Characterizing Liouvillian Exceptional Points Through Newton Polygons and Tropical Geometry

TL;DR

Liouvillian exceptional points (EPs) arise in open quantum systems described by the Lindblad equation, and their order and perturbation sensitivity encode rich non-Hermitian physics. The authors develop a framework combining Newton polygons (via Puiseux-series valuations) and tropical geometry (via tropicalization and amoebas) to identify EPs and their scaling laws, capturing anisotropy in how EPs respond to perturbations. Applying the method to a dissipative spin-$1/2$ system and a dissipative superconducting qubit, they obtain explicit EP regimes, quantify square-root and cube-root scalings, and demonstrate eigenvalue exchange under encircling, all validated by numerical diagonals. The approach provides a practical route to design Liouvillian EPs of desired order in experimental platforms and opens avenues for exploring non-linear and non-Markovian variants of EPs.

Abstract

The dynamics of open quantum systems described by the Lindblad master equation follows according to non-Hermitian operators. As a result, such systems can host non-Hermitian degeneracies called Liouvillian exceptional points (EPs). In this work, we show that Newton polygons and tropical geometric approach allow identification and characterization of Liouvillian EPs. We use two models -- dissipative spin$-1/2$ system and dissipative superconducting qubit system -- to illustrate our method. We demonstrate that our approach captures the anisotropy and order of the Liouvillian EPs, while also revealing the subtle dependence on the form of the perturbation. Our analytical analysis is supplemented by direct numerical calculations of the scaling and exchange of eigenvalues around Liouvillian EPs. Our analytical approach could be useful in understanding and designing Liouvillian EPs of desired order.

Figures (3)

• Figure 1: Second-order Liouvillian exceptional point in dissipative spin$-1/2$ model. (a) The Newton polygon corresponding to the Liouvillian of the dissipative spin$-1/2$ system. It shows a slope of $-\frac{1}{2}$, thus indicating a second-order Liouvillian EP. (b) The amoeba of the characteristic polynomial at the Liouvillian EP. A tentacle branches out with a slope value of 2, which matches the prediction by Newton polygons and thus further indicates a square root topology. The (c) real and (d) imaginary parts of eigenvalues near the Liouvillian EP as a function of the perturbing parameter $\epsilon$. (e) The encircling of the Liouvillian EP reveals the characteristic exchange of the eigenvalues as a function of the parameter $t$, where we used $\mathcal{L}_0+0.01\mathcal{L}_1 e^{it}$. (f) Plot of $\log(\omega-\omega_0)$ with respect to $\log (\epsilon)$, where the data points are taken from (c) and (d), illustrating the square root behaviour near the Liouvillian EP. The other parameters are chosen to be $(\gamma_-=0,\gamma_x=1,\gamma_y=2, \Omega=1)$.
• Figure 2: Hybrid Liouvillian exceptional points in dissipative qubit system with $\gamma_f$ perturbation. (a) The Newton polygon exhibiting two segments with non-trivial slopes $-1$ and $-\frac{1}{3}$, respectively. (b) The amoeba of the shifted characteristic polynomial at the Liouvillian EP. We note two tentacles branching out to negative infinity below with slopes 1 and 3, precisely normal to the segments of the Newton polygon. The (c) real and (d) imaginary parts of the numerically evaluated energy eigenvalues near the Liouvillian EP showing three eigenvalues (green, orange, and blue) coalescing. As seen in panel (c), two of the eigenvalues (blue and orange) have the same real parts, while in panel (d) we see that one of them (blue) remains purely real. (e) The encircling of the Liouvillian EP, i.e., a plot of eigenvalues of the matrix $\mathcal{L}_{eff}+0.01\mathcal{L}_1 e^{it}$ with varying $t$. (f) Plot of $\log(\omega-\omega_0)$ vs $\log (\epsilon)$, where the data points are taken from the orange curve in (c) and (d). A slope of $\frac{1}{3}$ indicates a cube root scaling. The model parameters are chosen to be ($\gamma_f=0,\gamma_e=1,J=0.25$), and the perturbation matrix is $\frac{\partial}{\partial \gamma_f}\mathcal{L}_{eff}$ to consider small variations in $\gamma_f$. Note that $\mathcal{L}_{eff}$ is linear in $\gamma_f$.
• Figure 3: Hybrid Liouvillian exceptional points in dissipative qubit system with $J$ perturbation. (a) The Newton polygon showing one segment with the slope of $-\frac{1}{2}$. The only other segment present is a vertical one. (b) The amoeba of the shifted characteristic polynomial at the Liouvillian EP. We see a tentacle branching downwards in the third quadrant with the slope of 2 and a purely horizontal tentacle branching to negative infinity along the horizontal axis. Amoebas therefore can predict invariant solutions as well, that is, zero roots. The (c) real and (d) imaginary part of the numerically evaluated energy eigenvalues near the Liouvillian EP showing two eigenvalues (green, orange) coalescing. Panel (e) shows the encircling of the Liouvillian EP, i.e., a plot of the eigenvalues of the matrix $\mathcal{L}_{eff}+0.01\mathcal{L}_J e^{it}$ with varying $t$. (f) Scaling plot of $\log(\omega-\omega_0)$ vs $\log (\epsilon)$, where the data points are taken from panels (c) and (d). A slope of $\frac{1}{2}$ indicates the square root scaling as predicted in (a) and (b). The parameters were chosen to be ($\gamma_f=0,\gamma_e=1,J=0.25$), and the perturbation matrix $\mathcal{L}_J$ is $\frac{\partial}{\partial J}\mathcal{L}_{eff}$ to consider small variations in $J$. Note that the behavior of the Liouvillian EP changes drastically by simply changing the perturbing parameter.