Topological theory of non-hermitian photonic systems

Transcription

1 Toological theory of non-hermitian hotonic systems Mário G. Silveirinha * () University of Lisbon Instituto Suerior Técnico and Instituto de Telecomunicações, Avenida Rovisco Pais,, Lisboa, Portugal, mario.silveirinha@co.it.t Abstract Here, we develo a gauge-indeendent Green function aroach to characterize the Chern invariants of generic non-hermitian systems. It is shown that analogous to the Hermitian case, the Chern number can be exressed as an integral of the system Green function over a line arallel to the imaginary-frequency axis. The aroach introduces in a natural way the band-gas of non-hermitian systems as the stris of the comlexfrequency lane wherein the system Green function is analytical. We aly the develoed theory to nonrecirocal electromagnetic continua, showing that the toological roerties of gyrotroic materials are strongly robust to the effect of material loss. Furthermore, it is roven that the sectrum of a toological material cavity terminated with oaque-tye walls must be galess. This result suggests that the bul-edge corresondence remains valid for a class of non-hermitian systems. * To whom corresondence should be addressed: mario.silveirinha@co.it.t --

2 I. Introduction Toological matter and toological systems have quite unique and often intriguing roerties, which can lead to novel hysical effects and henomena [-9]. The toological classification of materials was initially develoed for Hermitian systems described by some self-adjoint oerator. Recently, it was shown that non-hermitian systems, for examle systems with material absortion or material gain, can also have toological roerties [0-4]. The toological Chern invariants of non-hermitian systems are usually found from the bi-orthogonal set formed by the Bloch eigenstates of the Hamiltonian and by the eigenstates of the adjoint oerator [4, 5]. Other nonstandard toological invariants and non-bloch Chern numbers have also been ut forward recently [3, 6-8]. So far, most of the wors in the literature deal with idealized and abstract Hamiltonians (e.g., extensions of the D Su-Schrieffer-Heeger model or the D Rice- Mele model, or others [3, 6-]), that do not always connect in a straightforward way to realistic hysical structures. In contrast, here we show that the toological hases of standard disersive and lossy hotonic materials can be characterized using Green function methods. The roosed theory generalizes the results of our earlier article [6] to non-hermitian hotonic systems. Even though we shall focus on otical materials, our analysis alies to both fermionic and bosonic latforms. While it is well-nown that the Chern invariant of fermionic systems can be found from the system Green function [7-9], to our best nowledge, so far the alication of this technique to non-hermitian systems remains unexlored. --

3 Our theory enables the calculation of the Chern invariants of non-hermitian systems without relying on a gauge-deendent bi-orthogonal set of eigenstates, and thus may be advantageous and simler from a comutational oint of view. More imortantly, the Green function aroach sheds light over several unsettled roblems in the theory of toological non-hermitian systems. In articular, it maes clear that the band-gas corresond to stris of the comlex lane wherein the system Green function is analytic (with no oles). Furthermore, by extending the arguments of Ref. [30], we show that for eriodic systems described by differential-equations the sectrum must become galess when the system is enclosed with oaque-tye boundaries (often referred to as oen in the literature). Thus, our theory settles in art the controversy about the alication of the bul-edge corresondence to non-hermitian systems [0, 5, 6, 7, 9], and uncovers that a toological non-hermitian cavity terminated with oaque-tye walls must suort edge states in the bul band-gas. The article is organized as follows. In Sect. II, we resent a motivation examle that illustrates the alication of the develoed concets to a magnetized electric lasma. In Sect. III we develo the general theory that enables finding the toological invariants of non-hermitian systems from the system Green function. Then, in Sect. IV the theory is alied to hotonic systems, i.e., systems whose dynamics is described by the Maxwell equations. In Sect. V, we generalize some of the ideas of Ref. [30] to non-hermitian latforms and demonstrate that the Chern number integral deends critically on the Green function boundary conditions. Based on this result, we rove that consistent with the standard bul-edge corresondence, the sectrum of a toologically nontrivial hase -3-

4 terminated with oaque-tye boundaries must be galess. Finally, Sect. VI contains a brief summary of the main findings. II. Motivation Examle To illustrate the alication of the Green function methods that will be develoed later in the article, next we comute the toological invariants of a lossy magnetized electric lasma (e. g., a semiconductor biased with a static magnetic field [3, 3]). It is assumed that the material is non-magnetic and that the relative ermittivity tensor is of the form: i zˆ zˆz, ˆ () t t g t a with ˆ ˆ ˆ ˆ t xxyy. The ermittivity elements are ( t, 33 a and i ): g i / t i 0, g 0 0 i, a i. () Here, qb / m 0 0 is the cyclotron frequency determined by the bias magnetic field B B 0 0ˆ z, is the collision frequency, q e is the negative charge of the electrons, m is the effective mass, and is the lasma frequency [33]. We focus on transverse magnetic (TM) waves with H H z z, ˆ E E ˆ ˆ xx Eyy and / z 0. The electrodynamics is non-hermitian when the collision frequency is nonzero 0. In this case the energy is not conserved as a function of time because of the material absortion, and hence the natural modes must have a finite lifetime, i.e., are associated with comlex valued frequencies i and 0 (the mode lifetime is lf [34]). -4-

5 As extensively discussed in Refs. [35, 36], the characterization of the toological hases of a generic electromagnetic continuum requires the introduction of a highfrequency satial cut-off in the material resonse. The satial cut-off needs to ensure that the material resonse is asymtotically (in the limit ) analogous to that of a recirocal and non-bianisotroic dielectric. For a magnetized lasma the ermittivity disersion may be modified as [6, 35, 36]: t, t,loc / max,, g, (3) / g,loc max where is the satial cut-off. In the above, t,loc, g,loc stand for the ermittivity max elements of a material with no satial cut-off and are defined as in Eq. (). The element a is irrelevant for TM-olarized waves and so it is not discussed here. The disersion of the TM-olarized bul modes (lane waves roagating in the xoy lane with roagation factor ef i i xx y y e e ) is given by [6, 35]: / c, ef t g t, (4) with. It will be seen later that the Green function singularities in the x y comlex frequency lane are determined by the solutions (with resect to ) of Eq. (4) with, real-valued. Thus, the sectrum of the gyrotroic material is determined by x y the lane waves (Bloch modes) with, real-valued. x y Figure reresents the locus of the natural frequencies i in the comlex lane (showing both the ositive and negative frequency branches) associated with a realvalued wave vector for different values of the collision frequency. Since the material resonse is invariant to rotations around the z-axis, it is evident that with -5-

6 0. Therefore, in this examle, the locus of i is tyically formed by 5 disconnected curves, i.e., 5 disconnected bands. As seen, the bands are searated by vertical stris in the comlex lane where the material does not suort Bloch waves. These vertical stris corresond to the hotonic band-gas of the non-hermitian material. Curiously, the band-gas are nearly (but not exactly) indeendent on the value of in the range Evidently, when the material is lossless ( 0 ) the natural modes lie on the real-frequency axis. Note that when 0 there is a band of modes with frequencies along the imaginary frequency axis (blac lines in Fig. ). For sufficiently large values of the band-gas close (not shown). Fig. Locus of the natural frequencies of the gyrotroic material with in the comlex-frequency lane ( i) for a real-valued wave vector. Solid lines: 0 (lossless case). Dot-dashed lines: 0.. Dotted lines: 0.5. The shaded gray vertical stris reresent the band-gas in the comlex lane. The left anel (a) shows the root locus for a local material and the right anel (b) for a material with a high-frequency cut-off max 0 / c. It is interesting to comare the sectrum of a local material with no high-frequency satial cut-off (Fig. a) with the sectrum of a nonlocal material with max 0 / c -6-

7 (Fig. b). The two sectra are almost coincident for the natural modes with a small, but may differ somewhat for eigenmodes with (comare the lower end of the blue max curves in Figs. a and b). Figure deicts the exlicit deendence of both (anel a) and (anel b) on the real-valued wave vector for a generic roagation direction in the xoy-lane. In articular, Fig. a shows that the real-art of the eigenfrequencies, insensitive to the value of the daming arameter. x, is rather Fig. Comlex band structure of a gyrotroic material with and max 0 / c. (a) Real art of the natural oscillation frequencies as a function of. Solid blue lines: 0 (lossless case). Dashed green x lines: 0.5. The shaded gray horizontal stris reresent the band-gas. (b) Imaginary arts of the natural frequencies of the st, nd and 3 rd bands with non-negative frequency bands (ordered such that 0 ) as a function of 3 x for 0.5. Similar to our revious article [6], the Chern invariants of the non-hermitian material can be found by integrating the hotonic Green function along a vertical line Re ga arallel to the imaginary frequency axis. The integration ath Re must be contained in the relevant band-ga, i.e., in one of the shaded gray ga -7-

8 regions in Fig. b. Moreover, the analysis of the following sections shows that for electric gyrotroic media (with the ermittivity tensor ()) with a high-frequency satial cut-off max the ga Chern number can be exlicitly written as with: g d d ef 0 max t c ga i 3 g g g d d ef ef 0 max ef t t t. (5a) g g g g g c g ef ef ef t t t t t ef t ga i c, ef ef t g t where / / c, (5b), and, are defined as in Eq. (3). Equation (5) coincides exactly with Eq. (43) of Ref. [6], which was originally derived for lossless systems, but remains valid in the non-hermitian case. The justification of this roerty will be given in Sect. IV. Note that the integrand of Eq. (5) is singular in the comlex lane when is singular, i.e., when satisfies Eq. (4) for some real-valued wave vector (Bloch eigenfrequency). Using Eq. (5) the ga Chern numbers of the gyrotroic material were numerically found as a function of the collision frequency (see Fig. 3). The ga Chern numbers are labeled as in Fig. a. Due to the article hole-symmetry of Maxwell s equations, the ga Chern numbers of the negative frequency gas are identical to those of the corresonding ositive frequency gas. Note that the ga Chern numbers tae into account the contributions of all bands below the ga (modes with Re ga t g ), including the negative frequency bands [6]. As seen, the ga Chern numbers are indeendent of the -8-

9 value of because the comlex-band ga remains oen, i.e., the bands remain searated by a vertical stri in the comlex lane when loss is introduced into the material resonse. In articular, it follows that nonrecirocal hotonic materials can be toologically nontrivial even in resence of rather strong material absortion. The bandga associated with the Chern number ga, closes for a collision frequency slightly larger than. Fig. 3 Ga Chern numbers of a gyrotroic material with and max 0 / c as a function of. The ga Chern numbers are labeled as in Fig. a. III. General Theory In the following, we develo a general Green function formalism to toologically classify the hases of non-hermitian (fermionic or bosonic) systems. A. Comlex band structure We consider generic oerators ˆL and M g that may be non-hermitian. The oerator ˆL is arameterized by the real-valued wave vector xˆ y ˆ. For clarity, for now x y -9-

10 the family of oerators ˆL is assumed eriodic in with the irreducible cell (tyically a Brillouin zone) denoted by BZ. For examle, in lattice-tye models of hysical systems (e.g., tight-binding models) the sace is discretized into a grid of oints and ˆL is a matrix eriodic in. In general, the eriodicity in is not mandatory and more relaxed conditions may be enforced (these will be made recise later). The oerator M g is indeendent of and invertible. Here, If We introduce the following generalized eigenvalue roblem: Lˆ Q M Q (n=,, ). (6) n n g n Q n are the generalized eigenstates of ˆL and n are the generalized eigenvalues. M g is taen as the identity oerator, Eq. (6) becomes a standard eigenvalue roblem. The oerators ˆL and M g may not be Hermitian, and thereby the n are generally comlex-valued. Thus, Q n and n determine the comlex band structure of ˆL. Let us suose that ˆL has no eigenvalues in some vertical stri, Re, L U of the comlex-frequency lane, analogous to the examle of Fig.. Then, we say that the oerator ˆL has a comlete band-ga in the relevant stri. The band-ga searates the bands into two classes: (i) those formed by eigenvectors with Ren ga, which we shall refer as the filled (F) bands, and (ii) those formed by eigenvectors with Re n ga, which we refer as the emty (E) bands. Here, ga is a generic (realvalued) frequency in the ga ( ). L ga U -0-

11 B. Chern toological number We introduce a Green function oerator defined by: il ˆ. (7) M g Evidently, the Green function oerator has oles at the eigenfrequencies, but n otherwise is an analytic function of frequency. In articular, it is analytic over the vertical stri of the comlex lane that determines a comlete band-ga ( Re ). The L U inverse oerator is such that i L. For convenience, we denote ˆ M g Lˆ j (j=,) with x and y. Notice that i j j is j indeendent of frequency. The Chern toological number associated with the band ga is defined by means of the Green function oerator as: ga d d Tr, (8) BZ.. i ga i where /. The integral in is over the line Re ga arallel to the comlex-frequency imaginary axis and Tr... stands for the trace oerator. In our revious article [6], it was shown that the hotonic ga Chern number can be exressed as in Eq. (8) when the relevant system is formed by lossless materials, i.e., when the oerators are Hermitian (the lin between the theory of Ref. [6] and Eq. (8) is further discussed in Sect. IV). In the following, it is shown that for non-hermitian oerators remains a toological integer. Secifically, let us suose that the non-hermitian oerator of interest may be regarded as a smooth deformation of some Hermitian oerator. For examle, consider --

12 that L is the non-hermitian oerator of interest, with ˆL L ˆ ˆ a smooth function of the arameter (0 ), such that ˆ 0 L is Hermitian. In hotonic systems the arameter is tyically related to a daming factor, e.g., a collision frequency, analogous to the examle of Sect. II. Thereby, the Green function oerator and the Chern number are functions of. Insired by Ref. [7], we rove in Aendix A that rovided the deformation does not close the relevant band-ga (so that ga in Eq. (8) is fixed), then the derivative of with resect to the arameter is given by: ga ijl d d Tr i j l BZ.. i ga i. (9) In the above, ijl is the Levi-Civita symbol, 0 /, and the summation over all i, j, l 0,, is imlicit. Notice that the integrand is a sum of derivatives of the trace oerator. Thus, if the term and at the right-hand side of Eq. (9) vanishes. Tr... satisfies suitable conditions at the boundary of the BZ A system is toological when it is ossible to guarantee that the integral in Eq. (9) vanishes. The eriodicity of ˆL in is a sufficient (but not a necessary) condition to ensure that. Indeed, if ˆL is a eriodic function of then also is. In these conditions, the contribution from the boundary of BZ vanishes, and therefore it follows that 0. But since 0 is an integer (because by hyothesis ˆ 0 L is Hermitian) it follows that also is. Therefore, the ga Chern number is an integer totally insensitive to any --

13 ossible non-hermitian deformation that does not close the band-ga; this roerty unveils the toological nature of non-hermitian latforms. In summary, if the oerators ˆL guarantee that the integral in Eq. (9) vanishes (e.g., if ˆL is eriodic) and if ˆL may be regarded as a deformation of some Hermitian system then is a toological integer. C. Berry otential and Berry curvature The aroach of the revious subsection is rather owerful and relies on the well established toological roerties of Hermitian systems. It is instructive to rove that the ga Chern number is an integer following a different ath, namely by diagonalizing the Green function oerator. This second method is related to the theory of Ref. [4] and enables us to introduce in a natural manner the notions of Berry otential and curvature for non-hermitian systems. To begin with, it is convenient to define ˆ M Lˆ M / / g g. From Eq. (6) it is seen that ˆ with n n n / n Mg Qn the eigenvectors of ˆ. It will be assumed throughout that the n are ordered in such a way that the filled bands corresond to the indices n,,... NF, and the emty bands to the indices n N F,..., with N F the number of filled bands. When ˆ is Hermitian the sectral theorem guarantees that its eigenfunctions form a comlete set of the relevant vector sace. However, for general non-hermitian oerators, e.g., when the oerator has excetional oints [0, 37], this roerty does not necessarily hold. Here, we focus on the class of diagonalizable oerators ˆ whose n -3-

14 eigenfunctions san the entire sace, even when ˆ is non-hermitian. For examle, a ˆ that is a sufficiently wea erturbation of a Hermitian oerator must have that roerty. The oerator ˆ can thus be reresented by a diagonal matrix n m, n mn,,,... with resect to the basis determined by n. It is convenient to introduce a fixed (but otherwise arbitrary) basis of the relevant vector sace, e, e,..., with elements indeendent of the wave vector. Let S be the matrix with the coordinates of n in the basis e, e,... (secifically, the first column of S has the coordinates of in the e n basis, etc). Then, ˆ is reresented by the matrix the matrix S deends on the considered eigenfunctions S S in the e n basis. Evidently, n arbitrarily (both in amlitude and hase). Thus, S is gauge deendent., which are normalized Using the roosed matrix reresentation of ˆ it is roven in Aendix B that can be written as: ˆ, (0) d d z BZ.. BZ.. where by definition itrs S F F n n Ren ga is the Berry otential. Here, uˆ u ˆ reresents a diagonal matrix (indeendent of the wave vector) with diagonal elements identical to for n,,... NF (the filled bands) and equal to zero otherwise. It should be noted that is generally comlex-valued. We will see below that given by Eq. (0) is necessarily a real-number, and hence it is ossible to tae -4-

15 Re itr S S F without affecting the value of. It can be shown that the latter definition of is consistent with that of Ref. [4]. Clearly, when S is globally defined as a smooth eriodic function over BZ the Stoes theorem imlies that the ga Chern number vanishes. Hence, similar to the Hermitian case [7, 35], a nontrivial indicates the imossibility of choosing a globally defined smooth basis of eigenfunctions n, and imlies that there is an obstruction to the alication of the Stoes theorem to the entire wave vector domain. As usual, the Berry otential is gauge deendent. For examle, a gauge i n transformation of the form e, with n 0 an amlitude factor and n n n n a hase factor, leads to a different S, and thereby to a different Berry otential. It is shown in Aendix A, that for two generic bases of eigenfunctions corresonding Berry otentials are lined by: n and n the iln g, () with g some smooth (single-valued) function of (in the domain wherein both and are smooth). In articular, it follows that the Berry curvature zˆ n is gauge invariant also for non-hermitian oerators. n In order to demonstrate that is really an integer, we mimic the arguments of Ref. [35]. For simlicity, it is suosed that there is some globally defined smooth basis of eigenvectors, n, excet at a finite number of singular oints ( Si, ) in the BZ. Then, using Stoes theorem in Eq. (0), it follows that the ga Chern number can be written as a line integral of the Berry otential around the singular oints: dl. i C i -5-

16 Here, C i is a circle of infinitesimal radius centered at Si,. The contribution of each singular oint is necessarily an integer. The reason is that it is ossible to ic another gauge n that is smooth in the vicinity of Si,. The corresonding Berry otential is lined to as in Eq. (). The line contour of around circle of infinitesimal radius vanishes because is a smooth function. Therefore, dl Ci iln g dl n, Ci for some integer n. The latter identity follows from the fact that the logarithm is a multivalued function with the different branches differing by i n (note that g is continuous over C i but ln g may not be). This confirms that given by Eq. (8) is really a toological integer for non-hermitian systems. D. Systems described by differential-equations Let us now focus on roblems where ˆL and M g are differential oerators of the form L ˆ L ˆ r, i and g g M M r that act on functions defined over some volumetric region of interest (denoted in the following by cell ). In this context, it is convenient to trace out the degrees of freedom associated with the satial coordinates. This can be done considering a basis of ets r normalized such that r r r r, so that 3 d r r r. Then, the trace oerator is of the form 3 d Tr... tr... r r r where tr... is the trace over degrees of freedom unrelated to the satial coordinates (e.g., associated with the olarization of the electromagnetic field). Introducing rr,, r r it can be shown that Eq. (8) reduces to: -6-

17 ga i d d dvdv ˆ ˆ L L BZ ga i cell cell tr,,,, r r r r.() To obtain the above formula we used r r i rr Lˆ r, i i and 3 d r r r. The gradient oerator of ˆL acts exclusively on the r coordinate of i r, r, whereas the gradient oerator of ˆL From Eq. (7) one has L ˆ of the system, i.e., the solution of: g M i. acts on the r coordinate of rr, and thereby,, L ˆ, i,, i r M r g r r r r rr is the Green function. (3) Tyically,,, is a finite dimension matrix and thereby the rr () is the standard matrix trace oerator. tr... oerator in Eq. IV. Alication to Photonic Systems It is straightforward to aly the develoed ideas to non-hermitian hotonic latforms. Our analysis is focused on lossy systems (all the materials are assive), but the formalism can be extended in a trivial manner to systems with gain elements (with active materials). We follow the notations of our revious wors [6, 35], so that the frequency-domain source-free Maxwell s equations can be written in the comact form as f Mr f with T Nˆ, f E H the electromagnetic field (reresented by a sixvector), Mr, is the material matrix, and ˆN is a differential oerator defined by -7-

18 Nˆ 0 i i i , (4) with 33 the identity matrix of dimension three. For examle, for standard non-magnetic 0 0 media the material matrix is of the form M Without loss of generality, we focus on the case wherein all the oles of the material matrix lie either on the real axis or in the lower-half comlex lane (assive material). Similar to the lossless case [6, 35, 38, 39, 40], it is shown in Aendix C that when the material matrix is a meromorhic function of frequency the electrodynamics of a disersive lossy system can always be reduced to a standard Schrödinger-tye time-evolution roblem. In such a framewor, the state vector is of the form Q f Q... T. The first comonent of Q determines the electromagnetic fields f ; the remaining comonents ( Q ) are related to the internal degrees of freedom of the materials. The eigenmodes of the generalized roblem are the solutions of The oerators ˆL and Lˆ r, i QMg r Q. (5) M g can be exlicitly written as a function of the original material matrix, as detailed in the Aendix C. The toological classification of a hotonic system is based on the generalized eigenvalue roblem (5). A. Periodic systems As a first examle, we consider eriodic (fully three-dimensional) waveguide-tye hotonic crystals, such that the material matrix is eriodic in the coordinates x and y: xa, y, z x, y, z x, ya, z M M M. Furthermore, the system is assumed to be -8-

19 closed along the z-direction, e.g., it can be terminated with metallic lates laced at z 0 and z d (bottom and to walls, resectively), or with any other boundary conditions that force the waves to flow along directions arallel to the xoy lane. There is a common misconcetion that the Chern toological classification only alies to two-dimensional systems. This is not correct. Generally seaing, the sace dimension can be arbitrary (greater or equal than two), but the wave roagation should be constrained to directions arallel to some lane (e.g., the xoy lane). In these conditions, the electromagnetic modes can be classified as Bloch waves labeled by a two-dimensional wave vector xˆ y, ˆ even if the sace dimension exceeds. For a 3D waveguide-tye hotonic x y crystal the Bloch waves corresond to waveguide modes that roagate in the region delimited by the to and bottom walls. Let n ir Q r be the enveloe of a generic (Bloch) waveguide mode ( Q Q ) e n associated with the eigenfrequency n. From Eq. (5) it readily follows that Q n satisfies the general Eq. (6) with, ˆ, Lˆ r i L r i. (6) Even though ˆL is not eriodic (it is tyically linear in ) the toological classification remains feasible [4]. In articular, when the hotonic system has a full hotonic band ga in some vertical stri of the comlex lane, the corresonding ga Chern number can be evaluated using Eq. () with the Green function,, rr defined as in Eq. (3) rr satisfies and BZ the hotonic crystal Brillouin zone. The Green function,, eriodic boundary conditions over (the lateral walls) of a unit cell. In a band-ga the non- Hermitian waveguide does not suort any guided mode with a real-valued wave vector. -9-

20 For media without satial disersion ˆL is linear in, and thereby the oerators ˆ L i are indeendent of the wave vector (and of the oerator). Furthermore, in such conditions ˆ L i only acts on the electromagnetic degrees of freedom (f ). Therefore, the ga Chern number can be written in terms of the electromagnetic comonent, the Green function,, shown that: rr G, of. Secifically, similar to our revious article [6], it can be ga i d d dvdv ˆ ˆ N N BZ ga i cell cell tr G r, r, G r, r,. (7) In the above, N ˆ (i=,) stands for the constant matrix 0 uˆ i 33 i uˆ i 33 0 with ˆ ˆ u x and uˆ y ˆ. The Green function G corresonds to the 6 6 uer-bloc of, and it can be shown that it satisfies: N ˆ i G r, r, M r, G r, r, i r r, (8) subject to eriodic boundary conditions in a unit cell. Imortantly, G only deends on the electromagnetic degrees of freedom of the roblem and on the disersive material matrix. Thus, the ga Chern number can be found directly from the hotonic Green function [6], even for non-hermitian systems. In Aendix D, it is shown that Eq. (7) agrees with the result of Ref. [6]. In summary, the theory of Ref. [6] was extended to non-hermitian systems. The final result is unmodified: the ga Chern number is exressed exactly by the same formulas as in the Hermitian case. Nevertheless, different from the case of Hermitian systems [6] the domain of integration in Eq. (7) (over a line arallel to the imaginary -0-

21 frequency axis) cannot be reduced to a semi-straight line in the uer-half frequency lane because the introduction of loss breas the mirror symmetry with resect to the real-frequency axis. The ga Chern number includes the contribution of all bands below the ga, including the negative frequency bands. For comleteness, we note that the toological charge associated with only the ositive frequency bands is given by d dvdv d d ga i 0 i BZ cell cell ga i 0 i..., where the integrand (i.e., the term inside bracets) is the same as in Eq. (7). Possible oles on the imaginary frequency axis should be avoided by deforming slightly the integration contour so that it is contained in the semi-lane Re 0. The total toological charge of the negative frequency bands may be nonzero [6]. This haens, for examle, when the nonrecirocal resonse is so strong in the static limit that it leads to an exchange of toological charge between the ositive and negative frequency bands [4, 43]. Thus, in general, the two Chern numbers are different. The most relevant quantity is not : the bul-edge corresondence lins the ga Chern number with the net number of unidirectional edgestates, with the ga Chern number understood as the sum of the Chern numbers of all the individual bands below the ga [30]. B. Continuum case In the continuum case, the oerators ˆL and M g are indeendent of all the satial ir coordinates. Hence, for lane wave tye solutions ( Q Q ) Eq. (5) reduces to e n generalized matrix eigenvalue roblem, Lˆ Qn Mg Q n, with n Q a constant --

22 vector. The toological classification is done using taing the set BZ as the entire wave vector sace: Lˆ Lˆ directly in Eq. (8) and ga i d d Tr ˆ ˆ L L ga i, (9) where ˆ i L g M is a matrix. As already discussed in Sect. II, in order that is really toological it is necessary to enforce a high-frequency satial-cut off in the electromagnetic resonse [6, 35]. It can be checed that can be written in terms of the hotonic Green function inˆ, G M as in Eq. (34) of Ref. [6], which thereby remains valid for non-hermitian hotonic systems. Furthermore, for a generic electromagnetic continuum with a high-frequency cut-off the Chern number can be comuted using Eq. (4) of Ref. [6], even in resence of material loss. In articular, Eq. (5) of Sect. II (Eq. (43) of Ref. [6]) gives the ga Chern numbers of a non-hermitian electric gyrotroic material. V. Edge states In regular Hermitian toological systems, the bul edge corresondence establishes a recise lin between the Chern numbers of two toological materials and the number of edge states suorted by a material interface [7, 0, 30, 36, 44, 45, 46]. An illuminating roof of the bul edge corresondence was recently obtained relying on a lin between toological hotonics and fluctuation-electrodynamics [30]. Remarably, it turns out that the thermal (quantum) fluctuation induced light angular momentum sectral density is recisely quantized in a closed toological cavity and that its quantum is recisely the --

23 hotonic Chern number of the bul region [30, 47]. The nontrivial angular momentum of thermal light in a closed cavity is due to the circulation of electromagnetic energy in closed orbits. This effect may occur in nonrecirocal hotonic systems in thermal equilibrium with a large reservoir [48-50]. The roof of the bul edge corresondence in Ref. [30] relies on the assumtion that the material loss is vanishingly small, so that the system dynamics is effectively Hermitian. Recently, there has been some controversy about the alication of the bul edge corresondence to non-hermitian systems [0, 5, 6, 7, 9]. Several articles have underlined that the sectrum of a non-hermitian system with eriodic-tye boundaries may differ dramatically from the sectrum of the same system with oaque-tye boundaries (tyically referred to as oen boundaries in the condensed matter literature), i.e., with boundaries that are imenetrable by the wave [0, 5, 6]. Furthermore, some non-hermitian systems terminated with oaque-tye boundaries can have all the states anomalously localized at the boundary, and thereby the closed system states are aarently disconnected from the (Bloch) extended states of the associated eriodic system. This roerty is designated as the non-hermitian sin effect [0, 6]. To overcome this roblem, a non-bloch bul-boundary corresondence was recently develoed in Refs. [6, 7, 8], relying on toological invariants defined in a generalized Brillouin zone with a comlex-valued wave vector. In contrast, here we argue that at least for non-hermitian Chern-tye toological insulators described by differential-equations as in Sect. III.D with ˆL as in Eq. (6) (e.g., lossy hotonic crystals) the standard bul-edge corresondence holds. To begin with, we note that similar to our revious article [30] the ga Chern number integral in -3-

24 Eq. () can be written in terms of the Green function of a cavity that encomasses many unit cells of the hotonic crystal as follows (see Aendix D for the details and for the exact definition of ): ga i d dvdv ˆ ˆ L L tot ga i lim A tr,,,, tot A rr r r. (0) Here, A tot is the area of the cavity cross-section arallel to the xoy-lane, which should be large enough so that the discrete sectrum of the cavity aroaches the continuum result. The Green function satisfies eriodic boundary conditions at the cavity walls. The volume integrals are over the entire cavity domain. The ey oint is that the integral in Eq. (0) deends critically on the boundary conditions imosed on the cavity walls [30]. Secifically, in Aendix E it is demonstrated that when the Green function satisfies oaque-tye boundary conditions (e.g., for a erfect electric conductor PEC boundary) the value of calculated with Eq. (0) vanishes. At first sight this result is at odds with the fact that Eq. (0) gives the ga Chern number when the boundary conditions are taen as eriodic. Indeed, in a bandga (i.e., in a vertical stri of the comlex frequency lane wherein the bul region does not suort hotonic states) the Green function calculated for source ( r ) and observation (r ) oints interior to the cavity must be nearly indeendent of the boundary conditions imosed on the cavity walls. Note that the Green function rr,, must decay exonentially with r r in the bul region due to the absence of states with a realvalued wave vector. The only sensible exlanation for the critical deendence of the integral (0) on the boundary conditions (eriodic vs. oaque) is that the sectrum becomes galess for -4-

25 oaque-tye boundaries, i.e., the oaque boundaries must close the band-ga and lead to the emergence of edge states at the cavity walls [30]. This roerty suggests that the standard bul-edge corresondence holds at least for non-hermitian systems described by differential-equations. A more detailed discussion of the bul-edge corresondence in non-hermitian systems is left for future wor. In the following, we illustrate the alication of the bul edge corresondence to an interface ( y 0) between a gyrotroic material with (in the semi-sace y 0) and a erfect electric conductor. The satial cut-off of the gyrotroic material is taen equal to max 0 / c and the edge states are comuted using the formalism of Ref. [36]. Figure 4 reresents the edge states disersion x for x real-valued and toological materials with collision frequency 0.5 or 0. For simlicity, only the ositive-frequency modes (with 0 ) are shown. Figure 4a deicts the real art of the edge states natural frequency ( of the natural frequency ( x x ) while Fig. 4b shows the imaginary art ) for 0.5. For comarison, the disersion of the bul states of the gyrotroic material with 0.5 is also reresented in Fig. 4a (solid blac lines). Consistent with the bul-edge corresondence rincile, the edge states disersion (dot-dashed green lines in Fig. 4a) san entirely the two ositive frequency band gas. -5-

26 Fig. 4 Disersion of the comlex edge states suorted by an interface of a gyrotroic material and a erfect electric conductor. (a) Real art of the edge wave oscillation frequency as a function of. The x figure only shows the edge-states disersion in the band-gas. Dashed blue lines: 0 (lossless gyrotroic medium). Dot-dashed green lines: disersion ( vs. 0.5 (lossy gyrotroic medium). The solid blac lines reresent the ) of the lossy gyrotroic medium and the gray dashed horizontal lines delimit the x corresonding band-gas. (b) Imaginary art of the edge wave oscillation frequencies as a function of x for the case 0.5. The arameters of the gyrotroic material are and max 0 / c. Figure 5 show the locus of the edge states natural frequencies ( x with x real-valued) in the comlex lane (dashed green lines). This alternative reresentation further highlights that the edge states san the entire ga (i.e., the vertical stris of the comlex lane with no bul modes) and finally merge with the locus of the bul-material natural frequencies (blac curves) in the comlex lane. Note that similar to the examle of Sect. II, the low frequency ga has toological number, while the high-frequency ga has toological number. Thus, the bul edge corresondence redicts correctly the number of edge states in the comlex-frequency gas. -6-

27 Fig. 5 Locus of the edge states (dashed green lines) and of the bul modes (blac solid lines) of the gyrotroic material in the comlex-frequency lane for the examle of Fig. 4 with 0.5. The figure only shows the ositive-frequency art of the sectrum. VI. Summary We develoed a gauge-indeendent Green function formalism to calculate the toological invariants of non-hermitian (fermionic or bosonic) systems, with a focus on hotonic latforms. Our analysis shows that the standard Green function methods develoed for toological Hermitian systems [6-9] can be extended in a straightforward manner to non-hermitian latforms, and maes clear that the band gas in the comlex frequency lane must be understood as the regions wherein the system Green function is analytic (e.g., vertical stris of the comlex lane that searate the comlex eigenfrequencies into two disjoint sets). The Chern number may be found by integrating the Green function along a curve lying in the band-ga that searates the relevant bands in the comlex-frequency lane. Furthermore, it was shown that similar to -7-

28 the Hermitian case, the Chern number integral can be exressed in terms of the Green function of a large cavity, and that its value is highly sensitive to the boundary conditions (eriodic vs. oaque) imosed on the cavity walls. This roerty imlies that the sectrum of a large toological cavity terminated with oaque-tye walls must be galess. Thus, our analysis suggests that the standard bul-edge corresondence remains valid in non-hermitian systems described by differential-equations, i.e., that the number of edge-states can be lined to the Chern invariant. Using the develoed theory we characterized the toological hases of electromagnetic continua. In articular, it was shown that magnetized electric lasmas retain their toological roerties even in the resence of strong material loss, and that toological edge states with comlex-valued frequencies emerge at the interface of the magnetized lasma and a metal wall. To conclude, it is relevant to note that while in the Hermitian case the hotonic Chern number can be understood as the quantum of the fluctuation-induced angular momentum [30, 47], it is not obvious how to generalize such a result to toological latforms with strong material loss. The main difficulty is that different from the Hermitian case [6], for lossy systems the Chern number integral (7) cannot be exressed as an integral over a semi-infinite straight line contained in the uer-half frequency lane. Thereby, it does not seem ossible to lin the thermal (quantum) fluctuation induced angular momentum of a toological non-hermitian cavity with the Chern number of the bul region. The hysical meaning of the Chern number of hotonic latforms with strong material absortion remains thus an oen roblem. -8-

29 Acnowledgements: This wor is suorted in art by the IET under the Harvey Engineering Research Prize, by Fundação ara a Ciência e a Tecnologia grant number PTDC/EEI-TEL/4543/04 and by Instituto de Telecomunicações under roject UID/EEA/50008/07. Aendix A: Effect of erturbations on the ga Chern number In this Aendix, we calculate (Eq. (9) of the main text) with the ga Chern number given by Eq. (8), and some arameter associated with a deformation of the system Hamiltonian ( / ). The roof resented here extends a result reorted in [7,. 66] to non-hermitian systems. To begin with, we note that and hence Eq. (8) may be rewritten as: ga i d d Tr. (A) BZ.. ga i By integrating by arts in frequency Eq. (8), it is seen that exchanging the indices and flis the sign of the integral. From this roerty, it follows the Chern number is given by the symmetrized formula: BZ.. ga i ijl d Tr i j l ga i d 6. (A) Here, ijl is the Levi-Civita symbol and the summation over,, 0,, i j l is imlicit. Furthermore, by definition 0 and to clean u the notations we droed the index. Let us now suose that the Hamiltonian varies continuously with some generic arameter, so that the oerator, and thereby also the Chern number, can be -9-

30 regarded as functions of. The ga frequency ga is fixed and should lie in a comlete band-ga (vertical stri of the comlex lane with no eigenfrequencies), indeendent of the value of. From we find that:. (A3) i i i Using the cyclic roerty of the trace, it is straightforward to show that the derivative with resect to of a generic term of the integrand of Eq. (A) is: i j l Tr Tr,, i, j, l, jli,,, li,, j,, i, j, l, jli,,, li,, j Tr (A4) The sums are over 3 terms: i, j, l, j,, li and li,, j. From here it follows that: i j l Tr Tr,, i, j, l, jli,,, li,, j,, i, j, l, jli,,, li,, j Tr. (A5) Next, we differentiate both members of Eq. (A) with resect to and use the above result. Noting that ijl a 3 aijl for arbitrary coefficients a ijl, we are ijl i, j, l,, i, j, l, i, j, l jli,,, li,, j left with (the summation over all i, j, l 0,, is imlicit): BZ.. ga i ijl d itr j l ga i d Tr i j l (A6) -30-

31 Now, we observe that i j j i l i j l j i l i j l (A7) Using we may further write: j i j i i j j i l i j l j i l i j l (A8) The first and second terms on the right-hand side are invariant under a ermutation of the indices i, j, whereas the third term is invariant under a ermutation of the indices il., 0, and therefore the ijl These roerties imly that i j l i, j, l second term in the integrand of Eq. (A6) vanishes. This observation yields (restoring the index) Eq. (9) of the main text, which is the desired result. Aendix B: The ga Chern number integral In the following, we derive Eq. (0) of the main text relying on the matrix reresentation ˆ S S of the oerator ˆ. To begin with, we note that from ˆ M Lˆ M the Green function oerator can / / g g be written as with i ˆ M M / / g g. Taing into account that M g is indeendent of the wave vector and using the cyclic roerty of the trace oerator AB BA) it can be easily checed that Eq. (8) still holds with in the ( Tr Tr lace of. Clearly, ˆ i is reresented by the matrix -3-

32 S is. Substituting this result into Eq. (8) (with in the lace of ) and noting that BZ.. is a diagonal matrix it is found that: ga i Tr ga i d d S S S S (B) Straightforward calculations show that is j S S js, j with AB, ABBA the commutator of two oerators. Hence, the ga Chern number may be written as: BZ.. d ga i ga i d S S S S Tr,, (B) The oerator inside the trace can be written as a sum of four terms. The 3 terms that deend exlicitly on j vanish. For examle, the term S S Tr, can be rewritten as (using the cyclic roerty of the trace and noting that the matrices and hence commute) Tr, and m are diagonal j S S 3. The only factor that deends on frequency is 3. The integral d ga i 3 vanishes ga i because the residues of all oles vanish. Thus, the considered term does not contribute to the Chern number. Hence, we are left with: -3-

33 BZ.. ga i Tr,, ga i d d S S S S (B3) To roceed further we use the auxiliary result [6] ga i i d sgn ga Ren ga i m n m, (B4) n which holds when, are on different semi-lanes relative to the vertical line m n ga (e.g., Rem ga and Ren ga Re ). The same integral vanishes identically when m, n lie on the same semi-lane. Writing u ˆ ˆ n n u with ˆ n u n a vector whose n-th element is and with all the other elements identically to zero, it is found that:.. n n m ga i Tr, ˆ ˆ, ˆ ˆ d d S S un u n S S u m um BZ ga i m d Tr i S S, ˆ ˆ S S, ˆ ˆ u u u u n n m m BZ.. nf, me n m Tr i S S, S S, ˆ ˆ ˆ ˆ un un um um ne, mfn m n (B5) In the above E and F reresent the sets of emty and filled bands, resectively. After some simlifications we may simly write: d itr S S F S S E S S E S S F BZ.., (B6) where uˆ u ˆ and uˆ uˆ are diagonal matrices. Secifically, F n n nf E n n F ne since the eigenvectors are ordered so that the filled bands are associated with the indices -33-

34 n,,... NF, the matrix F has diagonal elements identical to for n,,... NF and identical to 0 otherwise. Note that E and F are indeendent of the wave vector. The term associated with the trace can be written in a more comact manner as S S F S S E Tr... Tr where reresents the first term with the indices and exchanged. Noting that Tr S S S S 0 one finds that: F F d i S S F S S BZ.. Tr. (B7) Using the cyclic roerties of the trace and S S S S, one gets Eq. (0) of the main text. Notice that is fully indeendent of the eigenvalues ( ) of the oerator: it only deends on the matrix ( S ) that reresents the eigenvectors on a fixed basis. In the rest of this Aendix, we discuss how a gauge transformation affects the Berry otential itrs S F eigenfunctions. Secifically, consider two generic bases of n and n, with the elements n,,..., NF generating the eigensace of filled bands, and the remaining elements generating the eigensace of emty bands. Then, the coordinates of sf s E with F F F F n in the n basis are determined by a matrix of the form s s and se EsE E (note that the only nontrivial n elements of the matrix F, F E F E s are those with mn,,,..., NF ; this roerty imlies that mn s s s s ; for non-degenerate eigenvalues the matrix sf s E is tyically -34-

35 diagonal). From here, it follows that the matrix S with the coordinates of e e is S S basis,,... E F n in the s s. Therefore, the Berry otential is transformed as: itr se sf S S se sf F. (B8) Noting that s 0 s it is readily found that: E F E F i Tr sf sf Using the general roerty Tr. (B9) s s ln det s (this result can be readily roven F F F when s F is a diagonal matrix, i.e., in case of non-degenerate eigenfunctions; the formula holds true even if s F is not diagonal) it is found that: i ln det sf. (B0) This yields Eq. () of the main text with g det s F. Aendix C: Electrodynamics of non-hermitian disersive systems In this Aendix, it is shown that the electrodynamics of generic non-hermitian disersive systems can be reformulated as a Schrödinger-tye time-evolution roblem. We consider generic bianisotroic materials with a frequency domain resonse determined by a 6 6 material matrix M that lins the electromagnetic field vectors f E H T and T g D B as follows:,,, g r M r f r. (C) It is assumed that M is a meromorhic function in the comlex lane, so that it has a artial-fraction decomosition of the form: -35-

36 Res Mr, M M. (C) Here,, M lim M gives the asymtotic high-frequency resonse of the material,, are the (comlex-valued) oles of M, all in the lower-half lane for assive systems, and Res lim, M M r,, gives the residue of the, ole. The reality of the electromagnetic fields imoses the additional constraint * * M M. Let us introduce the auxiliary variables with s sgn Re, /, s, Q r A fr,, (C3), and s Res / A M. Then, similar to the lossless case [6, 35, 38], it is ossible to show that the time-evolution of the state-vector Q f Q... Q... T is determined by the differential equation, with ˆ LQr, ti Mg Qr, tijg r, t, (C4) t / / Nˆ s s, s,... A A A / ˆ L s,,... A 0, / s, A 0, M g M (C5) In the above, j j T is a generalized current written in terms of the electric g and magnetic current densities, T j j j. The differential oerator ˆN is defined as e m -36-

37 in the main text [Eq. (4)]. For simlicity, the same symbols are used to denote the frequency domain and the time domain fields. Aendix D: Chern number as a function of the Green function of a large cavity In this Aendix, we rove that analogous to the case of Hermitian systems [6], the Chern number can be comuted from the Green function of a large cavity satisfying eriodic boundary conditions. Secifically, we will show that the ga Chern number in Eq. () with ˆL given by Eq. (6) can be written as: ga i d dvdv ˆ ˆ L L tot ga i lim A tr,,,, tot A rr r r. (D) Here, stands for the Green function that satisfies r r r M r r r rr Lˆ, i,,,, i (D) g in a domain that encomasses many (let us say Nx Ny ) unit cells of the hotonic crystal (referred from here on as the cavity ) and A tot is the area of the domain cross-section arallel to the xoy-lane. The function satisfies eriodic boundary conditions on the boundaries of the cavity. Similar to Eq. (), the oerator ˆL ( ˆL ) acts on the rimed (unrimed) coordinates, and by definition Lˆ x, Lˆ x Lˆ i Lˆ i x i i j j j j It is also ossible to write ˆ ˆ ilil 0 with ˆL defined as in Eq. (6)., j=,. (D3) -37-

38 To begin with, let us introduce r r r with r iaˆ ˆ x ia y a generic lattice oint inside the relevant cavity ( i 0,..., N x, i 0,..., N y ), and understood as the -function for the large cavity. Clearly, ri I I r is a eriodic function and its irreducible domain is coincident with the unit cell of the hotonic crystal. It is i x y J J r straightforward to verify that e NN r r where j xˆ j y ˆ ( j 0,..., N x, j 0,..., N y ). Using this identity we J Na x Na x NN J where see that rr, rr, x y J i J as but with e r r r r relacing J satisfies the same differential equation r r. By comarison with Eq. (3), it follows that. Hence, Eq. (D) is equivalent to (note that ij e r r J J ir ˆ ir e ˆ Le L and ir ˆ ir ˆ e Le L ): lim Atot Atot NN J, J x y ga i ga i d rr ˆ r r ijj rr dvdv e tr ˆ L,, L,, J J (D4) The term in rectangular bracets is eriodic over the unit cell (sace domain). Thus, after integration over the entire cavity only the terms with J survive. Therefore, after J reducing the integration to a single unit cell we are left with: ga i d dvdv ˆ ˆ L L tot J ga i cell cell J J rr r r lim A tr,,,, tot A (D5) -38-