Characterizing and Tuning Exceptional Points Using Newton Polygons

The study of non-Hermitian degeneracies -- called exceptional points -- has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning exceptional points. These polygons were first described by Isaac Newton in 1676 and are conventionally used in algebraic geometry, with deep roots in various topics in modern mathematics. We have found their surprising connection to non-Hermitian physics. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order exceptional points, using a recently experimentally realized optical system. Using the paradigmatic Hatano-Nelson model, we demonstrate how our Newton Polygon method can be used to predict the presence of the non-Hermitian skin effect. As further application of our framework, we show the presence of tunable exceptional points of various orders in $PT$-symmetric one-dimensional models. We further extend our method to study exceptional points in higher number of variables and demonstrate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand and tune exceptional physics.


INTRODUCTION
Energy non-conserving and dissipative systems are described by non-Hermitian Hamiltonians [1]. Unlike their Hermitian counterparts, they are not always diagonalizable and can become defective at some unique points in their parameter space -called exceptional points (EPs) -where both the eigenvalues and the eigenvectors coalesce [2,3]. Around such an EP, the complex eigenvalues lie on self-intersecting Riemann sheets. This means that upon encircling an EP once, the system does not return to its initial state, but to a different state on another Riemann sheet, manifesting in Berry phases and topological charges [4][5][6][7][8][9][10][11].
While early studies focused on second order EPs (where only two eigenvectors coalesce), very recently, the focus has shifted to higher order EPs, where more than two eigenvectors coalesce [46][47][48]. Apart from interesting fundamental physics, they show promise for several fascinating applications [49,50]. Higher order EPs and their unconventional phase transitions have been experimentally realized in various acoustic and photonic systems [49,[51][52][53].
Here, we introduce a new algebraic framework for characterizing such non-Hermitian degeneracies using Newton polygons. These polygons were first described by Isaac Newton, in 1676, in his letters to Oldenburg and Leibniz [54]. They are conventionally used in algebraic geometry to prove the closure of fields [55] and are intimately connected to Puiseux series -a generalization of the usual power series to negative and fractional exponents [56,57]. Furthermore, Newton polygons have deep connections to various topics in mathematics, including homotopy theory, braid groups, knot theory and algebraic number theory [54].
We develop the Newton polygon method to study EPs and illustrate its utility in predicting higher order EPs in experimentally realized systems, as well as predicting the non-Hermitian skin effect. We also present parity and time reversal (P T ) symmetric one-dimensional models to demonstrate how this method can provide an elegant way of tuning different system parameters to obtain a higher order EP, or to choose from a spectrum of EPs of various orders. The Newton polygon method can also be naturally extended to higher number of variables. Using such an extension we show rich anisotropic behaviour around such EPs.
We hope that our results stimulate further exploration of non-Hermitian degeneracies and their applications.

THE NEWTON POLYGON METHOD
We consider a system at an EP described by the Hamiltonian H 0 (t 1 , t 2 , ...), where t 1 , t 2 , ... are system-dependent parameters. If a perturbation of the form ϵH 1 (t 1 , t 2 , ...) is now added, one can write the eigenvalues of the perturbed Hamiltonian H(ϵ) = H 0 + ϵH 1 as a Puiseux series in ϵ.
To leading order, they have the form ω ∼ ϵ 1/N where N is the order of the EP (see The smallest convex shape that contains all the points plotted is called the Newton polygon. 3. Select a segment of the Newton polygon such that all plotted points are either on, above or to the right of it. The negative of the slope of this line-segment gives us the lowest order dependence of ω on ϵ. can be made to vanish (which occurs for g = √ 2κ), we can expect to observe exceptional behaviour.
Right panel shows the Newton polygon for this case. Here, a slope of −1/3 implies ω ∼ ϵ 1/3 , or the presence of a third order exceptional point. Here n(ω) and n(ϵ) denote the exponents of ω and ϵ, respectively.

Predicting higher-order EP
To illustrate our method, we consider a recently realized optical system, consisting of three coupled resonators, that exhibits a higher order EP and an unprecedented sensitivity to changes in the environment [49]. The system can be described by a remarkably simple non-Hermitian Hamiltonian where g accounts for gain and loss, κ is the coupling between the resonators and ϵ is the external perturbation. The characteristic equation, p(ω, ϵ) = 0, reads Figure 1 shows the Newton polygon for p(ω, ϵ), where each point on the graph corresponds to a term in the characteristic equation. The line-segment that contains all points on, above or to the right of it is shown in blue. This line has a slope of −1 implying that the lowest order dependence of the eigenvalues on ϵ has the form ω ∼ ϵ. Notice, however, that if we set g = √ 2κ, the coefficient of ω vanishes and the point (1, 0) is no longer present in the Newton polygon. The slope of the desired line-segment is now −1/3 which, in turn, means that ω ∼ ϵ 1/3 , or, we have a third order EP (EP3). The Newton polygon method could thus predict the presence of an EP3 for g = √ 2κ. This is indeed what has been found in the experiments by Hodaei et al. [49].
Here, we have illustrated the use of the Newton polygon method to evaluate the degree of an EP. In addition, it also provides an algebraic way of evaluating the expansion of the where J R/L = t ± γ/2 are the right and left hopping amplitudes. If we now add a perturbation, ϵ, coupling the first and the last sites, the Hamiltonian matrix takes the form The characteristic equation, in turn, is p(ω, ϵ) where each z M is a constant, z M ∈ Z. The Newton polygon for p(ω, ϵ) is shown in the left panel of Figure 2. We note that remarkably at t = γ/2, the coefficients of all the terms in p(ω, ϵ) vanish other than the ϵ 1 and the ω N terms. The Newton polygon for this case is plotted in the right panel in Figure 2. The slope of the relevant line-segment is 1/N . This implies the presence of an N -th order EP and correspondingly the presence of non-Hermitian skin effect at t = γ/2, which is physically the condition for unidirectional hopping (J R = γ, J L = 0). If the perturbing term ϵ is added to the bottom left element of the Hamiltonian, the other limit of purely unidirectional hopping with J R = 0, J L = −γ is obtained. This is also predicted from our Newton polygon approach by constructing the corresponding characteristic equation. Thus, our Newton polygon method can elegantly predict the occurrence of non-Hermitian skin effect. We note that our Newton polygon approach is able to characterize the non-Hermitian skin effect in the scenario where higher order EPs appear with an algebraic multiplicity scaling with system size while the geometric multiplicity becomes unity. In this situation, all the bulk modes may align to one state and result in the non-Hermitian skin effect.

Application to P T -symmetric 4-site model
We next consider a 4-site P T -symmetric system as shown schematically in Figure 3(a).
We consider a general form consistent with the P T -symmetry. Due to parity symmetry, two hopping parameters p and q are sufficient to describe the couplings between the four sites.
The balanced gain and loss for the outer and inner sites are given by δ and γ respectively. We will show that when such a system is perturbed, one can get an EP2, or an EP4 depending on the tuning of various parameters. As we shall demonstrate below, the Newton polygon method elegantly predicts the required tuning. The Hamiltonian for the 4-site system can be written as As the overall energy scale does not affect the behaviour, we shall set γ = 1 from hereon.
If the system is perturbed, say by slightly varying one of the couplings by ϵ, the perturbed Hamiltonian reads We present the Newton polygon of the characteristic polynomial in Figure 3  The 4-site model we presented here can show tunable second and fourth order EPs. Analogously, we have devised a 5-site P T -symmetric model. Using the Newton polygon framework, we have shown that it can exhibit tunable third and fifth order EPs [58]. Such P T -symmetric models have been experimentally realized in magnetic multi-layers [61], waveguides [62,63], topoelectric circuits [64], and photonic lattices [65][66][67][68]. The Newton polygon method can serve as a useful tool for tuning to EPs in these experimental platforms.

Extension to higher number of variables
The Newton polygon approach can be naturally extended to study exceptional behaviour for higher number of variables. Remarkably, it has been recently shown that if a system at an EP can be perturbed in two different ways such that H(ϵ, λ) = H 0 + ϵH 1 + λH 2 , then it possible to observe different exceptional behaviours along different directions in the ϵ-λ plane [51,69]. The Newton polygon framework can predict such anisotropic variations, as we show next. We consider the Hamiltonian given in Equation 2 and add a second perturbation, λ, to obtain We can set g = √ 2κ to obtain exceptional behaviour, as discussed earlier. If we pick any direction in the ϵ-λ plane making an angle ϕ with the ϵ-axis, then along that direction λ = ϵ tan ϕ. We can now use the Newton polygon method to determine, in one go, the order of the EPs along all directions (i.e., for all values of ϕ). The characteristic equation reads The corresponding Newton polygon is shown in Figure 4(a). Notice that ω ∼ ϵ 1/3 along all directions unless tan ϕ + 1 = 0 for which the point (0, 1) vanishes from the Newton polygon.
In this case we instead obtain ω ∼ ϵ 1/2 . The Newton polygon approach thus predicts the presence of an anisotropic EP, which shows second order behaviour along ϕ = −π/4, but third order behaviour along all other directions. The anisotropic nature of the EP manifests in the Newton polygon as coefficients which depend on ϕ. We note that a similar model along two special directions has been studied before [70], although a complete picture along all possible directions was lacking. Our method provides a unified way of obtaining the exceptional properties along all directions.
We again numerically verify the predictions from the Newton polygon method in multiple ways (see Figure 4). We first verify our analytical expressions for the eigenvalues by fitting them to the numerical band structure. Next, we study the variation of the phase rigidity, r, with the perturbation. Physically, phase rigidity is a measure of the bi-orthogonality of the eigenfunctions, and helps identify EPs [46,69,71]. We present further details in Methods.
Here, we find that the phase rigidity vanishes at the location of the EP at ϵ = 0 [ Figure 4 (c)]. Importantly, we observe that the phase rigidity scales as ϵ 2/3 for ϕ ̸ = −π/4, while it scales as ϵ 1 for ϕ = −π/4, thus confirming our predictions of the anisotropic nature of the EP.

DISCUSSIONS
We put forward a new algebraic framework using Newton polygons for characterizing EPs.
Using several examples, we illustrated how the Newton polygon method enables prediction and selection of higher order EPs. Using the celebrated Hatano-Nelson model, we showed how this method allows prediction of the non-Hermitian skin effect. We also proposed an extension to higher number of variables and used it to reveal rich anisotropic behaviour around such non-Hermitian degeneracies.
Looking ahead, our analytical approach could be useful for tuning to EPs in different experimental platforms, whether it be for enhanced performance of sensors or exploring unconventional phase transitions, especially as the dimensionality and complexity of the non-Hermitian Hamiltonians in question increases. Newton polygons play a natural role in homotopy theory, braid groups, knot theory and algebraic number theory [54,56]. Our work lays the foundation for exploring such connections in the context of non-Hermitian physics.
With the recent growing interest in knotted and linked exceptional nodal systems [72][73][74][75][76][77][78], it would be worthwhile to find a Newton polygon based characterization of such systems. In conclusion, we hope that our results inspire further exploration of EPs and their applications.

Computing the coefficients and higher-order terms
Our Newton polygon approach also provides a straightforward algorithmic method to evaluate the expansion of the eigenvalues beyond just the leading order. We describe here the steps for finding the coefficients and higher order terms in the Puiseux series expansion of eigenvalues.
The steps to evaluate the other unknowns in the expansion are as follows.
1. Collect the lowest order terms in ϵ in the polynomial p(ϵ γ 1 (c 1 + ω 1 ), ϵ). They must cancel each other as p(ω, ϵ) = 0. The first coefficient c 1 can be extracted from this requirement.
3. Next, calculate the Newton polygon for p 1 (ω, ϵ) and repeat the steps to find γ 2 and c 2 , and so on.
The eigenvectors form a bi-orthogonal basis consisting of both right |ψ⟩ and left ⟨ϕ| eigenvectors. They can be normalized using a bilinear product of the left and right eigenvectors, where i, j correspond to distinct states. Interestingly, both the eigenvectors and eigenvalues can split and follow a directional parameter dependence while approaching an EP from different parametric directions [46]. The phase rigidity, r α , can characterize these striking features due to extreme skewness EPs [6,[79][80][81]. It is defined as whereψ α andφ α are the normalized biorthogonal right and left eigenvectors of a state α. The phase rigidity quantitatively measures the eigenfunctions' bi-orthogonality. At an EP the states coalesce and thus phase rigidity vanishes (|r α | → 0). This enables defining a critical exponent around an EP in the parameter space [82,83]. For example, around an EP3, the scaling exponents for the phase rigidity are given by (N − 1)/N and (N − 1)/2, where N = 3 is the order of the EP. Note that different forms of the perturbation may result in different scaling of the phase rigidity. Our Newton polygon approach will allow diagnosing these EPs. The phase rigidity is also experimentally measurable, making it a very relevant quantity to look at.