Theory for Thermal Geometric Phases: Exceptional Point Encirclement

Abstract

In this chapter, we experimentally demonstrate that the geometric phase can also emerge in a macroscopic thermal convection-conduction system. Following Li et al. [Science 364, 170–173 (2019)], we study two moving rings with equal-but-opposite velocities, joined together by a stationary intermediate layer. We first confirm an exceptional point of velocity that separates a stationary temperature profile and a moving one. We then investigate a cyclic path of time-varying velocity containing the exceptional point, and an extra phase difference of $\pi$ appears (say, the geometric phase). These results broaden the scope of the geometric phase and provide insights into the thermal topology.

1 Opening Remarks

The Berry phase [1] was found in a quantum mechanical system, and now it has become a fundamental concept in various systems, including the classical one [2]. The Berry phase has significant impacts on electronic properties [3] and phononic properties [4, 5]. In addition to waves, diffusion is a widespread method for transferring energy or mass, such as heat conduction and the Brownian motion of classical particles. Geometric phase was also revealed in diffusion systems [6,7,8,9,10,11], which led to novel phenomena such as heat pumping [12] and geometric heat flux [13].

However, the related physics has not yet been established in a macroscopic thermal convection-conduction system, mainly resulting from the difficulty of defining phase. Inspired by pioneering studies, we introduce phase-related properties with thermal convection [14,15,16,17,18,19,20,21], and study two moving rings with equal-but-opposite velocities, joined together by a stationary intermediate layer. This macroscopic system is different from the microscopic one where phonon is the carrier of heat transfer [22,23,24], but it can also be effectively described by a non-Hermitian Hamiltonian [25,26,27,28]. Here, the Hamiltonian can be understood as a matrix, and non-Hermitian means that the conjugate transpose is not the matrix itself. As a result, an exceptional point of velocity appears, similar to optics and photonics [29, 30]. The exceptional point is related to anti-parity-time symmetry [15, 16], which is also widely explored in other systems [31,32,33]. The exceptional point further leads to the geometric phase for a cyclic path of time-varying velocity. If the cyclic path contains the exceptional point, a moving temperature profile can accumulate an extra phase difference of $\pi$ (the geometric phase).

As revealed by a recent study [35] which provides an alternative method to explain the findings reported in Ref. [15], the competition between convection and conduction is the key to the exceptional point. Therefore, the two moving rings are the protagonists, and the intermediate layer plays a supporting role in allowing the competition.

2 Exceptional Point

As shown in Fig. 21.1a, we investigate two moving rings with equal-but-opposite velocities ($+u$ and $-u$), joined together by a stationary intermediate layer with inner radius $r_1$ and outer radius $r_2$. For the convenience of theoretical discussion, we unfold the three-dimensional model along $y-z$ plane, and expand the interior surface along x axis [with length $l=2\pi r_1$, see Fig. 21.1b, where the left and right ends are applied with periodic boundary condition. This theoretical simplification does not affect the final conclusions, just for discussion convenience. We denote the temperature distributions in the upper ring, lower ring, and intermediate layer as $T_1$, $T_2$, and $T_i$, respectively. The macroscopic thermal convection-conduction process is dominated by

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{\partial T_1}{\partial t}=D_1\left( \frac{\partial ^2T_1}{\partial x^2}+\frac{\partial ^2T_1}{\partial z^2}\right) -u\frac{\partial T_1}{\partial x}, &{}w_i/2\le z\le w_i/2+w\\ \\ \frac{\partial T_i}{\partial t}=D_i\left( \frac{\partial ^2T_i}{\partial x^2}+\frac{\partial ^2T_i}{\partial z^2}\right) ,&{}-w_i/2< z< w_i/2\\ \\ \frac{\partial T_2}{\partial t}=D_2\left( \frac{\partial ^2T_2}{\partial x^2}+\frac{\partial ^2T_2}{\partial z^2}\right) +u\frac{\partial T_2}{\partial x},&{}-w_i/2-w\le z\le -w_i/2 \end{array} \right. , \end{aligned}$$

(21.1)

where $D_1\left( =D+d\right)$, $D_2\left( =D-d\right)$, and $D_i$ are the diffusivities of the upper ring, lower ring, and intermediate layer, respectively. The two moving rings and the intermediate layer thicknesses are denoted as w and $w_i$, respectively. For generality, we extend Li et al.’s theory [15] by setting the two moving rings with different diffusivities. Meanwhile, we follow the Hamiltonian description of Li et al.’s theory, and one can also use a dimensionless description of Zhao et al.’s theory [35].

Fig. 21.1

Adapted from Ref. [34]

Basic properties of the macroscopic thermal convection-conduction system. a Three-dimensional model. b Simplified two-dimensional model. c Decay rate (-Im$\omega$) and frequency (Re$\omega$) as a function of velocity u. Parameters: $w=2.5$ mm, $w_i=0.5$ mm, $r_1=50$ mm, $r_2=52$ mm, $D=10^{-5}$ m$^2$/s, $\rho c=10^6$ J m$^{-3}$ K$^{-1}$, and $\kappa _i=0.1$ W m$^{-1}$ K$^{-1}$. These parameters lead to $u_\textrm{EP}=4$ mm/s. The velocity of $\boldsymbol{\psi }_1$ and $\boldsymbol{\psi }_1'$ is $2\sqrt{2}$ mm/s. d Five representative eigenstates. The phase difference of $\boldsymbol{\psi }_1$ (or $\boldsymbol{\psi }_1'$) is $\pi /4$ (or $3\pi /4$), and that of $\boldsymbol{\psi }_2$, $\boldsymbol{\psi }_3$, and $\boldsymbol{\psi }_3'$ is $\pi /2$. The left and right temperature profiles of each eigenstate correspond to the lower and upper rings, respectively.

As a quasi one-dimensional model ($l\gg w$), it is reasonable to suppose that the temperature variance along z axis is negligible ($\partial ^2 T/\partial z^2=0$). The intermediate layer allows energy exchange between the two moving rings, which can be regarded as two source terms. Therefore, the middle equation in Eq. (21.1) is replaced with two source terms, i.e., $s_1$ for the upper ring and $s_2$ for the lower ring. We can then obtain

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial T_1}{\partial t}=D_1\frac{\partial ^2T_1}{\partial x^2}-u\frac{\partial T_1}{\partial x}+\frac{s_1}{\rho c}, &{}w_i/2\le z\le w_i/2+w\\ \\ \frac{\partial T_2}{\partial t}=D_2\frac{\partial ^2T_2}{\partial x^2}+u\frac{\partial T_2}{\partial x}+\frac{s_2}{\rho c},&{}-w_i/2-w\le z\le -w_i/2 \end{array} \right. . \end{aligned}$$

(21.2)

We take the same density and heat capacity product of the upper and lower rings, i.e., $\rho c$.

The boundary conditions are given by the continuities of temperature and heat flux,

$$\begin{aligned} \left\{ \begin{array}{llll} T_1=T_i,&{}z=w_i/2\\ T_2=T_i,&{}z=-w_i/2\\ \\ j_1=-\kappa _1\frac{\partial T_1}{\partial z}=-\kappa _i\frac{\partial T_i}{\partial z},&{}z=w_i/2\\ \\ j_2=\kappa _2\frac{\partial T_2}{\partial z}=\kappa _i\frac{\partial T_i}{\partial z},&{}z=-w_i/2 \end{array} \right. , \end{aligned}$$

(21.3)

where $j_1$ and $j_2$ are the heat fluxes from the intermediate layer to the upper and lower rings, respectively. $\kappa _1$, $\kappa _2$, and $\kappa _i$ are the thermal conductivities of the upper ring, lower ring, and intermediate layer, respectively. Since we have neglected the higher-order terms $\left( \partial ^2 T/\partial z^2=0\right)$, $T_i$ is linear along z axis, thus yielding $\partial T_i/\partial z=\left( T_1-T_2\right) /w_i$. The width of the two moving rings (w) is small enough, so we can assume that the two sources ($s_1$ and $s_2$) are uniformly distributed along the ring width, i.e., $s_1=j_1/w=-\kappa _i(T_1-T_2)/\left( ww_i\right)$ and $s_2=j_2/w=-\kappa _i(T_2-T_1)/\left( ww_i\right)$. Equation (21.2) can then be reduced to

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial T_1}{\partial t}=D_1\frac{\partial ^2 T_1}{\partial x^2}-u\frac{\partial T_1}{\partial x}+h\left( T_2-T_1\right) ,&{}w_i/2\le z\le w_i/2+w\\ \\ \frac{\partial T_2}{\partial t}=D_2\frac{\partial ^2 T_2}{\partial x^2}+u\frac{\partial T_2}{\partial x}+h\left( T_1-T_2\right) ,&{}-w_i/2-w\le z\le -w_i/2 \end{array} \right. , \end{aligned}$$

(21.4)

where $h=\kappa _i/(\rho c w w_i)$. Since h describes the heat exchange rate between the two moving rings, which is vertical to the velocity direction, it is independent of the velocity.

We use plane-wave solutions to introduce phase-related properties,

$$\begin{aligned} \left\{ \begin{array}{ll} T_1=A_1\textrm{e}^{\textrm{i}\left( kx-\omega t\right) }+T_0\\ T_2=A_2\textrm{e}^{\textrm{i}\left( kx-\omega t\right) }+T_0 \end{array} \right. , \end{aligned}$$

(21.5)

where $A_1$ (or $A_2$) is the temperature amplitude in the upper (or lower) ring, k is wave number, $\omega$ is complex frequency, and $T_0$ is reference temperature which is set to zero for brevity. Only the real parts of Eq. (21.5) make sense. By substituting Eq. (21.5) into Eq. (21.4), we can obtain

$$\begin{aligned} \left\{ \begin{array}{ll} \omega A_1=-\textrm{i}k^2D_1A_1+kuA_1+\textrm{i}h\left( A_2-A_1\right) ,&{}w_i/2\le z\le w_i/2+w\\ \omega A_2=-\textrm{i}k^2D_2A_2-kuA_2+\textrm{i}h\left( A_1-A_2\right) ,&{}-w_i/2-w\le z\le -w_i/2 \end{array} \right. . \end{aligned}$$

(21.6)

Equation (21.6) can also be expressed as

$$\begin{aligned} \boldsymbol{\hat{H}}|\boldsymbol{\psi }\rangle =\omega |\boldsymbol{\psi }\rangle , \end{aligned}$$

(21.7)

where $|\boldsymbol{\psi }\rangle =\left[ A_1,\,A_2\right] ^{\tau }$ is eigenstate, and $\tau$ denotes transpose. The Hamiltonian $\boldsymbol{\hat{H}}$ reads

$$\begin{aligned} \boldsymbol{\hat{H}}= \left[ \begin{matrix} -\textrm{i}\left( k^2D_1+h\right) +ku &{} \textrm{i}h\\ \textrm{i}h &{} -\textrm{i}\left( k^2D_2+h\right) -ku \end{matrix} \right] . \end{aligned}$$

(21.8)

Equation (21.8) is a general expression. Here, we discuss the case of $d=0$ (i.e., $D_1=D_2=D$), and Eq. (21.8) becomes

$$\begin{aligned} \boldsymbol{\hat{H}}= \left[ \begin{matrix} -\textrm{i}\left( k^2D+h\right) +ku &{} \textrm{i}h\\ \textrm{i}h &{} -\textrm{i}\left( k^2D+h\right) -ku \end{matrix} \right] , \end{aligned}$$

(21.9)

where $D=\kappa _1/\left( \rho c\right) =\kappa _2/\left( \rho c\right)$.

The eigenvalues of the Hamiltonian (Eq. (21.9)) take on the form

$$\begin{aligned} \omega _\pm =-\textrm{i}\left[ \left( k^2D+h\right) \pm \sqrt{h^2-k^2u^2}\right] , \end{aligned}$$

(21.10)

which are complex numbers. The system exhibits two different properties as u varies. The point $u_\textrm{EP}=h/k$ determines the transition of two different properties, thus serving as an exceptional point. As required by the periodic boundary condition, wave numbers can only take on discrete values, i.e., $k=2\pi n/l=nr_1^{-1}$ with n being positive integers. We discuss the fundamental modes with $n=1$ because their decay rates are the lowest.

In the region $u<u_\textrm{EP}$, the complex frequencies ($\omega _\pm$) exhibit two different branches with purely imaginary values (Fig. 21.1c), indicating that the waves described by Eq. (21.5) only decay but do not propagate. The difference between $\omega _+$ and $\omega _-$ is the decay rate: the decay rate of $\omega _-$ is smaller than that of $\omega _+$. Therefore, $\omega _+$ is also observable, but it decays much faster than $\omega _-$. The corresponding eigenstates are

$$\begin{aligned} |\boldsymbol{\psi }_+\rangle =\left[ 1,\,\textrm{e}^{\textrm{i} \left( \pi -\alpha \right) }\right] ^{\tau },\,|\boldsymbol{\psi }_-\rangle =\left[ 1,\,\textrm{e}^{\textrm{i} \alpha }\right] ^{\tau }, \end{aligned}$$

(21.11)

where $\alpha =\sin ^{-1}\left( ku/h\right)$. Therefore, the temperature profiles of the two moving rings maintain a constant phase difference ($\pi -\alpha$ for $\omega _+$ and $\alpha$ for $\omega _-$) and decay motionlessly (see $\boldsymbol{\psi }_1$ and $\boldsymbol{\psi }_1'$ in Fig. 21.1d).

When the velocity reaches the exceptional point, the difference between $\omega _+$ and $\omega _-$ disappears (Fig. 21.1c). The two eigenstates have the same phase difference of $\pi /2$ and decay motionlessly (see $\boldsymbol{\psi }_2$ in Fig. 21.1d).

When $u>u_\textrm{EP}$, the complex frequencies ($\omega _\pm$) take on real components (Fig. 21.1c), indicating that the waves described by Eq. (21.5) not only decay but also propagate. The corresponding eigenstates become

$$\begin{aligned} |\boldsymbol{\psi }_+\rangle =\left[ \textrm{e}^{-\delta },\,\textrm{e}^{\textrm{i} \pi /2-2\delta }\right] ^{\tau },\,|\boldsymbol{\psi }_-\rangle =\left[ \textrm{e}^{-\delta },\,\textrm{e}^{\textrm{i} \pi /2}\right] ^{\tau }, \end{aligned}$$

(21.12)

where $\delta =\cosh ^{-1}\left( ku/h\right)$. Therefore, the two eigenstates maintain the same phase difference of $\pi /2$ but decay with motion. The moving direction follows the ring with a larger temperature amplitude (see $\boldsymbol{\psi }_3$ and $\boldsymbol{\psi }_3'$ in Fig. 21.1d).

Fig. 21.2

Adapted from Ref. [34]

Evolutions of the temperature profiles. Two types of color maps denote temperature (a–c, g–i and m–o) and time (d–f and j–l), respectively. The temperature scale is the same as that in Fig. 21.1. The initial states are presented in a–c with the form in the Cartesian coordinates as $T_1=Ay/\sqrt{x^2+y^2}+B$ and $T_2=Ay/\sqrt{x^2+y^2}+B$ for $\boldsymbol{\psi }_0$, $T_1=Ay/\sqrt{x^2+y^2}+B$ and $T_2=-Ay/\sqrt{x^2+y^2}+B$ for $\boldsymbol{\psi }_0'$, $T_1=Ay/\sqrt{x^2+y^2}+B$ and $T_2=-Ax/\sqrt{x^2+y^2}+B$ for $\boldsymbol{\psi }_2$, and $T_i=B$, where $A=100$ and $B=400$. The trajectories of Max($T_1$) and Max($T_2$) along interior edges are shown in d–i with $u=2\sqrt{2}$ mm/s and (j)-(o) with $u=6$ mm/s. The parameters are the same as those for Fig. 21.1. The meshing parameters are as follows. The maximum and minimum element sizes are $5\times 10^{-4}$ and $10^{-5}$ m, respectively. The maximum element growth rate is 1.3, the curvature factor is 0.2, and the resolution of narrow regions is 1.

We use COMSOL Multiphysics to perform finite-element simulations based on a three-dimensional model (Fig. 21.2). We define $T_1$ and $T_2$ as the temperature distributions along the upper and lower interior edges of the two moving rings. We track the evolutions of temperature profile by following maximum-temperature points, i.e., Max($T_1$) and Max($T_2$). The initial states are set as the three eigenstates $\boldsymbol{\psi }_0$, $\boldsymbol{\psi }_0'$, and $\boldsymbol{\psi }_2$ (Fig. 21.2a–c).

If we set the velocity to $2\sqrt{2}$ mm/s ($<u_\textrm{EP}$), the initial state moves to a certain position and remains stationary thereafter (Fig. 21.2d–f). All three final states (with $\alpha \approx \pi /4$) are the eigenstate corresponding to eigenvalue $\omega _-$ (i.e., $\boldsymbol{\psi }_1$, see Fig. 21.2g–i). This phenomenon occurs because of the non-orthogonality of the eigenstates at different branches [25]. Meanwhile, the decay rate at the upper branch is much higher than that at the lower branch, so the eigenstate at the lower branch becomes the observable one associated with eigenvalue $\omega _-$.

Moreover, we also care about how these initial states evolve to final states (i.e., evolution routes). One principle is that the evolution routes should try to keep away from the eigenstate corresponding to eigenvalue $\omega _+$ (i.e., $|\boldsymbol{\psi }_+\rangle$, with a much higher decay rate) to survive longer. For this purpose, the moving direction of the maximum-temperature point in Fig. 21.2f is even against the background velocities of respective moving rings.

If we set the velocity to 6 mm/s ($>u_\textrm{EP}$), the trajectories of Max($T_1$) and Max($T_2$) are always moving because the eigenvalue has a real component (Fig. 21.2j–l). The corresponding states at 70 s are presented in Fig. 21.2m–o. Here, the duration for $u=6$ mm/s (70 s) is shorter than that for $u=2\sqrt{2}$ mm/s (160 s) because the decay rate for $u=6$ mm/s is higher than that for $u=2\sqrt{2}$ mm/s.

The results presented in Fig. 21.2d–i can help us draw some conclusions. When the velocity is smaller than the exceptional point, the final eigenstate prefers to stay at the lower branch corresponding to eigenvalue $\omega _-$. The evolution route should keep away from the eigenstate corresponding to eigenvalue $\omega _+$. The evolution routes should ensure that the temperature profiles decay as slowly as possible before reaching the final states.

3 Thermal Geometric Phase

After understanding the exceptional point and evolution routes, we can reveal the geometric phase in our system. The Hamiltonian (Eq. (21.9)) is a function of multiple parameters. We resort to a time-varying velocity $u\left( t\right)$ which is experimentally controllable, instead of other parameters such as the thickness of the intermediate layer. Although the Hamiltonian $\hat{\boldsymbol{H}}$ is not Hermitian, we can check that

$$\begin{aligned} \hat{\boldsymbol{H}}^{\dagger }|\overline{\boldsymbol{\psi }}_{\pm }\rangle = \overline{\omega }_{\pm }|\overline{\boldsymbol{\psi }}_{\pm }\rangle , \end{aligned}$$

(21.13)

where $\hat{\boldsymbol{H}}^{\dagger }$ is the Hermitian transpose of $\hat{\boldsymbol{H}}$. $|\overline{\boldsymbol{\psi }}_{\pm }\rangle$ and $\overline{\omega }_{\pm }$ are the complex conjugates of $|\boldsymbol{\psi }_{\pm }\rangle$ and $\omega _{\pm }$, respectively. The eigenstates satisfy

$$\begin{aligned} \left\langle \overline{\boldsymbol{\psi }}_{\pm }|\boldsymbol{\psi }_{\mp } \right\rangle =0. \end{aligned}$$

(21.14)

$\left\langle \overline{\boldsymbol{\psi }}_{\pm }|\boldsymbol{\psi }_{\mp } \right\rangle$ denotes the complex inner product of the vectors $|\overline{\boldsymbol{\psi }}_{\pm }\rangle$ and $|\boldsymbol{\psi }_{\mp }\rangle$. As discussed in Fig. 21.2, the final states always go back to the eigenstate associated with eigenvalue $\omega _-$ (i.e., the initial state) after experiencing a cyclic evolution, thus ensuring an adiabatic process. Therefore, we can write down the complex geometric phase under an adiabatic approximation as

$$\begin{aligned} \varphi _{\pm }=\textrm{i} \int \frac{\left\langle \overline{\boldsymbol{\psi }}_{\pm }(u)|d \boldsymbol{\psi }_{\pm }(u)\right\rangle }{\left\langle \overline{\boldsymbol{\psi }}_{\pm }(u)|\boldsymbol{\psi }_{\pm }(u)\right\rangle }, \end{aligned}$$

(21.15)

which is in accordance with the results of non-Hermitian quantum systems [36]. We find that $\left\langle \overline{\boldsymbol{\psi }}_{\pm }(u)|\boldsymbol{\psi }_{\pm }(u)\right\rangle =0$ at the exceptional point because the two eigenstates coalesce. Therefore, the exceptional point is a pole in the complex integral. We can rewrite Eq. (21.15) in a closed loop around the exceptional point as [37]

$$\begin{aligned} \varphi _{\pm }=\dfrac{\textrm{i}}{2} \oint d \ln \left\langle \overline{ \boldsymbol{\psi }}_{\pm }(u)|\boldsymbol{\psi }_{\pm }(u)\right\rangle . \end{aligned}$$

(21.16)

According to the residue theorem, we know that $\varphi _{\pm }=\pi$ or $-\pi$, and the sign depends on the direction of the closed loop. If the evolution route does not contain the exceptional point, the geometric phase in a cyclic evolution equals zero.

Fig. 21.3

Adapted from Ref. [34]

Finite-element simulations of the geometric phase. The parameters are the same as those for Fig. 21.1. a Five evolution routes. b Initial state. The trajectories of Max($T_1$) and Max($T_2$) corresponding to the five evolution routes are presented in c, d, g, h, and i. The final states are presented in e, f, j, k, and l, respectively.

We also perform finite-element simulations to observe the geometric phase. For this purpose, we apply a cyclic path of time-varying velocity, which is governed by the Hamiltonian $\hat{H}\left[ u\left( t\right) \right]$. Note that a fiduciary marker on the surface of the ring would not necessarily return to its original location after the cyclic evolution of velocity. We explore five different paths of velocity, as shown in Fig. 21.3a. The initial velocity is $u=0$ mm/s, and the initial state is set to the eigenstate associated with eigenvalue $\omega _-$ (i.e., $\boldsymbol{\psi }_0$, see Fig. 21.3b).

In Route 1 or 2, the eigenvalue is purely imaginary because the velocity is smaller than the exceptional point, indicating no extra phase difference accumulated. As a result, the two evolution routes bring the final state back to the initial state exactly (see Fig. 21.3c, d or Fig. 21.3e, f).

However, the situation is different when the cyclic evolution of velocity contains the exceptional point (Fig. 21.3g, h). As the velocity increases and exceeds the exceptional point, the eigenvalue obtains a real component, indicating that an extra phase difference accumulates. This property means that the initial state moves smoothly from one branch to another. When the state goes through the eigenstate corresponding to another eigenvalue $\omega _+$ (i.e., $\boldsymbol{\psi }_0'$), the state cannot return, as discussed in Fig. 21.2. The phase difference then continuously increases to reach a different place with eigenvalue $\omega _-$ (i.e., $\boldsymbol{\psi }_0$). Therefore, the temperature profile stops at a different position with a phase difference of $\pi$ compared with the initial position (Fig. 21.3j, k). This phenomenon is evidence of the geometric phase.

Finally, we repeat Route 4 twice (indicated as Route 5). Since the state goes through the eigenstate corresponding to eigenvalue $\omega _+$ (i.e., $\boldsymbol{\psi }_0'$) twice, Route 5 brings the state back to the initial state without any global phase change (Fig. 21.3i, l).

Although the macroscopic thermal convection-conduction system can also be effectively described by a non-Hermitian Hamiltonian (Eq. (21.9)), its properties are distinct from wave systems. In the case of non-Hermitian wave systems, the cyclic evolution around the exceptional point may interchange the instantaneous eigenvectors [30]. However, the present eigenstates at the upper branch (especially with small velocities) are metastable because of their large decay rates. Therefore, they do not dominate the end of evolutions. In contrast, these metastable eigenstates are more similar to a “wall”, which should be avoided when the velocity is smaller than the exceptional point, thus determining the evolution route and final position. When the velocity is larger than the exceptional point, it provides an opportunity to cross the eigenstate corresponding to the eigenvalue $\omega _+$, once to accumulate an extra phase difference of $\pi$. Therefore, the geometric property is reflected on $|\boldsymbol{\psi }_+\rangle$ to some extent.

Fig. 21.4

Adapted from Ref. [34]

Experimental demonstration of the geometric phase. a Experimental setup. The sizes of the two moving rings are $r_1=60$, $r_2=65$, and $w=2$ mm, respectively. The rings are made of polycaprolactam with $D=7.3\times 10^{-7}$ m$^2$/s and $\rho c=1.8\times 10^6$ J m$^{-3}$ K$^{-1}$. The intermediate layer is 1-mm-thick grease with $\kappa _i=0.1$ W m$^{-1}$ K$^{-1}$. These parameters cause the exceptional value of rotation speed ($\Omega _\textrm{EP}$) to be approximately 0.25 rmp. b and c Initial and final states with rotation speed not crossing the exceptional point. d and e Initial and final states with the rotation speed crossing the exceptional point.

We also conduct experiments to validate the theoretical analyses and finite-element simulations. The different views of the experimental setup are shown in Fig. 21.4a. The aluminum frames are wrapped with black tape to avoid the environmental reflection of thermal radiation and ensure the accuracy of temperature profiles. We also use small woodblocks (with thermal conductivity of 0.1 W m$^{-1}$ K$^{-1}$) to separate the rings and aluminum frames (with thermal conductivity of 218 W m$^{-1}$ K$^{-1}$) to reduce thermal dissipation. With these preparations, the environmental dissipation is small enough to be neglected. The two rings are driven by two motors with different rotation directions, as shown by blue arrows. The two motors are controlled by one circuit, thus ensuring the equal-but-opposite rotation speed. The slide rheostat can manually adjust the rotation speed. We use the Flir E60 infrared camera to detect temperature profiles and find maximum temperatures in a particular region.

We use a copper plate with one end heated to generate the initial temperature profiles of the two moving rings (see Fig. 21.4b, d). We quickly remove the copper plate and push the two rings together to observe the evolution. We further tune the rotation speed ($\Omega$) from zero to a small value ($\sim$0.1 rpm) or a large one ($\sim$1 rpm) and then set it back to zero. If the rotation speed does not cross the exceptional point, the final state (Fig. 21.4c) is precisely the same as the initial state (Fig. 21.4b). However, if the rotation speed crosses the exceptional point, the final state (Fig. 21.4e) exhibits a phase difference of $\pi$ compared with the initial position (Fig. 21.4d).

We also analyze the experimental results and compare them with finite-element simulations. For this purpose, we track the maximum temperature of the front ring. The initial position is set as a rotation angle of 0, and $\varphi$ is the rotation angle of the maximum temperature of the front ring, which denotes the geometric phase of our interest. When applying a cyclic evolution avoiding the exceptional point (Fig. 21.5a), the final state goes back to the initial state exactly, resulting in $\varphi =0$ (Fig. 21.5b). When applying a cyclic evolution containing the exceptional point (Fig. 21.5c), the final state has a geometric phase compared with the initial state, resulting in $\varphi =\pi$ (Fig. 21.5d).

Fig. 21.5

Adapted from Ref. [34]

Detailed information of Fig. 21.4. In a and c, dashed lines represent the exceptional value of rotation speed ($\Omega _\textrm{EP}=0.25$ rmp), and solid lines exhibit the cyclic evolutions of rotation speed ($\Omega$). The measured and simulated results are presented in b and d. The phase difference is $\varphi =0$ for the cyclic evolution shown in a, whereas it is $\varphi =\pi$ for the cyclic evolution shown in c.

4 Conclusion

Besides various wave systems, macroscopic thermal convection-conduction systems (essentially non-Hermitian systems) can also exhibit the geometric phase by encircling an exceptional point. More relevant properties such as topological invariance (or winding number) in thermotics can be further explored with a similar method applied in non-Hermitian systems [38,39,40,41,42]. These results may also provide insights into heat regulation with exceptional points.

5 Exercise and Solution

Exercise

• 1. Calculate the eigenvalues and eigenvectors of Eq. (21.8).

Solution

• 1. The eigenvalues of Eq. (21.8) can be expressed as

  $$\begin{aligned} \omega _\pm =-\textrm{i}\left[ \left( k^2D+h\right) \pm \sqrt{h^2+k^4d^2-k^2u^2+2k^3du\textrm{i}}\right] , \end{aligned}$$

  (21.17)

  and the corresponding eigenstates are

  $$\begin{aligned} |\boldsymbol{\psi }_\pm \rangle =\left[ 1,\frac{-h}{k^2d+ku\textrm{i}\pm \sqrt{h^2+k^4d^2-k^2u^2+2k^3du\textrm{i}}}\right] ^{\tau }. \end{aligned}$$

  (21.18)