Characterizing and Tuning Exceptional Points Using Newton Polygons

Rimika Jaiswal,1 Ayan Banerjee,2 and Awadhesh Narayan2, ∗ 1Undergraduate Programme, Indian Institute of Science, Bangalore 560012, India 2Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India (Dated: August 3, 2021) 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, and develop its connection to Puiseux expansions. 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. As an 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 Isaac Newton, in 1676, in his letters to Oldenburg and systems are described by non-Hermitian Hamiltoni- Leibniz [54]. They are conventionally used in algebraic ans [1]. Unlike their Hermitian counterparts, they are not geometry to prove the closure of fields [55] and are in- always diagonalizable and can become defective at some timately connected to Puiseux series – a generalization unique points in their parameter space – called excep- of the usual power series to negative and fractional ex- tional points (EPs) – where both the eigenvalues and the ponents [56, 57]. Furthermore, Newton polygons have eigenvectors coalesce [2, 3]. Around such an EP, the com- deep connections to various topics in mathematics, in- plex eigenvalues lie on self-intersecting Riemann sheets. cluding homotopy theory, braid groups, knot theory and This means that upon encircling an EP once, the system algebraic number theory [54]. We develop the Newton does not return to its initial state, but to a different state polygon method to study EPs and illustrate its utility on another Riemann sheet, manifesting in Berry phases in predicting higher order EPs in experimentally realized and topological charges [4–11]. systems. We also present parity and time reversal (PT ) While the notion of EPs was known theoretically for symmetric one-dimensional models to demonstrate how several decades, their controllable realization has only this method can provide an elegant way of tuning differ- been possible recently. This has led to enormous inter- ent system parameters to obtain a higher order EP, or est and, by now, EPs are ubiquitous in acoustic [12, 13], to choose from a spectrum of EPs of various orders. The optical [8], photonic [14–17], mechanical [18], and con- Newton polygon method can also be naturally extended densed matter systems [19–23]. They also appear in to higher number of variables. Using such an extension the study of atomic and molecular physics [24], elec- we show rich anisotropic behaviour around such EPs. tronics [25], superconductivity [26–28], and quantum We hope that our results stimulate further exploration phase transitions [29]. They can lead to a variety of of non-Hermitian degeneracies and their applications. intriguing phenomena such as uni-directional invisibil- Newton polygon method– We consider a system at an ity [30, 31], double refraction [32], laser mode selectiv- EP described by the Hamiltonian H0(t1, t2, ...), where ity [17, 33], non-reciprocal energy transfer [34], non- t1, t2, ... are system-dependent parameters. If a per- Hermitian skin effects [35–43] and interesting quantum turbation of the form H1(t1, t2, ...) is now added, one dynamics of exciton-polaritons [44, 45], to name just a can write the eigenvalues of the perturbed Hamiltonian few. H() = H0 + H1 as a Puiseux series in . While early studies focused on second order EPs ω() = α 1/N + α 2/N + ... (1) (where only two eigenvectors coalesce), very recently, the 1 2 arXiv:2107.11649v2 [cond-mat.mes-hall] 2 Aug 2021 focus has shifted to higher order EPs, where more than Note that, to leading order, they have the form ω ∼ two eigenvectors coalesce [46–48]. Apart from interest- 1/N where N is the order of the EP (see Supplementary ing fundamental physics, they show promise for several Materials [58] for a detailed discussion). The Newton fascinating applications [49, 50]. Higher order EPs and polygon method gives us an algorithmic way of deter- their unconventional phase transitions have been exper- mining the order of an EP by evaluating the power of imentally realized in various acoustic and photonic sys- the leading order term in the eigenvalue expansion start- tems [49, 51–53]. ing from the characteristic equation. This can be done Here, we introduce a new algebraic framework for char- though the following steps: acterizing such non-Hermitian degeneracies using New- ton polygons. These polygons were first described by 1. Given a characteristic equation p(ω, ) = 2 √ 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 ∼ 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 FIG. 1. Newton polygon method enabled prediction experiments by Hodaei et al. [49]. of higher order exceptional points. Left panel shows the Here, we have illustrated the use of the Newton poly- Newton polygon for the characteristic polynomial of H(). gon method to evaluate the degree of an EP. In addition, The line-segment which has all points either on, above, or to it also provides an algebraic way of evaluating the ex- the right of it (shown in blue) has a slope of −1 implying that pansion of the eigenvalues beyond just the leading order, ω ∼ 1, or, no exceptional behaviour. However, if the term including the coefficients of the terms at various orders. corresponding to√ the point (1, 0) can be made to vanish (which occurs for g = 2κ), we can expect to observe exceptional We present a detailed discussion in Supplementary Ma- behaviour. Right panel shows the Newton polygon for this terials [58]. case. Here, a slope of −1/3 implies ω ∼ 1/3, or the presence Application to PT -symmetric 4-site model– We next of a third order exceptional point. Here n(ω) and n() denote consider a 4-site PT -symmetric system as shown the exponents of ω and , respectively. schematically in Figure 2(a). We consider a general form consistent with the PT -symmetry. Due to parity symmetry, two hopping parameters p and q are sufficient to det H − ω = 0, write p(ω, ) in the form I describe the couplings between the four sites. The bal- P a (t , t , ...)ωmn. m,n mn 1 2 anced gain and loss for the outer and inner sites are given m n by δ and γ respectively. We will show that when such 2. For each term of the form amnω in the poly- 2 a system is perturbed, one can get an EP2, or an EP4 nomial, plot a point (m, n) in R . The smallest convex shape that contains all the points plotted is depending on the tuning of various parameters. As we called the Newton polygon. shall demonstrate below, the Newton polygon method el- egantly predicts the required tuning. The Hamiltonian 3. Select a segment of the Newton polygon such that for the 4-site system can be written as 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 . iδ p 0 0  p iγ q 0  H4 =   . (4) To illustrate our method, we consider a recently re-  0 q −iγ p  alized optical system, consisting of three coupled res- 0 0 p −iδ onators, that exhibits a higher order EP and an unprece- dented sensitivity to changes in the environment [49]. As the overall energy scale does not affect the be- The system can be described by a remarkably simple haviour, we shall set γ = 1 from hereon. non-Hermitian Hamiltonian If the system is perturbed, say by slightly varying one of the couplings by , the perturbed Hamiltonian reads

ig + κ 0   iδ p + 0 0  H() =  κ 0 κ  , (2) 0 κ −ig p + i q 0  H4() =   . (5)  0 q −i p  where g accounts for gain and loss, κ is the coupling 0 0 p −iδ between the resonators and is the external perturbation. The characteristic equation, p(ω, ) = 0, reads We present the Newton polygon of the characteristic polynomial in Figure 2(b). Observe that if the point at (0, 0) is absent, we would get an EP2. The part of the − ω3 + ω2 + (2κ2 − g2) ω + igω − κ2 = 0. (3) three-dimensional parameter space which shows a second order EP is thus the surface where the coeﬃcient of ω00 Figure 1 shows the Newton polygon for p(ω, ), where vanishes, which is shown in red in Figure 2(c). Now, if each point on the graph corresponds to a term in the the point (2, 0) was also absent, we would get an EP4. characteristic equation. The line-segment that contains The parameter space for which this vertex vanishes is all points on, above or to the right of it is shown in blue. shown in blue in Figure 2(c). The locus of EP4 is then This line has a slope of −1 implying that the lowest order the curve formed by the intersection of these two surface dependence of the eigenvalues on has the form ω ∼ . [dashed line in Figure 2(c)]. Note that this system cannot 3

FIG. 2. Tunable higher order exceptional points in the PT -symmetric model via Newton polygons. (a) Illustration of the 4-site PT -symmetric model which has the Hamiltonian H4. (b) The Newton polygon for H4(). The red and blue surfaces shown in (c) correspond to the regions in the parameter space where the red and blue points in the Newton polygon vanish. This predicts that one would get an EP2 if the system parameters fall on the red surface, while one would get an EP4 if they fall on the black curve (which depicts the intersection of the two surfaces). Numerical verification of these predictions are shown in middle and right panels for EP2 and EP4, respectively. (d) The numerically obtained band structure near the exceptional point EP2 (green) closely matches the predicted behaviour of 1/2 (red). Inset shows the plot on a logarithmic scale. (f) The numerical band structure near the exceptional point EP4 (green) closely matches the predicted behaviour of 1/4 (red). Panels (e) and (g) show the behaviour of the four bands as one encircles the exceptional points once (choosing the contour = 0.05 exp(iθ)). As expected, two of the bands swap for the middle panel√ (e) while the four bands undergo a cyclic permutation√ for the right panel (g). Parameter values for middle panel: δ = 2, p = 2, q = 2. Parameter values for right panel: δ = 3, p = 3, q = 2. give us an EP3 because of the PT -symmetry. Our New- the bands swap [Figure 2 (e)], while for the EP4, all four ton polygon framework, thus, enables the identification bands undergo a cyclic permutation [Figure 2 (g)]. and selection of higher order EPs in a straightforward We remark here that Zhang et al. have studied a model manner. for supersymmetric arrays showing an EP4√ [59]. Their We verify these predictions from our method in sev- model is a special case of ours with p = 3, δ = 3, and eral ways. First, we show that close to the EPs, our q = 2. bands scale exactly as ω ∼ 1/N , where N is the order The 4-site model we presented here can show tunable of the EP as determined from our approach [Figure 2(d) second and fourth order EPs. Analogously, we have de- and (f)]. The Newton polygon method also allows cal- vised a 5-site PT -symmetric model. Using the New- culating the coefficients of the Puiseux expansions. For ton polygon framework, we have shown that it can ex- q 2p(δ+p2) 1/2 example, we obtain ω = 2p2+q2−δ2−1 to first order hibit tunable third and fifth order EPs [58]. Such PT - for EP2, which has an excellent match to the numerical symmetric models have been experimentally realized in fit. Derivations of such analytical expressions are pre- magnetic multi-layers [60], waveguides [61, 62], topoelec- sented in the Supplementary Materials [58]. Second, we tric circuits [63], and photonic lattices [64–67]. The New- numerically show that if we encircle the EP, the bands ton polygon method can serve as a useful tool for tuning swap among themselves as expected. For an EP2, two of to EPs in these experimental platforms. 4

ig + κ 0  H(, λ) =  κ 0 κ  . (6) 0 κ −ig + λ √ We can set g = 2κ to obtain exceptional behaviour, as discussed earlier. If we pick any direction in the - λ plane making an angle φ with the -axis, then along that direction λ = 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 (for κ = 1)

− ω3 + (tan φ + 1) ω2 − tan φ ω2+ √ (7) i 2(tan φ − 1) ω − (tan φ + 1) = 0.

The corresponding Newton polygon is shown in Fig- ure 3(a). Notice that ω ∼ 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 ∼ 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. We note that a similar model along two special directions has been studied before [69], although a complete picture along all possible directions was lack- ing. Our method provides a unified way of obtaining the FIG. 3. Determination of anisotropic behaviour exceptional properties along all directions. around an exceptional point using Newton polygons. We again numerically verify the predictions from the For most directions in the -λ plane making an angle φ with Newton polygon method in multiple ways (see Figure 3). the -axis, the Newton polygon has the form in (i.a) implying We first verify our analytical expressions for the eigen- ω ∼ 1/3. However, the (0, 1) term can vanish for φ = −π/4 giving us the Newton polygon (ii.a) which gives ω ∼ 1/2 values by fitting them to the numerical band structure. instead. The Newton Polygon method thus predicts second Next, we study the variation of the phase rigidity, r, with order behaviour along φ = −π/4, but third order behaviour the perturbation. Physically, phase rigidity is a mea- along all other directions. Numerical verification of these pre- sure of the bi-orthogonality of the eigenfunctions, and dictions are shown in (b) and (c). Note that the numerical helps identify EPs [46, 68, 70]. We present further de- band structures along a chosen direction (in green) closely tails in Supplemental Materials [58]. Here, we find that match the predicted behaviours (in red) along that direction. the phase rigidity vanishes at the location of the EP at Panels (i.c) and (ii.c) show the behaviour of the phase rigid- 2/3 = 0 [Figure 3 (c)]. Importantly, we observe that the ity, r. The phase rigidity scales as for φ 6= −π/4 as is 2/3 characteristic for an EP3, while it scales as 1 for φ = −π/4 phase rigidity scales as for φ 6= −π/4, while it scales 1 confirming an EP2. For plots in the left panel, φ = π/6 was as for φ = −π/4, thus confirming our predictions of chosen as a representative direction. the anisotropic nature of the EP. Summary and Outlook– We put forward a new algebraic framework using Newton polygons for characteriz- Extension to higher number of variables– The Newton ing EPs. Using several examples, we illustrated how the polygon approach can be naturally extended to study ex- Newton polygon method enables prediction and selec- ceptional behaviour for higher number of variables. Re- tion of higher order EPs. We also proposed an extension markably, it has been recently shown that if a system at to higher number of variables and used it to reveal rich an EP can be perturbed in two different ways such that anisotropic behaviour around such non-Hermitian degen- H(, λ) = H0 + H1 + λH2, then it possible to observe eracies. different exceptional behaviours along different directions Looking ahead, our analytical approach could be use- in the -λ plane [51, 68]. The Newton polygon framework ful for tuning to EPs in different experimental platforms, can predict such anisotropic variations, as we show next. whether it be for enhanced performance of sensors or We consider the Hamiltonian given in Equation 2 and exploring unconventional phase transitions, especially as add a second perturbation, λ, to obtain the dimensionality and complexity of the non-Hermitian 5

Hamiltonians in question increases. Newton polygons Acknowledgments – We acknowledge Madhusudan play a natural role in homotopy theory, braid groups, Manjunath (Department of Mathematics, Indian Insti- knot theory and algebraic number theory [54, 56]. Our tute of Technology, Bombay) and Chinmaya Kausik work lays the foundation for exploring such connections (Department of Mathematics, University of Michigan) in the context of non-Hermitian physics. With the re- for illuminating discussions. R.J. thanks the Kishore cent growing interest in knotted and linked exceptional Vaigyanik Protsahan Yojana (KVPY) for a fellowship. nodal systems [71–77], it would be worthwhile to ﬁnd a A.B. is supported by the Prime Minister’s Research Newton polygon based characterization of such systems. Fellowship (PMRF). A.N. acknowledges support from In conclusion, we hope that our results inspire further the startup grant of the Indian Institute of Science exploration of EPs and their applications. (SG/MHRD-19-0001) and DST-SERB (project number SRG/2020/000153).

SUPPLEMENTARY MATERIAL

In this Supplement, we elaborate on the following aspects: (i) the connection between the order of exceptional points and the Puiseux series, (ii) relation of the Riemann sheet structure to holonomy, (iii) the method to calculate coeﬃcients and higher order terms using Newton polygons, (iv) further details about the 4-site model, (v) 5-site PT symmetric model to generate and tune third and ﬁfth order exceptional points, (vi) details of the bivariable model, and (vii) extension of Newton Polygon method to higher number of variables.

ORDER OF EXCEPTIONAL POINTS AND PUISEUX SERIES

Let us start by examining the behaviour of the eigenvalues and eigenvectors of a Hamiltonian H (of dimension m) in the vicinity of an exceptional point (EP), when it is subjected to a small perturbation H1 such that H() = H0 +H1. It turns out that the eigenvalues and eigenvectors of H() are not always holomorphic functions of [3, 78], which means that they cannot be expressed in a simple power series in . Next, consider the characteristic equation corresponding to the perturbed Hamiltonian

p(ω, ) = det H0() − ωIm = 0. (S1)

Pm n If we write it as p(ω, ) = n=1 an()ω , we see that the characteristic equation is an algebraic equation in ω (of degree m) such that the coeﬃcients are holomorphic functions in . This tells us that the roots are (branches of) analytic functions of with only algebraic singularities. Note that since holomorphic functions comprise of only a small subset of the larger set of analytic function, our eigenvalues need not be holomorphic. It follows that there are m distinct eigenvalues (roots) for all values of other than at a ﬁnite number of points 0, which are called the EPs. In the simpler case, if ∈ D where D is a region excluding any EPs, then the eigenvalues,

ω1(), ω2(), ..., ωm(), (S2)

are just m holomorphic functions in D. In the neighbourhood of an EP, say at 0 = 0, the behaviour is more rich. If we consider D to be a small disk near, but excluding = 0, then the eigenvalues can again be expressed as m holomorphic functions similar to Equation S2. Now if D is rotated continuously around = 0, these m functions can be continued analytically. When D has been brought to its initial position after one rotation, they will have undergone a permutation among themselves. As a result, these functions may therefore be grouped in the manner

[ ω1(), ω2(), ..., ωp()], [ ωp+1(), ωp+2(), ..., ωp+q()], ··· (S3)

such that each group undergoes a cyclic permutation upon one rotation of D. From this it is now clear that the elements of a group (with p elements) constitute a branch of an analytic function with a branch point at = 0 (if p ≥ 2) [3], and we have a Puiseux series of the form

s 1/p 2s 2/p ωs() − ω = α1ζ + α2ζ + ..., s = 0, 1, ..., p (S4) 6

where ζ = exp i2π/p. A similar expansion can also be written for the eigenvectors. Since |ωs() − ω| is in general 1 of the order || p for small , the rate of change of the eigenvalues at an EP is inﬁnitely large compared to the change in the Hamiltonian itself. This is, in fact, what results in interesting physical properties, such as increased sensitivity, around EPs.

CALCULATION OF COEFFICIENTS AND HIGHER ORDER TERMS USING NEWTON POLYGONS

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. Any solution of the characteristic equation p(ω, ) = 0 has the form

γ1 γ1+γ2 γ1+γ2+γ3 ω = c1 + c2 + c3 + .... (S5)

The steps for computing γ1 were described in the main text. Once γ1 has been determined, we can write ω = γ1 γ2 γ2+γ3 (c1 + ω1) where ω1 = c2 + c3 + .... The steps to evaluate the other unknowns in the expansion are as follows.

γ1 1. Collect the lowest order terms in in the polynomial p( (c1+ω1), ). They must cancel each other as p(ω, ) = 0. The ﬁrst coeﬃcient c1 can be extracted from this requirement.

−β γ1 2. To get the next order term, ﬁnd the polynomial p1(ω, ) = p( (c1 + ω1), ) where β is the y-intercept of the line segment whose slope gave us −γ1.

3. Next, calculate the Newton polygon for p1(ω, ) and repeat the steps to ﬁnd γ2 and c2, and so on.

As an illustration, let us evaluate the coeﬃcients of Puiseux expansion for the√ experimentally realizable optical system, with the Hamiltonian H(), which was discussed in the main text. At g = 2κ we get a third order EP with γ1 = 1/3. Then

1/3 3 2 p( (c1 + ω1), ) = −(c1 − κ ) + ··· . (S6) 3 2 2/3 To ensure p(ω, ) = 0, we must have c1 − κ = 0 or c1 = κ . Thus, to lowest order around the EP3, we have ω = κ2/31/3 + ··· . This matches the expansion which has been experimentally veriﬁed in the literature [49]. We can follow similar steps to obtain the coeﬃcients for the other models discussed in the main text. We list these next.

• For the 4-site model H4() with EP2: If parameter values fall on the EP2 surface, we calculate using the procedure outlined above 2p(p2 + δ) c2 = . (S7) 1 2p2 + q2 − δ2 − 1 √ For the plots in the main text, one representative point was chosen from the surface → δ = 2, p = 2, q = 2. q √ 8 2 1/2 At this point, we thus have c1 = 3 = 1.94. Around the EP2, ω = 1.94 perfectly ﬁts the numerical band structure [refer Figure 2(d) of the main text].

4 2 • For the 4-site model H4() with EP4: If parameter values fall on the EP4 locus, calculation√ gives c1 = 2p(p +δ). Again, one representative√ point from the curve was chosen for the plots → δ = 3, p = 3, q = 2. At this point, 1/4 we have c1 = (12 3) = 2.13, which is one of the ﬁt parameters used to match the numerical band structure [refer Figure 2(f) of the main text].

1/4 • For the bivariable model H(, λ): Along φ = −π/4, we get c1 = 2 = 1.19, which exactly matches the numerical band structure [refer Figure 3(b) of the main text].

3 • For the bivariable model H(, λ): Along all directions with φ 6= −√π/4, we instead get c1 = 1 + tan φ. The 1/3 representative plots were shown for θ = π/6, wherein c1 = (1 + 1/ 3) = 1.164, which agrees with the ﬁt parameters used in Figure 3(b) of the main text. 7

RELATION TO HOLONOMY

In this section, we relate the evolution of the eigenmodes to holonomy, which we have used to characterize the topology of EPs in the main text. As discussed above (see Eq. S3), the eigenvalues and eigenvectors undergo a cyclic permutation as one encircles an EP. This property is captured by the holonomy [79]. In particular, one can deﬁne a holonomy matrix which describes the evolution of eigenbands in the parameter space along a contour enclosing the EP [80]. To study the holonomy, we numerically trace out the evolution of the complex eigenmodes around a loop param- eterized by θ. We then continuously tune the parameter from θ to θ + 2π such that the loop encircles the EP. As presented in the main text, we found that that eigenvalues cycle among each other as expected and show a complete revolution after p-th cycle [see Figure 2 (d) and (e) of the main text]. We note that the number of turns (p cycle) depends on fractional exponent in the Puiseux series (see Eq. S4). The number of eigenbands undergoing such a cyclic permutation then indicates the order of the EP. Interestingly, two equivalent loops enclosing the same EP are connected by homotopy, which reﬂects the global features of EPs [81]. This idea is related to parallel transport [78, 82] in algebraic geometry. Consequently, one obtains a quantized geometric phase of 2π/p while encircling a p-th order EP [83, 84].

DIAGNOSING EXCEPTIONAL POINTS USING PHASE RIGIDITY

In this section, we elaborate on the diagnosis and identiﬁcation of EPs in the parameter space using phase rigidity. The phase rigidity, rα, is deﬁned as [6, 85]

hψα|φαi rα = , (S8) hφα|φαi

where ψα and φα are the normalized biorthogonal left and right eigenvectors of a state α. The phase rigidity quantitatively measures the eigenfunctions’ bi-orthogonality. At an EP the states coalesce and thus |rα| vanishes. The phase rigidity is also experimentally measurable. It enables deﬁning a critical exponent around an EP in the parameter space [86, 87]. For example, around an EP3, the scaling exponents are given by (N − 1)/N and (N − 1)/2, where N = 3 is the order of the EP. We numerically evaluate phase rigidity for the bivariable model with directional anisotropy, which shows that the phase rigidity scales as 2/3 and 1 for φ 6= −π/4 and φ = −π/4, respectively [see Fig. 3 (c) in the main text].

DETAILS OF THE 4-SITE MODEL

In this section we provide more details about the 4-site model discussed in the main text. For a small change in the couplings H4, the characteristic equation reads

ω4 − ω22 − 2pω2 + (1 + δ2 − 2p2 − q2)ω2 − I(1 + δ)(2 + 2p)ω + (δ + p2)(2 + 2p) + (p4 + δ2 + 2δp2 − δ2q2) = 0. (S9) Thus, we get an EP2 when

p4 + δ2 + 2δp2 − δ2q2 = 0, (S10) while we get EP4 when additionally the parameters also satisfy

1 + δ2 − 2p2 − q2 = 0. (S11)

EP3 AND EP5 IN A 5-SITE PT -SYMMETRIC MODEL

Next, we present a 5-site PT-symmetric model to illustrate our Newton polygon method for tuning of higher order EPs. We consider a general form consistent with the PT -symmetry [refer Figure S1(a)]. Due to parity symmetry, 8

FIG. S1. Tunable higher order exceptional points in a 5-site PT -symmetric model via Newton polygons (a) Illustration of the 5-site PT -symmetric model which has the Hamiltonian H5. (b) The Newton polygon for H5(). The red and blue surfaces shown in (c) correspond to the regions in the parameter space where the (1,0) and (3,0) points in the Newton polygon vanish. This predicts that one would get an EP3 if the system parameters fall on the blue surface, while one would get an EP5 if they fall on the black curve (which depicts the intersection of the two surfaces). Numerical verification of these predictions are shown in middle and right panels for EP3 and EP5, respectively. (d) The numerically obtained band structure near the exceptional point EP3 (green) closely matches the predicted behaviour of 1/3 (red). (f) The numerical band structure near the exceptional point EP5 (green) closely matches the predicted behaviour of 1/5 (red). Panels (e) and (g) show the behaviour of the five bands as one encircles the EPs once (choosing the contour = 0.05 exp(iθ)). As expected, three of the bands permute for the middle panel (e) while all five bands undergo a cyclic permutation for the right panel (g). Parameter values for middle panel: a = 3, p = 1, q = 1. Parameter values for right panel: a = 2, p = 1, q = p3/2.

two hopping parameters p and q are sufficient to describe the couplings between the five sites. The balanced gain and loss for the outer and inner sites are given by a and γ respectively. We employ the Newton polygon method to find the tuning of parameters required order to obtain third and fifth order EPs. The Hamiltonian reads

ia p 0 0 0   p iγ q 0 0    H5 =  0 q 0 q 0  . (S12)    0 0 q −iγ p  0 0 0 p −ia

As the overall energy scale does not aﬀect the behaviour, we shall set γ = 1 from hereon. We can slightly vary one of the couplings in order to perturb the system and then observe the frequency response to the perturbation, . The perturbed Hamiltonian reads 9

 Ia p + 0 0 0  p + I q 0 0    H5() =  0 q 0 q 0  . (S13)    0 0 q −I p  0 0 0 p −Ia

For a small change in the couplings as for H5(), the characteristic equation reads:

ω5 − ω32 − 2pω3 + ω3(1 + a2 − 2p2 − 2q2) − I(1 + a)ω2(2 + 2p) + (a + p2 + q2)ω2 + 2(ap + p3 + pq2)ω (S14) +(a2 + 2ap2 + p4 − 2a2q2 + +2p2q2)ω + Iaq2(2 + 2p) = 0.

Thus, we get EP3 when

a2 + 2ap2 + p4 − 2a2q2 + 2p2q2 = 0, (S15)

while we get EP5 when additionally the parameters also satisfy

1 + a2 − 2p2 − 2q2 = 0. (S16)

The Newton polygon of its characteristic polynomial is shown in Figure S1. The part of the three dimensional parameter space which shows a third order EP is the surface where the coefficient of ω10 vanishes, which is shown in red in Figure S1(c). Additionally, if the point (3, 0) was also absent, we would get an EP5. The parameters for which this vertex vanishes has been shown in blue in the same figure. The locus of EP5 is just the curve formed by the intersection of these two surface. As before, we verify these predictions in several ways [see Figures S1(d) -(g)]. The topological nature of the EPs are reflected in the homotopy properties of EPs when they are encircled. Thus Newton polygon method thus provides an elegant way of manipulating EPs of various orders by tuning system parameters. Note that our model is a generalised version of the 5-site supersymmetric model in Ref. [59].

DETAILS ABOUT THE BIVARIABLE MODEL

We next present additional details on the bivariable model discussed in the main text. The characteristic equation for H(, λ) reads

√ − ω3 + (tan(θ) + 1) ω2 + i 2κ(tan(θ) − 1) ω − tan(θ) 2ω − κ2(tan(θ) + 1) = 0. (S17)

Note that our predictions for an anisotropic EP do not depend on the value of κ. Thus, in the main text we choose κ = 1 for simplicity.

EXTENSION TO MORE THAN TWO VARIABLES

Our approach can be generalized to study multiple perturbations to reach an EP – we summarize the approach next. Consider a Hamiltonian at an EP which can be perturbed in n diﬀerent ways such that

H(1, 2, 3, ··· ) = H0 + 1H1 + 2H2 + 3H3 + ··· + nHn. (S18)

Diﬀerent exceptional behaviours could then be possible along diﬀerent direction in this n-dimensional space. Anal- ogous to the bivariable case discussed in the main text, the Newton polygon method can be employed to predict exceptional behaviour along all direction in one go. We can shift to spherical coordinates as follows 10

1 = cos φ1

2 = sin φ1 cos φ2 3 = sin φ1 sin φ2 cos φ3 (S19) . .

n = sin φ1 sin φ2 ··· cos φn−1.

P p q The characteristic equation then can be written in the form p(ω, ) = p,q apq(φ1, φ2, ··· , φn−1)ω and the Newton polygon method can be used. Some of the coefficients apq can vanish along specific directions in the n- dimensional space, leading to varying exceptional behaviours along different directions. The specific case of two variables was presented in the main text.

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