Adiabaticity enhancement in the classical transverse field Ising chain, and its effective non-Hermitian description

We analyse the near-adiabatic dynamics in a ramp through the critical point (CP) of the classical transverse field Ising chain. This is motivated, conceptually, by the fact that this CP -- unlike its quantum counterpart -- experiences no thermal or quantum fluctuations, and technically by the tractability of its effective model. For a `half-ramp' from ferromagnet to CP, the longitudinal and transverse magnetization scale as $\tau^{-1/3}$ and $\tau^{-2/3}$, respectively, with $1/\tau$ the ramp rate, in accord with Kibble-Zurek theory. For ferro- to paramagnetic ramps across the CP, however, they stay closer, $\tau^{-1/2}$ and $\tau^{-1}$, to adiabaticity. This adiabaticity enhancement compared to the half ramp is understood by casting the dynamics in the paramagnet in the form of a non-hermitian Dirac Hamiltonian, with the CP playing the role of an exceptional point, opening an additional decay channel.