Temporal anti-parity–time symmetry in diffusive transport

Nature Physics (2025)Cite this article Parity–time symmetry has revolutionized wave and energy transport control in non-Hermitian systems, yet has so far been mostly explored in static phases, where a system’s behaviour is locked into a fixed-symmetric or broken-symmetry phase. The vast potential of time-domain dynamics has remained largely untapped. Here we introduce the concept of temporal anti-parity–time symmetry, a principle that allows the transport dynamics of a system to be actively shaped in real time. Rather than designing static phases, we influence the timing of non-Hermitian phase transitions, making the system’s temporal evolution itself a programmable degree of freedom. Through the dynamic control of material properties and convective flow, we dictate the exact moments these transitions occur, thereby controlling the entire transport history of the system. This temporal control achieves highly tunable field localization and realizes counterintuitive thermal transport, enabling temperature profiles to move forwards with convection, backwards against it or remain trapped at arbitrary locations. Our findings extend non-Hermitian physics into the time domain and establish a framework for on-demand wave and energy transport.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutSource data are provided with this paper.Gross, D. J. The role of symmetry in fundamental physics. Proc. Natl Acad. Sci. USA 93, 14256–14259 (1996).Article ADS MathSciNet Google Scholar Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).Article ADS MathSciNet Google Scholar Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 15005 (2021).Article ADS MathSciNet Google Scholar Heiss, W. D. The physics of exceptional points. J. Phys. A: Math. Theor. 45, 444016 (2012).Article ADS MathSciNet Google Scholar Miri, M. A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).Article MathSciNet Google Scholar Özdemir, S. K., Rotter, S., Nori, F. & Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).Article ADS Google Scholar Chen, W. J., Özdemir, S. K., Zhao, G. M., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192–196 (2017).Article ADS Google Scholar Feng, L., Wong, Z. J., Ma, R. M., Wang, Y. & Zhang, X. Single-mode laser by parity–time symmetry breaking. Science 346, 972–975 (2014).Article ADS Google Scholar Zeng, Y. Q. et al. Electrically pumped topological laser with valley edge modes. Nature 578, 246–250 (2020).Article ADS Google Scholar Zhang, J. et al. A phonon laser operating at an exceptional point. Nat. Photon. 12, 479–484 (2018).Article ADS Google Scholar Zhang, X. Y. et al. Symmetry-breaking-induced nonlinear optics at a microcavity surface. Nat. Photon. 13, 21–24 (2019).Article ADS Google Scholar Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008).Article ADS Google Scholar Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photon. 11, 752–762 (2017).Article ADS Google Scholar El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).Article Google Scholar Zhu, X. F., Ramezani, H., Shi, C. Z., Zhu, J. & Zhang, X. PT-symmetric acoustics. Phys. Rev. X 4, 031042 (2014).

Google Scholar Zhang, Z., Delplace, P. & Fleury, R. Superior robustness of anomalous non-reciprocal topological edge states. Nature 598, 293–297 (2021).Article ADS Google Scholar Han, Y. C. et al. Bound chiral magnonic polariton states for ideal microwave isolation. Sci. Adv. 9, eadg4730 (2023).Article Google Scholar Wu, J. H., Artoni, M. & La Rocca, G. C. Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices. Phys. Rev. Lett. 113, 123004 (2014).Article ADS Google Scholar Peng, P. et al. Anti-parity–time symmetry with flying atoms. Nat. Phys. 12, 1139–1145 (2016).Article Google Scholar Konotop, V. V. & Zezyulin, D. A. Odd-time reversal symmetry induced by an anti-PT-symmetric medium. Phys. Rev. Lett. 120, 123902 (2018).Article ADS MathSciNet Google Scholar Azizi, P. et al. Lattice materials with topological states optimized on demand. Proc. Natl Acad. Sci. USA 122, e2506787122 (2025).Article MathSciNet Google Scholar Qi, M. H. et al. Geometric phase and localized heat diffusion. Adv. Mater. 34, 2202241 (2022).Article Google Scholar Qi, M. H. et al. Observation of high-decay-rate topological corner states in diffusive thermal metamaterials. Phys. Rev. Lett. 135, 096604 (2025).Article ADS Google Scholar Li, Y. et al. Anti-parity–time symmetry in diffusive systems. Science 364, 170–173 (2019).Article ADS MathSciNet Google Scholar Xu, G. et al. Non-Hermitian chiral heat transport. Phys. Rev. Lett. 130, 266303 (2023).Article ADS Google Scholar Cao, P. C. et al.Observation of parity–time symmetry in diffusive systems. Sci. Adv. 10, eadn1746 (2024).Article Google Scholar Xu, G. et al. Hydrodynamic moiré superlattice. Science 386, 1377–1383 (2024).Article ADS MathSciNet Google Scholar Xu, G. et al. Diffusive topological transport in spatiotemporal thermal lattices. Nat. Phys. 18, 450–456 (2022).Article Google Scholar Xu, G. Q. et al. Observation of Weyl exceptional rings in thermal diffusion. Proc. Natl Acad. Sci. USA 119, e2110018119 (2022).Article MathSciNet Google Scholar Xu, G. Q., Li, Y., Li, W., Fan, S. H. & Qiu, C.-W. Configurable phase transitions in a topological thermal material. Phys. Rev. Lett. 127, 105901 (2021).Article ADS Google Scholar Yang, F. B. et al. Controlling mass and energy diffusion with metamaterials. Rev. Mod. Phys. 96, 015002 (2024).Article ADS MathSciNet Google Scholar Li, Y. et al. Transforming heat transfer with thermal metamaterials and devices. Nat. Rev. Mater. 6, 488–507 (2021).Article ADS Google Scholar Zhu, C. L., Bamidele, E. A., Shen, X. Y., Zhu, G. M. & Li, B. W. Machine learning aided design and optimization of thermal metamaterials. Chem. Rev. 124, 4258–4331 (2024).Article Google Scholar Liu, Z. F. et al. Topological thermal transport. Nat. Rev. Phys. 6, 554–565 (2024).Article Google Scholar Jin, P. et al. Tunable liquid-solid hybrid thermal metamaterials with a topology transition. Proc. Natl Acad. Sci. USA 120, e2217068120 (2023).Jin, P. et al. Deep learning-assisted active metamaterials with heat-enhanced thermal transport. Adv. Mater. 36, 2305791 (2024).Article Google Scholar Li, Y., L. Xu & Qiu. C.-W. Thermal Metamaterials: Controlling the Flow of Heat (World Scientific, 2025).Tirole, R. et al. Double-slit time diffraction at optical frequencies. Nat. Phys. 19, 999–1002 (2023).Article Google Scholar Sapienza, R. Splitting light pulses. Nat. Photon. 19, 551–552 (2025).Article ADS Google Scholar Liu, Z. F. et al. Topology in thermal, particle, and plasma diffusion metamaterials. Chem. Rev. 125, 8655–8730 (2025).Article Google Scholar Fan, C. Z., Wu, C.-L., Wang, Y., Wang, B. & Wang, J. Thermal metamaterials: from static to dynamic heat manipulation. Phys. Rep. 1077, 1–111 (2024).Article MathSciNet Google Scholar Dai, G. L. et al. Controlling transient and coupled diffusion with pseudoconformal mapping. Proc. Natl Acad. Sci. USA 122, e2511708122 (2025).Article Google Scholar Lei, M. et al. Reconfigurable, zero-energy, and wide-temperature loss-assisted thermal nonreciprocal metamaterials. Proc. Natl Acad. Sci. USA 121, e2410041121 (2024).Article Google Scholar Tan, H. H. et al. Bioinspired energy-free temperature gradient regulator for significant enhancement of thermoelectric conversion efficiency. Proc. Natl Acad. Sci. USA 122, e2424421122 (2025).Article Google Scholar Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).Article Google Scholar Rudner, M. S. & Lindner, N. H. Band structure engineering and non-equilibrium dynamics in Floquet topological insulators. Nat. Rev. Phys. 2, 229–244 (2020).Article Google Scholar Lei, M. et al. Quantum thermalization and Floquet engineering in a spin ensemble with a clock transition. Nat. Phys. 21, 1196–1202 (2025).Article Google Scholar Ashkin, A., Dziedzic, J. M., Bjorkholm, J. E. & Chu, S. Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 11, 288–290 (1986).Article ADS Google Scholar Download referencesWe thank H. Fang for the constructive suggestions on improving the quality of the manuscript. This work was supported by the National Natural Science Foundation of China to J.H. (12035004 and 12320101004), the Innovation Program of Shanghai Municipal Education Commission to J.H. (2023ZKZD06), the Singapore Ministry of Education to C.-W.Q. (A-8002978-00-00) and the Singapore Ministry of Education under the Academic Research Fund Tier 2 (FY2023) to G.W.H. (T2EP50124-0007).These authors contributed equally: Peng Jin, Chengmeng Wang, Yuhong Zhou, Shuihua Yang.College of Science, University of Shanghai for Science and Technology, Shanghai, ChinaPeng Jin & Jinrong LiuDepartment of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai, ChinaPeng Jin, Chengmeng Wang, Yuhong Zhou, Pengfei Zhuang, Yiyang Zhang & Jiping HuangDepartment of Electrical and Computer Engineering, National University of Singapore, Singapore, SingaporePeng Jin, Shuihua Yang, Ya Sun, Yi Zhou, Ghim Wei Ho & Cheng-Wei QiuGraduate School of China Academy of Engineering Physics, Beijing, ChinaFubao Yang & Liujun XuState Key Laboratory of Metal Matrix Composites, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, ChinaYa SunSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarP.J., C.-W.Q. and J.H. conceived the ideas. C.W. developed the theory. P.J. designed the algorithm and performed the simulations. P.J., C.W., Yuhong Z. and J.L. conducted the experiments. P.J., C.W. and Yuhong Z. performed the visualization. P.J., C.W., Yuhong Z. and S.Y. wrote the manuscript. G.W.H., C.-W.Q. and J.H. supervised the project. All authors contributed to the discussion and finalization of the manuscript.Correspondence to Cheng-Wei Qiu or Jiping Huang.The authors declare no competing interests.Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Top panels: peak temperature Tpeak(K) vs time t; bottom panels: Position (normalized distance of the instantaneous peak relative to the initial location, expressed as cycle fraction) vs time t. a, Material-property jitter: thermal conductivity of the middle ring varied by \(\pm 10 \%\) (160, 180, 200 W m−1 K−1). b, Stochastic flows: the programmed velocity of Ring 1 is perturbed by \(\pm 10 \%\) (\(v^{\prime} =0.9v,\,1.0v,\,1.1v\)). c, Uncertain boundary conditions: ambient 293.15 K with convective heat-transfer coefficients \({h}_{c}=\) 0, 50, 100 W m−2 K−1. In all cases, the protocol preserves the Tpeak and Position trajectories within numerical tolerances, confirming stable transport and trapping under these perturbations. For this model, the switching time is \({t}_{0}=\) 2.8 s, with all other parameters identical to those in Fig. 2.Source dataa, Three-ring model (inner radius R = 0.2 m) and concentric architecture used for tangential transport; all other parameters match Fig. 2. b, Ring angular velocities before and after the switching time \({t}_{0}\). c, At \({t}_{0}\), the hollow interior is replaced by a structured copper disk with radially graded properties, \(\kappa \left(r\right)={\kappa }_{0}{e}^{-100(1-r/R)}\) and \(\rho \left(r\right)={\rho }_{0}{e}^{-100(1-r/R)}\), to emulate radial drift. Here, \({\kappa }_{0}=\) 400 W m−1 K−1 and \({\rho }_{0}=\) 8900 kg m−3 are the thermal conductivity and density of copper. d-e, Two representative scripts (\({t}_{0}=\) 0.7 s and \({t}_{0}=\) 2.0 s): polar plots of the peak location over time and early-stage temperature profiles show controlled tangential motion, directed radial steering, and trapping at the selected polar coordinates.Source dataPassive obstacles are placed between the rails. The upper rail centerline is parameterized by \(S\in [\mathrm{0,1}]\), with \(S=0\) at the left end and \(S=1\) at the right end. The initial temperature distribution along the track at \(t=\) 0 s is visualized by the color map. a, Normalized peak position Speak (\(0\to 1\)) along the upper rail centerline vs time for \({t}_{0}=\) 2.1 s, showing evolution from \(\approx 0.25\) to \(\approx 0.50\). b, Temperature snapshots (\(t=\) 0–3 s) on both semicircular rails with obstacles, illustrating guided motion around obstacles from a cosine-modulated initial condition. c, Same as (a) for \({t}_{0}=\) 3.8 s; the peak advances from \(\approx 0.25\) to \(\approx 0.75\). d, Snapshots (\(t=\) 0–3 s) for \({t}_{0}=\) 3.8 s. Geometry: two semicircular rails concatenated into an S-track; inner radius \(R=\) 0.2 m; rings rotate at \(|\omega |=\) 1 rad s−1; all other parameters match Fig. 2.Source dataa, Time-dependent temperature evolution at points 1 and 2 on two copper rings (with thermal grease). b, Time-dependent temperature evolution at points 1 and 2 on two copper rings (without thermal grease). The insets display the temperature profiles (comes from the second measurement) of both copper rings (with/without grease) over time. Error bars (shaded areas) represent one standard deviation from the mean. The middle inset displays the picture of our experimental setup.Source dataa, Finite‑element simulation of the experimental stack with identical geometry and parameters but zero interfacial resistance (heating on Ring 1 with power of \(q(\theta )=-230\cos (8\theta -\pi )\) W, where \(-\pi /16\le \theta \le \pi /16\), for 15 s, then natural dissipation). Temperatures at two representative locations on the upper and lower layers (Points 1 and 2) become isothermal essentially immediately after heating is switched off, whereas the experiment with thermal grease shows cross‑layer equilibration only after ~20 s (see Fig. S22). Insets: tangential temperature profiles at \(t=\) 0, 15, and 60 s (295-305 K), illustrating relaxation. Because both our initialization and waiting windows exceed 20 s, the residual resistance in the experiment does not affect the protocol timing. b-c, Robustness of temporal APT to interlayer transport variations: varying the effective thermal conductivity of the middle layer by \(\sim \pm 10 \%\) (\(\kappa =\) 160, 180, 200 W m−1 K−1) leaves the propagation trajectory and trapping unchanged, as seen from the overlapping peak‑position evolution for 1/3‑cycle capture (b; \({t}_{0}=\) 1.8 s) and 2/3‑cycle capture (c; \({t}_{0}=\) 4.3 s).Source dataThe dashed line and the solid line denote the position of the EP and system’s evolution trajectory, respectively. Black and white dots indicate two distinct coupling coefficients, respectively. In the upper panel, the phase diagram depicts the scenario in which the system evolves along their original temporal sequence. In contrast, the lower panel illustrates the phase diagram resulting from swapping the temporal order of the two coupling coefficients.a, Experimental setup. The wafer is positioned in thermal contact at the 1/3 circumferential location of the ring—precisely where thermal energy is trapped by the heat trapping. The locally trapped cooling energy is obtained from the wafer, enabling active cooling. Additionally, the silicon wafer acts as a temperature-sensitive resistive component in an electrical circuit. As the wafer temperature decreases, its electrical resistance changes, thus modulating the current and adjusting the brightness of connected light bulbs. b, The left panels illustrate the temporal evolution of the silicon wafer’s temperature (bottom panel, T in °C) and the corresponding illuminance (top panel, E in lx) of a light bulb, whose brightness is modulated by the wafer’s temperature-dependent resistance. These profiles compare the cooling performance of the temporal APT (dashed curves), convection (solid curves), and natural dissipation (dotted curves). The right panels provide photographic snapshots of the light bulb’s brightness and the simultaneously measured wafer temperature (displayed on the monitor, unit: °C) at selected time intervals: \(t=1\,{\rm{s}}\) (images ①, ④, ⑦), \(t=15\,{\rm{s}}\) (images ②, ⑤, ⑧), and \(t=30\,{\rm{s}}\) (images ③, ⑥, ⑨) for the trapping, convection, and dissipation methods, respectively. Error bars (shaded regions) indicate one standard deviation based on three independent measurements.Source dataFinite‑element simulation replicating the experimental hot‑air initialization: a tangentially localized boundary heat flux \(q(\theta )=-230\cdot \cos (8\theta -\pi )\) W, where \(-\pi /16\le \theta \le \pi /16\), is applied for 15 s. Curves show the tangential temperature profile \(T(\theta ,t)\) at \(t=\) 0, 3, 6, 9, 12, and 15 s. Fast advection and diffusion broaden the initially localized heating into a first‑harmonic-dominated waveform, approaching a cosine‑like envelope by ~10-15 s. This validates the use of a \(\cos \theta\) initial condition in simulations and is consistent with the demonstrated robustness to waveform in Supplementary Section 11. Position is plotted as \(\theta\) (rad) over \([-\pi ,\,\pi ]\).Source dataSupplementary Sections 1–14, Figs. 1–34 and references.Source data for Fig. 2Source data for Fig. 3.Source data for Fig. 4.Source data for Extended Data Fig. 1.Source data for Extended Data Fig. 2.Source data for Extended Data Fig. 3.Source data for Extended Data Fig. 4.Source data for Extended Data Fig. 5.Source data for Extended Data Fig. 7.Source data for Extended Data Fig. 8.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsJin, P., Wang, C., Zhou, Y. et al. Temporal anti-parity–time symmetry in diffusive transport. Nat. Phys. (2025). https://doi.org/10.1038/s41567-025-03129-8Download citationReceived: 17 July 2025Accepted: 04 November 2025Published: 10 December 2025Version of record: 10 December 2025DOI: https://doi.org/10.1038/s41567-025-03129-8Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative