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Characterizing and Tuning Exceptional Points Using Newton Polygons

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Characterizing and Tuning Exceptional Points Using Newton Polygons 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 excep- tional 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 demon- strate 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 !() = 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 ! ∼ 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(!; ) = 2 p Notice, however, that if we set g = 2κ, the coefficient of ! 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 ! ∼ 1=3, or, we have a third order EP (EP3). The Newton polygon methodp 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, ! ∼ 1, or, no exceptional behaviour. However, if the term including the coefficients of the terms at various orders. corresponding top 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 ! ∼ 1=3, or the presence Application to PT -symmetric 4-site model{ We next of a third order exceptional point. Here n(!) and n() denote consider a 4-site PT -symmetric system as shown the exponents of ! and , respectively. schematically in Figure 2(a). We consider a general form consistent with the PT -symmetry. Due to parity sym- metry, two hopping parameters p and q are sufficient to det H − ! = 0, write p(!; ) in the form I describe the couplings between the four sites. The bal- P a (t ; t ; :::)!mn. 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 amn! 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 2 3 gives us the lowest order dependence of ! on . iδ p 0 0 6 p iγ q 0 7 H4 = 6 7 : (4) To illustrate our method, we consider a recently re- 4 0 q −iγ p 5 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 2ig + κ 0 3 2 iδ p + 0 0 3 H() = 4 κ 0 κ 5 ; (2) 0 κ −ig 6p + i q 0 7 H4() = 6 7 : (5) 4 0 q −i p 5 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(!; ) = 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 − !3 + !2 + (2κ2 − g2) ! + ig! − κ2 = 0: (3) three-dimensional parameter space which shows a second order EP is thus the surface where the coefficient of !00 Figure 1 shows the Newton polygon for p(!; ), 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.
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