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

We analyze 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 a ferromagnet to the CP, the longitudinal and transverse magnetizations 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.

Figure 1

Numerical data for deviations from the adiabatic value for spin components $x, y$, and $z$ (blue, red, and green) from top to bottom after quenching to the critical point ($g=1$) from the FM ($g_0=0$) phase. The black dashed lines depict the ${\tau }^{-1/3}$ and ${\tau }^{-2/3}$ scalings.

Figure 2

Temporal data collapse of ${\delta }^z\left(t\right)$ close to the CP for various ramp rates after a ramp from the FM ($g_0=0$). The data are obtained from the numerical solution of Eq. (4). Deviations from zero start to appear at times $\sim {\tau }^{1/3}$ before reaching the CP. A similar data collapse characterizes the other spin components.

Figure 3

Numerical solution of the deviations from the adiabatic value for spin components $x, y$, and $z$ (blue, red, and green) from top to bottom after quenching from FM ($g_0=0$) to PM ($g=3$). The black dashed lines depict the ${\tau }^{-1/2}$ and ${\tau }^{-1}$ scalings obtained analytically.