An exceptional surface and its topology

TL;DR

This paper addresses the emergence of exceptional topology for higher-order NH singularities by introducing an exceptional surface (ES) formed by EP3s in a NH 3D system, embedded in a four-dimensional parameter space. The ES is treated as a two-dimensional extension of a tensor monopole, and its topology is characterized by the Dixmier-Douady invariant and a NH Berry phase, with a Riemann-surface-based interpretation of eigenenergy trajectories. The authors demonstrate that the ES is encircled by a four-dimensional bulk Fermi arc and provide both a bulk-boundary perspective via NH SSH3 toy models and a concrete experimental proposal using a circuit-QED platform with postselection to realize and measure the DD invariant and Berry phase. Together, these results establish a framework for studying exceptional higher-order topology in NH systems and outline practical routes to probe such phenomena experimentally.

Abstract

Non-Hermitian (NH) systems can display exceptional topological defects without Hermitian counterparts, exemplified by exceptional rings in NH two-dimensional systems. However, exceptional topological features associated with higher-dimension topological defects remain unexplored yet. We here investigate the topology for the singularities in an NH three-dimensional system. We find that the three-order singularities in the parameter space form an exceptional surface (ES), on which all the three eigenstates and eigenenergies coalesce. Such an ES corresponds to a two-dimensional extension of a point-like synthetic tensor monopole. We quantify its topology with the Dixmier-Douady invariant, which measures the quantized flux associated with the synthetic tensor field. We further propose an experimentally feasible scheme for engineering such an NH model. Our results pave the way for investigations of exceptional topology associated with topological defects with more than one dimension.

Figures (6)

• Figure 1: The projection of the ES from the four-dimensional space onto the three-dimensional case, which is defined by the coordinates $\{q_1,q_2,q_3 \}$$(q_4=0)$ in (a) and $\{q_1,q_3,q_4 \}$$(q_2=0)$ in (b). Real (c) and imaginary (d) parts of the eigenenergies with respect to $\Omega_1=q_1+iq_2$ and $\Omega_2=q_3+iq_4$.
• Figure 2: The DD invariant characterized in the NH three-dimensional system, with respect to $R/r_0$, where $R$ is the radius of the parameter sphere and $r_0$ is the shortest distance between the sphere's center and the ES. The left and right sides of the broken shaft indicate the two cases of the parameter sphere being inside and outside the ES as the radius $R$ increases, respectively. The blue and red lines characterize the DD invariant calculated by the quantum metric and the Berry curvature described in Eq. (\ref{['eq04']}), respectively. The grey dashed lines are the theoretical horizontal lines of the values 0 and 1.
• Figure 3: Energy spectra and the Riemann surface of the NH model (\ref{['eq1']}). The real (a) and imaginary (b) parts of the Riemann surface with respect to $q_3$ and $q_{\perp}$. The red arrowhead tube represents the travel path from $\theta=0$ to $\theta=6\pi$.
• Figure 4: The Berry phase characterized in the NH model. The parameter loop on the $\{\vec{q_3},\vec{q_\perp}\}$ plane travels encircling (upper-left) or separated from (upper-right) the ER on the $\{\vec{q_3},\vec{q_4}\}$ plane. The Berry phase versus $(\Delta-R)/r$, a sharp transition happens when the parameter loop shrinks to pass through the ER.
• Figure 5: Topological boundary modes in the trimer SSH model. (a) and (b) are two distinct toy models of the NH SSH3 system, each defined by a unique bulk Hamiltonian. For the first model, the energy spectra are under PBC (c) and OBC (d), respectively. Under OBC, the NHSE is clearly observed. In the second model, the EP3s are observed under PBC (e), whereas they vanish under OBC (f).