/1367-2630/aa8022.
URL http://dx.doi.org/10.1088/1367-2630/aa8022

  * [29] A. Pires, Magnon spin Nernst effect on the antiferromagnetic checkerboard lattice, Phys. Lett. A 383 (32) (2019) 125887.  doi:10.1016/j.physleta.2019.125887.
URL http://dx.doi.org/10.1016/j.physleta.2019.125887

  * [30] A. S. T. Pires, Topological magnons in the antiferromagnetic checkerboard lattice, Physica E 118 (2020) 113899\.  doi:10.1016/j.physe.2019.113899.
URL https://doi.org/10.1016/j.physe.2019.113899

  * [31] A. Pires, Topological magnons on the checkerboard lattice, Physica B 602 (2021) 412490.  doi:10.1016/j.physb.2020.412490.
URL http://dx.doi.org/10.1016/j.physb.2020.412490

  * [32] D.-S. Ma, Y. Xu, C. S. Chiu, N. Regnault, A. A. Houck, Z. Song, B. A. Bernevig, Spin-orbit-induced topological flat bands in line and split graphs of bipartite lattices, Phys. Rev. Lett. 125 (2020) 266403.  doi:10.1103/PhysRevLett.125.266403.
URL https://link.aps.org/doi/10.1103/PhysRevLett.125.266403

  * [33] Z. Zhang, W. Feng, Y. Yao, B. Tang, Photoinduced Floquet topological magnons in a ferromagnetic checkerboard lattice, Phys. Lett. A 414 (2021) 127630.  doi:10.1016/j.physleta.2021.127630.
URL http://dx.doi.org/10.1016/j.physleta.2021.127630

  * [34] H. Kabbour, E. Janod, B. Corraze, M. Danot, C. Lee, M.-H. Whangbo, L. Cario, Structure and magnetic properties of oxychalcogenides ${A}_{2}{F}_{2}{F}e_{2}{O}{Q}_{2}$ (A = Sr, Ba; Q = S, Se) with ${F}e_{2}{O}$ square planar layers representing an antiferromagnetic checkerboard spin lattice, J. Am. Chem. Soc. 130 (26) (2008) 8261–8270.  doi:10.1021/ja711139g.
URL http://dx.doi.org/10.1021/ja711139g

  * [35] X. Hu, R.-W. Zhang, D.-S. Ma, Z. Cai, D. Geng, Z. Sun, Q. Zhao, J. Gao, P. Cheng, L. Chen, K. Wu, Y. Yao, B. Feng, Realization of a two-dimensional checkerboard lattice in monolayer ${C}u_{2}{N}$, Nano Lett. 23 (12) (2023) 5610–5616.  doi:10.1021/acs.nanolett.3c01111.
URL http://dx.doi.org/10.1021/acs.nanolett.3c01111

  * [36] A. Sufyan, M. Sajjad, J. Andreas Larsson, Evaluating the potential of planar checkerboard lattice ${C}u_{2}{N}$ monolayer as anode material for lithium and sodium-ion batteries using first-principles methods, Appl. Surf. Sci. 654 (2024) 159474.  doi:10.1016/j.apsusc.2024.159474.
URL http://dx.doi.org/10.1016/j.apsusc.2024.159474

  * [37] J. M. Midtgaard, Z. Wu, Y. Chen, Constraints on the energy spectrum of non-Hermitian models in open environments, Eur. Phys. J. B 92 (11) (Nov. 2019).  doi:10.1140/epjb/e2019-100393-5.
URL http://dx.doi.org/10.1140/epjb/e2019-100393-5

  * [38] D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A: Math. Theor. 47 (3) (2013) 035305.  doi:10.1088/1751-8113/47/3/035305.
URL http://dx.doi.org/10.1088/1751-8113/47/3/035305

  * [39] S. Groenendijk, T. L. Schmidt, T. Meng, Universal Hall conductance scaling in non-Hermitian Chern insulators, Phys. Rev. Res. 3 (2) (Apr. 2021).  doi:10.1103/physrevresearch.3.023001.
URL http://dx.doi.org/10.1103/PhysRevResearch.3.023001

  * [40] K. Ishikawa, T. Matsuyama, Magnetic field induced multi-component ${QED}_{3}$ and quantum Hall effect, Z. Phys. C - Particles and Fields 33 (1) (1986) 41–45.  doi:10.1007/bf01410451.
URL http://dx.doi.org/10.1007/BF01410451

  * [41] K. Ishikawa, T. Matsuyama, A microscopic theory of the quantum Hall effect, Nucl. Phys. B 280 (1987) 523–548.  doi:10.1016/0550-3213(87)90160-x.
URL http://dx.doi.org/10.1016/0550-3213(87)90160-X

  * [42] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (6) (1982) 405–408.  doi:10.1103/physrevlett.49.405.
URL https://doi.org/10.1103/physrevlett.49.405

  * [43] A. S. T. Pires, A brief introduction to topology and differential geometry in condensed matter physics, IOP Concise Physics, Morgan & Claypool, San Rafael, CA, 2019.
  * [44] H. Shen, B. Zhen, L. Fu, Topological band theory for non-Hermitian Hamiltonians, Phys. Rev. Lett. 120 (14) (Apr. 2018).  doi:10.1103/physrevlett.120.146402.
URL http://dx.doi.org/10.1103/PhysRevLett.120.146402

  * [45] X.-L. Zhang, C. T. Chan, Hybrid exceptional point and its dynamical encircling in a two-state system, Phys. Rev. A 98 (3) (Sep. 2018).  doi:10.1103/physreva.98.033810.
URL http://dx.doi.org/10.1103/PhysRevA.98.033810

  * [46] L. Jin, H. C. Wu, B.-B. Wei, Z. Song, Hybrid exceptional point created from type-iii Dirac point, Phys. Rev. B 101 (4) (Jan. 2020).  doi:10.1103/physrevb.101.045130.
URL http://dx.doi.org/10.1103/PhysRevB.101.045130

  * [47] H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, B. Zhen, Observation of bulk Fermi arc and polarization half charge from paired exceptional points, Science 359 (6379) (2018) 1009–1012.  doi:10.1126/science.aap9859.
URL http://dx.doi.org/10.1126/science.aap9859

  * [48] S. Dey, A. Banerjee, D. Chowdhury, A. Narayan, Hall conductance of a non-Hermitian Weyl semimetal, New J. Phys. 26 (2) (2024) 023057.  doi:10.1088/1367-2630/ad2b0e.
URL http://dx.doi.org/10.1088/1367-2630/ad2b0e