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Topological Angular Momentum and Radiative Heat Transport in Closed Orbits

Here, we study the role of topological edge states of light in the transport of thermally generated radiation in a closed cavity at a thermodynamic equilibrium. It is shown that even in the zero temperature limit - when the field fluctuations are purely quantum mechanical - there is a persistent flow of electromagnetic momentum in the cavity in closed orbits, deeply rooted in the emergence of spatially separated unidirectional edge state channels. It is highlighted the electromagnetic orbital angular momentum of the system is nontrivial, and that the energy circulation is towards the same direction as that determined by incomplete cyclotron orbits near the cavity walls. Our findings open new inroads in topological photonics and suggest that topological states of light can determine novel paradigms in the context of radiative heat transport.

The Chern number can only vary when some topological charge is exchanged between different photonic bands, i.e., when a bandgap closes and reopens [1][2][3]. Thus, a topological phase transition must occur when some material is continuously transformed into another material with a different Chern number (e.g., through a continuous deformation of the structural unities or a variation of a biasing field). Notably, this property implies that if two topologically distinct materials that share a common bandgap are joined to form an interface, then topologically protected unidirectional edge states will appear in the bandgap, a property which is known as the "bulk edge correspondence principle" [3,20,21].
A nonzero Chern number requires breaking the time-reversal symmetry and thus a nonreciprocal material response. This can be achieved using a static magnetic field that originates a gyrotropic response [1,7]. Nonreciprocal effects may also occur in systems with moving parts [17,22,23], when the material response is time-modulated [24], or in presence of strong nonlinearities [25]. Furthermore, in some antiferromagnetic materials the time-reversal symmetry may be spontaneously broken due to magnetic ordering [18,26]. -3-It is natural to wonder whether the intriguing properties of topologically protected unidirectional edge states can have any exotic consequences in the context of fluctuational electrodynamics, i.e., when the electromagnetic fields are either generated by the random jiggling of atoms due to thermal agitation or due to purely quantum fluctuations. Specifically, we are interested in scenarios where the system of interest is closed (e.g., a closed cavity) and is in a thermal equilibrium. The presence of topologically protected "one-way" channels raises puzzling questions: do these channels lead to a transport of the thermal (or zero-point) energy? Does a heat transport prevent that a thermal equilibrium is reached or does it breach the second law of thermodynamics due to the energy concentration in some location of the cavity? Here, it is theoretically shown that the emergence of topological edge states does not lead to any paradoxical settings and that standard fluctuation electrodynamics provides a conclusive answer to the enunciated questions. Specifically, it is found that the topological edge states induce a circulation (with no net sinks or sources) of thermal (or zero-point) energy in closed orbits, such that the angular momentum of the electromagnetic field is nonzero.
Remarkably, it is shown that the heat flux circulates towards the direction determined by incomplete electron cyclotron orbits. Notably, the energy flow persists even in the zero temperature limit, wherein the system is in its (parametric) "ground" state.
It is important to connect our findings with previous studies. The circulation of electromagnetic momentum in nonreciprocal systems in a thermal equilibrium has been discussed in a few different contexts. In Ref. [27], we demonstrated that in a system with parts in a shear motion the quantum expectation of the zero-point electromagnetic momentum associated with a moving body is typically nonzero, even though the -4-expectation of the total momentum of the system vanishes. Unfortunately, in such systems the effect is negligibly small for nonrelativistic velocities. More recently [28], it was shown that in a nonreciprocal many-body system formed by magnetized nanoparticles near-field interactions enable a persistent directional heat current at a thermodynamic equilibrium, in qualitative agreement with our conclusions. This article builds on these previous works and looks at the problem from a different angle: we investigate the heat circulation in a system with topological properties, showing that the topological unidirectional edge states lead to the formation of spatially separated channels wherein the energy flows in closed orbits, so that the angular electromagnetic momentum is nontrivial at equilibrium.
It is relevant to mention that the influence of a nonreciprocal electromagnetic response in the context of radiative heat transport in nonequilibrium situations (e.g., in presence of temperature gradient) was discussed in other recent works. For example, it was shown that nonreciprocal effects enable a giant radiative thermal rectification [29], a thermal Hall effect [30] and a tunable near-field heat transfer [31]. Asymmetric radiative heat flows can also occur in reciprocal structures with a temperature dependent response [32][33] or in anisotropic materials [34]. More generally, there is a great interest in the recent literature in novel approaches to tame the radiative heat flow at the nanoscale due to its applications in thermophotovoltaics [35][36][37][38][39][40][41], thermal radiation based microscopy [42][43][44][45][46] or radiative cooling [47].

II. Gyrotropic Waveguide
We consider a closed cavity formed by a parallel plate waveguide filled with an electric gyrotropic material (Fig. 1a). The widths of the structure along the x and y -5-directions are much larger than the waveguide height ( , x y L L d  ) and, for simplicity of modeling, the corresponding ends are terminated with periodic boundary conditions. Thus, the structure can be assumed invariant to translations along the x and y directions.
The waveguide walls are modeled as perfect electric conductors (PEC), which is a good approximation up to terahertz frequencies.

A.
Topological properties of the bulk gyrotropic material    [19,20,[49][50][51]. In particular, it is seen that TMpolarized waves have two different bandgaps for propagation in the xoz plane. Here, we will focus on the high-frequency bandgap determined by  There is a single band above the gap with Chern number 3  where k  is the Berry potential [19]. The calculation of the topological numbers in an electromagnetic continuum requires imposing a wave vector cut-off in the nonreciprocal part of the electromagnetic response [19,20]. Note that the low-frequency band ( 0 However, if the propagation direction is slightly tilted so that it gains a small ycomponent the dark mode becomes bright, as can be inferred from the band structure in Evidently, the associated isofrequency surface has revolution symmetry around the y-axis. As seen in Fig. (2b), the isofrequency surface is slightly elongated along the y-direction due to the anisotropy of the material. For propagation in the x-direction the wave is a linearly polarized (LP) TEmode, whereas for propagation along the y-direction the wave is CP polarized.

B. Topological waveguide modes
Let us now consider the waveguide of  Notably, the numerical results reveal that for propagation along the +x direction ( Fig. 3c-A) the energy tends to flow near top wall of the waveguide, whereas for propagation along the -x direction (Fig. 3c-D) the bottom wall is preferred. On the other hand, for propagation along y-direction the energy distribution is symmetric with respect to the center of the waveguide (Fig. 3c-C). The locking between the spatial region wherein the light flows and its momentum parallels precisely the electron flow induced by a static magnetic field due to incomplete cyclotron orbits (skipping orbits) near the metallic walls ( Fig. 1b). This suggests that the two phenomena are deeply linked.
From another perspective, the spatially asymmetric energy transport along the xdirection has its origin in the topological properties of the bulk gyrotropic material.
Indeed due to the bulk edge correspondence principle [19,20]  when the metallic wall stands below, in the region 0 z  (above, in the region 0 z  ), the gyrotropic material. In particular, the dispersion of the topological edge mode supported by the single-interface waveguide obtained when the top PEC plate is removed is given by 0 This unidirectional topological mode propagates exclusively towards the -x direction and has the dispersion depicted in Fig. 3b with a black line. Notably, this dispersion is precisely coincident (in the region with negative abscissa) with that of the mode supported by the parallel plate gyrotropic waveguide (blue line in Fig. 3b). Indeed, it can be checked that in the limit 0 y k  the dispersion of the waveguide modes [Eq.
(B5)] reduces to 0   To further illustrate the discussion, we used a commercial electromagnetic simulator [53] to compute the fields radiated by a short vertical electric dipole inside the gyrotropic waveguide. A density plot of a time snapshot of the z-component of the radiated electric field is depicted in Fig. 5. Consistent with Fig. 3c, it is seen that in the xoz plane (Fig. 5a) the emission in spatially asymmetric, such that the emitted fields flow nearer the top/bottom metallic plates depending if the wave propagates along +x or -x, respectively.
In contrast, in the yoz plane the field distribution exhibits an even symmetry with respect

III. Fluctuational Electrodynamics
Having discussed in detail the topological origin of the wave phenomena in the gyrotropic waveguide, next we investigate correlations of the electromagnetic fields induced by either thermal or quantum fluctuations. It is supposed that all the points of the waveguide are held at a fixed temperature T.

A. Field Correlations
To begin with, it is convenient to introduce some notations and write the Maxwell's equations in the frequency domain in the following compact form: Here, 3 is the zero-point energy of an harmonic oscillator. In Appendix C it is proven that this result is fully consistent with the fluctuation-dissipation theorem [59], and that the field correlations may also be written in terms of a retarded Green-function.
Furthermore, such a link shows that Eq. -18- The subscript T indicates that the expectation is taken at the temperature T, and in accordance, 0,  was replaced by ,

B. Poynting vector
Using Eq. (7) [28]. It can be shown -using directly the fluctuation-dissipation theorem [Eq. (C3)] -that this result remains valid when the material absorption is nonnegligible (i.e. when reciprocity is not equivalent to time reversal). Therefore, the system reciprocity needs to be broken to have a radiative heat flow in a stationary state (with the system temperature constant at all points of space) [28].
It is evident that when the waveguide is invariant to translations along x and y nk S  depends exclusively on the z coordinate, so that

IV. Heat Flow in Closed Orbits
To illustrate the implications of the theory, we used Eq. (10) to determine the Poynting vector spectral density. The calculation is restricted to the frequency interval   As seen in Fig. 6b, the Poynting vector lines form closed orbits oriented in the clockwise direction (the direction opposite to the applied magnetic field). Because the waveguide width is assumed very large along the x-direction, the orbits of the Poynting vector are closed at infinity. Remarkably, the heat flow near the top plate is towards the +xdirection, imitating thus the electron flow due to incomplete cyclotron orbits near the same metallic wall (see Fig. 1b  if the ratio of 0 / p   is kept fixed. In practice, the strength of the heat current is limited by the cyclotron frequency and thereby by the static magnetic field, which realistically can hardly be made larger than a few Tesla. A possible experiment may be based on a directional coupler (e.g., created by inserting slots on the top wall of the waveguide [63]) that can be used to detect the imbalance between the heat flows along the +x and x directions near the top wall.

V. Discussion and conclusion
In summary, it was demonstrated that in a gyrotropic topological waveguide in thermal equilibrium there is a permanent heat flow in closed orbits. The effect persists even in the limit of a zero temperature wherein the field fluctuations are purely quantum mechanical. This phenomenon is deeply rooted in the nontrivial topological properties of -25-the gyrotropic material, which lead to the formation of unidirectional edge states near each metallic wall. In particular, it was highlighted that in a thermal equilibrium the electromagnetic field has a nontrivial angular momentum, and that the Poynting vector orbits mimic the electron skipping orbits due to an incomplete cyclotron motion. The effect appears to be strong enough to allow for an experimental verification.
We would like to highlight that the considered system is equivalent to a metallic cavity with closed walls in all directions of space, and hence is electromagnetically closed. One may wonder what happens in a system with a single infinitely-wide metallic plate. A conceptual problem with such a configuration is that it is not electromagnetically closed (note also that the gyrotropic material has no full bandgap), and for open systems it is not reasonable to assume that there is a thermal equilibrium to begin with.
Nevertheless, the structure can be mathematically closed with periodic boundaries  that impose a cyclic variation of the fields  and which would make the single-plate geometry effectively equivalent to a torus waveguide with a very large radius of curvature. In that case, it is expected that similar to the example studied in the article there is a heat flow in closed orbits (the orbits of the torus) at equilibrium. For a single-plate with finite dimensions along x and y the same conclusion holds, and the heat flux is expected to circulate around the plate in closed orbits (note that in this scenario the topological modes propagating to the left on the top face of the metallic plate are forced to propagate in the opposite direction on the bottom face). Thus, we believe that quantum or thermally induced fluctuations in "one-way" topological systems may lead to exciting new physics, and we hope that this work may stimulate further studies in related directions.