Universal critical timescales in slow non-Hermitian dynamics

Abstract

Non-Hermitian systems driven along slow parametric loops undergo non-adiabatic transitions whose outcome depends sensitively on the driving speed, yet no explicit formula has been available for the critical timescale $T_{\mathrm{cr}}$ at which these transitions develop. Using a $2\times 2$ Hamiltonian with circular parameter trajectories, we derive $T_{\mathrm{cr}} = \mathcal{G}\,\ln(1/|Δ|)$ in closed form for non-encircling loops, phase-shifted loops, offset loops, and loops encircling exceptional points, where $\mathcal{G}$ is a geometry-dependent growth factor and $Δ$ is the instability seed. This formula sharply separates the regime where the system remains in the averagely dominant eigenstate ($T< T_{\mathrm{cr}}$) from the superadiabatic regime where the instantaneous dominant eigenstate takes over ($T> T_{\mathrm{cr}}$), resolving the apparent tension between the previous literature. We identify two competing seeds: a geometric Stokes multiplier and the finite-precision floor. When the geometric seed vanishes, precision alone governs the transition, yielding $T_{\mathrm{cr}} \propto m\lnβ$, linear in the number of precision bits $m$. This provides a purely forward-evolution manifestation of precision-induced irreversibility (PIR)~\cite{PIR}, demonstrating that the fundamental limit identified through echo protocols also controls the outcome of slow non-Hermitian dynamics without requiring time reversal. For PT-symmetric energy spectra, $T_{\mathrm{cr}}$ additionally determines the onset of chirality: the dynamics is non-chiral for $T< T_{\mathrm{cr}}$ and chiral for $T> T_{\mathrm{cr}}$.

Figures (6)

• Figure 1: (a) Parameter loop $z(t)=-r e^{-i f t}$, with exceptional points at $z=\pm1$. On the right the image in energy space, $E(t)=\sqrt{1-z^2(t)}$. The energy trajectory is traversed twice per period. Red markers denote the initial point ($t=0$), crossed again at $t=T/2$ and $t=T$. With blue we highlight the segment of the circle where $R_{ad}(t)$ is stable and with red when it is unstable. (b) Time evolution of $\ln |R_+(t)|$ for the parameter loop of Fig. 1a and three periods: $T=100$, $850$, and $1500$. Dashed curves show the instantaneous adiabatic fixed points $|R_{\mathrm{ad}}(t)|$; the dotted curve shows $|R_{\mathrm{nad}}(t)|$ for $T=1500$. The Schrödinger equation is solved using a fourth-order Runge--Kutta scheme with time step $dt = 10^{-2}$; the coefficients $c_{\pm}(t)$ are obtained by projecting $\psi(t)$ onto the left eigenvectors $u_{\pm}$ of Eq. \ref{['basis']}. (c) $\ln |R_+(T)|$ as a function of $T$ for three loop radii $r = 0.3$, $0.5$, and $0.7$, with $z(t) = -r e^{-i f t}$. The horizontal purple line marks $|R_+| = 1$, defining $T_{\mathrm{cr}}$. (d) Fit of $\text{const.}/r^2$ to the numerically extracted $T_{\mathrm{cr}}$ vs $r$ (red markers). Inset: rescaling the horizontal axis to $r^{2}T$ collapses the transition regions. (e) Parameter loop $z(t)=-r e^{-i(f t+\phi_0)}$ for $t\in[0,T]$ and on the right its image under $E(t)=\sqrt{1-z^2(t)}$ where with blue/red we highlight the segments of the circle where $R_{ad}(t)$ is stable/unstable. (f) Fitted curve to the numerically extracted $T_{\mathrm{cr}}$ vs $\phi_0$ for $r=0.3$. The functional form is $\text{const.}/(r\sin\phi_0)^2$ [Eq. \ref{['res2']}].
• Figure 2: (a) Parametric curve $z(t)=g_0-re^{-ift}$ for $g_0<r$. (b) Same for $g_0\geq r$; red markers indicate the start/end point. (c) Fitted curve to the numerically extracted $T_{\mathrm{cr}}$ vs. $g_0$ for $g_0<r$ and $r=0.2$; the functional form is $\text{const.}\,(1-g_0^2)^{1/2}/(r-g_0)^2$ [Eq. \ref{['res3']}]. (d) Same for $g_0\geq r$; the functional form is $\text{const.}\,(1-g_0^2)^{1/2}/(g_0 r)$ [Eq. \ref{['res4']}]. (e) Parametric curve $z(t)=1+re^{-ift}$ (EP-encircling). (f) $|R_-(T)|$ vs. $T$ for $\phi_0=0$ and three loop radii: $r=0.1$ (blue), $0.3$ (red), and $0.5$ (black). Dashed curves show $|R_{\mathrm{ad}}(T)|$ and $|R_{\mathrm{nad}}(T)|$ for each $r$.
• Figure 3: $|R_+(T)|$ for clockwise (red) and anticlockwise (blue) dynamics, confirming Eq. \ref{['chiral']}.
• Figure 4: Critical timescale $T_{\mathrm{cr}}$ as a function of precision bits $m$ for EP-encircling loops. Data points are obtained from simulations using arbitrary-precision arithmetic for different radii. The nearly linear scaling with slope $\ln(2)\,\pi/(2\sqrt{r})$ is consistent with precision-limited behavior.
• Figure 5: Critical timescale $T_{\mathrm{cr}}$ as a function of precision bits $m$ for symmetric non-encircling loops. Data points (circles) are obtained from simulations using arbitrary-precision arithmetic. The dashed line shows the theoretical prediction $T_{\mathrm{cr}} = \mathcal{G}\, m \ln\beta$ from Eq. \ref{['Tcr_precision']} with $\mathcal{G} = 2\pi/r^2$, yielding excellent agreement and confirming the linear scaling.