The Topological Phase Transitions near the Exceptional Point of a Non-Hermitian Electromagnetic Lattice Model

In this paper, we take the double-layer three-component non-Hermitian Lieb lattice system as an example to discuss the topological properties of the non-Hermitian lattice point electromagnetic field system. The topological phase transitions of the Lieb lattice system near the exceptional point are analyzed by calculating the topological invariants. We find that there are abundant topological phase transitions near the exceptional point. This study will help us explore more topological physical phenomena of electromagnetic fields of lattice systems in condensed-matter physics.


Introduction
In the realm of condensed-matter physics, the topological state is a hot topic of current research [1][2][3][4][5][6].The topological phases of electronic materials exhibit a list of interesting phenomena [7][8][9][10][11].For electronic systems, there are abundant topological phase states in matter behind the seemingly complex wave function that describes how do electrons move in matter [12,13].The study of topological properties in matter can help us better understand the intrinsic properties of various states of matter, and also help us to explore more physical phenomena in condensates.
It is shown that the continuous Maxwell equations are topologically nontrivial and the appearance of the surface Maxwell waves at interfaces with different parameters of permittivity ε and permeability μ of the medium has topological properties [1].On the contrary, we want to understand better the topological properties of condensed-matter discrete systems in this paper.We extend the discussion of electron lattice problems to periodic lattice electromagnetic field systems.The topological phase transition near exceptional points of a double-layer three-component non-Hermitian system is therefore proposed, and the topological characteristic of this problem is analyzed as a model described in terms of topological quantities calculated using wave functions [14][15][16][17][18].
Interestingly, topological classification can also exist in non-Hermitian systems, which, like Hermitian systems, can also define topological invariants [18][19][20][21].We discuss the non-trivial phase diagram of the lattice electromagnetic field theory of the non-Hermitian systems, which can further help us adequately understand the non-trivial phase figure near the exceptional point.In this paper, the non-Hermitian theory based on the discrete lattice model is used to explain why the apparent Maxwell modes of the transverse electric and transverse magnetic field polarization are present in the relevant areas of the parameter space.

The Hamiltonian of the non-Hermitian electromagnetic lattice mode
In this section, we consider a double-layer three-component non-Hermitian Lieb lattice system, one of which contains ABC three lattice points.The Hamiltonian of the system can be described in the following way: , where , , is the creation (annihilation) operator on the three lattice points; is the index of spin components, and ,   are the matrix elements' index.The first item is the transition between different lattice points within the same layer, as shown in Figure 1(a) below.In the following set of formulas: , and , , , ( )  along the and y axes are the closest neighborhood hoppings on the axes, and the relative hopping range is also plotted in Figure 1(a).We also constructed a 3D Lieb lattice structure, which not only features intra-layer hoppings, but also has interlayer hoppings structure, as displayed in Figure 1(b).The Hamiltonian is written in a real space, after the Fourier transform.The Hamiltonian of the system can be simply given as follows: † ( ) , where the Fourier transform is expressed as ) , and V is the periodic boundary system with unit volume; the wave function in momentum space, i.e.
1 2 , is a six-component wave function, and the Hamiltonian has the form of Equation (3) below: where 2 I is the 2 2  identity matrices, and 3 I is the 3 3  identity matrices, respectively.
where 2 sin( ), is the Bloch vector, and the distance between the lattices is a .For each momentum k  , the eigenenergy of the system is defined as ( ) 0 E k    , and the system has a flat band of zero energy in the middle of the three bands.Through the wave function of the energy band, we can define the topological quantity Berry phase to describe its topological state.Integrate momentum space into obtain the Chen number related to helicity, i.e. the Berry phase defined as ( ) , where c is the closed path, ( ) , the unipolar charge on the momentum space is Berry curvature ( ) , and k  is the intrinsic wave vector of the system at parameter k .Next we will consider the topological phase transition of the system in the near of the exceptional point 0 E  .The eigen-equation of the system is expressed as follows: ( ) .

Topological phase transtions near the exceptional point of non-Hermitian lattice magnetic
( ) H k  can be written as the following: , and is the momentum between layers of particles without external potential; c is the speed; kc  is the interlayer hoppings energy; Z is the strength coefficient of the interlayer hoppings and is also a complex number.12 21 g g  indicates that the first layer term second layer hoppings intensity is different.It is further defined that the interlayer hoppings intensity coefficient is regulated by two parameters:  and  , as in of the interlayer hoppings is defined, then: The characteristic frequency and characteristic velocity of the interlayer hoppings are kc , respectively.The corresponding Berry curvature, i.e. the field strength, is expressed , where the Berry phase is ( ) . When the wave vector is real, the topological invariant with Chen number is defined as follows: The helicity operator further characterizes the system as: The spin topology invariant is defined as It is revealed that the Maxwell's equations in their most basic form are topologically non-trivial, which only involves homogeneous isotropic media characterized by dielectric constant and permeability.The corresponding topological invariant is a number that describes the helicity dispersion in the medium relating to the phase of the energy gap.
This phenomenon and the physical meanings of topological invariants can be illustrated in  when, likewise, both the relative permittivity r  and the relative permeability r  are complex.

Conclusions
In conclusion, we have investigated the topological phase transitions of a non-Hermitian electromagnetic system near the exceptional point using a double-layer three-component non-Hermitian Lieb lattice system.By calculating the topological invariants, we have revealed the rich topological phases of the non-Hermitian lattice electromagnetic system.In the experiment, the non-Hermitian Lieb lattice system was realized by a cold-atom platform, and the topological phase transition could be observed through cold-atom manipulation.Our study expands the scope of electromagnetic system research to lattice systems, and provides an alternative perspective for studying the non-Hermitian topological phase transitions.
the nearest collar transition strength within and between the lattice primary envelope.The first term is the Hermitian copula of the second term.The third and fourth items are the coupling between the same index lattice points of different layers, i.e. the interlayer transition, as shown in Figure1(and fourth items do not have Hermitian conjugation.This is a non-Hermitian system of spin-1 spinor matrix Ŝ .

Figure1.
Figure1.Double-layer three-component non-Hermitian Lieb lattice system.(a)Hoppings between different lattice points within the same layer.A, B, and C are three different lattice points forming a single unit cell, as shown in the dashed line.a is the lattice constant, and t is the hopping magnitude.(b) The coupling between the interlayer hoppings.The black lines represent the intralayer hoppings, and the red line represents the interlayer hoppings.The double-layer three-component Lieb lattice model shown in Figure 1 is composed of three lattice points: A, B, and C, forming a protocell of the pseudospin-1 and showing the relative spin-flip hoppings intensity.Analogous to the scheme in a squarish optical lattice, this hopping can be implemented by using a laser beam.This corresponds to three spin states, with each form indicated by A   , 0 B  ,and C

Figure 2 .   and 2 
Figure 2. Topological phase diagram of surface waves (The yellow area is the range where transverse electric (TE) mode exists, and the blue area is the range where transverse magnetic (TM) mode exists).(a) The surface area where zero helicity mode exists.(b) The phases of the transverse electric and transverse magnetic modes are divided and described by equation .The blue region is the range where 2 0 z k  addition to the Berry phase, we can also define another expression of topological invariant through the winding number of the model, as shown below:

Figure 2 .r   and 1 r
Based on topological invariants of helicity, the topological structure of Maxwell surface waves can be divided into different phases according to two parameters -permittivity 0 lines.The four simple quadrants correspond to four different optical media respectively in Figure2.The topological phase diagram of the interface wave can be drawn according to Equation .Figure2(a) shows the surface pattern propagating on the medium dividing surface.TE and TM surface waves are present at each point of the dual polarization zone, but there is only one mode, as seen in Figure2(b).All surface patterns are displayed in Figure2(c).The real and imaginary frequency regions are divided by the lines 1    , where corresponds to Surface plasmon resonance at the planar interface[1].During the replacement process, the imaginary and real frequency regions are subsequently swapped.This causes the phase diagram of Figure2(c) to be divided into those shown in both Figure2(d) and 2(e).

Figure 2 (
d) is the image of the fine division of the magnetic (TM) mode and transverse electric (TE) mode by 2 0  when both the relative permittivity r  and the relative permeability  are complex.Finally, Figure2(e) shows the fine division of the phase diagram of transverse magnetic mode and transverse electric mode by 0