The analytically tractable zoo of similarity-induced exceptional structures

Abstract

Exceptional points (EPs) are non-Hermitian spectral degeneracies marking a simultaneous coalescence of eigenvalues and eigenvectors. Despite the fact that multiband $n$-fold EPs (EP$n$s) generically emerge as special points on manifolds of EP$m$s, where $m<n$, EP$n$s as well as their topological properties have hitherto been studied as isolated objects. In this work we address this issue and carefully map out the emerging properties of multifold exceptional structures in three and four dimensions under the influence of one or multiple generalized similarities, revealing diverse combinations of EP$m$s in direct connection to EP$n$s. We find that simply counting the number of constraints defining the EP$n$s is not sufficient in the presence of similarities; the constraints can also be satisfied by the EP$m$-manifolds obeying certain spectral symmetries in the complex eigenvalue plane, reducing their dimension beyond what is expected from counting the number of constraints. Furthermore, the induced spectral symmetries not always allow for any EP$m$-manifold to emerge in $n$-band systems, making the plethora of exceptional structures deviate further from naive expectations. We illustrate our findings in simple periodic toy models. By relying on similarity relations instead of the less general symmetries, we simultaneously cover several physically relevant scenarios, ranging from optics and topolectrical circuits, to open quantum systems. This makes our predictions highly relevant and broadly applicable in modern research, as well as experimentally viable within various branches of physics.

Figures (6)

• Figure 1: Exceptional structures of the pseudo-Hermitian four-band model in 3D given by Eqs. \ref{['eq:PSHHam1']}-\ref{['eq:PSHHam3']} in panels (a)-(e), and their concomitant three-level Fermi surfaces in for $k_x=0$ in panels (f)-(j). In panels (a)-(e), the blue surfaces are similarity-induced EP2s ($\mathcal{D}_4=0$), the red arcs are similarity-induced EP3s ($\alpha^2+\gamma=\beta^2+4\alpha^3=0$), and the black dots are similarity-induced EP4s $(\alpha=\beta=\gamma=0$). The green and yellow arcs, however, denote special EP2s of codimension 2. The yellow (green) arcs arise as solutions to $\beta=3\alpha^2-\gamma=0$ with $\alpha>0$ ($\alpha<0$). Panels (a)-(e) display how the EP4s are pairwise created/annihilated as a function of the real parameter $M$, a process that affects the appearance of the similarity-induced EP3 arcs, but also the special EP2 arcs. In panels (f)-(j), the behavior of the three-level Fermi surfaces (green and orange arcs), where three of the four eigenvalues share identical real parts, and the discriminant (blue arc) is displayed. The green and orange arcs correspond to three-level Fermi surfaces given by Eqs. \ref{['eq:3FS1']} and \ref{['eq:3FS2']}, respectively. For illustrative purposes, this is displayed for a 2D cut of the Brillouin zone given by $k_x=0$. The intersection of the three-level Fermi surfaces with the discriminant forms EP3s (red dots), and when there are no intersections, the EP3s are gapped out, as shown in panel (j).
• Figure 2: Illustration of how pseudo-Hermitian similarity constrains the position of exceptional structures of $n$-band systems (red dots) in the complex eigenvalue plane. (a) Similarity-induced EP$m$s of codimension $m-1$ are, when $m \leq n$, forced to appear at real eigenvalues. (b) Despite the similarity, EP$m$s of order $m \leq \lfloor \frac{n}{2} \rfloor$, with $\lfloor x \rfloor$ denoting the integer part of $x$, of generic codimension $2m-2$ may still appear in the spectrum, but they are constrained to appear pairwise at complex conjugate eigenvalues.
• Figure 3: Exceptional structures in part of the Brillouin zone of the model given by Eqs. \ref{['eq:SSSHam1']}-\ref{['eq:SSSHam3']}. The blue (green) surfaces correspond to the real (imaginary) part of the discriminant $\mathcal{D}_4'$ of $P_4'(\lambda)$ being zero, while the red (magenta) arcs illustrate the EP2 2D surfaces in 4D given by $\nu=\eta^2$ ($\nu=0$) as 1D arcs in 3D. Panels (a)-(e) show different cuts in $k_w$, given by $k_w = \left(-\frac{3k_0}{2},-k_0,0,k_0,\frac{3k_0}{2}\right)$, respectively, with $k_0 = \cos^{-1}\left[\sum_{i=x,y,z}\cos(k_i)-M\right]$. When $k_w=-k_0$ [panel (b)] and $k_w=k_0$ [panel (d)], the system hosts EP4s appearing as intersections between the different EP2 structures, which do not appear when $k_w=-\frac{3k_0}{2}$ [panel (a)], $k_w = 0$ [panel (c)], or $k_w = \frac{3k_0}{2}$ [panel (e)]. Notable is further that the direction of winding of the magenta arc around the red arc changes when passing through the EP4s, something that can only occur through such a transition.
• Figure 4: Illustration of how self-skew-similarity constrains the position of exceptional structures of $n$-band systems (red dots) in the complex eigenvalue plane. (a) Similarity-induced EP$m$s are, when $m<n$, forced to appear at the origin and have codimension $2k$ when $m=2k+\delta$, where $\delta=0$ ($\delta=1$) for $n\in\text{even}$ ($n\in\text{odd}$). (b) Despite the similarity, EP$m$s of generic codimension $2m-2$ may still appear in the spectrum when $m<\frac{n}{2}$, but they are constrained to appear pairwise in adjacent quadrants of the complex eigenvalue plane.
• Figure 5: Exceptional structures in part of the Brillouin zone of the model given by Eqs. \ref{['eq:MSHam1']}-\ref{['eq:MSHam3']}. As predicted, the EP6s (black dots) are connected by various exceptional structures. The blue surfaces denote EP2s corresponding to $\kappa^3+3\eta\kappa-2\nu=0$, while the green (purple) surfaces are EP2s corresponding to $\eta^3+\nu^2=$ for $\kappa^3+\nu<0$ ($\kappa^3+\nu>0)$. On these surfaces are special EP2 arcs, one of which is located at $\lambda=0$, defined by the additional constraint $\kappa^3-8\nu$ (yellow arcs). Along these arcs, all eigenvalues are twofold degenerate. Additionally, if instead $\kappa^3+\nu=0$ on the EP2 surfaces, arcs of EP4s at $\lambda=0$ appear (cyan arcs). When $\eta = \nu = 0$, arcs of either purely real or complex conjugated EP3s emerge (red arcs). All of these structures are possibly connected to EP6s, but as the figure shows, they are not always. For $M=0,0.5$ [panels (a) and (b), respectively], the EP4 arcs connect the EP6s, while the EP3 and special EP2 arcs do not. When this EP6 pair is annihilated at $M=1$ [panel (c)], another pair is created. These EP6s are connected by all different EP arcs; see panel (d) for $M=1.5$. At $M=3$, these EP6s are annihilated, resulting in the EP arcs to disconnect.