Topological Phase Transitions of the Non-Hermitian Electromagnetic Media with Magneto-electric Effect

The non-Hermitian topological properties of electromagnetic media have attracted much attention. In the present work, we reveal the properties of the topology of non-Hermitian electromagnetic media with a magneto-electric effect. By renormalizing the permittivity, we acquire the electromagnetic media topological phase diagram under the influence of magneto-electric effects. We find that the topological phase transition in the system is driven by the renormalization of the permittivity and permeability. This work provides an alternative way to explore the various topological phases of the electromagnetic media.


Introduction
In recent years, non-Hermitian physics [1] has attracted great interest in the fields of Maxwell electromagnetism [2][3][4][5] and topological insulators [6][7][8][9] .In Maxwell electromagnetism, the real-valued permittivity and permeability describe the bulk-boundary correspondence at the interface of an isotropic media and the topological classification of surface Maxwell waves [10,11] , leading to a simple phase diagram [8,12] .In the present work, the non-Hermitian electromagnetic media with magneto-electric effects are investigated for topological phase transitions.We indicate that the phase transition of the topology of the system is driven by the renormalized permittivity and permeability.

Mode
The Hamiltonian for electromagnetic media is provided by the following equation.
where ε is the permittivity of the media, μ is the permeability of the media, H denotes the magnetic field, and E denotes the electric field.While the first term is the energy density of the electromagnetic field in the dielectric system, the second term is the magnetoelectric effect, which indicates that the dielectric has a magnetic field in the electric field proportional to the electric field.
The second term with a minus sign is to show that the appearance of the magneto-electric effect is conducive to the reduction of system energy to make the media more stable.Where θ denotes the coupling constant of the magnetic field and the electric field.The non-Hermitian property of the electromagnetic media is mainly reflected in the permeability and permittivity of the media.In order to facilitate the analysis of the non-Hermitian property of the electromagnetic media, we hope to integrate the magneto-electric effect into the dynamics term of the system.By redefining effective velocity v  , the effective magnetic field, and the effective electric field  , the Lagrange quantity expression containing energy dimension can be written as: The effective Maxwell equations in a media can be written as the Weyl-type equation with six components: ( ) where ψ represents the six-component wave function, Ŝ represents the matrix vector with spin-1, ˆi = − ∇ p represents the momentum operator, and the Cartesian component acting on the field is ˆ⋅ =∇× S p , while the matrix ( ) ˆm θ σ describes the media properties and operates on the "electromagnetic" degrees of freedom.That means the mixture of fields  and  .

Bulk-boundary correspondences and helicity winding number
Helicity is uncertain when 0 θ ε = and 0 μ = separate different phases (as well as diverging eigenvalue ω ).Thus, there is always a "helicity gap" in optical media (except in the case of singular where η represents the phase of refractive index.In the four types of media shown in Figure 1(a), the 4  value w is 0 , 1 ± , 2 , and 2


, TM TE w is 0 , 1 .Most importantly, the surface electromagnetic modes of the interfaces in different media [13][14] are in accordance with the difference in the number of topological structures at the interfaces, as shown in Equation (4).Second, the number of transverse electric and transverse magnetic domain of the surface modes in the interface is given by the difference in the topological number in Equation ( 4): Subscripts 1 and 2 indicate two types of bulk media parameters, and subscripts r indicate two types of bulk media relative parameters: ( ) ( ) Equations ( 5) and ( 6) determined the topological number Equation (4) and bulk-boundary correspondences of surface Maxwell waves.Simply put, Equations ( 5) and (6) show that there are single transverse magnetic fields (transverse electric fields) surface modes in the interface with only permittivity (permeability) changed its symbol and that transverse magnetic fields (transverse electric fields) surface modes in the interface with changes.This is explained in the topological phase diagram of Figure 1(a), which corresponds exactly to the known property for Maxwell waves on surfaces in metamaterials [13] and plasmas [13][14][15] .
In free space ( 1 ε μ = = ), Berry curvature of photon is the charge at the original point of the momentum space monopole σ : related to helicity.This reveals that the helicity winding number of Equation ( 4) is closely related to the phase and helicity of the topological Chern number of the photon.

Phase diagram
We obtained the topological phase diagram of electromagnetic media under the influence of the magneto-electric effect.The electromagnetic media containing magnetoelectric effects can be topologically classified into four categories, described by topological bulk invariants, denoted by ( , ) θ ε μ , i.e., a pair of 2  numbers (a 4  number) in Figure 1(a).From the spatial inversion symmetry between the transverse magnetic modes and transverse electric modes, the two media exchange, which leads to the diagram (Figure 1 One mode

Conclusions
We have examined the topological transformations of electromagnetic systems with magneto-electric effects, which are characterized in terms of helicity winding number and bulk-boundary correspondence.We also show that the electromagnetic system with magneto-electric effect is non-Hermitian, and refine its phase diagram, which provides fresh ideas into the study of the topological properties of electromagnetic systems.


. The electromagnetic media containing magnetoelectric effects can be topologically classified into four categories, described by topological bulk invariants, denoted by ( , ) θ ε μ , i.e., a pair of 2

1 σ
photon.Integrating over the momentum space sphere gives the Chern number 2 C σ σ = related to helicity.This reveals that the helicity winding number of Equation (4) is closely related to the phase and helicity of the topological Chern number of the photon.

Figure 1 .
Figure 1.Topological phase diagram of surface waves in a non-Hermitian electromagnetic media containing magneto-electric effects.a.The presence of 0, 1, and 2 surface zero-helicity modes in the region is described by Equations (4), (5), and (6).b. and c.The two topological phase diagrams indicate when 2 0 surf k > is the propagating mode, and 2 0 surf k < is the evanescent mode, respectively.The transverse magnetic and transverse electric modes swap positions in the third quadrant ( 0, 0) r r θ μ ε < < , indicating that the electromagnetic modes containing magnetoelectric effects have spatial inversion symmetry.d. and e. Partition the topological phase diagram ( 2 0 surf k > ) b into regions

1 r
fact, direct calculations of surf kshow that the wave vectors of the surface modes haveTransverse magnetic fields and transverse electric field polarizations.Thus, one of these is always the real (Figure1(b)) propagating mode, while the other is the evanescent surface mode for imaginary (Figure1(c)).These evanescent surface modes are observable, although they have never been discussed before.Since the model we are discussing is non-Hermitian, the frequency surf ω of the surface modes can be real or imaginary, and these zones are separated by the lines , as illustrated in Figure1 (d and e).