Observation of Non-Hermitian Skin Effect in Thermal Diffusion

The paradigm shift of the Hermitian systems into the non-Hermitian regime profoundly modifies the inherent topological property, leading to various unprecedented effects such as the non-Hermitian skin effect (NHSE). In the past decade, the NHSE effect has been demonstrated in quantum, optical and acoustic systems. Besides in those non-Hermitian wave systems, the NHSE in diffusive systems has not yet been explicitly demonstrated, despite recent abundant advances in the study of topological thermal diffusion. Here we first design a thermal diffusion lattice based on a modified Su-Schrieffer-Heeger model which enables the observation of diffusive NHSE. In the proposed model, the periodic heat exchange rate among adjacent unit cells and the asymmetric temperature field coupling inside unit cells can be judiciously realized by appropriate configurations of structural parameters of unit cells. The transient concentration feature of temperature field on the boundary regardless of initial excitation conditions can be clearly observed, indicating the occurrence of transient thermal skin effect. Nonetheless, we experimentally demonstrated the NHSE and verified the remarkable robustness against various defects. Our work provides a platform for exploration of non-Hermitian physics in the diffusive systems, which has important applications in efficient heat collection, highly sensitive thermal sensing and others.

In the past years, the intriguing non-Hermitian skin effect (NHSE) has been discovered in open systems, whereby all the eigenmodes decay exponentially and localize at the open boundaries [31][32][33][34][35][36][37][38] .This feature indicates interesting physics of NHSE in classical and quantum systems.The NHSE reflects the existence of peculiar gap topology that is unique in non-Hermitian systems [39][40][41] and alters the bulk-boundary correspondence in the conventional wisdom 31,33,42 , expanding the research scope in topological phases of matter.Recently, NHSE was discovered to have vibrant applications in different areas such as enhanced sensing 43,44 , topological lasers 45 , and light funneling 32 owing to its unique properties.The experimental verifications of NHSE hitherto have been achieved in various systems, including the phononic crystals, 46,47 synthetic lattices 32 , and electric ciruits 48 .However, NHSE have not yet been experimentally observed in diffusive systems.Unlike those wave systems governed by Hermitian Hamiltonians with real eigenvalues, the diffusive systems are dissipative and the coupling coefficients between separate meta-atoms are purely imaginary, providing a natural platform for studying the non-Hermitian physics 21,49,50 .For example, the researchers have demonstrated many interesting phenomena in the diffusiveconvective systems, such as the anti-parity-time symmetric phase transition at an exceptional point 23,51 , the topological heat localization 52 and chiral heat transport 50 .One recent theoretical proposal has shown the possibility of NHSE in thermal diffusion 53 , which renders an automatic concentration of the temperature field towards the boundary regardless of initial heat excitation.
Due to the existing challenges in realizing precise modulation of material parameters, such as thermal conductivity and mass density, the experimental demonstration of diffusive NHSE has remained elusive hitherto.
In this article, we design a one-dimensional (1D) thermal diffusion lattice based on the modified Su-Schrieffer-Heeger (SSH) model and report the first experimental demonstration of NHSE in diffusive systems for the transient directional temperature field concentration.The required modulation of material parameters is implemented by the judicious configuration of structural parameters of site rods and coupling sticks, so as to flexibly manipulate temperature field couplings between the neighboring sites.By tailoring the asymmetric couplings inside each unit cell, we find that the thermal system can change from anti-Hermitian to non-Hermitian.
When the asymmetry coupling factor deviates from unitary, all eigenmodes take the form of localized skin modes on the boundary.Here we first verify the theoretical prediction of diffusive NHSE in numerical simulations with effective material parameters.The temperature field is proved to concentrate to the designated boundary in a transient process, regardless of the initial excitations.Then we fabricated the sample with the structural parameters satisfying the coupling parameters and observed the temperature field evolution to concentrate towards the target boundary in a vacuum chamber.We further demonstrated the robustness of diffusive NHSE against different defects.Our work provides a platform for a distinctive type of thermal metamaterials and functional thermo-devices based on asymmetric temperature field couplings.
The experimental realization of transient diffusive NHSE opens the door for the realization of high-precision thermal sensing and robust heat harvesting.

NHSE based on asymmetric coupling dimer unit-cells
In nature, thermal diffusion intrinsically enables heat to spontaneously spread and the temperature field will distribute evenly in materials according to the zeroth law of thermodynamics, as shown in Fig. 1a.However, in the transient process, it is possible to break the symmetry via thermal metamaterials to manipulate the temperature field preferentially concentrating at the target boundary, as sketched in Fig. 1a, denoting the proposed diffusive skin effect.The key recipe for the diffusive skin effect here is the non-Hermitian asymmetric coupling in the unit-cell, for instance, a coupled cavity dimer.As shown in Fig. 1b, a simple cavity dimer model is presented, where the temperature field coupling can be asymmetric, with the heat exchange efficiencies between  and  cavities (  or between  and  cavities (  being / and , respectively.Here a non-zero parameter  is utilized in combination with the heat transfer efficiency  to construct the asymmetric heat exchange.According to the Fourier's law in heat conduction, the coupling equations can be written as where   and   are the temperature fields in  and  cavities, and  denotes the time. We consider that the Eq.(1 has a wavelike solution, which takes the form of  =  − +  0 , where  and  are the amplitude and complex frequency of the temperature field, and  0 is a constant 21,51,54  , for which the corresponding eigenstates are  1 = (1, 1)  and  2 = (− 2 , 1)  .For  1 = 0, it can be found that the temperature field is symmetrical in the cavity dimer unit cell.
However, there also exists an eigenvector  2 that satisfies an asymmetric temperature field coupling in a dimer with an asymmetrically coupling factor  .When the heat exchange efficiencies   ≠ , we will obtain the asymmetric temperature field distributions in the cavity dimer unit cell, as shown in Fig. 1b.The full wave simulations in Fig. 1c further present that the degree of asymmetry for temperature field coupling increases as the factor  becomes smaller.

Topology and skin effect in the diffusive SSH model
The cavity dimer with asymmetric temperature field coupling inside can further be aggregated as a 1D chain to form a two-band Su-Schrieffer-Heeger (SSH model, as illustrated in Fig. 2a, where the NHSE can be readily observed.Specifically, the couplings from left to right are for the intra-cell coupling coefficient and  1 for the inter-cell coupling coefficient, as indicated by the red and blue arrows in the upper part of the lattice.In contrast, the couplings from right to left as marked by the arrows in the lower part are  2  for the intra-cell coupling coefficient and  1 for the inter-cell coupling coefficient, respectively.Following the Fourier's law, the temperature field evolution at each lattice site can be written as where    (   ) is the temperature field at site () in the th period.Then applying the periodic boundary condition (PBC , the corresponding Hamiltonian in the reciprocal space can be obtained as where  is the wave number and only the nearest coupling is considered for the tight binding model.The eigenvalues solved by Eq. (4 have the following form To facilitate analysis, we show the energy spectra for  1 =  2 = 0.0072 and different values of  in Figs.2b-2d.In each figure, the blue solid line corresponds to PBC, and the orange dots correspond to OBC for the lattice with 10 unit-cells.For the case of  = 1 in Fig. 2b, the intracell and inter-cell coupling coefficients are symmetrical as where () is the eigenspectrum over the Brillouin zone with PBC, and  0 is reference point of frequency given arbitrarily.If there exists NHSE in the system and the skin mode is localized at the right (or left) boundary, then  is calculated to be  = −1 (or + 1) at the reference frequency of  0 .In our case, Eq. ( 6) gives out  = −1, which means that eigenstates exhibit the skin mode at the rightside boundary (see Supplementary Note 3).In addition, the area size of closed loops reflects the strength of skin effect in accumulating the temperature fields.In Fig.  , where the inter-cell coupling coefficient is . In Fig. 3, we simulate the 1D thermal lattice consisting of 10 unit cells and with the asymmetric factor of  = 0.5 by using a commercial finite element solver (COMSOL Multiphysics 6.0 .The theoretical and simulation results of eigenvalues are shown in Fig. 3b, represented by the red and blue dots, respectively (see Supplementary Note 4 .Note that the lower band and the band gap agree well with each other.However, the higher band in simulations deviates much from the theoretically calculated one.The discrepancy primarily roots in the reason that as −Im() decreases, the tight-binding model of 1D thermal lattice is no longer appropriate to be used for the description of continuous heat diffusion process.We further simulate the eigenmode profiles in Fig. 3c, which clearly present the temperature field concentration towards the target boundary, which is consistent with theoretical analyses above.
In Figs.

Experimental demonstration of transient diffusive NHSE
To experimentally observe diffusive NHSE, the main challenge roots in the realization of effective volume specific heat capacity and thermal conductivity given in Eq. (7 .Here we overcome the challenge by resorting to the thermal metamaterials and utilizing the structural parameters to engineer the effective coupling coefficients.For example, the designed sample for experiments, as shown in Fig. 4a, is made of Aluminium Alloy (AlSi10Mg with thermal conductivity  = 180 W/m • K, density  = 2700 kg/m 3 , and heat capacity  = 900 J/kg • K.
The length of site rod is equal to the length of coupling stick with  =  = 16 mm.The volume of cylindrical site rod is   =   2  and the cross-section area of cylindrical coupling stick is Therefore, the coupling coefficients between site rods n and n+1 take the form of for the left-to-right coupling and  +1, = for the right-to-left coupling, respectively.In order to make the adjacent site coupling of the sample meet the requirements given in Eq. (3 , we construct the thermal lattice with the structural parameters following where the parameters  0 and  0 are set to 8 mm and 4 mm, respectively.In Fig. 4b, we show the structural parameter settings of each site rod (blue bars and coupling stick (red bars of the fabricated thermal lattice.In our case, the factor  is set to be 0.7 for the convenience of 3D metal printing.The experimental measurement was conducted in a vacuum chamber, where we used a resistance wire as the heat source and the silicone grease was covered to improve the contact between the resistance wire and the sample for better heat transfer.A current was loaded to the resistance wire to generate a large amount of heat in a short time, which was transferred to the target site rod as marked by the red asterisk in Fig. 4a.More details of the experiments are appended in Supplementary Note 5. We present the starting and final states of temperature field distributions in Fig. 4c.The experimental results are highly consistent with the simulation ones, except for the heated region where the resistance wires and silicone grease cover.Due to the very different emissivity, the heated region marked by a gray bar has an abrupt temperature variation in the thermograph.However, the discrepancy at the heated region does not affect the overall evolution of temperature field.For example, at  = 8 s, the temperature field is localized in the vicinity of heated region, as shown by the red (experiment and blue (simulation curves.
After the evolution at  = 120 s, we observed a significant change of temperature field, which was concentrated at the rightside boundary, as indicated by the green (experiment and purple (simulation curves.In Fig. 4d, we show three freeze-frames of thermographs taken at  = 8, 60, 120 s.The direction of temperature field flows is marked by a blue dashed arrow line. The experimental results clearly show that driven by the asymmetric couplings, the temperature field gradually concentrates towards the rightside end, confirming the transient diffusive NHSE in the fabricated thermal lattice.Experimental Methods We used the Aluminium alloy (AlSi10Mg for fabricating the thermal lattice, for which the experimental samples were fabricated by selective laser melting (SLM) technology of 3D metal printing.The manufacturing precision of SLM technology can reach ± 0.1 mm/100 mm.We placed the sample in a vacuum chamber and suspended it with plastic stands of poor thermal conductivity.The resistance wire interwining one site rod was connected to an adjustable power supply outside the vacuum chamber by a circuit for loading the heat source.At the beginning of experiment, the circuit was switched on and a large current passed through the resistance wire to generate a large amount of heat in a short time.Then we turned off the power supply and stopped heating for the following observation of temperature field evolution.The evolution of temperature field was recorded by using an infrared camera (Fotric 347).

Fig. 1 |
Fig. 1 | Diffusive skin effect and asymmetric coupling.a The concept of transient diffusive 2e, we present the eigenfield distributions of eigenmodes at the OBC for  = 0.5.The result clearly shows that the corresponding temperature fields of eigenmodes are finely localized at one boundary (or nearby site 20 , representing a typical feature of NHSE.The position of temperature localization can also be verified numerically by solving the Eq.(6 to check out the sign of winding numbers.

Fig. 2 | 7
Fig. 2 | 1D NHSE in the diffusive SSH model.a The analytical model of diffusive NHSE with 3d and 3e, we conduct the transient simulation of temperature field evolution to verify the existence of transient diffusive NHSE.In the first case of Fig.3d, we impose a heat source to the 7th cylindrical rod and heat it to 343.15 K, while other site rods maintain the room temperature of 293.15 K.The freeze-frames of temperature fields at six different times ( = 1, 100, 200, 300, 400, 500 s are shown in Fig.3d.For convenience, the temperature field is normalized by   = ( −   )/(  −   ), where   is the temperature at the leftmost site and   (  ) is the maximum (minimum temperature on the thermal lattice.The result shows that the region where the maximum temperature field locates keeps on moving towards the rightside boundary in a transient process under the drive of asymmetric temperature field coupling.At around  = 500 s, the temperature field concentrates at the right boundary of the thermal lattice.In the second case of Fig.3e, we further impose a random heat source on the thermal lattice.For example, we heat the 5th, 9th and 13th site rods up to 338.15 K, 333.15 K, and 318.15 K, while we cool the 4th, 6th, and 11th site rods down to 323.15 K, 333.15 K, and 333.15 K, with the remaining site rods at the room temperature of 293.15 K.The transient temperature field evolution corresponding to different times is presented in Fig.3e.In close resemblance to the result in Fig.3d, the temperature field in Fig.3eevolves to concentrate at the rightside boundary of the thermal lattice ( = 500 s .The numerical simulation indicates that the thermal lattice we proposed in this article can be used as a platform for realizing the transient diffusive NHSE, for which temperature field will be localized at the targeted boundary over time regardless of the initial heat excitation conditions.

Fig. 3 |
Fig. 3 | The designed thermal lattice for simulating transient diffusive NHSE. a A schematic