Proof of the bulk-edge correspondence through a link between topological photonics and fluctuation-electrodynamics

The bulk-edge correspondence links the Chern-topological numbers with the net number of unidirectional states supported at an interface of the relevant materials. This fundamental principle is perhaps the most consequential result of topological photonics, as it determines the precise physical manifestations of nontrivial topological features. Even though the bulk-edge correspondence has been extensively discussed and used in the literature, it seems that in the general photonic case with dispersive materials it has no solid mathematical foundation and is essentially a conjecture. Here, I present a rigorous demonstration of this fundamental principle by showing that the thermal fluctuation-induced light-angular momentum spectral density in a closed cavity can be expressed in terms of the photonic gap Chern number, as well as in terms of the net number of unidirectional edge states. In particular, I highlight the rather fundamental connections between topological numbers in Chern-type photonic insulators and the fluctuation-induced light-momentum

One of the most significant and far-reaching results in topological photonics is the socalled "bulk-edge correspondence" [3,5,15]. This fundamental principle links the Chern invariants of two photonic insulators with the net number of unidirectional edge states supported by an interface of the two materials. A recent work reported a proof of the bulk-edge correspondence for a bosonic Bogoliubov-de Gennes Hamiltonian over a tightbinding Hilbert space [16]. However, notwithstanding the bulk-edge correspondence has been extensively discussed and used in the recent literature, it seems that so far, for photonic crystals formed by dispersive materials, it has no solid mathematical foundation and is mainly a conjecture (see a discussion in [8]). Indeed, the arguments in favor of the bulk-edge correspondence are mainly heuristic, e.g., that a continuous transformation of one photonic "mirror" into another topologically distinct mirror requires closing the band gap, and that thereby a material interface must support edge-states. Alternatively, they rely on analogies with the electronic case, for which there are compelling physical reasons to avow that the bulk-edge correspondence holds [10,12], and mathematical derivations for some two-dimensional systems [17][18][19][20]. However, given the different nature of fermionic and bosonic systems, the extrapolation of the condensed-matter arguments to optics is at least questionable. Furthermore, the analogies between electronics and optics are typically valid in a limited quasi-momentum range, e.g., in the framework of some tight-binding approximation limited to some section of the Brillouin zone, whereas the topological invariants are determined by the global properties of the Hamiltonian. Some photonic systems may be mapped onto either a single-particle fermionic system (see Refs. [21,22]) or possibly onto a Bogoliubov-de Gennes lattice model [16] in the entire Brillouin zone. The use of the bulk-edge correspondence is evidently justified in such cases, but these are arguably the exceptions rather than the rule.
Different from most studies of topological photonics, the configuration under analysis here consists of topological material enclosed in a cavity (Fig. 1a). In closed systems the edge states are forced to circulate around the cavity walls and this may lead to novel physical effects. In the recent work [23], I showed that the thermal fluctuation-induced light angular momentum density per unit of area is precisely quantized in the photonicinsulator cavity, and that its "quantum" is determined by the net number of unidirectional edge-states circulating around the cavity. This rather universal property holds even when the system has no topological classification. In particular, in nonreciprocal platforms the thermal equilibrium condition is compatible with a persistent energy circulation in closed orbits [23][24][25][26].
The quantization of the fluctuation-induced angular momentum can be explained in a simple and intuitive manner. Specifically, it may be observed that in the band-gaps of the bulk region the allowed photonic states are necessarily edge waves; thus the topological cavity may be regarded as a circular multi-mode transmission line (Fig. 1b). The system is closed along z so that the energy is forced to flow along directions parallel to the xoy plane.
In this study there are two cases of interest: (i) the fields satisfy opaque-type (e.g., perfectly electric conducting) boundary conditions at the lateral-walls (not shown) or (ii) the fields satisfy periodic boundary conditions at the lateral walls. (b) For opaque-type lateral walls, the photonic insulator cavity is equivalent to a circular one-dimensional transmission line.
As it is well-known, thermal and quantum fluctuations can induce energy flows in a transmission line terminated with resistive loads [28].  [23]. In Ref. [23], the "bulk-edge correspondence" was used to link the angular momentum "quantum" (N) with the gap Chern number.
Here, I follow precisely the inverse path. It is demonstrated -never making use of the bulk-edge correspondence-that there is an intimate connection between the Chern number and the fluctuation-induced Abraham angular momentum. In particular, both the Chern number and the angular momentum spectral density can be expressed in terms of an integral of the photonic Green function along a semi-straight line parallel to the imaginary frequency axis [27]. By exploiting this connection, I show that in a band-gap the angular momentum "quantum" is the Chern number. This result together with the theory of Ref. [23], establish the formal link between the Chern number and the netnumber of unidirectional edge states, and thereby demonstrate the bulk-edge correspondence in photonics. I numerically illustrate the application of the developed concepts to a gyrotropic photonic crystal cavity.
In short, the key idea that drives the analysis of the article is that the Chern number is determined by the Green function of a system terminated with periodic boundaries, while the fluctuation-induced angular momentum density is determined by the Green function of a cavity terminated with opaque-type walls, i.e., walls impenetrable by the electromagnetic radiation. In a photonic band-gap, the two Green functions are essentially identical in the bulk region. I prove that due to this property the Chern number and the fluctuation-induced angular momentum are profoundly related.
The article is organized as follows: In Sect. II, I present a quick overview of the topological classification of photonic platforms using the system Green function [27] and of the Hamiltonian-type description of a dispersive photonic system (e.g., a topological cavity). Both the fluctuation-induced angular momentum and the Chern number can be written either in terms of the normal modes of the equivalent Hamiltonian, or alternatively, in terms of the system Green function. The two different formalisms are used in my analysis. In Sect. III it is demonstrated that the Green function boundary conditions (periodic vs. "opaque") play a critical role in the Chern number calculation. In Sect. IV, the angular momentum expectation is related to the system Green function, and in particular it is shown that its spectral density is given by an integral of the Green function over a semi-straight line parallel to the imaginary frequency axis. The proof that the Chern number is the angular momentum "quantum" is given in Sect. V. The bulkedge correspondence is formally demonstrated in Sect. VI and is applied to a photonic crystal cavity. A short summary of the main findings is given in Sect. VII.

II. Topological classification
The topological classification of Chern-type photonic materials is typically based on the normal modes (eigenstates) of the problem. The Maxwell's equations, on their own, do not provide for a Hermitian-type description of dispersive systems [1,2,27,29,30].
However, the electrodynamics of lossless systems can be modeled by a generalized (1) The state vector The precise definition of the multiplication operator   0 L r can be found in Refs. [27,29,30]. It depends on the poles and residues of the 6×6 material matrix M that links the frequency domain fields as follows: The integration is over the volume of the relevant "cavity".
The topological classification of a periodic system is typically done by introducing a The summation is over all the "filled" photonic bands (F) below the gap ( In photonic systems the summation must include both positive and negative frequency branches, because the sum of the Chern numbers of negative frequency bands may be nonzero [27,35,36].
Importantly, the topological classification of a photonic system can be done without any detailed knowledge of the photonic band structure or of the Bloch waves, and without introducing a gauge dependent-Berry potential [27,37]. Indeed, the gap Chern number can be simply expressed as an integral in the complex frequency plane of the photonic Green function   , ,  G r r , as follows [27,38]: .. is the trace operator and î N  stands for the 6×6 matrix stands for the first term in rectangular brackets with the indices "1" and "2" interchanged.
This notation will be used throughout the article. It is worth pointing out that in the electronic case the Chern number is also determined by the fermionic Green function [39]. The integration in Eq. (8) with 6 6   1 1 and periodic boundary conditions over the lateral walls. -11-

III. Boundaries matter
The Chern number written as in Eq. (7) depends on the Berry curvature, and thereby on the Bloch eigenmodes envelopes. Thus, it is essential that the considered system is periodic in the x and y-directions. In contrast, Eq. (8) gives the Chern number in terms of the photonic Green function. As previously mentioned, it is implicit (and necessary so that Eqs. (7) and (8) give the same result) that the system Green function satisfies periodic boundary conditions over the cavity lateral-walls [27].
Is the value of  given by Eq. (8)  ,  r r are interior to the cavity. Thus, at first sight it seems that the integral in the right-hand side of (8) should be insensitive to the lateral boundary conditions. -12-Surprisingly, it is shown in Appendix A that this heuristic understanding is wrong. In particular, for a nontrivial topology one has: where Berry  is calculated with Eq. (7) vanish, i.e., that then the lateral-walls are "opaque". Note that because ˆg H is a Hermitian operator one might think that   0 j R  is a universally valid property. However, as discussed in Appendix A, such an understanding is wrong and for periodic-type boundaries   0 j R  .
Besides the PEC walls already mentioned, another simple example of an opaque-type boundary is the case of perfect magnetic conductor (PMC) walls. As further detailed in Appendix A, the physical reason for the dissimilar results in Eq. (11)  principle. Later, the outlined arguments will be made rigorous.

IV. Angular momentum
Let us now consider a nontrivial Chern-type insulator cavity with opaque-type lateral walls. For now, it is assumed that the system is perfectly isolated from the external environment so that there is no dissipation, i.e., both the cavity walls and the topological -14-material are lossless. Furthermore, it supposed that the electromagnetic fields have no quanta so the system is in the ground state ("quantum vacuum"). A discussion of the thermal states in a weakly dissipative cavity will be presented in Sect. V.B.
The unidirectional nature of the edge states in topological materials implies that the fluctuation-induced light may be characterized by a nontrivial angular momentum. This property was first discussed in Ref. [24], where it was argued that topological systems in equilibrium with a thermal bath (or in the quantum vacuum state) may enable the circulation of a heat current in closed orbits. Furthermore, such an effect can in principle be observed in other nonreciprocal (but not necessarily topological) systems [23,25,26,41].
Importantly, it was demonstrated in Ref. [23] that in the continuum limit ( tot A   ) the spectral-density of the Abraham light-angular momentum ( , the mean energy of a quantum-harmonic oscillator at temperature T [40]. This is the result discussed in the Introduction and heuristically justified using an analogy with a circular transmission line. In particular, the formula holds with 0 T  in the "quantum vacuum case" when the material and the cavity walls are lossless. The finite temperature result assumes vanishingly small (but nonzero) material absorption, e.g., in the cavity walls (see Sect. V.B). It is implicit that at the frequency of interest the system does not support bulk states (electromagnetic band-gap). The sum in Eq. (14) is over all the edgemodes and simply counts the difference between the number of edge modes associated  may be directly written in terms of the topological Chern number [23]. Next, I develop the theoretical formalism necessary to prove that that is indeed the case.

A. Classical states
There are two relevant light momenta: the Abraham momentum (kinetic momentum of light) and the Minkowski momentum (canonical momentum of light) [41][42][43][44]. This article uses the Abraham formalism, which leads to a quantization of the angular momentum spectral density in the topological cavity. The fluctuation-induced Minkowski angular spectral density is not quantized. For discussion on the detailed meaning of each momenta the reader is referred to Refs. [41,44].
The Abraham (kinetic) light momentum in the cavity is determined by 2 1 dV   [45] is adopted here). In particular, the zcomponent of the angular momentum (perpendicular to the plane of propagation) of a given state vector Q is: The angular momentum can be expressed explicitly in terms of the electromagnetic fields as 2 1 dV c    r S  [23]. The angular momentum of light is extensively discussed in Refs. [45][46][47][48]. As further detailed in Appendix A, ˆg H  k is given by the commutator of the position operator and the pseudo-Hamiltonian: 1, Thus, the angular momentum may also be written as:

B. Quantum vacuum state
In a lossless dispersive material cavity the electromagnetic field can be quantized by letting each normal classical mode become a quantum harmonic oscillator [49]. The angular momentum of the "quantum vacuum" ( one finds that the expectation of the light angular momentum in the cavity is: It is supposed that the modes are normalized such that , | n m nm gives the stored energy [27,30] and thus the parameter   n  has unities of angular momentum per Joule. Similar to Ref. [24], it is simple to show that Eq. (18) can be directly obtained from the fluctuation-dissipation theorem using a modal expansion of the system Green function.
In open systems, the total light-angular momentum may depend on the origin of the coordinate axes [45]. Importantly, for a closed cavity the expectation of the total light momentum vanishes (even though locally the light momentum density is generally nontrivial [24]). Indeed, for opaque-type walls (   0 i R  ), Eq. (A6) shows that a generic cavity mode satisfies: Combining this result with Eq. (16), it follows that   n  is origin independent. Thereby, the expectation of z  , i.e., of the total angular momentum, is also origin independent.
I introduce a (unilateral) "quantum vacuum" angular momentum spectral density   such that Clearly, it has the modal expansion [23]: -18-Using the fluctuation-dissipation theorem [40] with 0 T   , (which is applicable to the ground state of lossless closed systems, see [24]) it is proven in Appendix B that   can alternatively be written in terms of the system Green function of the lossless cavity:

C. "Partial" angular momentum
It is useful to introduce a partial "quantum-vacuum" angular-momentum, Clearly, the quantum-vacuum angular momentum satisfies . Thus, the spectral density is: By integrating the fluctuation-dissipation theorem result (21) The integration path is along the section of the straight line in the upper-half plane.
Interestingly, the partial-angular momentum can be expressed in terms of the function f defined by Eq. (9). Specifically, for a cavity with opaque-type lateral walls one has: can be written as an integral of the system Green function over the imaginary frequency axis. This is analogous to Casimir's theory where the zero-point energy of a system is determined by an integral over "imaginary" frequencies [50,51]. The angular momentum expectation is manifestly independent of the coordinate system origin.

D. Cavity with periodic lateral walls
Up to now, in this section it was assumed that the lateral walls of the cavity are "opaque". However, Eqs. (23) and (25) (14)]. Equation (26) is consistent with the fact that the bulk material does not support any states in the band-gap and hence its angular momentum density must vanish.

A. Quantum fluctuations
Let us introduce    defined such that the expectation of the "quantum-vacuum" angular momentum spectral density per unit of area satisfies: The function    is dimensionless and from Eqs. (23) and (25), it may be generally written as:  1 Re 2 The second identity is a consequence that the term with the periodic Green function Thus, in agreement with Ref. [23], I conclude that the quantum fluctuation-induced angular momentum in a topological system is precisely quantized. Its "quantum" is exactly the gap Chern number. It is underlined that the derivation of Eq. (31) is fully independent of Ref. [23].

B. Thermal fluctuations
The results of Sect. IV.B can be readily generalized to weakly dissipative systems, e.g., a cavity filled with the topological material with the cavity walls slightly absorptive.
Such systems are coupled to the external environment (e.g., through the cavity walls) and hence support thermal states. The thermal states of weakly dissipative systems can be studied perturbatively simply by considering that the mean energy of the n-th mode is This result may be further generalized to give the number of edge modes propagating at the interface of two topological materials: "the bulk-edge correspondence". To that end, I consider the geometry depicted in Fig. 2, which shows a cavity half-filled with two photonic insulators (the two materials share a photonic band-gap). The cavity lateral walls are assumed "opaque". Let 1  and 2  be the gap Chern numbers for material 1 and 2, respectively. Formula (33) implies that 1  and 2  determine the number of modes propagating in the clockwise direction around the cavity walls (see Fig. 2). Hence, the number of modes propagating at the interface of the two materials (along the 1 x  direction) must be precisely 2 1    , i.e., the gap Chern number difference. Fig. 2 Illustration of the bulk-edge correspondence principle. A cavity (terminated with "opaque-type" lateral walls) is filled with two photonic insulators. The points A and B represent the two junctions.
The reason why this needs to be so is that otherwise the system would be unstable and a steady-state could not be reached. Indeed, suppose that the net number of unidirectional modes propagating at the interface of the two materials is different from 2 1    . In this situation, for one of the junction points (let us say point B of Fig. 2) the number of edge modes arriving at the junction is larger than the number of edge-modes propagating away from the junction. Since the system response is linear, this implies that it would be possible to choose the complex amplitudes of the incident waves in such a way that the edge waves propagating away from the junction are not excited. But then, since by assumption there is no loss and there are not scattering channels available, the energy incident in the junction must remain stored in it. Hence, it is impossible to reach a steadystate for a time-harmonic excitation: the energy stored at the junction grows linearly with time similar to a lossless LC circuit excited at the resonance. Physically this is not -25-acceptable, and hence the net number of unidirectional edge-modes propagating at the interface of the two-materials must be precisely 2 1    . This concludes the proof of the bulk-edge correspondence principle. To illustrate the application of the developed theory, I consider a two-dimensional photonic crystal (the condition / 0 z    in enforced) formed by squared shaped nonreciprocal inclusions organized in a square lattice with period a (Fig. 3a). The inclusions stand in air and are spaced by d. Furthermore, the analysis is restricted to transverse-magnetic (TM) polarized waves with nontrivial field components , , The inclusions electric response is assumed to be gyrotropic with the same dispersion model as a lossless magnetized plasma [53] (e.g., a magnetized semiconductor [54]) and ˆˆˆt     1 x x y y. In the above, p  is the plasma frequency, . For p    the air-gaps are deeply subwavelength, and hence in the long-wavelength limit it seems reasonable to approximate the photonic crystal by a continuum with the same permittivity as the inclusions, as illustrated in Fig. 3a. This approximation greatly simplifies the calculation of the band structure and of the gap Chern numbers. In order that the electromagnetic continuum is topological it is necessary to impose a highfrequency spatial cut-off, max k [29]. For the physical reasons discussed in detail in Ref. [15], the spatial cut-off should be taken on the order of max 1/ k d  . The photonic band structure obtained with the continuum approximation is depicted in Fig. 3b  Ref. [27] and takes into account the contribution of the negative frequency bands (not shown in Fig. 3b).
Next, I focus on the low-frequency band-gap for which the continuum approximation is arguably more accurate. Its gap Chern number is thus it is topologically nontrivial. Hence, if the material is paired with a PEC boundary, the bulk-edge correspondence predicts that there is a single edge-state propagating along the +x-direction. To confirm this prediction, I used the continuum approximation to compute the edge-states dispersion. The spatial cut-off max k is taken into account using the spatially dispersive model described in Ref. [15]. The calculated dispersion (for a material biased with 0 0 B  and 0.8 ) is plotted with a green-dotted line in Fig.   3b, and yields the unidirectional gapless edge-mode.
It is also interesting to analyze the case in which two topologically distinct plasmas are paired to form an interface (inset of Fig. 3b). In this scenario, the top region ( 0 y  ) is   [56] show that the edge-state excited at the interface of the two photonic crystals ( 0 y  ) by the vertical dipole is a forward wave (Fig. 4), whereas the edge state excited by the horizontal dipole is a backward wave (Fig. 5). Hence, in agreement with the dispersion of the edge-states obtained with the continuum approximation (Fig. 3b), the interface 0 y  supports two unidirectional edge-modes: a forward wave and a backward wave. Furthermore, as seen in Figs. 4 and 5, z H has even (odd) symmetry with respect to 0 y  for the forward (backward) mode, respectively.
The edge-mode profiles obtained with the continuum theory have the same symmetries, which further reinforces the validity of this approximation. holds. Thus, in agreement with Ref. [23], it follows that the spectral density of the angular momentum per unit of area is quantized in the bulk band-gaps, and that the Chern number is the "quantum" of the fluctuation-induced angular momentum. Furthermore, using the findings of Ref. [23], which link

A. Alternative formula for the Chern number
To begin with, I use the fact that independent of the boundary conditions enforced on the lateral walls, the value of  calculated with Eq. (8)   As discussed in the main text, the lateral-walls are said to be "opaque" when the boundary conditions guarantee that   0 j R  . For opaque boundaries, one has 0 n   [Eq.
(A7)]. In electronics, the Berry curvature n  can be understood as the normalized electric conductivity contribution from an electron in the n-th state when it is excited by a staticelectric field [34]. In a system with periodic-type boundaries the electric current can be nonzero because the boundaries are cyclic, and hence it is possible to have 0 n   . In contrast, for opaque-type boundaries it is unfeasible to have a steady electric current because it cannot go through the boundaries, and therefore 0 n   . Translating these ideas to optics, one may regard n  as the (linear) response of the light momentum due to some steady external stimulus (some analogue of the static-electric field excitation in fermionic systems), when the initial state of the system is determined by the n-th cavity mode. For cyclic (periodic) boundaries the induced momentum can be nontrivial, but for opaque-type boundaries it must vanish.