Metric-induced non-Hermitian physics

Abstract

I consider the longstanding issue of the hermiticity of the Dirac equation in curved spacetime. Instead of imposing hermiticity by adding ad hoc terms, I renormalize the field by a scaling function, which is related to the determinant of the metric, and then regularize the renormalized field on a discrete lattice. I found that, for time-independent and diagonal (or conformally flat) coordinates, the Dirac equation returns a pseudo-Hermitian (i.e., PT-symmetric) Hamiltonian when properly regularized on the lattice. Notably, the PT-symmetry is unbroken, ensuring a real energy spectrum and unitary time evolution. This establishes stringent conditions for the existence of complex spectra in 1D non-Hermitian (NH) models. Conversely, time-dependent spacetime coordinates break pseudohermiticity, yielding NH Hamiltonians with nonunitary time evolution. Similarly, space-dependent coordinates lead to the NH skin effect (NHSE), i.e., the accumulation of localized states on the boundaries. Arguably, these NH effects are physical: time dependence leads to local gain and loss processes and nonunitary growth or decay. Conversely, space dependence leads to the NHSE with spatial decay of the fields in a preferential direction. In other words, the curvature gradients induce an imaginary gauge field, corresponding to a drift force acting in space and time, pushing the eigenmodes to the boundaries or forcing their probability density to increase or decrease over time. Hence, temporal curvature gradients produce nonunitary gain or loss, while spatial curvature gradients correspond to the NHSE, allowing for the description of these two phenomena in a unified framework. This also suggests a duality between NH physics and spacetime deformations, framing NH physics in purely geometric terms. This metric-induced nonhermiticity unveils an unexpected connection between the spacetime metric and NH phases of matter.