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 (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 non-Hermitian (NH) models in 1D. Conversely, time-dependent spacetime coordinates break pseudohermiticity, yielding nonunitary time evolution. Similarly, space-dependent spacetime coordinates lead to the NH skin effect, i.e., the accumulation of localized states on the system boundaries. Arguably, these effects are physical: The time dependence leads to local gain and loss processes on the lattice and nonunitary growth or decay over time. Conversely, space dependence leads to the NH skin effect with spatial decay of the fields in a preferential direction. In other words, the curvature gradients induce an imaginary gauge field on the lattice, 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 gradients produce nonunitary gain or loss, while spatial gradients correspond to the NH skin effect, allowing for the description of these two phenomena in a unified framework. This also suggests a duality between NH phenomena and spacetime deformations, framing NH physics in purely geometric terms, and unveils an unexpected connection between the spacetime metric and NH phases of matter.

Figures (4)

• Figure 1: Local density of states (LDOS) of the lattice Hamiltonians corresponding to the Dirac equation in curved spacetimes with conformally flat coordinates [see \ref{['tab:conformalexamplestable']}], calculated on a finite patch as a function of the energy and position. Different panels correspond to: Rindler metric in conformally flat coordinates $\dd s^2=e^{qx} (\dd t^2 - \dd x^2)$ in the massless (a) and massive (b) cases; anti-de Sitter metric in conformally flat coordinates $\dd s^2=(q x)^{-1} (\dd t^2 - \dd x^2)$ with coordinate singularity at $x=0$ in the massless (c) and massive (d) cases; de Sitter metric in conformally flat coordinates $\dd s^2=(r t)^{-1} (\dd t^2 - \dd x^2)$ with coordinate singularity at $t=0$ in the massless (e) and massive (f) cases; In the de Sitter case, the lattice Hamiltonian has no $\mathcal{PT}$ symmetry and thus exhibits complex energy eigenvalues, with the imaginary part of the LDOS (not shown) being time-dependent, while the real part of the LDOS remains time-independent. The skin effect is visible for the Rindler metric [(a) and (b)] and the anti-de Sitter metric [(c) and (d)] in the massless and massive cases. The mass in (b), (d), and (f) is $M=0.5$.
• Figure 2: Local density of states (LDOS) on the real and imaginary axes of the time-dependent and non-Hermitian lattice Hamiltonian corresponding to curved spacetime with the conformally flat metric $\dd s^2=(rt+qx)^{-1} (\dd t^2 - \dd x^2)$ [see \ref{['tab:conformalexamplestable']}], calculated on a finite patch as a function of the energy and position and at different time slices $t=0.25,0.5,0.75$ with $r=1$. For time-dependent Hamiltonians, the time-dependent local density of states shown here describes the adiabatic time-evolution of the energy spectra. Different panels correspond to the massless (a) and massive (b) cases ($M=0.5$). This metric has a coordinate singularity at $x=t=0$.
• Figure 3: Local density of states (LDOS) of the lattice Hamiltonians corresponding to the Dirac equation in curved and spacetimes with diagonal metric tensors [see \ref{['tab:diagonalexamplestable']}], calculated on a finite patch as a function of the energy and position. Different panels correspond to: Rindler metric $\dd s^2=(qx)^2 \dd t^2 - \dd x^2$ with a coordinate singularity at $x=0$ in the massless (a) and massive (b) cases; anti-de Sitter metric $\dd s^2=\alpha^2 \dd t^2 - \beta^{2}\dd x^2$ with $\alpha=1/\beta=\sqrt{1+(qx)^2}$ in the massless (c) and massive (d) cases; de Sitter metric $\dd s^2=\alpha^2 \dd t^2 - \beta^{2}\dd x^2$ with $\alpha=1/\beta=\sqrt{1-(qx)^2}$ with a coordinate singularity at $x=1/q=N$ in the massless (e) and massive (f) cases. The mass in (b), (d), and (f) is $M=0.5$.
• Figure 4: The duality between the unitary evolution of the field $\widetilde{\psi}$ with mass $\widetilde{M}$ in flat spacetime and the nonunitary evolution of the field $\psi$ with mass $M$ in curved spacetime. The relation between the two fields is determined by the metric tensor by $\widetilde{\psi}=\sqrt{\Omega} \,\psi$ and $\widetilde{M}=M\alpha$. For simplicity, the plots show only the real part of the field in the case $\widetilde{M}=M=0$, with a curved spacetime corresponding to the Weyl metric with $\Omega={e}^{rt+qx}$, giving $\widetilde{\psi}={e}^{rt+qx} \,\psi$ with $\psi\propto {e}^{\mathrm{i} (\omega t + k x)}$ (plane waves). Note that the field $\psi$ shown above shows nonunitary time evolution for $r\neq0$ (with the probability density decaying exponentially in time) and the non-Hermitian skin effect $q\neq0$ (with the probability density decaying exponentially in space). In the massive case $M\neq0$, the renormalized mass $\widetilde{M}$ depends explicitly on the spacetime coordinates.

Theorems & Definitions (4)

• Proposition 1
• proof
• Proposition 2
• proof