Topological Charge of Causality at a PT-Symmetric Exceptional Point

TL;DR

This paper reveals that causality in a PT-symmetric open dimer has a topological charge at an exceptional point, affecting the reflection response and measurable via a proposed spectroscopy protocol.

Contribution

It introduces the concept of topological charge in causality, linking pole migration to measurable topological phenomena at an exceptional point.

Findings

Pole of reflection coefficient migrates into upper half-plane at exceptional point

Kramers-Kronig relation breaks down with a Lorentzian residual at the transition

Proposed THz spectroscopy protocol detects topological charge through residuals

Abstract

Causality in linear response is conventionally treated as a binary property: a response function is either analytic in the upper half-plane or it is not. We show that in a PT-symmetric open dimer it instead carries a topological charge. As the gain-loss parameter crosses the exceptional point, a single pole of the reflection coefficient migrates into the upper half-plane, the Blaschke winding number jumps from 0 to 1, and standard Kramers-Kronig (KK) reconstruction acquires a Lorentzian residual fixed by the pole residue. The transition is sharp, protected by the codimension-one structure of the exceptional point, and directly measurable in a one-port reflection experiment. Most strikingly, the violation magnitude scales as Delta_KK ~ |gamma - gamma_c|^nu with nu ~ -1.08 in the single-port geometry: the breakdown of standard KK is strongest at threshold and weakens deeper in the broken phase. We derive the exact reflection coefficient, verify the residue-corrected dispersion relation, and propose a THz time-domain spectroscopy protocol that detects the topological charge through the residual itself.