Winding around Non-Hermitian Singularities: General Theory and Topological Features
Qi Zhong, Mercedeh Khajavikhan, Demetrios Christodoulides, Ramy El-Ganainy
WWinding around Non-Hermitian Singularities: General Theoryand Topological Features
Q. Zhong , M. Khajavikhan , D.N. Christodoulides , and R. El-Ganainy , ∗ Department of Physics, Michigan TechnologicalUniversity, Houghton, Michigan, 49931, USA College of Optics & Photonics-CREOL,University of Central Florida, Orlando, Florida, 32816, USA and Center for Quantum Phenomena,Michigan Technological University, Houghton, Michigan, 49931, USA
Abstract
Non-Hermitian singularities are ubiquitous in non-conservative open systems. These singularitiesare often points of measure zero in the eigenspectrum of the system which make them difficult toaccess without careful engineering. Despite that, they can remotely induce observable effects whensome of the system’s parameters are varied along closed trajectories in the parameter space. Todate, a general formalism for describing this process beyond simple cases is still lacking. Here,we bridge this gap and develop a general approach for treating this problem by utilizing thepower of permutation operators and representation theory. This in turn allows us to reveal thefollowing surprising result which contradicts the common belief in the field: loops that enclose thesame singularities starting from the same initial point and traveling in the same direction, do notnecessarily share the same end outcome. Interestingly, we find that this equivalence can be formallyestablished only by invoking the topological notion of homotopy. Our findings are general with farreaching implications in various fields ranging from photonics and atomic physics to microwavesand acoustics. ∗ [email protected] a r X i v : . [ qu a n t - ph ] M a y ntroduction Non-Hermitian singularities arise in multivalued complex functions [1, 2] as points wherethe Taylor series expansion fails. In the context of non-Hermitian Hamiltonians, these points,commonly referred to as exceptional points (EPs) feature special degeneracies where two ormore eigenvalues along with their associated eigenfunctions become identical [3, 4]. An EPof order N (EPN) is formed by N coalescing eigenstates. Recently, the exotic features ofEPs have been subject of intense studies [5–8] with various potential applications in laserscience [9–12] , optical sensing [13–15], photon transport engineering [16, 17] and nonlinearoptics [18, 19] just to mention few examples. For recent reviews, see Refs [20, 21].Very often, EPs are points of measure zero in the eigenspectra of non-Hermitian Hamil-tonians which makes them very difficult to access, even with careful engineering. Yet,their effect can be still felt globally. Particularly, an intriguing aspect of non-Hermitiansystems is the eigenstate exchange along loops that trace closed trajectories around EPs.In this regard, stroboscopic encircling of EP2 has been studied theoretically [22, 23] anddemonstrated experimentally in various platforms such as microwave resonators [24, 25] andexciton-polariton setups [26]. Complementary to these efforts, the dynamic encircling ofEPs was shown to violate the standard adiabatic approximation [27–30]. These predictionswere recently confirmed experimentally by using microwave waveguides platforms [31] andoptomechanical systems[32].Notably, the aforementioned studies focused only systems having only one EP of or-der two. Richer scenarios involving multiple and/or higher order EPs have been largelyneglected, with rare exceptions that treated special systems (admitting simple analyticalsolutions) on a case by case basis [33, 34]. This gap in the literature is probably due tothe complexity of the general problem and its perceived experimental irrelevance. However,recent progress in experimental activities that explore the physics of non-Hermitian systemsare quickly changing the research landscape, and controlled experiments that probe morecomplicated structures with multiple EPs will be soon within reach. These developmentsbeg for a general approach that can provide a deeper theoretical insight into these complexsystems.In this work, we bridge this gap by introducing a general formalism for treating the eigen-state exchange along arbitrary loops enclosing multiple EPs. More specifically, our approach2tilizes the power of group theory together with group representations to decompose the fi-nal action of any loop into more elementary exchange processes across the relevant branchcuts (BCs). This formalism simplifies the analysis significantly, which in turn allows us togain an insight into the problem at hand and unravel a number of intriguing results: (1)Trajectories that encircle the same EPs starting from the same initial point and having thesame direction do not necessary lead to an identical exchange between the eigenstates; (2)Establishing such equivalence between the loops (i.e. same eigenstate exchange) is guaran-teed only by invoking the topological notion of homotopy. As a bonus, our approach canalso paint a qualitative picture of the dynamical properties of the system. General Formalism for Encircling Multiple Exceptional Points
Before we start our analysis, we first describe the simple case of EP2. These are specialpoints associated with the multivalued square root function in the complex plane. TheRiemann surface of this function is shown in the top panel of Fig. 1(a). Clearly, as twoparameters are varied in the complex plane to trace a closed loop, the initial point on thesurface ends up on a different sheet. This process can be also viewed by considering theprojection on the complex plane after adding a BC (lower panel). As we mentioned before,this simple scenario has been studied in the literature in both the stroboscopic and dynamicalcases. Consider however what happens in more complex situations where there are morethan one EP. For instance, Fig. 1(b) depicts a case with three EPs. One can immediatelysee that this scenario exhibits an additional complexity that is absent from the previouscase. Namely, there are now different ways for encircling the same EPs (as shown by thesolid and dashed loops in the figure). This in turn raises the question as whether theseloops lead to the same results or not. These are the type of questions that we would like toaddress in this work. As we will see, in resolving these questions, our analysis also revealseveral peculiar scenarios.
Permutation operators and the exchange of eigenstates—
Consider an n -dimensional non-Hermitian discrete Hamilton. The Riemann surface associated with the real (or imaginary)part of its eigenvalues will consist of n sheets corresponding to different solution branches.We will label these n branches as b , b , ..., b n . In the complex plane, these branchesare separated by BCs. Thus, an initial point on any trajectory in the complex plane will3 P s s (b) =? (a) EP EP EP FIG. 1.
Different ways of encircling multiple EPs. (a) Illustration of Riemann surface associ-ated with the square root function associated with an archetypal 2 × s will map it onto s and vice versa. In the complex plane (lower panel), this is represented byadding a BC. (b) A scenario that exhibit three EPs. In this case, loops can encircle EPs in differentways as illustrated by the two loops (solid/dashed lines) that enclose EP , starting from the samepoint (gray dot). correspond to n initial eigenstates, which we will label as s , s , ..., s n . The eigenvalue foreach state s i will be denoted by λ i . As the encircling parameters are varied, the eigenstateswill move along the trajectory, crossing from one branch to another across the BCs. Thecrucial point here is that, we will always fix the initial subscript of the state as it changes.We now describe the initial configuration on the trajectory by the mapping: C = ˜ s ˜ b , (1)where ˜ s = ( s , s , ..., s n ) and ˜ b = ( b , b , ..., b n ) are two ordered sets. In our notation, C maps (or associates) every element of ˜ s to the corresponding element in ˜ b . Note that wecan change the orders of the elements in both ˜ s and ˜ b identically without changing C . Inother words, we have several different ways for the same configuration. As the loop crossesBCs, the exchange between the eigenstates will result in new configurations which, again,can be described in different ways. Two particular choices are interesting here. In the firstone, we always fix ˜ s and allow the elements of ˜ b to shuffle, effectively creating a new ˜ b . Inthe second, we just do the converse. We will call these two equivalent notations the s- and4 b ( ) π s b sb ( ) π sb ( ()( ) ) ..π π π. K s b ( ) First branch c π ut: ... ( ) ( ) ( ) π π π.... K sb
01 1 101 2 ... ( ) π ( ) π K s-frameb-frame (b) (a) s b s-frameb-frame FIG. 2.
Different permutation frames. (a) A simple illustration of the two different framesused for representing the same configuration. (b) The mathematical formulation of the concept in(a) in terms of permutation mappings as discussed in details in the text. b-frames, respectively. This is explained by the cartoon picture in Fig. 2(a).The first step in our analysis is to choose a scheme for sorting the eigenstates and locatingthe BCs accordingly. We will discuss the details of the sorting later but for now we assumethat we have a certain number of BCs and we label each one with a unique integer value(positive for a crossing in certain direction and negative for reverse crossing). Next wedetermine how the eigenstates are redistributed across an infinitesimal trajectory across eachBC (see discussion later on sorting schemes). For every loop, we then create an ordered list σ that contains the number of the crossed BCs in the order they are crossed by the loop.In other words, the element σ ( j ) is the number of the j -th crossed BC. Clearly the set σ will be in general different from loop to another and even can be different for the same loopdepending on the initial point or the encircling direction. Then the final configuration inboth the s- and b-frames is given by: C s σ = ˜ s ˜ b σ ≡ ˜ s P [ (cid:81) π σ ( j ) ] ◦ ˜ b , C b σ = ˜ s σ ˜ b ≡ {P [ (cid:81) π σ ( j ) ] } − ◦ ˜ s ˜ b , (2)5here P denotes the ordering operator which arranges the multiplication of the permutationoperators π σ ( j ) from right to left according to the order of crossing the BCs; and the productruns across the index j . For example, if σ = (3 , , P [ (cid:81) π σ ( j ) ] = π σ (3) ◦ π σ (2) ◦ π σ (1) = π ◦ π ◦ π . The permutation operator π k associated with BC k is the standardpermutation mapping that, which when applied to a set will shuffles the order of its elements[35]. Here it is used to describe how the eigenstates are redistributed when a trajectorycrosses a BC. For instance, if the permutation exchange the order of the first two elementsof ˜ b across a BC k , then π k ( b , ) = b , , and π k ( b i ) = b i for i >
2. Figure 2(b) illustratesthe relation between the s- and b-frame calculations as expressed by Eqs. (2).
From permutations to matrices—
The above discussion can be directly mapped into linearalgebra by using representation theory. To do so, we define the vectors (cid:126)s = ( s , s , ..., s n ) T and (cid:126)b = ( b , b , ..., b n ) T . In the s-frame, we will fix (cid:126)s and allow (cid:126)b to vary in order torepresent the change in configuration. In the b-frame, we just do the opposite. For instance,if after crossing a BC, eigenstate 1 moves to branch n , eigenstate 2 moves to branch 1and eigenstate n moves to branch 2, this will be expressed as (cid:126)b = ( b n , b , ..., b ) T in the s-frame; and (cid:126)s = ( s , s n , ...s ) T in the b-frame. After a loop completes its full cycle, the finalvector is then compared with the initial one to determine the exchange relations betweenthe eigenstates. For instance, if the above vector was the final result, the exchange relationswill be: { s , s , ..., s n } → { s n , s , ..., s } , which means that after the evolution s became s n , s became s and s n became s .We can now express the action of the permutation operators π k by the matrices P π k whoseelements are obtained according to the rule P π k ( m, l ) = 1 if b l = π k ( b m ), and 0 otherwise[36]. In the s- & b-frames, the redistribution of the eigenstates across the branches in Eq.(2) can be then described by: (cid:126)b σ = {P [ (cid:89) M σ ( j ) ] } − (cid:126)b ,(cid:126)s σ = P [ (cid:89) M σ ( j ) ] (cid:126)s , (3)where M k = P − π k . In arriving at the above equation, we have used standard results fromgroup theory: P π ◦ π = P π P π and P π − = P − π .In the rest of this manuscript, we adopted the b-frame with matrices M . This approachoffers a clear advantage: the order of the matrices acting on the state vectors (cid:126)s is consistent6ith the order of crossing the BCs. As we will see shortly, this will allow us to developthe topological features of the equivalent loops in a straightforward manner. Finally, wenote that if crossing a BC from one direction to another is associated with a matrix M , thereverse crossing will be described by M − . In some cases (such as with EP2), we can have M − = M but this is not the general case. Sorting of the eigenstates—
The discussion so far focused on developing the general formal-ism by assuming that the eigenstates of the system are somehow classified according to acertain criterion. This is equivalent to say that we divide the associated Riemann surfaceinto different sheets, each harboring a solution branch. Of course, one can pick any suchcriterion to classify the solutions. In previous studies that involved one EP of order twoor three, the eigenstates were classified based on the analytical solution of the associatedcharacteristic polynomial. This however has two drawbacks: (1) It generates relatively com-plex branches on the Riemann sheet; (2) It cannot be applied for discrete Hamiltonianshaving dimensions larger than four since analytical solutions do not exist for polynomials oforder five or larger. Thus our analysis above is useful only if one can find a sorting schemethat circumvents the above problems. Interestingly, such a sorting scheme is easy to find.Particularly, we can sort the eigenstates based on the ascending (or descending) order ofthe real or imaginary parts of their eigenvalues. This scheme can be easily applied to anysystem of arbitrarily high dimensions. Moreover, it lends itself to straightforward numericalimplementations. To compute the a permutation operator π k and its associated matrix M k across a BC k , one choses an infinitesimal trajectory that crosses the BC and calculate howthe eigenvalues evolve along this trajectory, comparing their order before and after crossingthe BC. That will immediately provide information about the permutations. We illustratethis using concrete example in the Methods. Equivalent Loops and Homotopy
In this section, we employ the predictive power of our formalism to address the followingquestion: are there any global features that characterize the equivalence between differentloops regardless of their geometric details? In answering this question, we will first focus onthe stroboscopic case and later discuss the implication for the dynamical behavior.Here, two loops are called equivalent if they lead to identical static eigenstates exchange.7t is generally believed that two similar loops starting at the same point and encircling thesame EPs in the same direction are equivalent. Surprisingly, we will show below that thiscommon belief is wrong.In general two loops will be equivalent if they have the same matrix product in Eq. (3).This can occur for two unrelated loops which we will call accidental equivalence. However,We are particularly interested in establishing the conditions that guarantee this equivalence.To do so, we invoke the notion of homotopy between loops. In topology, two simple paths,having the same fixed endpoints in a space S , are called homotopic if they can be continu-ously deformed into each other [37]. Here the word “simple” means injective, that is, eachpath does not intersect itself. If the two endpoints of a path are identical, this path is aloop with the identical endpoint as a basepoint. The space S here will be a two dimensionalpunctured parameter space (for example, the space spanned by Re[ κ ]–Im[ κ ] in the examplesdiscussed in the Methods) after removing all the EPs. Based on these definitions, we cannow state the main results of this section: (a) Homotopy is a sufficient condition forequivalence between loops; (b) Loops that are connected by free homotopy (con-tinuous deformation between loops without any fixed points) can be equivalentfor some starting points and inequivalent for others .In order to validate this statement, we consider a generic Hamiltonian having a number ofEPs and, without any loss of generality, we focus only on a subset of the spectrum as shownin Fig. 3. The axes on the figures represent any two parameters of the Hamiltonian. Wedefine the space S to be the two dimensional parameter space excluding the EPs. Figure 3(a)depicts a loop a (cid:13) that encircles two EPs starting from point z in the counterclockwise(CCW)direction. Consequently, its final permutation matrix is given by M p M o . Consider now whathappens when loop a (cid:13) is deformed continuously to a new loop. Here different scenarios canarise: (1) The deformation can take place only by crossing additional EP any number oftimes. This case is shown in Fig. 3(b), where it is clear that the new matrix product ofloop b (cid:13) ( M p M r M o M − r ) is in general different than the initial one. In this case, the twoloops are not equivalent (unless accidental equivalence takes place). (2) The deformation canoccur without changing the number or order of the crossed BCs, in which case the loops areequivalent. (3) The deformation can change the number of the crossed BCs in pairs traversedconsecutively back and forth as shown in Fig. 3(c). Here the two loops a (cid:13) and c (cid:13) are alsoequivalent because the matrix product is still the same: M p M − q M q M o = M p M o . (4) The8 M q M EP (c) z o M r M A p M q M EP (d) z o M r M p M q M EP (b) z o M r M p M q M EP (e) z o M r M p M q M EP (a) z o M r M z a b c d e FIG. 3.
Homotopy between loops.
Illustration of equivalence between homotopic loops in theparameter space of a generic Hamiltonian. (a) Loop a (cid:13) encloses two EPs associated with matrices M o and M p . (b) Loop b (cid:13) encloses the same two EPs yet it cannot be deformed into a (cid:13) withoutcrossing EP associated with M r . Consequently it has different matrix product (assuming notaccidental equivalence). On the other hand, loops c (cid:13) and d (cid:13) in (c) and (d) can be deformed intoa (cid:13) without crossing any EP. As a result, they are equivalent (have the same matrix product) asshown in the text. (d) Presents a peculiar case of free homotopy. Loop e (cid:13) is homotopic with a (cid:13) forthe starting point z but not for z (cid:48) . As a result, the two loops are equivalent for the former pointbut not for the latter. The discussion here is very generic and can be extended easily to any otherconfiguration of EPs and BCs. deformation can occur without crossing any EP but it changes the number of the crossed BCsin pairs traversed back and forth but not consecutively as shown in Fig. 3(d). In this case, thefinal matrix product is given by M p M − q M o M q . It is not immediately clear if this productis equivalent to M p M o . However, since the intersection point of the BCs (point A ) is notan EP, then by definition, encircling point A with a loop that does not enclose any EP mustgive the identity operator. In terms of matrices, this translates into M o M q M − o M − q = I ,or [ M o , M q ] = 0. Consequently, M p M − q M o M q = M p M o M − q M q = M p M o , i.e. loops d (cid:13) and a (cid:13) are equivalent. (5) Finally we can also have a loop similar to e (cid:13) as shown in Fig.9(e). This probably the most intriguing situation. For a starting point at κ , both loops a (cid:13) and e (cid:13) have the same matrix product M p M o which is consistent with the fact that they canbe deformed into one another without crossing any EP. On the other hand, for a differentstarting point such as z (cid:48) , the matrix product of loop e (cid:13) is given by M − r M o M p M r , i.e.different than that of loop a (cid:13) , which is given by M o M p . Note that for this starting point,the two loops cannot be deformed into each other without crossing any EP. In topology,continuous deformation that do not involve fixed points are called free homotopy. Thiscompletes our argument.The above discussion focused only on the stroboscopic case. However, as we will showin the explicit example presented in Methods, homotopy is also relevant to the dynamicalencircling of EPs. Particularly, our numerical calculations show that homotopic loops tend tohave the same outcome, despite the failure of the adiabatic perturbation theory. Intuitively,this interesting result can be roughly understood by noting that homotopic loops explorevery similar landscape in the complex domain. However a deeper understanding of thisbehavior requires further investigation. Conclusion
In conclusion, we have introduced a general formalism based on permutation groups andrepresentation theory for describing the stroboscopic encircling of multiple EPs. By usingthis tool, we uncovered the following counterintuitive results: trajectories that enclose thesame EPs starting from the same initial parameters and traveling in the same direction,do not necessarily result in identical exchange between the states. Instead, we have shownthat this equivalence can be established only between homotopic loops. Finally we have alsodiscussed the implication of these results for the dynamic encircling of EPs. Our work mayfind applications in various fields including the recent interesting work on the relationshipbetween exceptional points and topological edge states [39, 40].
Method
Illustrative Examples —
We now discuss a concrete numerical example to demonstrate theapplication of our formalism and confirm the various predictions presented in the main text.10 c) (a) (b) FIG. 4.
Numerical illustration of our approach. (a) The branches of Riemann surfaceof the real part of eigenvalues of H in Eq. (4) are distinguished by different colors accordingto the magnitude of Re[ λ ]. The EPs and their corresponding BCs (red lines) are illustrated in(b). Each BC is related with a permutation matrix M k in Eq. (5). One closed loop (blueline) encircles EP and EP CCW, starting from the gray points (solid or hollow) on the loop.Loops intersecting with BCs would lead to eigenvalues moving from one branch to another, andresult in the swap of eigenstates finally. (c) The stroboscopic evolution of complex eigenvaluesare plotted as a parametric function of κ when it moves along the loop CCW. The eigenvaluesat the starting point are labeled as gray points on their trajectory. The colors in the eigenvaluetrajectory represent which branch the eigenvalues are located at instantaneously. The joints of twocolors are where the κ crosses the BCs. The gray points (solid or hollow) and arrows illustratethe evolution of eigenvalues for starting from κ or κ (cid:48) , and therefore the evolution of eigenstatesis { s , s , s , s } → { s , s , s , s } and → { s , s , s , s } , respectively. Model—
Consider the following Hamiltonian: H = iγ J J κ κ J J − iγ , (4)where i is the imaginary unit, κ & J are coupling coefficients and γ is the non-Hermitianparameter. In what follows, the four eigenvalues of H will be investigated as a function ofthe complex κ by fixing J = γ = 1 (in certain physical platforms such as optics, it might bepractically easier to fix all the parameters and change γ , but that will not affect the mainconclusions of this work).Under these conditions, H has three pairs of EPs at κ = ± ± (cid:112) √ − ± i (cid:112) √ , EP (cid:48) , EP , EP (cid:48) , EP , EP (cid:48) , respectively. In each group, EP (cid:48) , , has same properties as EP , , . The Riemann surface and the distribution of the EPs in thecomplex κ plane are shown in Fig. 4(a) and (b), respectively.11s discussed in the main text, the first step in our approach is to identify a simple sortingmethod. Here we chose to sort the eigenvalues according to the magnitude of their real partsas shown in Fig. 4(a) where every branch is distinguished by a distinct color. From thisfigure, we can also identify the features of the EPs as follows: EP & EP (cid:48) are of secondorder and connect branches 2 and 3; EP & EP (cid:48) are of second order and connect branches 1and 2 on one hand, and branches 3 and 4 on the other; and finally EP & EP (cid:48) are of secondorder and connect branches 1 and 3 as well as branches 2 and 4 (In fact all the four surfacesof Re[ λ ] are connected at EP & EP (cid:48) and one has to look at the Im[ λ ] surface, which is notshown here, to infer the connectivity). Equivalently, the surface connectivity across the EPscan be characterized by using a two dimensional plane spanned by the real and imaginaryparts of κ along with the lines that separate the different solution branches (BCs) and theinformation on the transition between the different branches across each line. The lattercan be expressed in terms permutation matrices. Our sorting scheme of the eigenvaluesof H results in six BCs as shown in Fig. 4(b), but one can identify only three differentpermutation matrices: M = , M = , M = . (5)The correspondence between these matrices and the BCs is depicted in Fig. 4(b). It is notdifficult to see that the above matrices have the following properties: (1) M = M = M = I ; (2) [ M , M ] = [ M , M ] = 0. Stroboscopic encircling of EPs—
We now focus on the loop encircling both EP and EP ,as shown in Fig. 4(b). Clearly, the final exchange relation is determined by the product of M and M . Since [ M , M ] (cid:54) = 0, one has to be more specific about the starting point anddirection. For sake of illustration, let us choose counterclockwise direction, and κ or κ (cid:48) asthe starting point. In the first case, the loop intersects the BC associated with M first beforeit crosses that of M . As such, we have M M ( s , s , s , s ) T = ( s , s , s , s ) T , which in turnimplies the exchange { s , s , s , s } → { s , s , s , s } . Similarly, the starting point κ (cid:48) willgive M M ( s , s , s , s ) T = ( s , s , s , s ) T which leads to { s , s , s , s } → { s , s , s , s } .These exchange relations are also evident from the eigenvalues trajectories in Fig. 4(c).Another important consequence for the absence of commutation between M and M is12 (a) ② ① (b) ② (c) (d) ③ ④ ③ ④ (e) (f) FIG. 5.
Numerical example of homotopic relations between loops. (a) Depicts two similarloops 1 (cid:13) and 2 (cid:13) that encircle EP and EP . The two loops are non-homotopic for any starting pointincluding κ (which is considered for the example), since they cannot be deformed into one anotherwithout crossing EP . Their corresponding matrix product is M M M M and I , respectively.This is confirmed by their eigenvalue trajectories as shown in (b) and (c). (d) The two similarloops 3 (cid:13) and 4 (cid:13) are non-homotopic for the starting point κ but homotopic for κ (cid:48) . This is alsoreflected in the exchange relations of the eigenvalues as shown in (e) and (f). that M M M M (cid:54) = I . Hence encircling the loop in Fig. 4(b) twice still lead to nontrivialexchange. For example, the state s will evolve into s , s and s after encircling the looptwo, three and four times, respectively. Topological features of equivalent loops—
Here, we further elucidate on the topological fea-tures of equivalent loops in the context of the example given by Eq. (4). In this case, thespace ¯ S would be the space spanned by Re[ κ ] and Im[ κ ] after removing the points EP , , and EP (cid:48) , , . By inspecting the two loops 1 (cid:13) and 2 (cid:13) in Fig. 5(a), it is clear that they are nothomotopic for the starting point κ . Indeed the net permutation matrix associated with loop1 (cid:13) is M M M M , resulting in { s , s , s , s } → { s , s , s , s } . However, the permutationmatrix associated with loop 2 (cid:13) is M M M M = I . Consequently their exchange relationsare in general different as shown in Fig. 5(b) and (c).Next, we investigate a scenario that highlight the case of free homotopy. The two loops3 (cid:13) and 4 (cid:13) in Fig. 5(d) are similar (enclose the same EPs), yet they are not homotopic13or the starting point κ , i.e. they cannot be transformed into one another while keepingthe starting point fixed and without crossing EP . Thus the two loops are not necessarilyequivalent. Indeed the net redistribution matrix associated with loop 3 (cid:13) is M , resultingin { s , s , s , s } → { s , s , s , s } ; while for loop 4 (cid:13) , the permutation matrix is M M M ,which gives { s , s , s , s } → { s , s , s , s } . On the other hand, if we consider the sameloops 3 (cid:13) and 4 (cid:13) but with a different starting point κ (cid:48) , they are homotopic and the netpermutation matrix is M for both loops. Figures 5(e) and (f) confirm these results. Implications for dynamical evolution—
So far we have discussed the stroboscopic (or static)exchange between the eigenstates as a result of encircling EPs. Whereas this type of “evo-lution” can be in general accessed experimentally (see Refs. [24–26] for the case of secondorder EPs), recent theoretical and experimental efforts are painting a different picture forthe dynamic evolution, showing that the interplay between gain and loss will inevitablybreak adiabaticity [27–32]. It will be thus interesting to investigate whether the homotopybetween the loops (or its lack for that matter) has any impact on the dynamic evolution.Here we do not attempt to answer this question rigorously but will rather consider illustra-tive example. To do so, we focus again on the same loops 3 (cid:13) and 4 (cid:13) shown in Fig. 5(d), andwe perform numerical integration to compute the dynamical evolution around these loopsstarting from either κ or κ (cid:48) . As we discussed before, the loops are similar for both initialconditions but homotopic only for the later one. The computational details are presentedbelow but the main results confirm our conclusion in the main text: (1) When the two loopsare homotopic (i.e when the initial point on the the loop is κ (cid:48) ) any initial state s i , with i = 1 , , ,
4, will end up at state s regardless of the considered loop; (2) For similar butnon-homotopic loops (i.e when the initial point on the the loop is κ ), the initial states onloop 3 (cid:13) always evolve to s while those on loop 4 (cid:13) will evolve to s . These results suggestthat homotopy between the loops plays a much greater role than just describing the staticexchange between the states. Particularly, it might be also useful in classifying the dynamicevolution. We plan to investigate this interesting direction in future work. Numerical calculation of dynamic evolution—
Here we present the details of the numericalcalculations for the dynamic evolution. First, we choose the point κ = (0 . , − .
15) in Fig.14. Next, choose the loop 4 (cid:13) in Fig. 6(a) as:Re[ κ ( t )] = c + r cos( ωt ) , t ∈ [0 , T / c + r cos( ωt ) , t ∈ [ T / , T / c − r cos( ωt ) , t ∈ [ T / , T / c + r cos( ωt ) , t ∈ [3 T / , T ] , Im[ κ ( t )] = r sin( ωt ) , t ∈ [0 , T / r sin( ωt ) , t ∈ [ T / , T ] , (6)where c = 0 . c = 0 . c = 1. Note that the centers of the semicircles associatedwith loop 4 (cid:13) in Fig. 6(a) are given by the coordinates ( c , , , r = 0 .
45 and r = 0 .
15. The quantity T = 4 π/ | ω | is the time needed to complete one cycle.The exact position of point κ (cid:48) can be now chosen to be the intersection between the linepassing through κ and EP and the top large semi-circle, and κ (cid:48) ≈ (1 . , . (cid:13) in Fig. 6(b) was chosen to be a titled ellipse with the line connecting κ and κ (cid:48) as the major axis. This ellipse has semi-major axis a ≈ . c = a − .
002 and a rotating angle θ = arctan . Therefore the parametric function of loop3 (cid:13) is: Re[ κ ( t )] = c x + a cos( ωt ) cos θ − b sin( ωt ) sin θ, Im[ κ ( t )] = c y + a cos( ωt ) sin θ + b sin( ωt ) cos θ, (7)where b = √ a − c is the semi-minor axis of the ellipse and ( c x , c y ) = ( c + a sin θ, − r + a cos θ ) is the center of the ellipse.In all simulations, we chose the encircling speed ω = ± − (the positive/negative signsCCW/CW respectively). 15 c c c r ( , ) x y c c (a) (b) ④ ③ r r r FIG. 6.
Trajectories of dynamical evolutions.
The details of loops 3 (cid:13) and 4 (cid:13) used in thenumerical simulation of dynamic evolution of eigenstates in the main text are illustrated in (a) and(b). Loop 3 (cid:13) is a titled ellipse with the line connecting κ and κ (cid:48) as the major axis. Loop 4 (cid:13) is acombination of one large semi-circle and three identical small semi-circles. Data availability—
The data that support the findings of this study are available from thecorresponding author upon reasonable request.
Author Contribution
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