# Non-Hermitian physics and topological phenomena in convective thermal
metamaterials

Zhoufei Liu<EMAIL_ADDRESS>Department of Physics, State Key Laboratory
of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures
(MOE), Fudan University, Shanghai 200438, China

###### Abstract

Non-Hermitian physics and topological phenomena are two hot topics attracted
much attention in condensed matter physics and artificial metamaterials.
Thermal metamaterials are one type of metamaterials that can manipulate heat
on one’s own. Recently, it has been found that non-Hermitian physics and
topological phenomena can be implemented in purely diffusive systems. However,
conduction alone is not omnipotent due to the missing of degrees of freedom.
Heat convection, accompanying with conduction, is capable of realizing a large
number of phases. In this review, we will present some important works on non-
Hermitian and topological convective thermal metamaterials. In non-Hermitian
physics, we will first discuss the implementation of exceptional point (EP) in
thermal diffusion, followed by high-order EP and dynamic encirclement of EP.
We then discuss two works on the extensions of EP in diffusion systems,
namely, the chiral thermal behavior in the vicinity of EP and the Weyl
exceptional ring. For topological phases, we will discuss two examples: a one-
dimensional topological insulator and a two-dimensional quadrupole topological
insulator. Finally, we will make a conclusion and present a promising outlook
in this area. Besides the scientific values, non-Hermitian and topological
convective thermal metamaterials have great potentials for industrial
applications.

##  I I. Introduction

The effective non-Hermitian Hamiltonian provides a simple and intuitive method
to understand the dynamics of open systems [1, 2]. In recent decades, the
study of non-Hermitian physics has been a frontier and attracted many
researchers from condensed matter physics, optics, quantum information, etc.
Among this field, parity-time (PT) symmetry is one of the most important
symmetries because the system will exhibit a real spectrum [3, 4]. When the
eigenstate of PT-symmetric Hamiltonian preserves (violates) PT symmetry, the
system is in the PT unbroken (broken) phase. The critical point between these
two phases is called the exceptional point (EP). Besides, many interesting
phenomena without the Hermitian counterpart are proposed in non-Hermitian
physics, such as non-Hermitian skin effect [5, 6, 7] and non-Hermitian
topological classification [8, 9].

The quantum Hall effect discovered in the 1980s is the first proposed
topological phase transition beyond the conventional Landau paradigm [10, 11].
In the last twenty years, it witnessed the flourishing of topological physics,
and plenty of topological phases of matter have been theoretically predicted
and experimentally discovered [12, 13]. Apart from condensed matter physics,
topology has been incorporated into classical wave systems, giving rise to
many new fields. One most typical and significant example is topological
photonics [14, 15]. The basic principle for topological photonics is the
mapping between Maxwell’s equations in electrodynamics and
Schr${\rm{\ddot{o}}}$dinger equation in quantum mechanics. Other examples,
such as topological acoustics [16, 17] and topological mechanics [18], have
also been good platforms for realizing topological states.

Thermal metamaterials have become the focus of growing interest in physics,
materials science, and thermophysics [19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32]. Based on transformation thermotics, thermal cloaking was
first proposed in 2008 [33]. After that, more thermal functions with different
artificial structures became possible, such as thermal illusion [34, 35, 37,
36, 38, 39, 40, 41, 42], thermal transparency [43, 44], thermal coding [45,
46, 47, 48, 49, 50, 51], thermal chameleonlike metashells [52, 53, 54, 55, 56,
57, 58], thermal sensor [59, 60, 61, 62, 63], and others [64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]. However, the above works
are all constrained to purely thermal conduction. Thermal convection, as
another fundamental mechanism of heat transfer, is usually dominant in moving
media such as fluids. After considering it, hybrid diffusive systems can
revive many interesting phenomena in dissipative diffusion beyond pure
conduction [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97,
98, 99]. Nonreciprocal heat transfer can be easily implemented with the help
of convection due to its directional nature, such as spatiotemporal modulation
[100, 101, 102, 103, 104, 105, 106] and angular momentum bias [107, 108].
Transformation thermotics has been extended to thermal radiation [109, 110,
111, 112], nonlinear thermal conductivities [113, 114, 115, 116, 117, 118,
119, 120, 121, 122], and thermoelectrics [123, 124, 125, 126, 127, 128, 129,
130]. Besides, the concept of thermal metamaterials has also been generalized
to other diffusion processes, such as particle diffusion [131, 132, 133] and
plasma transport [134].

Recently, diffusion system has become a brand new platform to realize the
topological phase and non-Hermitian physics. For heat conduction, there are
two paradigms for implementing these phases. The first paradigm is spatially
discretizing the diffusion equation [135]. Though with some imperfections in
theory [136], the proposal has been experimentally confirmed to observe the
edge state of one-dimensional (1D) Su-Schrieffer-Heeger model [137]. As a
natural generalization, this proposal can be extended to higher-dimensional
systems, such as higher-order topological insulators [138, 139, 140]. The
other paradigm is constructing the coupled ring chain structure. The structure
consists of several rings vertically coupled in one direction to form a chain
through interlayers. It has been proposed that the non-Hermitian skin effect
[141, 142] and the quasicrystal [143] can be realized within this structure.
Besides, the thermal analogue of coherent perfect absorption may support the
establishment of non-Hermitian diffusive scattering theory [144].

However, when convection is introduced as a new degree of freedom into heat
transfer, increasingly non-Hermitian phenomena and topological physics can be
realized in thermal diffusion beyond purely conduction. In this review, we
present some important works in this area, organized as follows. In Section
${\rm{II}}$, we will discuss the work about the realization of EP in diffusion
systems. In addition, we will present how to implement high-order EPs and
dynamically enclose EPs in thermal diffusion. In Section ${\rm{III}}$, we will
discuss two works about the extensions of EP in convective thermal
metamaterials, i.e., chiral thermal behaviour in the vicinity of EP and Weyl
exceptional ring. In Section ${\rm{IV}}$, we will introduce the topological
physics in convection-diffusion systems. Two topological phases, a 1D
topological insulator and a 2D quadrupole insulator, have been realized with
the help of thermal convection. A concluding discussion is presented in
Section ${\rm{V}}$, accompanying a reasonable outlook that may come true in
the near future.

##  II II. Non-Hermitian physics in convective thermal metamaterials: The
implementation of EP

Figure 1: Two paths to realize the PT-related phenomenon: the introduction of
gain and loss in wave systems and the introduction of the forward and backward
wave-like field in diffusion systems. Adapted from Ref. [145]. With permission
from the Author.

The research on non-Hermitian convective thermal metamaterials starts from the
implementation of anti-parity-time (APT) symmetry and EP in diffusion systems
[145]. This work lays the foundation of non-Hermitian convective thermal
metamaterials because of the importance of EP in non-Hermitian physics. In the
wave systems, realizing the PT symmetry is very intuitive by introducing
balanced gain and loss [blue dots in Fig. 1]. However, it is not easy to
realize the PT-related phenomenon in diffusion systems that are intrinsically
dissipative (anti-Hermitian). This problem can be solved by introducing the
forward and backward wave-like field, i.e., heat convection [red dots in Fig.
1], which acts as a Hermitian component. After the introduction of thermal
convection, the diffusive system will become APT symmetric and its effective
Hamiltonian anti-commutes with the PT operator.

Figure 2: APT symmetry and EP in convective thermal metamaterials. (a) The
double ring model in the 2D plane (upper panel) and its 3D diagram (lower
panel). (b) Decay rates and eigenfrequencies of double ring model. The blue
squares are the simulation results for the fundamental mode, and the red lines
are the theoretical eigenvalues of the effective Hamiltonian. (c) Infrared
images of the temperature profiles on ring 1 at $t$ = 0 s and $t$ = 18 s with
rotating velocity 0.5 rpm, 1.0 rpm, and 1.5 rpm. Adapted from Ref. [145]. With
permission from the Author.

The basic structure is two coupled rings with equal but opposite rotating
velocities [see the schematic diagram in Fig. 2(a)]. The governing heat
transfer equations for this double ring model can be written as

$\displaystyle\frac{{\partial}T_{1}}{{\partial}t}=D\frac{{\partial}^{2}T_{1}}{{\partial}x^{2}}-v\frac{{\partial}T_{1}}{{\partial}x}+h(T_{2}-T_{1})$
(1)
$\displaystyle\frac{{\partial}T_{2}}{{\partial}t}=D\frac{{\partial}^{2}T_{2}}{{\partial}x^{2}}+v\frac{{\partial}T_{2}}{{\partial}x}+h(T_{1}-T_{2})$
(2)

where $D$ is the diffusivity of rings, $h$ is the heat exchange rate between
two rings, and $v$ is the rotating velocity of rings. Using the plane-wave
solution, the effective APT symmetric Hamiltonian is obtained as:

${\hat{H}}=\left(\begin{matrix}-i(k^{2}D+h)+kv&ih\\\
ih&-i(k^{2}D+h)-kv\end{matrix}\right)$ (3)

where $k$ is the wave number. The corresponding eigenvalues of the effective
Hamiltonian are

$\displaystyle\omega_{\pm}=-i\left[(k^{2}D+h){\pm}\sqrt{h^{2}-k^{2}v^{2}}\right]$
(4)

Then the EP emerges when $h^{2}=k^{2}v_{\rm{EP}}^{2}$. The fundamental mode
$k=1/R$ is chosen because only the slowest decaying mode can be observed. The
theoretical decay rates and eigenfrequencies of the effective Hamiltonian are
shown in Fig. 2(b), which are in accordance with the simulation results. The
two branches of decay rates coalesce at the EP, and the nonzero eigenfrequency
appears as the $v$ increases. In the APT unbroken phase, the temperature
profile of ring 1 remains motionless, and the maximum temperature point is
fixed after evolving a certain time [see the middle two subfigures in Fig.
2(c)]. However, the temperature profile keeps moving because of nonzero
eigenfrequency in the APT broken phase [see the right subfigure in Fig. 2(c)].
The key point behind these thermal behaviours is the competition between
thermal convection and coupled thermal conduction.

Further increasing the number of rings, high-order EP can be achieved in
convection-diffusion systems, which is robust against perturbations and phase
oscillations [146]. The model is the quadruple ring structure with opposite
but equal velocities between adjacent channels [see in Fig. 3(a) and (b)].
Similar with the derivation in Ref. [145], the effective Hamiltonian is a four
by four matrix:

${\hat{H}}=\left(\begin{matrix}-i(k^{2}D+h)-kv&ih&0&0\\\
ih&-i(k^{2}D+h)-ih+kv&ih&0\\\ 0&ih&-i(k^{2}D+h)-ih-kv&ih\\\
0&0&ih&-i(k^{2}D+h)+kv\end{matrix}\right)$ (5)

After diagonalizing the Hamiltonian, a third-order EP can be obtained at the
critical velocity when APT phase transition occurs [point $A$ in Fig. 3(c)].
In the APT symmetric phase, the temperature profile remains stationary with a
phase-locked difference despite the velocity perturbation. While in the APT
broken phase, the temperature field starts to dynamically evolve with phase
oscillations regardless of the initial conditions.

Figure 3: High-order EP in convective thermal metamaterials. Schematic diagram
of (a) 3D quadruple ring model and (b) its corresponding 2D model. (c) The
imaginary part and real part of eigenvalues vs. the background flow velocity.
Red dots at $A$ and $B$ mark the EPs. Adapted from Ref. [146]. With permission
from the Author.

In order to reveal the nontrivial topology of EPs in non-Hermitian physics, a
sufficiently large parameter space is essential for dynamical EP encircling.
However, the fixed and positive conductivity and structural parameters prevent
the encircling from occurring in thermal diffusion. To address this problem, a
recent work has created an orthogonal convection space to provide enough
degrees of freedom [147]. The first step is to mimic two pairs of orthogonal
oscillations as a complex parameter space [see Fig. 4(a)]. In order to easily
implement such periodic convection, it is necessary to devise a multiple ring
system consisting of two orthogonal pairs of counter-convective rings. One
pair is rotating in $r-{\theta}$ space, and the other pair is translating
along the $z$ direction [see the 3D diagram in Fig. 4(b) and the corresponding
cross-section view in Fig. 4(c)]. Substituting the wave-like solution into the
heat transfer equation, the effective Hamiltonian can be written as

$\hat{H}=i\left(\begin{matrix}-\left(\dfrac{\kappa}{{\rho}c}k_{r{\theta}}^{2}+2m\right)-k_{\rm{eff}}v_{\rm{eff}}\left[{\rm{cos}}(n)+i{\rm{sin}}(n)\right]&2m\\\
2m&-\left(\dfrac{\kappa}{{\rho}c}k_{r{\theta}}^{2}+2m\right)+k_{\rm{eff}}v_{\rm{eff}}\left[{\rm{cos}}(n)+i{\rm{sin}}(n)\right]\end{matrix}\right)$
(6)

where $\kappa$, $\rho$, and $c$ are the thermal conductivity, mass density,
and heat capacity of rings. $m$ is the heat exchange rate, which is fixed and
positive. For clarity, the orthogonal convection space is denoted as
$\left[k_{\rm{eff}}v_{\rm{eff}}{\rm{cos}}(n),k_{\rm{eff}}v_{\rm{eff}}{\rm{sin}}(n)\right]$,
where
$k_{\rm{eff}}v_{\rm{eff}}=\sqrt{(k_{r{\theta}}v_{r{\theta}})^{2}+(k_{z}v_{z})^{2}}$
and $n={\rm{arctan}}(k_{r{\theta}}v_{r{\theta}}/k_{z}v_{z})$. $v_{r{\theta}}$
and $v_{z}$ are the rotating and translating velocities of rings. The
effective wave numbers are $k_{r{\theta}}=1/r$ in $r-{\theta}$ plane and
$k_{z}=1/d$ along the $z$ direction, where $r$ is the radius of ring and $d$
is the thickness of the medium layer. The complex spectrum of this effective
Hamiltonian is calculated in Fig. 4(d), where two EPs can be found.

Figure 4: Dynamic encircling around the EP in convective thermal
metamaterials. (a) Schematic of the periodic convection-conduction system with
two orthogonal pairs of advections. (b) Schematic diagram of the multiple ring
system. (c) The cross section view labelled in the red dashed border of Fig.
4(b). (d) Real part and imaginary part of the eigenvalues of the effective
Hamiltonian. Adapted from Ref. [147]. With permission from the Author.

The topological properties of the EP are determined by the geometric phase
generated by a dynamically encircling trajectory. It is related to two
quantized topological invariants: the eigenstate winding number $\omega$ [148]
and the eigenvalue vorticity $\nu$ [149]. When the evolutionary trajectory of
convections encircles two EPs, both topological invariants are integer
numbers, revealing a non-Hermitian thermal topology. The temperature profile
of the surface of the central medium exhibits a dynamic-equilibrium
distribution without much deviation, and so does the location of the maximum
temperature point [see the top subfigure in Fig. 5)]. In this case, the
initial state undergoes a full Riemann sheet during the adiabatic evolution
and returns to itself at the end of one period. By comparison, when the
trajectory encloses only one EP, both topological invariants are half-integer.
The temperature profile and the location of maximum temperature show a step-
like $\pi$-phase transition after one period [see the second top subfigure in
Fig. 5. The eigenstate will exchange to another state with a $\pi$-phase
winding after one period and return to its initial position after two periods.
Furthermore, both topological invariants are zero when the EPs are not
enclosed in the locus. Then the system becomes a conventional thermal
metamaterial. For large advection modulations, the maximum temperature
locations continuously change with the convection [see the third top subfigure
in Fig. 5. While for small convection, the maximum temperature locations keep
unchanged [see the fourth top subfigure in Fig. 5. In this way, the dynamical
encircling of the EP in thermal diffusion is realized and the configurable
topological phase transition of thermal systems is observed for the first
time.

Figure 5: Measured temperature profiles of four cases on the surface of the
central medium. The black stars indicate the locations of the maximum
temperature points. The white and red arrows indicate the initial and measured
azimuthal angles. $\omega$ and $\nu$ are the eigenstate winding number and
eigenvalue vorticity of EP. Adapted from Ref. [147]. With permission from the
Author.

Meanwhile, a separate work shows the geometric phase induced by heat
convection in diffusion systems [150]. Based on the double ring model, the
temperature field will accumulate an extra $\pi$ phase when the cyclic
evolution of time-varying rotating velocity contains the EP.

##  III III. Non-Hermitian physics in convective thermal metamaterials: The
extension of EP

In this section, we will introduce two extended researches based on the
previous works about EP in thermal diffusion, namely the chiral behaviour in
the vicinity of EP [151] and the Weyl exceptional ring [152].

Figure 6: Chiral behaviour in convective thermal metamaterials. (a) Schematic
diagram of the chirality in heat transport under constant thermal
perturbations and advections. Two wave-like advections are imposed on the
upper and lower boundaries. The eigenstates (gray wave packets) exhibit the
same directions (green arrows). (b) Schematic experimental setup for studying
the chiral heat transport in a non-Hermitian thermal system with counter-
spinning plates. (c) Real and imaginary part of eigenvalues of the effective
Hamiltonian. The normalized velocity is $|v_{\rm{up}}|+|v_{\rm{low}}|$. Region
A (B) denotes the APT broken (unbroken) phase. Adapted from Ref. [151]. With
permission from the Author.

The chiral behaviour at the EP has been proposed to achieve asymmetric wave
propagation and unidirectional invisibility in carefully designed whispering-
gallery-mode resonators [153, 154, 155]. However, the chirality has not been
demonstrated in thermal diffusion so far. The reason is that the moving
temperature field only follows the dominating advection introduced into the
system [145, 147, 161, 152]. A recent work has shown the chiral heat transport
after imposing the thermal perturbation [151]. The asymmetric thermal profile
can be observed regardless of the directions and magnitudes of advections [see
the schematic diagram in Fig. 6(a)]. The proposed experimental fluid system is
shown in Fig. 6(b). Two thin plates are employed on the central fluid to
impose the counter-advection at its upper and lower boundaries. The heating
and cooling sources are directly imposed as perturbations on the boundary of
the circular fluid region. These sources are retained after observing the
steady-state thermal distribution, and advection is subsequently activated.
After substituting the initial perturbed temperature field into the heat
transfer equation, the effective Hamiltonian for the global system can be
written as:

$\hat{H}=i\left(\begin{matrix}-\left(\dfrac{k^{2}\kappa_{0}}{\rho_{0}c_{0}}+\dfrac{h}{\rho_{0}c_{0}b}\right)-ik|v_{\rm{up}}|&\dfrac{k^{2}\kappa_{0}}{\rho_{0}c_{0}}+\dfrac{h}{\rho_{0}c_{0}b}\\\
-\dfrac{k^{2}\kappa_{0}}{\rho_{0}c_{0}}+\dfrac{h}{\rho_{0}c_{0}b}&-\left(\dfrac{k^{2}\kappa_{0}}{\rho_{0}c_{0}}+\dfrac{h}{\rho_{0}c_{0}b}\right)+ik|v_{\rm{low}}|\end{matrix}\right)$
(7)

where $k$, $h$, and $b$ denote the effective wave number, the convective heat
transfer coefficient, and the thickness of the spinning plate. $v_{\rm{up}}$
and $v_{\rm{low}}$ are the rotating velocities of the upper and lower plates.
An EP can be found in the spectrum of this Hamiltonian [see in Fig. 6(c)].

Figure 7: Temperature field of chiral and nonchiral heat transport. (a) The
temperature distribution in region A on the measured line (purple dashed line
in the inset). The insets present the calculated (left) and experimental
(right) thermal distributions. (b) The temperature distribution in region B on
the measured line (the same location as in the inset of Fig. 6(a)). (c) The
unidirectional thermal twisting profile induced by the chiral and nonchiral
heat transport. Adapted from Ref. [151]. With permission from the Author.

Then the temperature field of the central fluid is explored in different
regions. In the vicinity of EP in region A (APT broken phase), the thermal
field at the fluid interface moves towards the CW direction, though the
rotating velocity at CW is smaller than the one at CCW [see in Fig. 7(a)].
These positive phase deflections are caused by the velocity difference of the
imposed advection. However, when the advection is extremely large, that is,
away from the EP in region A, the temperature field becomes homogeneous
without any chiral behaviour. When in region B (APT symmetric phase), the
temperature field is unbiased with nonchiral heat transport [see Fig. 7(b)].
Moreover, the chirality vanishes at the EP due to purely imaginary eigenvalues
and zero phase deflections in the eigenstates. Chiral thermal transport refers
to an effectively anisotropic thermal conductivity, whereas the nonchiral one
can be treated as isotropic. Based on these facts, the combination of
nonchiral and chiral heat transports within one fluid may attribute to a
twisted thermal profile in Fig. 7(c). Thus, the chiral temperature field
exhibits a large power in the free manipulation of the heat current.

Weyl point is a band degeneracy point whose dispersion satisfies the Weyl
equation in high-energy physics [156, 157]. It can be regarded as a magnetic
monopole in 3D momentum space and possesses a quantized Chern number.
Moreover, due to the fermion doubling theorem, the Weyl points must pair with
opposite topological charges. When the non-Hermitian term is introduced into
Weyl Hamiltonian, the Weyl point will evolve into the Weyl exceptional ring
composed of EPs [158]. The Weyl exceptional ring has been implemented in
topological photonics [159] and topological acoustics [160].

Figure 8: Weyl exceptional ring in convective thermal metamaterials. (a)
Schematic diagram of diffusive Weyl exceptional ring model with four counter
convections in orthogonal surfaces. The green arrows indicate the horizontal
convections imposed on the upper and bottom surfaces. The purple arrows denote
the vertical convections imposed at the lateral surfaces. (b) Schematic
diagram of experimental setup which exhibits the Weyl exceptional ring. The
heating and cooling sources are imposed periodically and alternatively on the
entire system. The green and red strips indicate horizontal and vertical
convection. The arrows denote the initial direction of motion of these strips.
The upper and lower insets are enlarged views of the central strip and its
decompositions. (c) Real and imaginary part of the eigenvalues of diffusive
Weyl exceptional ring Hamiltonian. Adapted from Ref. [152]. With permission
from the Author.

A recent work demonstrates the realization of Weyl exceptional ring in
diffusion systems [152]. Inspired by helical waveguides with helicoidal
properties in optics [159], a central strip is surrounded by four counter-
convection components in orthogonal spaces [see the schematic diagram in Fig.
8(a)]. To realize the periodic wave-like field, the initial temperature
profile on the combined structure is imposed periodically and alternatively by
the heating and cooling sources [see the experimental setup in Fig. 8(b)].
Substituting the periodic wave-like solution into a heat transfer process, the
diffusive Weyl exceptional ring Hamiltonian is:

$\hat{H}=\left(\begin{matrix}-i\left[\dfrac{\kappa}{{\rho}c}(k_{\rm{hor}}^{2}+ik_{\rm{ver}}^{2})+m_{\rm{hor}}+im_{\rm{ver}}\right]+k_{\rm{hor}}v_{\rm{hor}}&i(m_{\rm{hor}}+im_{\rm{ver}}+k_{\rm{ver}}v_{\rm{ver}})\\\
i(m_{\rm{hor}}+im_{\rm{ver}}-k_{\rm{ver}}v_{\rm{ver}})&-i\left[\dfrac{\kappa}{{\rho}c}(k_{\rm{hor}}^{2}+ik_{\rm{ver}}^{2})+m_{\rm{hor}}+im_{\rm{ver}}\right]-k_{\rm{hor}}v_{\rm{hor}}\end{matrix}\right)$
(8)

where $\kappa$, $\rho$, and $c$ are the central strip’s thermal conductivity,
mass density, and heat capacity. $v_{\rm{hor}}$ and $v_{\rm{ver}}$ are the
velocities of horizontal and vertical strips. The effective wave numbers can
be indicated as $k_{\rm{hor}}=k_{\rm{ver}}=1/L$ where $L$ is the length of the
central strip in one period. $m_{\rm{hor}}$ and $m_{\rm{ver}}$ denote the heat
exchange rate in the horizontal and vertical plane. Because of the large
vertical thickness of the central strip and material parameters of vertical
strips, $m_{\rm{ver}}$ is proportional to the square root of $v_{\rm{ver}}$.
So a 3D synthetic parameter space
$\left(m_{\rm{ver}},k_{\rm{ver}}v_{\rm{ver}},k_{\rm{hor}}v_{\rm{hor}}\right)$
can be generated, and it provides enough degrees of freedom to encircle the
Weyl exceptional ring. When $m_{\rm{ver}}$ keeps unchanged, a Weyl exceptional
ring can be found in the spectrum as shown in Fig. 8(c).

Figure 9: Temperature field of Weyl exceptional ring and Fermi arc. (a) The
experimental evolutions of maximum temperature point in the horizontal plane
when the integration surface encloses the Weyl exceptional ring. The
temperature profiles at specific moments are shown in the side. (b) Schematic
diagram of the experimental setup with an inserted system that shows the Fermi
arc state. (c) The experimental maximum temperature evolution in the
horizontal plane of the inserted system for the Fermi arc. Adapted from Ref.
[152]. With permission from the Author.

Next the thermal behaviour of Weyl exceptional ring is investigated. If the
integration surface encircles the whole Weyl exceptional ring, the temperature
profile and the location of the maximum temperature point remain stationary
[see Fig. 9(a)], which is a signature of nontrivial topology. Besides, the
Fermi arc state still exists but is suppressed in the Weyl exceptional ring
system [158]. To observe the Fermi arc in diffusion systems, an additional
subsystem should be inserted into the initial system, and it contributes to an
opposite topological charge with an opposite integral direction [see the
schematic diagram in Fig. 9(b)]. The evolution of maximum temperature
positions for the inserted system also remains stationary with robustness [see
in Fig. 9(c)], and so does one for the original system.

##  IV IV. Topological phenomena in convective thermal metamaterials

A problem that limits the development of thermal topology is that only the
temperature field of the slowest decaying branch can be observed in purely
diffusive systems. But the introduction of thermal convection can evade this
disadvantage. Essentially, thermal convection is a wave-like field that
induces a wave topology rather than a diffusive one. Based on this fact, many
interesting and important topological states can be realized in diffusion
systems. In this section, we will introduce two works about topological
physics in convective thermal metamaterials, i.e., 1D topological insulator
[161] and 2D quadrupole topological insulator [162].

Figure 10: Coupled ring chain structure. (a) Schematic diagram of a planar
fluid surface with periodic convections in the virtual space
($x_{p},y_{p},z$). These periodic convections are imposed along the $x_{p}$
direction and arranged along the $z$ direction to form a lattice with four
units. (b) Schematic diagram of four units for coupled ring chain structure in
the transformed space ($x,y,z$). Adapted from Ref. [161]. With permission from
the Author.

The topological edge state can be probed by tracing the temperature evolution
in pure conduction systems [135, 137]. However, the temperature field for the
edge states decays rapidly and is not clearly observed, which is unfavorable
for localized thermal management. With the help of convection modulation, a
recent work states that the edge state for a 1D system has been observed with
a robust temperature profile [161]. A planar fluid surface is considered in
the virtual space ($x_{p},y_{p},z$) [see in Fig. 10(a)]. The periodic heat
convections along the $x_{p}$ direction are imposed on the corresponding
regions marked by different colours. One lattice with four units along the $z$
direction has been generated. Furthermore, the two ends for one unit along the
$x_{p}$ direction should be connected to form a ring in the transformed space
($x,y,z$) for practical implementation [see the ring structure in Fig. 10(b)].
Substituting the wave-like solution into the heat transfer process, the Bloch
Hamiltonian for this coupled ring chain structure can be written as

$\hat{H}=i\left(\begin{matrix}ikv_{\rm{i}}&m&0&me^{ik_{t}4a}\\\
m&-ikv_{\rm{ii}}&m&0\\\ 0&m&-ikv_{\rm{i}}&m\\\
me^{-ik_{t}4a}&0&m&ikv_{\rm{ii}}\end{matrix}\right)-i\left(\frac{\kappa}{{\rho}c}k^{2}+m\right)I_{4{\times}4}$
(9)

where $\kappa$, $\rho$, and $c$ are the thermal conductivity, mass density,
and heat capacity of rings. $m=\kappa/{{\rho}cbd}$ is the heat exchange rate
between two adjacent rings. $b$ and $d$ are the thickness of the ring and
interlayer. $v_{\rm{i}}$ and $v_{\rm{ii}}$ are the modulation velocities and
$a$ is the length of one unit. $k$ denotes the fundamental wave number of ring
and $k_{t}$ indicates the effective Bloch wave number. $I$ is the identity
matrix. The topological properties of this system are nontrivial with a pair
of edge states under this convective arrangement.

Figure 11: Temperature field of topological edge state and interface state.
(a) The experimental locations of the maximum temperature point for each unit
at the specific moments with topological edge state. The corresponding
temperature profiles are shown on the sides. (b) Schematic diagram of one
domain wall consisting of five nontrivial (20 units) and five trivial (20
units) lattices. (c) The experimental locations of the maximum temperature
point for each unit at the specific moments with one domain wall. The
corresponding temperature profiles are shown aside. Adapted from Ref. [161].
With permission from the Author.

Coming to the temperature profile, the locations of maximum temperature points
at the specific moments are presented in Fig. 11(a) for this arrangement. In
this case, the thermal profile is robust and stationary, with some slight
deviations with respect to the initial position at the two boundaries of the
system, which is a signature of topological edge states. Besides, one domain
wall composed of five nontrivial (20 units) and five trivial (20 units)
lattices can be configured to implement an interface state [see Fig. 11(b)].
The temperature field shows a robust and stationary profile in the nontrivial
lattice while it shows large deviations to the initial state and becomes
homogenized finally in the trivial lattice [see Fig. 11(c)]. This exotic
thermal profile shows great potential for manipulating the thermal field.

Higher-order topological insulator, which is beyond the conventional bulk-
boundary correspondence, arouses much attention from the community of
condensed matter physics [163, 164, 165, 166]. Generally speaking, a
$d$-dimensional $n$th-order topological insulator exhibits ($d-n$)-dimensional
gapless boundary state ($n>1$). Except in condensed matter physics, the
higher-order topological insulator has also been realized in the classical
wave systems [167, 168, 169, 170]. There are two classes of higher-order
topological insulators in the physics community, those without multipole
moments, denoted by the 2D Su-Schrieffer-Heeger model, and those with
multipole moments, denoted by the Benalcazar-Bernevig-Hughes model. Though
there are some works which have realized the higher-order topological
insulator without multipole moment in thermal diffusion [138, 139, 140], the
quadrupole topological phase has not been implemented due to the absent bulk
quadrupole moment and undefined negative couplings in heat transfer. A recent
work has addressed these problems and realized the non-Hermitian quadrupole
topological insulator in diffusion systems [162].

Figure 12: 2D quadrupole topological insulator in convective thermal
metamaterials. (a) Schematic diagram of the square thermal lattice consisting
of 16 sites with different advections. The red dashed box indicates a four-
site unit-cell structure. $\Omega_{{I}/{II}}$ represents the magnitude of the
angular velocities imposed on each site. (b) The theoretical real spectrum
induced by Hermitian advection at the corner state. (c) Experimental
temperature distributions at the corner state induced by Hermitian advection.
(d) Schematic diagram of the square thermal lattice with different coupling
strengths. The intercell and intracell channels are fabricated in different
structures, enabling different thermal coupling strengths. (e) The theoretical
imaginary spectrum induced by non-Hermitian couplings at the corner state. (f)
Experimental temperature distributions at the corner state induced by non-
Hermitian couplings. Adapted from Ref. [162]. With permission from the Author.

A convective thermal lattice with multiple discrete sites is shown in Fig.
12(a). Each site represents a finite volume heat transfer process and the grid
lines between neighboring sites correspond to their thermal couplings. Then
the tunable advections $\Omega_{{I}/{II}}$ are adopted on each site to form a
unit-structure consisting of four neighboring sites, as shown in the red
dashed box of Fig. 12(a). The neighboring sites are coupled via the heat
exchanges induced by intracells and intercells. $\beta$ is denoted as the
ratio between the intercell and intracell thermal coupling strengths. Two
types of tilted connections (over-coupling and under-coupling) are designed to
mimic the coexisting positive and negative nearest-neighbor couplings with
opposite isotherms.

With this structure, Hermitian advection and non-Hermitian couplings provide
two modulations that enable a quadrupole topological phase. For the Hermitian
advections, the $\Omega_{II}$ is adjusted while the $\Omega_{I}$ is kept
unchanged. Besides, $\beta$ is set to be one to ensure the heat exchange areas
of the intracell and intercell components are the same. In this case, the
imaginary bands are gapless, so that in-gap corner states can be observed in
the real spectrum. When modulating the $\Omega_{II}$ at
$\Omega_{II}=-1.385\Omega_{I}$, the robust in-gap corner states can be found
in the theoretical real band [see in Fig. 12(b)]. In order to experimentally
observe these corner states, a thermal system consisting of
${\rm{12{\times}12}}$ sites is fabricated. All of the sites are hollowed out
to allow water to pass. The measured temperature profile at the corner state
is shown in Fig. 12(c), with a relatively high field intensities at four
corners. Then the non-Hermitian thermal couplings are modulated as shown in
Fig. 12(d). With the same advections on all sites and intercell coupling
channels, the intracell coupling channels are enlarged by inserting internal
fins to modulate the $\beta$. In this case, the real-valued band is gapless
and the imaginary-valued band is gapped. At $\beta=0.5$, the 0D corner states
exist in the imaginary spectrum [see in Fig. 12(e)]. The experimental
temperature field distribution for the corner state is shown in Fig. 12(f).
The higher-order topological state can be observed both in the real-valued
(modulated by advections) and imaginary-valued (modulated by thermal
couplings) bands, which breaks the restriction that only real-valued band can
exhibit corner states in classical wave systems.

Besides, a separate work exhibits the implementation of a Chern insulator with
the chiral edge state in diffusion systems [171]. The basic unit is a
hexagonal structure with thermal convection. Based on this structure, a
honeycomb lattice can be designed to mimic the Haldane model [172]. A steady
temperature field will propagate unidirectionally along the boundary of the
system without backscattering.

##  V V. Conclusion and outlook

As a conclusion, we discuss some interesting works on non-Hermitian physics
and topological phenomena in convection-conduction systems. From the point of
view of non-Hermitian physics, we first discuss the realization of EP in
thermal diffusion, followed by the high-order EP and the dynamically
encirclement of EP. As two extensions of the EP, we introduce chiral heat
transport around the EP and diffusive Weyl exceptional ring. From topological
physics, 1D thermal topological insulators and 2D thermal quadrupole
topological insulators have been introduced as two examples of diffusive
topological phases.

The promising future of topological and non-Hermitian convective thermal
metamaterials can be anticipated. Until now, only a small fraction of exotic
phases of matter have been realized in diffusive systems. The realization of a
wide variety of topological states is a prevailing trend in this field. From
the non-equilibrium dynamics, except for non-Hermitian physics, Floquet
thermal metamaterials can be devised by spatiotemporal modulating the
materials’ parameters [173]. From the higher-dimensional systems, many
peculiar configurations, such as fractal systems [174], moir${\rm{\acute{e}}}$
lattices [175], and disclinations [176], can be engineered in the thermal
lattices. The nonlinearity can also be introduced into non-Hermitian and
topological thermal metamaterials to explore more interesting physics [177,
178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
193, 194, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208,
209, 195]. In addition, machine learning and other algorithms are expected to
be valuable in studying topological effects in diffusion systems [210, 211,
212, 213, 214, 215]. The econophysics may inspire the study of non-Hermitian
and topological convective thermal metamaterials [216, 217, 218, 219, 220,
221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235,
236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248]. Besides,
this field can also be generalized to nanoparticles [249, 250, 251, 252, 253,
254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268,
269, 270, 271, 272, 273], complex fluids [274, 275, 276, 277, 278, 279, 280,
281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294], and
other electric behaviours [295, 296, 297, 298, 299, 300, 301, 301, 302, 303,
304, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317,
318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329]. Potential
applications in topological convective thermal materials are also to be
expected. Topological edge states may be useful for localized heat management
and robust heat transport. In brief, topological and non-Hermitian convective
thermal metamaterials are highly interdisciplinary and of great scientific and
technological values.

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