Can scaling analysis be used to interpret the anti-parity-time symmetry in heat transfer?

Abstract

In a previous work (Li et al. Science 364, 170) [1], we proposed a heat transfer system that preserves the anti-parity-time (APT) symmetry, and observe the rest-to-motion phase transition during the symmetry breaking. Recently, it was suggested (Zhao et al. arXiv:1906.08431) [2] that the behaviours of the system can be understood using scaling analysis based on the P\'eclet and Nusselt numbers (Pe and Nu). It was further proposed that there exists a third regime in the phase diagram in addition to the symmetric and symmetry broken phases. Although we appreciate the proposal to characterize the contributions of coupling, diffusion, and advection with dimensionless numbers, here we show that they do not help to predict or interpret the behaviours of the APT system. The dimensionless numbers do not provide enough details about the system to conclude that there is a motionless phase, a phase transition, to find the critical point, or to give the correct phase diagram with only two regimes.