⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 106601 Two-Dimensional Thermal Regulation Based on Non-Hermitian Skin Effect Qiang-Kai-Lai Huang (黄强开来)1,2,3,4, Yun-Kai Liu (刘云开)5, Pei-Chao Cao (曹培超)1,2,3,4, Xue-Feng Zhu (祝雪丰)5*, and Ying Li (李鹰)1,2,3,4* Affiliations 1Interdisciplinary Center for Quantum Information, State Key Laboratory of Extreme Photonics and Instrumentation, ZJU-Hangzhou Global Scientific and Technological Innovation Center, College of Information Science & Electronic Engineering, Zhejiang University, Hangzhou 310027, China 2International Joint Innovation Center, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining 314400, China 3Key Laboratory of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Jinhua Institute of Zhejiang University, Zhejiang University, Jinhua 321099, China 4Shaoxing Institute of Zhejiang University, Zhejiang University, Shaoxing 312000, China 5School of Physics and Innovation Institute, Huazhong University of Science and Technology, Wuhan 430074, China Received 25 July 2023; accepted manuscript online 31 August 2023; published online 26 September 2023 *Corresponding authors. Email: xfzhu@hust.edu.cn; eleying@zju.edu.cn Citation Text: Huang Q K L, Liu Y K, Cao P C et al. 2023 Chin. Phys. Lett. 40 106601    Abstract The non-Hermitian skin effect has been applied in multiple fields. However, there are relatively few models in the field of thermal diffusion that utilize the non-Hermitian skin effect for achieving thermal regulation. Here, we propose two non-Hermitian Su–Schrieffer–Heeger (SSH) models for thermal regulation: one capable of achieving edge states, and the other capable of achieving corner states within the thermal field. By analyzing the energy band structures and the generalized Brillouin zone, we predict the appearance of the non-Hermitian skin effect in these two models. Furthermore, we analyze the time-dependent evolution results and assess the robustness of the models. The results indicate that the localized thermal effects of the models align with our predictions. In a word, this work presents two models based on the non-Hermitian skin effect for regulating the thermal field, injecting vitality into the design of non-Hermitian thermal diffusion systems.
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 DOI:10.1088/0256-307X/40/10/106601 © 2023 Chinese Physics Society Article Text Thermal metamaterials[1-3] are heat transfer control systems that achieve novel characteristics by artificially designing system structures at macroscopic scales. This method can be applied to fields such as thermal cloak,[4-7] thermal transparency,[8-10] thermal concentration,[11-14] and thermal absorption.[15] However, effective modulation methods for thermal metamaterials are relatively scarce. In the research scope of thermal metamaterials, we only have strategies such as discrete control of material parameters,[16,17] reliance on topological theory[18-23] or spatiotemporal modulation.[24,25] However, these methods have limitations such as low modulation freedom or dependence on specific eigenmodes. Since the solution of the Fourier equation satisfies the form of the wave-like solution,[26] we can establish a coupled mode theory for heat diffusion.[27] This theory enables us to discretize designs of thermal superstructures. In other words, we can create a thermal control system by first designing periodic thermal coupling units and then performing array operations on them.[28] It is important to note that due to the diffusion characteristics of heat conduction, the Hamiltonian in diffusive systems is naturally anti-Hermitian.[29] As a result, non-Hermitian physics phenomena[30,31] such as non-Hermitian skin effect (NHSE)[32-37] can also be applied in designs of thermal regulation systems.[38] In this Letter, we present a combination of thermal diffusion coupled mode theory and NHSE. We propose two models: a two-dimensional edge state Su–Schrieffer–Heeger (SSH) model and a two-dimensional corner state SSH model. These models aim to achieve thermal field regulation. By leveraging NHSE, we can achieve directional regulation of the thermal field. Compared with other cases of thermal field regulation, our models have more modulation freedoms and achieve the required localized states in almost all eigenmodes. This work not only expands the strategies for thermal field regulation but also provides further validation of the non-Hermitian theory in the context of thermal diffusive systems. Let us start from the edge state model shown in Fig. 1(a). This structure is composed of cell arrays with two cavities in each cell. According to Fourier's law in heat conduction, the $x$ direction coupling equations of the first cell can be written as \begin{align} &\frac{\partial T_{1} }{\partial t}=D_{1} \frac{\partial^{2}T_{1} }{\partial x^{2}}+C_{12}, \nonumber\\ &\frac{\partial T_{2} }{\partial t}=D_{2} \frac{\partial^{2}T_{2} }{\partial x^{2}}+C_{21}, \tag {1} \end{align} where $T_{1}$ ($T_{2}$) represents the temperature field of the 1$^{\rm st}$ (2$^{\rm nd}$) cavity, while $C_{12}$ ($C_{21}$) is the coupling term from the 2$^{\rm nd}$ (1$^{\rm st}$) cavity to the 1$^{\rm st}$ (2$^{\rm nd}$) cavity, and $D_{1 }=\kappa_{1}$/($\rho_{1}C_{p 1}$) $[D_{2 }=\kappa_{2}/(\rho_{2}C_{p 2})$] is the diffusivity of the 1$^{\rm st}$ (2$^{\rm nd}$) cavity, with $\kappa_{1}$ ($\kappa_{2}$), $\rho_{1}$ ($\rho_{2}$), $C_{p 1}$ ($C_{p 2}$) referring to thermal conductivity, density and heat capacity at constant pressure, respectively. When our system is small, it can be considered that the temperature is uniformly distributed in the cavity and linearly distributed in the coupling rod. Thus, the Laplace operator term in Eq. (1) should be zero. Now, we take this simplification and coupling mode theory into consideration, Eq. (1) becomes \begin{align} &{\frac{\partial T_{1} }{\partial t}=C_{12} =\frac{\kappa_{i} }{\rho_{1} C_{p1} }\frac{S_{i} }{V_{1} L}(T_{2} -T_{1})=h_{12} (T_{2} -T_{1})}, \nonumber\\ &{\frac{\partial T_{2} }{\partial t}=C_{21} =\frac{\kappa_{i} }{\rho_{2} C_{p2} }\frac{S_{i} }{V_{2} L}(T_{1} -T_{2})=h_{21} (T_{1} -T_{2})}, \tag {2} \end{align} where $S_{i}$ is the cross-sectional area of the coupling rod, and $L$ is the length of coupling rod. $V_{1}$ ($V_{2}$) is the volume of the 1$^{\rm st}$ (2$^{\rm nd}$) cavity, and $h_{12}$ ($h_{21}$) is the coupling strength which depends on the material and geometric parameters. We can assume a wave-like solution $T_{1,2} = A_{1,2}e^{ik_{x}x-\omega t}$ for Eq. (2), where $A_{1,2}$, $k_{x}$, and $\omega$ are amplitude, Bloch wave number in the $x$ direction, and complex frequency, respectively. By substituting the wave-like solution into Eq. (2), the Hamiltonian of the first cell in Fig. 1(a) can be expressed as \begin{align} H_{1-2} =-i \begin{bmatrix} {h_{12}}&{-h_{12}}\\ {-h_{21}}&{h_{21}}\\ \end{bmatrix}. \tag {3} \end{align} Obviously, the Hamiltonian is naturally anti-Hermitian. According to Eq. (2), non-Hermiticity can be achieved by modulating material or geometric parameters, ensuring that the coupling strengths satisfy $h_{12}\ne h_{21}$. In terms of the experimental aspect, conductivity and mass density can be modified by creating air duty cycles through punching holes in the background material. The effective parameters can be determined by the Maxwell–Garnett formula.[16] Additionally, modulating geometric parameters could be a more straightforward approach for experiment.[39] In short, these modulations make the Hamiltonian non-reciprocal and cause NHSE, which manifests the directed aggregation of temperature.
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Fig. 1. (a) Schematic of the edge state model under the Bloch boundary condition applied in the $y$ direction. (b) GBZ of the edge state model. The radius of GBZ is always 1/$a$, irrespective of the Bloch wave number in the $y$ direction ($k_{y}$). (c) Energy spectra under PBC and OBC. The energy centers are shifted to (0,0). The blue and orange regions represent eigenvalues under PBC. The black dots represent the energies under OBC, calculated by a $64 \times 64$ Hamiltonian. However, in the case of OBC, the $x$ and $y$ axes do not represent $k_{x}$ and $k_{y}$. The drawing of PBC and OBC cases together is only for comparing the energy amplitude of its imaginary part. The energy bands exhibit two energy loops in each phase, and the rightmost image illustrates two loops in one of the phases.
Figure 1(a) illustrates the modulation of the non-reciprocal coupling coefficient within a single cell. In the positive $x$ direction, we set the coupling strength to $h_{0}/a$, while in the negative $x$ direction, it is set to $ah _{0}$. The coupling strength between cells remains as $h_{0}$. Here, $h_{0}$ represents the standard coupling strength determined by material and geometric parameters, and $a$ ($0 < a < 1$) denotes the non-reciprocal coefficient. We use Bloch boundary conditions in the $y$ direction, which achieves reciprocal coupling. The Bloch boundary conditions facilitate the creation of arrays in the $y$ direction for constructing a larger edge state model. Through such modulation, the non-Hermitian modulation depth of the system can be determined lonely by parameter $a$. By applying the Bloch theorem, the temperature field satisfies the equations \begin{align} &T_{2n+1} =\beta_{x}^{n}T_{1},~~~(n\in \mathbb{Z}), \nonumber \\ &T_{2n} =\beta_{x}^{n}T_{2},~~~(n\in \mathbb{Z}), \tag {4} \end{align} where $n$ represents the cavity ordinal number. We define $\beta_{x,y}$: = exp($ik _{x,y}$) as the Bloch phase factors in the $x$ or $y$ direction. In a Hermitian system, the real Bloch wave number $k$ makes $|\beta|=1$, while in a non-Hermitian system, $k$ can be a complex number resulting in $|\beta|$ not necessarily equal to 1. Combining Eqs. (3)and (4), we can obtain the Bloch Hamiltonian of the system depicted in Fig. 1(a) as \begin{align} H_{\rm edge} =-i \begin{bmatrix} {(a+1)h_{0}}&{-(\beta_{x}^{-1}+a)h_{0}}\\ {-(\beta_{x} +1/a)h_{0}}&{(1+1/a)h_{0}}\\ \end{bmatrix}. \tag {5} \end{align} The behavior of the temperature field can be predicted by examining the coincidence between the generalized Brillouin zone (GBZ)[40-42] and the Brillouin zone (BZ). Figure 1(b) displays the GBZ at $a = 0.1$, which is computed from Eq. (5). We prove that the GBZ in the $x$ direction of the system forms a circle with a radius of 1/$a$, which remains unaffected by the real Bloch wave number in the $y$ direction ($k_{y}$). This indicates that when the system is in non-Hermitian state ($0 < a < 1$), the GBZ radius exceeds that of BZ, i.e., $|\beta_{x}|>1$, resulting in thermal field accumulation in the positive $x$ direction.[43] We can further evaluate the behavior of temperature field through the energy distribution in the complex plane, i.e., the non-Hermitian band theory. When considering periodic boundary condition (PBC) of a non-Hermitian system, the presence of energy loops (curves) in the energy complex plane signifies the existence (nonexistence) of NHSE under the open boundary condition (OBC).[44] This conclusion can be proved by the unequal winding numbers of GBZ and BZ. For our real PBC case of $a = 0.1$, the system exhibits a significant extent of non-reciprocity, and there is always a gap between the blue band and the orange band as depicted in Fig. 1(c). This energy gap indicates that there will be a line gap in the imaginary part of the energy in energy complex plane. The projection of this energy band on the energy complex plane performs as loops form. To avoid phase masking, we draw only two energy loops in a single phase as depicted in the rightmost panels. As $a$ increases to 0.5, the PBC energy gap narrows but remains unclosed. When $a$ reaches 1.0, the system transforms into a Hermitian case, resulting in the closure of the energy gap. The black dots in Fig. 1(c) represent the energies in the OBC case, and their real parts are all 0. It can be observed that the bandgap opening position and the energies amplitudes of OBC and PBC are aligned. Notably, when $a = 0.5$, the band gaps of the system are not significant under OBC. This is due to the energy phase shift in $y$ direction, which masks the expected band gap. Therefore, to observe the non-Hermitian model's band gap under multi-dimensional model more intuitively, it should be examined under PBC. According to the previous non-Hermitian band theory, the appearance of energy loops will indicate the occurrence of NHSE under the OBC.
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Fig. 2. (a) Distribution of bulk states for all eigenmodes at $a = 1.0$. (b) Distribution of edge states for all eigenmodes at $a = 0.5$. (c) Thermal field distribution map when $a = 0.45$, $t = 300$ s, with the yellow region and the white dashed line representing the heat source cavities and observed heat transfer path, respectively. (d) Time-dependent evolution results of the thermal field along the white dashed line depicted in (c).
We have determined the occurrence of NHSE from the perspective of GBZ and energy bands. Now we study the temperature field distribution in the edge state model by simulating the model in Fig. 1(a) with COMSOL Multiphysics. In Figs. 2(a) and 2(b), we show the eigenmodes of normalized temperature field for $a = 1.0$ and $a = 0.5$. The results indicate that when $a = 1.0$, the system is Hermitian, and the temperature fields exhibits a uniform body distribution. However, when $a = 0.5$, the temperature fields localize in the 8$^{\rm th}$ cavity for most eigenmodes, which represents the occurrence of NHSE. Figure 2(c) displays the thermal field evolution map after 300 s when $a = 0.45$. An initial temperature of 373 K is applied to the 4$^{\rm th}$ and 5$^{\rm th}$ cavities of each row. From Fig. 2(c), we can see that the temperature field is localized at the right open boundary, confirming the previous prediction. Additionally, Fig. 2(d) presents a time-dependent evolution image, where the propagation path follows the white dashed line in Fig. 2(c). After 180 s, the temperature field localizes just at the last cavity. These time-dependent evolution results further validate the feasibility of using NHSE for achieving thermal field regulation.
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Fig. 3. (a) Schematic of the corner state model with similarly non-reciprocal modulation in both the $x$ and $y$ directions. The power function of $a$ represents the modulation depth of material parameters. (b) GBZs of the corner state model when $a = 0.1$. The radius of GBZ in both the $x$ and $y$ directions are 1/$a$. (c) Energy spectra under PBC and OBC. The energy centers are shifted to (0,0). The blue, orange, yellow and purple regions represent the four eigenvalues under PBC. The black dots represent the energies under OBC, calculated by a $64 \times 64$ Hamiltonian. However, in the case of OBC, the $x$ and $y$ axes do not represent $k_{x}$ and $k_{y}$. The drawing of PBC and OBC cases together is only for comparing the energy amplitude of imaginary part. The energy bands exhibit four energy loops in each phase, and the rightmost image illustrates four energy loops in one of the phases.
Next, let us consider the corner state model as shown in Fig. 3(a). We modulate non-reciprocal coupling with strengths of $ah _{0}$ and $h_{0}/a$ in both $x$ and $y$ directions, resulting in a corner NHSE in the system. In Fig. 3(a), we use thick and thin blue arrows to represent the intracell non-reciprocal coupling strengths of $h_{0}/a$ and $ah _{0}$, respectively. Additionally, we use black bidirectional arrows to represent the intercell reciprocal coupling strength. Due to the higher coupling strengths in the upper-right direction, it is foreseeable that a corner NHSE will occur in the upper-right open boundary corner. We can obtain the Bloch Hamiltonian of the system in Fig. 3(a) as follows: \begin{align} H_{\rm corner} =-ih_{0} \begin{bmatrix} {(a+1/a+2)}&{-(a+\beta_{x}^{-1})}&0&{-(1/a+\beta_{y})}\\ {-(1/a+\beta_{x})}&{(2/a+2)}&{-(1/a+\beta_{y})}&0\\ 0&{-(a+\beta_{y}^{-1})}&{(a+1/a+2)}&{-(1/a+\beta_{x})}\\ {-(a+\beta_{y}^{-1})}&0&{-(a+\beta_{x}^{-1})}&{(2a+2)}\\ \end{bmatrix}. \tag {6} \end{align} Similarly, we also calculate the GBZs of the system in both the $x$ and $y$ directions from Eq. (6), as shown in Fig. 3(b). We find that both GBZs are circles with radius of 1/$a$, and the GBZ in the $x$ and $y$ directions do not affect each other. When $a \ne 1$, GBZs do not coincide with BZs, and NHSE appears in both the $x$ and $y$ directions, resulting in corner state localization. We also analyze the energy distribution of this system. The energy bands under PBC and OBC are depicted in Fig. 3(c). We can observe a pattern of 4 bands and 2 band gaps. When $a$ approaches 1, the band gaps narrow. When $a = 1$ the band gaps close, and the system degenerates in to a Hermitian state. Regarding the energy bands in the energy complex plane, as depicted in the rightmost panel of Fig. 3(c), the band structure exhibits four loops in a single phase, indicating the presence of NHSE in OBC. Now, we consider the time-dependent evolution results of the corner state model. The simulation model with four cavities in both the $x$ and $y$ directions is shown in Fig. 4(a). The yellow area covers four heat source cavities, and the black dashed line represents the heat transfer path under study. We apply heat, cold, and alternating heat and cold sources to the heat source cavities, and their corresponding time-dependent evolution images are shown in Figs. 4(b)–4(d). It can be observed that under the conditions of heat, cold, and alternating heat and cold sources, the system forms a stable corner skin mode after 80 s, 80 s, and 120 s, respectively. The time-dependent evolution results indicate the model's good efficiency in regulating the thermal field.
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Fig. 4. (a) Structure diagram of the corner state model. The yellow region signifies four external heat source cavities and the black dashed line represents the observed heat transfer path in (b)–(d). (b) Time-dependent evolution results of the thermal field when a high temperature is applied at the heat source cavities. (c) Time-dependent evolution results of the thermal field when a low temperature is applied at the heat source cavities. (d) Time-dependent evolution results of the thermal field when an alternating heat and cold source is applied at the heat source cavities.
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Fig. 5. (a) Schematic of the corner state model with modulation gradient disorder. The dark blue cavities and varying shades of red arrows indicate the disordered modulation depth and coupling strength. (b) Time-dependent evolution results of the thermal field under modulation gradient disordered. The observing heat transfer path is the black dashed line in Fig. 4(a). (c) A comparison of energy bands before and after introducing noise, with the variation $a$. The noise used is Gaussian multiplicative noise, characterized by a mean of 1 and a variance of 0.1. (d) Time-dependent evolution results of the thermal field after incorporating Gaussian noise. The observing heat transfer path is the black dashed line in Fig. 4(a).
Finally, let us examine the model's robustness, as it serves as a crucial indicator of thermal regulation systems.[45,46] Firstly, we test the model's robustness when the modulation gradient is disturbed. The model with modulation gradient disordered is presented in Fig. 5(a). Comparing it with the model shown in Fig. 3(a), we observe that the disordered model disrupts the modulation depth of 7 cavities, thereby disturbing the modulation gradient. The time-dependent evolution results of the model with modulation gradient disordered are displayed in Fig. 5(b), and it is evident that the model still achieves a stable corner skin mode after 100 s. Therefore, we can conclude that the model exhibits good robustness under gradient disordered conditions. Secondly, we consider the robustness of the model under Gaussian multiplicative noise, where the noise has a mean of 1 and a variance of 0.1. The OBC band diagram of the model before and after adding noise is depicted in Fig. 5(c). It can be observed that the energy distribution of the model undergoes minimal changes before and after introducing Gaussian noise, indicating that its eigenstate distribution remains close to the ideal scenario. Subsequently, we present the time-dependent evolution results after incorporating Gaussian noise in Fig. 5(d). It is evident that the thermal field reaches a corner skin mode after 80 s, demonstrating the model's robustness against Gaussian multiplicative noise. In summary, we have presented an edge state model and a corner state model as means of regulating the thermal field through NHSE. When comparing these two models, it becomes apparent that these two models are independent, but they share the same design core. This core involves the introduction of non-reciprocal coupling in a specified direction to achieve temperature field aggregation. We have demonstrated the existence of NHSE in the designed system by examining its energy band structure and the noncoincidence of BZ and GBZ. Subsequently, we observe the time-dependent evolution results of the models and see that, under the initial conditions of heat, cold, and alternating heat and cold sources, a localized state emerges at the designated open boundary after a certain period of time. Finally, we investigate the robustness of the corner state model and determine that it displays good resilience in the presence of modulation gradient disordered, when subjected to Gaussian multiplicative noise. These findings provide a novel approach for designing thermal field regulation models. Acknowledgement. This work was supported by the Key Research and Development Program of China (Grant No. 2022YFA1405200), and the National Natural Science Foundation of China (Grant Nos. 92163123 and 52250191). 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