Robust Exceptional Points in Disordered Systems

TL;DR

This paper introduces a theory for topologically robust exceptional points (EPs) in disordered systems, proposing a method to achieve EPs that are insensitive to disorder, with potential applications in sensors and robust devices.

Contribution

The authors develop a new approach to realize disorder-robust EPs using asymmetric coupling, extending the concept of topological robustness to exceptional points in disordered systems.

Findings

Disordered systems typically lack EPs at the same parameters as ordered systems.

Incorporating asymmetric coupling can create robust, extended EPs insensitive to disorder.

Proposed photonic system can experimentally realize the robust EPs.

Abstract

We construct a theory to introduce the concept of topologically robust exceptional points (EP). Starting from an ordered system with $N$ elements, we find the necessary condition to have the highest order exceptional point, namely $N^\text{th}$ order EP. Using symmetry considerations, we show an EP associated with an order system is very sensitive to the disorder. Specifically, if the EP associated with the ordered system occurs at the fixed degree of non-Hermiticity $\gamma_{EP}$, the disordered system will not have EP at the same $\gamma_{PT}$ which puts an obstacle in front of the observation and applications of EPs. To overcome this challenge, by incorporating an asymmetric coupling we propose a disordered system that has a robust EP which is extended all over the space. While our approach can be easily realized in electronic circuits and acoustics, we propose a simple experimentally feasible photonic system to realize our robust EP. Our results will open a new direction to search for topologically robust extended states (as opposed to topological localized states) and find considerable applications in direct observation of EPs, realizing topological sensors and designing robust devices for metrology.

Figures

Equations

$$\begin{split}\mathcal{H}_{2\times 2}=&i\gamma\sigma_{z}+\sqrt{2}\sigma_{x}\\ \mathcal{H}_{3\times 3}=&\left(\begin{array}[]{ccccc}i\gamma&1&0\\ 1&0&1\\ 0&1&-i\gamma\end{array}\right)\\ \mathcal{H}_{4\times 4}=&\left(\begin{array}[]{cc}i\gamma\sigma_{z}+\sigma_{x}&-i\sigma_{y}\\ i\sigma_{y}&i\gamma\sigma_{z}+\sigma_{x}\end{array}\right)\end{split}$$

$${\mathcal{H}}^p|\lambda ⟩={\lambda }^p|\lambda ⟩$$

$$\mathcal{H}$$

$$j_n=\{\begin{array}{ll}{t}_n& \text{if }n\ne N-1\\ t_n-\gamma & \text{if }n=N-1\end{array}$$

Full text

Introduction– Non-Hermitian systems present new functionalities and features that are inaccessible in Hermitian systems. For instance the existence of exceptional points (EP) is a unique feature of the non-Hermitian systems that resulted to many fascinating features including unidirectional invisibility unitran2 ; unitran3 , unidirectional lasing unitran1 , lasing and anti-lasing in a cavity unitran4 , enhanced optical sensitivity sense1 ; sense2 and stopping of light listop . An EP is a topological singular point in the parameter space of a non-Hermitian Hamiltonian and occurs when at least two eigenvalues and the corresponding eigenvectors of the Hamiltonian coalesce EP1 ; EP2 ; EP3 meaning that the Hamiltonian becomes defective. It can be shown that during the encircling of an EP using a closed loop, the eigenstates of the corresponding Hamiltonian become swapped dynexcep1 ; dynexcep2 . The topological nature of the EP is a consequence of the fact that swapping of the eigenstates is independent of the loop and it occurs even if we deform the loop. In particular, in parity-time symmetric systems with carefully distributed balanced amplification and absorption mechanisms, an EP determines a phase transition from a real spectrum to a complex spectrum. Usually, EPs are associated with the states that are distributed all over the space and thus EPs are sensitive to the unintentional and yet unavoidable changes in the parameter space. As a result, direct observation of EPs is a difficult task. Despite the challenges, their influence on the dynamical properties of the system has been experimentally realized EPdeney1 ; EPdeney2 ; EPdeney3 ; EPdeney4 .

Due to their topological nature, EPs are believed to play an essential role in the theory of topological insulators in non-Hermitian systems. The extension of topological phases to non-Hermitian systems is a relatively new field of study and has recently attracted a great deal of attention. This includes theoretical studies of topological edge states in $1$-D non-Hermitian systems nonh2 ; PTop3 ; ekl56 ; PTop4 . These theoretical works are followed by the first experimental realization of dynamically and topologically stable zero energy state in $1$-D using waveguides sondeney1 . The main model studied in the context of non-Hermitian topological insulators is the $1$-D Su-Schrieffer-Heeger (SSH) model with gain and loss 1d1 ; 1d2 ; 1d3 . The $1$-D study of topological phases has been extended to higher dimensions such as $2$-D photonic honeycomb lattice with gain and loss 2d1 ; 2d3 and a non-Hermitian extension of the Bernevig, Hughes, and Zhang (BHZ) model 2d4 . A major conclusion of these works shows that non-Hermitian systems don’t respect the standard bulk-boundary correspondence of topological insulating systems bulkboun01 ; bulkboun02 ; bulkboun03 ; bulkboun04 ; bulkboun05 ; bulkboun06 ; bulkboun07 . For example, topological transition point of the open periodic systems can be different in non-Hermitian systems in contrast to their corresponding Hermitian ones. Furthermore, standard formulation of topological numbers such as winding and Chern numbers don’t perfectly work in non-Hermitian systems winding1 ; winding2 ; winding3 . Not only topological insulators but also other topological systems have been studied in the context of non-Hermitian systems. These are, for example, topological superconductors with gain and loss and Majorana modes majo1 ; majo2 ; majo3 ; majo4 ; majo5 ; majo6 , nodal surfaces formed by EPs nodalkun ; nodalkun2 and Floquet topological phase in non-Hermitian systems floquet1 ; floquet2 ; floquet3 ; floquet4 .

As mentioned earlier, from the experimental point of view, accessing EPs is a difficult task. Unfortunately, unavoidable disorders in an experiment shift the EPs in parameter space drastically which makes it almost impossible to access an EP. In this paper, we provide a general theory for the creation of a robust EP. While our theory can be applied to different systems including acoustic, phononic, and electronic systems, we provide a simple model for the realization of such robust EPs in a non-Hermitian photonic system.

Formalism– In a Hermitian system, the spectrum is purely real and degenerate points in the spectrum, the so-called diabolical points, are associated with orthogonal and distinct eigenstates. In contrast in a non-Hermitian system, degenerate points or EPs in the spectrum have a degenerate eigenstate due to the coalescing of the corresponding eigenstates EP1 . For instance, consider a finite system that is described by an $N{\times}N$ non-Hermitian Hamiltonian $\mathcal{H}_{N\times N}$. An $n$-th order exceptional point is formed by the simultaneous coalescence of $n$ eigenstates. It is obvious that an $N{\times}N$ non-Hermitian Hamiltonian can have at most $N$-th order exceptional point.

$N$-th order exceptional points– Apart from fundamental curiosity higher order exceptional points are shown to be good nominees for sensing and metrologylyang1 ; mkhaje . So it would be important to find the sufficient condition of realizing an $N$-th order EP. To answer this question without losing generality we search for Hamiltonians that have an $N$-th order exceptional point with zero energy eigenvalue. In other words we are looking for Hamiltonians with a spectrum given by $N$-th root of an expression that becomes zero at the EP. Let us consider the following non-Hermitian Hamiltonians with $N=2,3$ and $4$:

[TABLE]

where $\gamma$ is the degree of non-Hermiticity and $\sigma_{i}$ are the Pauli matrices. All the above Hamiltonians in Eq.(1) share a same feature namely all of them are nilpotent of order $N$ for some value of $\gamma$. A finite-dimensional Hamiltonian $\mathcal{H}$ is a nilpotent operator of order $p$ if $\mathcal{H}^{p}$ is equal to the zero operator for some positive integer $p$, namely $\mathcal{H}^{p}=\mathbf{0}$. A nilpotent Hamiltonian $\mathcal{H}$ is always noninvertible, namely $Det(\mathcal{H})=0$, and all of its eigenvalues are equal to [math]. The later can be easily shown when we operate the Hamiltonian $\mathcal{H}$ on its nonzero eigenvector $|\lambda\rangle$ with eigenvalue $\lambda$ for $p$ times

[TABLE]

However, the left hand side of Eq.(2) is zero operator $\mathbf{0}$ with eigenvalue zero. Thus, the $p$-th root of $\lambda$ is zero. Consequently, if a non-Hermitian Hamiltonian is not nilpotent of order $p<N$, then the order of (zero energy) exceptional states are less than $N$. The most simple example of a nilpotent matrix is an upper triangular matrix, the so-called Jordan block matrix. Now let us go back to the Hamiltonians in Eq.(1). The first two Hamiltonians have open edges while the edges for the last system are connected with antisymmetric tunneling amplitude. We observe that $\mathcal{H}_{2\times 2}^{2}=\mathcal{H}_{3\times 3}^{3}=\mathcal{H}_{4\times 4}^{3}=\mathbf{0}$ when $\gamma=\sqrt{2}$. Therefore, all these three Hamiltonians are nilpotent with zero energy exceptional eigenstate at $\gamma=\gamma_{EP}=\sqrt{2}$. Note that the exceptional point $\gamma_{EP}$ is also the transition point from real valued eigenvalues to complex ones.

Our above discussion depicts that a $N{\times}N$ non-Hermitian Hamiltonian must be nilpotent for the existence of $N$-th order zero energy exceptional states. On the other hand if the $N$-th order energy exceptional state has a nonzero eigenvalue it does not necessarily need to be nilpotent. Any nonzero exceptional point can be formed by $\mathcal{H}^{\prime}=\mathcal{H}+\alpha I$, where the Hamiltonian $\mathcal{H}$ is nilpotent, $\alpha$ is an arbitrary complex valued constant and $I$ is the $N{\times}N$ identity matrix. This new Hamiltonian is not nilpotent since $Det\left(\mathcal{H}^{\prime}\right)\neq 0$. One can easily see that $N$-th order exceptional eigenstate of $\mathcal{H}$ is the simultaneous eigenstate of $\mathcal{H}^{\prime}$ with eigenvalue $\alpha$. Therefore, we conclude that nonzero energy exceptional states appear in a system that is not nilpotent. As an example, consider the following Hamiltonian with an asymmetric tunneling amplitude and an on-site potential difference between the sites: $\mathcal{H}=\sigma_{z}+i\sigma_{y}$. This Hamiltonian is nilpotent of order $2$ and hence it has a second order exceptional state with zero energy. Now, let us consider $\mathcal{H}+\alpha\mathcal{I}$, which is not nilpotent. It is easy to see that both Hamiltonians have a simultaneous exceptional eigenstate \frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}-1&1\end{array}\right)^{T}. However, it is important to notice that such nonzero energy EP is a trivial shift of the levels and does not imply significant physics.

Symmetry properties– So far we provided the condition for obtaining an $N$-th order exceptional point. To illustrate the concept of topologically protected EP, it is constructive to understand the relation between the symmetry properties of a non-Hermitian Hamiltonian $\mathcal{H}(\gamma)$ and the reality of its energy eigenvalues, assuming $\mathcal{H}(\gamma)$ is non-Hermitian Hamiltonian that can become nilpotent for some value of $\gamma=\gamma_{EP}$. More specifically, the exceptional point $\gamma_{EP}$ is a transition point from real-valued to complex-valued energy eigenvalues for the nilpotent Hamiltonian $\mathcal{H}(\gamma_{EP})$. We stress that a nilpotent Hamiltonian is not necessarily a parity-time ($\mathcal{PT}$) symmetric operator and its energy eigenvalues are always zero. Specifically, $\mathcal{PT}$ symmetry is neither a necessary nor a sufficient condition for the existence of the nilpotent Hamiltonian $\mathcal{H}(\gamma)$. In other words, $\mathcal{PT}$ symmetry is not restored below or beyond the EP of $\mathcal{H}(\gamma)$.

Generalization of the above conclusion to higher dimension is straightforward, namely a non-Hermitian Hamiltonian $\mathcal{H}(\gamma_{1},\gamma_{2},...)$ with more than one parameter becomes nilpotent on a exceptional curve, a exceptional surface, and a exceptional hyper-surface for the $2$, $3$, and larger dimensional parameter spaces, respectively. Obviously, $\mathcal{PT}$ symmetry condition is not generally satisfied for this case, either. As a simple example, consider a two-level Hamiltonian $\mathcal{H}(\gamma_{1},\gamma_{2})=i\sigma_{z}+\frac{\gamma_{1}+\gamma_{2}}{2}\sigma_{x}+\frac{i(\gamma_{1}-\gamma_{2})}{2}$ with asymmetric tunneling amplitudes $\gamma_{1,2}$ and a balanced gain and loss coefficient equal to one. This Hamiltonian is parity-time $\mathcal{PT}=\sigma_{y}\mathcal{K}$ ($\mathcal{PT}=\sigma_{x}\mathcal{K}$) symmetry when $\gamma_{1}=-\gamma_{2}$ ($\gamma_{1}=\gamma_{2}$), where $\mathcal{K}$ is the complex conjugation operator. For any other values of $\gamma_{1,2}$, namely $\gamma_{1}\neq\mp\gamma_{2}$ this Hamiltonian is no longer $\mathcal{PT}$ symmetric. Despite the fact that $\mathcal{H}(\gamma_{1},\gamma_{2})$ might not be $\mathcal{PT}$ symmetric it admits real-valued eigenvalues for $\gamma_{2}\geq 1/\gamma_{1}$ and complex eigenvalues for otherwise. Furthermore, the exceptional curve with zero energy is determined by $\gamma_{2}=1/\gamma_{1}$ and the corresponding eigenstate \frac{1}{\sqrt{1+\gamma_{1}^{2}}}(\begin{array}[]{cc}i\gamma_{1}&1\end{array})^{T}. Note that the Hamiltonian $\mathcal{H}$ becomes nilpotent at $\gamma_{2}=\gamma_{1}$ and a $2^{\text{nd}}$ order EP appears.

Another example of non-Hermitian systems with EPs and without $\mathcal{PT}$ symmetry is a non-Hermitian disordered system. In general a disordered system can’t be described by a nilpotent Hamiltonian and hence they can’t have $N$-th order EP. However, second order EP can occur in a disordered system. To illustrate this statement, consider a disordered tight binding lattice with alternating gain and loss

[TABLE]

where $t_{n}$ is the tunneling amplitude drawn randomly from a uniform box distribution and $\gamma$ is the non-Hermitian degree. In Fig.(1) we are plotting the imaginary parts of the energy eigenvalues associated with the $\mathcal{H}$ in Eq.(3) for $N=12$ (top panel) and $N=13$ (lower panel) as a function of $\gamma$ for two different distributions $t_{n}\in[0.5,2]$ (left panel) and $t_{n}\in[0.5,3]$ (right panel). We observe that there exist $M\in\left[1,N\right]$ distinct $\gamma_{EP}$ values (branch points associated with the second order Eps). The number $M$ and the value of $\gamma_{EP}$ are sensitive to the disorder. Namely, every time that we set the tunneling amplitudes from a uniform distribution box and we vary the $\gamma$, EPs are found at different value with respect to another set of tunneling. However, as it can be seen from the lower panel of Fig.(1), there is a distinctive eigenvalue $E=i\gamma$ (highlighted with a orange line) for odd values of $N$. This arises in the systems with $N\equiv 2m-1$ (where $m\in\mathbb{N}$) since gain and loss are not balanced and they do not form a set of dimers with amplification and absorption. Furthermore, eigenenergy of this state is not affected by tunneling disorders, namely its exact energy eigenvalue $E=i\gamma$ does not disturb when we change the tunneling strength randomly. This is in contrast to other eigenenergies that are sensitive to tunneling disorder. Therefore, we can conclude that for odd values of $N$ this state is a topological state. It is important to notice that this state is not a localized state on either side of the lattice and is extended through the whole lattice with zero amplitude on the even sites.

From the above discussion, it is clear that while second order EPs are not destroyed by the disorder, in general they are not robust against the tunneling amplitude disorder. In other words the positions of them on the $\gamma$ axis, namely $\gamma_{EP}$ values, are changing with the disorder. The only robust point in the Fig.(1) is associated with $\gamma=0$ point, which is not an EP as the Hamiltonian becomes Hermitian. Thus in general, exceptional points are not robust in the presence of disorder. However, this does not mean that one cannot find a robust EP. Below we propose a system that has a robust EP in the presence of the disorders.

Robust exceptional points– Consider the following $1D$ non-Hermitian tight binding Hamiltonian with on-site potential $\beta$ and symmetric tunneling amplitude $t_{n}=1$ everywhere except at its right edge which has an asymmetric coupling.

[TABLE]

where $t_{n}$ and $j_{n}$ are forward and backward tunneling which in general they can be complex (random) numbers. If $t_{n}=J_{n}^{*}$, then the system becomes Hermitian otherwise it is non-Hermitian. Let us now assume that the chain support symmetric couplings except at the last coupling, viz.

[TABLE]

Where in Eq.(5) the parameter $\gamma$ denotes the strength of the asymmetry in the last coupling. Without loss of generality we assume that $N$ is an even number. The Hamiltonian in Eq.(4) becomes non-invertible when $\beta=0$. As depicted in the top panel of Fig.(2), this system for $t_{n}=1$ has a second order EP which appears when $\gamma=1$. More specifically, the energy eigenvalues are real (complex) valued when $\gamma<1$ ($\gamma>1$). This EP is robust against tunneling amplitude disorder as long as $\gamma=1$ and $t_{N-1}=1$. To show the robustness of this EP we perform numerical computation on the disordered system with $\beta\neq 0$ (which makes $\mathcal{H}$ invertible) note and $t_{n}\in 1+[-0.75,0.75]$ while we keep $t_{N-1}=1$. An example of the real and imaginary part of the eigenvalues of this disordered system is shown in the lower panel of Fig. (2). As can be seen, the energy eigenvalues are deformed due to the disorder but the EP at $\gamma=1$ remains the same, meaning it is robust against disorder. We stress that the eigenstate associated with the EP is an extended system as shown in Figs(2c,f).

The above intrinsic model can be realized in a simple passive photonic resonator chain. Especially, consider a chain of coupled-resonator optical waveguide which consists of identical symmetric coupled cavities with resonance frequency $\omega_{0}$ 52 ; 53 ; 54 . Now let us couple this chain to a single cavity with resonance frequency $\omega_{0}$ that is no longer symmetric and has a snail shell form as sketched in Fig.(2g) 55 . Each resonator features a clockwise (CW) and a counterclockwise (CCW) mode. While the CW and CCW modes of a symmetric cavity are decoupled there is a coupling $t$ between the CW (CCW) of a cavity to the CCW (CW) of its nearest neighbor cavities. Only in the last non-symmetric cavity there is asymmetric coupling between the CW and CCW mode, i.e. CW is coupled to CCW with coupling strength $a$ but CCW is coupled to CW mode with coupling $b-\alpha$. This structure has a similar band structure as the one depicted in Fig.(2). For example, for $a/t=1$ and $b/t=1.5$ we have an EP at $\alpha=1.5$ which is rebuts against disorder (not shown here).

Conclusion– We have introduced the concept of robust EPs. We have started from an ordered system and find a recipe for obtaining the highest possible EP. We showed that disorder reduces the order of an EP. Using asymmetric tunneling, we show that one can obtain a topologically robust EP in a disordered system. The eigenstate associated with the robust EP is not localized, namely, it is extended. Our finding challenges the classical view related to the topological systems where a topological state is protected by some symmetries such as time reversal, chiral and particle-hole symmetries.