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Emergence of imaginary time crystals in the non-Hermitian Su-Schrieffer-Heeger model

E. Slootman1,2,*, L. Eek1,*, C. Morais Smith1, and R. Arouca3,†,‡

  • 1Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands
  • 2MESA+ Institute for Nanotechnology, University of Twente, PO Box 217, 7500AE Enschede, The Netherlands
  • 3Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden

  • *These authors contributed equally to this work.
  • †Present address: Centro Brasileiro de Pesquisas Físicas, Rua Doutor Xavier Sigaud 150, Rio de Janeiro, 22290-180, Brazil.
  • ‡Contact author: rodrigo-arouca@cbpf.br
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Phys. Rev. B 112, 085149 – Published 26 August, 2025

DOI: https://doi.org/10.1103/ttny-t2gs

Abstract

Parity-time symmetry constrains the spectrum of non-Hermitian systems to be either real or come in complex conjugate pairs. The transition between a symmetry-preserving phase with real energies and a symmetry-broken phase with complex energies is marked by exceptional points, one of the hallmarks of non-Hermitian systems. Because of these properties, these systems are widely studied, both theoretically and experimentally. In this work, we investigate the thermodynamic properties of the gain and loss Su-Schrieffer-Heeger model for both bosons and fermions, and establish the existence of an imaginary time crystal phase, an imaginary time analog of a time crystal. This phase occurs when there is a resonance condition between the Matsubara frequencies and the spectrum of the system, making the Green's function of the system oscillate in imaginary time with the Matsubara frequency. We show that this phase appears in the symmetry-broken region. In particular, the topological edge states of this system exhibit oscillations that are present for fermions. Finally, we discuss the applicability of our results for experiments. We examine signatures of these phases in terms of correlation functions in real time and oscillations in thermodynamic potential in inverse temperature β, and explore possible experimental platforms to realize this system.

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Phys. Rev. B 112, 085149– Published 26 August, 2025
  • Received 29 April 2025
  • Revised 4 July 2025
  • Accepted 23 July 2025
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