Exceptional Cones in 4D Parameter Space

The notion of synthetic dimensions has expanded the realm of topological physics to four dimensional (4D) space lately. In this work, non-Hermiticity is used as a synthetic parameter in PT-symmetric photonic crystals to study the topological physics in 4D non-Hermitian synthetic parameter space. We realize a 3D exceptional hypersurface (EHS) in such 4D parameter space, and the degeneracy points emerge due to the symmetry of synthetic parameters. We further demonstrate the existence of exceptional degenerate points (EDPs) on the EHS that originates from the chirality of exceptional points (EPs), and the exceptional surface near EDPs behaves like a Dirac cone. We further show that a very narrow reflection plateau can be found near these EDPs, and their sensitivity towards the PT-symmetry breaking environmental perturbation can make these degeneracy points useful in optical sensing and many other nonlinear and quantum optical applications.

On the other hand, one-dimensional photonic crystals provide a very simple and practical experimental platform to investigate topological properties of energy bands up to optical frequencies. Recently, through defining geometry parameters as synthetic dimensions, Weyl points were obtained in 3D parameter space [15,16] and interesting phenomena such as reflection phase vortex and the related topological interface states ("fermi arc") were observed in optical frequencies.
In the meantime, non-Hermitian systems and in particular, PT-symmetric systems, are attracting more recent attention [17]. In optical structures, PT-symmetric photonic modes can be obtained through adding gain and loss in materials, and have been used to realize unidirectional waveguides [18,19] and lasing [20][21][22]. The PT-symmetric phase transition can also induce coalescence of both eigenvalues and eigenvectors, leading to degeneracies called exceptional points (EPs) [17,23,24]. The novel properties at and near EPs can be used for various applications such as optical sensing [23][24][25], nonlinear optics [26,27] and slow light devices [28,29]. Adding gain and loss to topological materials can create a new platform to observe new phenomena, in particular those associated with degeneracies and singularities [30][31][32][33][34][35]. In general, the degeneracy points of Hermitian topological bands can be changed to EP structures of non-Hermitian topological bands. For examples, diabolic points can be changed to two EPs [36], Weyl points can be changed to EP rings [37][38][39] and a nodal line can be changed to an EP surface (ES) [40,41].
In this work, we propose a simple platform that is rich in physics and can be used to investigate PT-symmetric topological physics in 4D. Here, employing a 1D photonic crystal with a seven-layer unit cell, we construct a 4D parameter space based on wavevector k, two geometric parameters p, q and one non-Hermitian coefficient . A 3D EHS is obtained in such a 4D space. Through varying the non-Hermitian coefficient, we found two kinds of exceptional degenerate points (EDPs) in the EHS emerge in the process.
As photonic gap is closed at EDPs, and a very narrow gap can be introduced near the EDP before the gap's closing. As a result, a narrow reflection peak with high Q factor can be produced near the EDP. A small change of environment which breaks the PT-symmetry will change reflection drastically. This interesting property of EDPs can potentially be used in optical sensing.

Structure and the nodal line
In previous work, when adding the non-Hermitian perturbation, a Weyl point will change into an exceptional ring [38]. Therefore, introducing non-Hermiticity to the nodal line could deform it to an exceptional surface or not is a worth pursuing topic, and this is not hard to study by introducing the parameter space. In this paper, we first realize the Nodal line in the parameter space. Then we introduce the non-Hermiticity but preserve the PT symmetry of the system, we can clearly see the nodal line will change into exceptional tours.
The unit cell consists of a stack of seven layers incorporating gain and loss, as shown in Fig. 1(a). The thickness of the layers is: . (1) To be experimentally doable, we set and , and the (1 ( )) (1 cos( )) / 2 (1 cos( )) / 2 (1 cos( )) period of the unit cell is . Real parts of refractive index of the seven layers are set as and which are mirror-symmetric, while the imaginary parts of refractive index are anti-mirror-symmetric ( Fig. 1(a)). Such a PC possesses PTsymmetry. The non-Hermitian parameter represent the optical gain and loss inside the material, which is a tunable parameter in this work. To acquire gain material, we can use the semiconductors, such as InGaAsP multiple quantum wells [22]; another way is to dope the rare earth element into the conventional material [42]. For the loss term, we can also dope metallic element to control. The real part of optical length is , which does not depend on the values of p and q, so the central frequency of the band gap is almost fixed. The synthetic parameters , together with the Bloch wavevector k of the PC, constitute a 3D parameter space . The parameter   Based on transfer matrix method (TMM), we can calculate the band dispersion of PC in the 3D parameter space, where n denotes the n-th band of the PC. For the 1D PC, the first band gap between the first and the second band is located at Brillouin zone boundary . Therefore, we show the dispersion of these two bands in the 2D subspace, as shown in Fig. 1(b). We see that these two bands indeed have a set of degeneracy points (white dashed line) which form a nodal line, as shown by the black line in Fig. 1(c). The effective Hamiltonian for this nodal line can be retrieved based on the data calculated from transfer matrix as , with , and . In principle, the Hamiltonian can be derived from TMM [15], but we can also simplify the procedure based on the symmetry of such system near the nodal line. Note that the Hamiltonian has square terms in p and q due to the inversion symmetry of the synthetic space. While in the k dimension, we only keep the linear term, as the crossing of 1D PCs near the nodal line is always linear as shown in Fig.1(b). In Eq. (2), we neglect the term, which only determines the frequency of the nodal line. The nodal line is obtained 0.020 cos cos 2.4 cos cos 1.4 tells the nodal line lies on the plane, and shows the nodal line should satisfy . It is worthy to mention that the result is quite different from that in the former four-layer PC [15], in which only one single degeneracy point is found between the first and the second band, which is obtained by breaking the inversion symmetry.

Exceptional hypersurface and exceptional degenerate point
We already show that a nodal line can exist in the parameter space, and now we study in which manner the non-Hermitian parameter will deform the nodal line. As reported recently [40,43], after introducing gain and loss, the nodal line likely changes to ES under the protection of symmetries. Therefore, we also use TMM to calculate the EPs in the parameter space for different . For instance, when , the numerical result is given in Fig. 1(c). We see that each point in the nodal line evolves into an exceptional ring encircling this point, and hence all the EPs form a torus surrounding the nodal line [37][38][39][40]44]. The torus separates the space into two parts, the volume inside the torus is the PT-broken phase, which means two bands share the same imaginary parts of  q-k (c, d) space, we can clearly see two exceptional rings highlighted by yellow lines, which is labelled with 1-4 as in Fig. 1(c). On the four exceptional rings, the real and imaginary part of the eigenfrequencies are both degenerate. If we treat as another parameter, then the ES becomes a 3D EHS in the 4D parameter space .  Fig. 1(e). The increase of will make the region III and IV gradually disappear, as shown in Fig. 1(f).
To see this, we plot the EHS in the 2D subspace at three different values of in

Figs. 3(a-c). When
, the EHS is a donut as shown in Fig. 1(c). Increasing will expand the donut, and the inner ring of EHS will approach the center . When , the inner ring will collapse at the center, shown by blue stars in Fig. 1(d) and 3(b), which we refer it as EDP-1: . As , the inner ring disappears, as shown in Fig. 3(c) at . During the process, the topology of EHS changes, which will be discussed later. The outer ring of EHS approaches to another point at the same value of , as shown by the yellow star in Figs  EDP-1, while for EDP-2, we can find the EPs only along q axis, but produce a gap along p axis.

Hamiltonian and topological properties
To study topological properties of EDPs, we discuss Hamiltonian near these EDPs and then investigate geometric phase associated with the point. Figure 3(d) shows the shape of EHS in the subspace near EDP-1 for . We see that the two EHSs linearly touch at one point, and form an exceptional cone. To describe its behavior, we expand the parameters near EDP-1 as , , , and , and then effective where , , and . Note here, we keep the square term of p and q due to the inversion symmetry of parameter space. While in 1D PCs, the dispersion near the degenerate point is linear, so here we keep the first order of k. Usually, we can get the Hamiltonian from TMM [15], however we can also simplify the procedure by the symmetry of the system. Equation (3) shows the dispersion near EDP-1 has the following form , where , , , and . When , the change of EHS at the EDP-1 can be seen from the right side of Eq. (4), and we can directly get . It can be seen that the EHS near EDP-1 is a linear cone.
For EDP-2 shown in Fig. 3(e), the effective Hamiltonian will have the same form as Eq. (3) with different parameters as , and . Straightforward, the dispersion near EDP-2 is also given by Eq. (4), but the coefficients are , , , and .
Comparing these with those in EDP-1, we see that the values of , , and are almost the same, but the sign of changes. This sign flip makes the direction of the cone switch to horizontal axis ( ) although the conic shapes of EHS near EDP-1 and EDP-2 are similar.
To see the difference between EDPs in Fig. 3 with the linear crossing in Fig. 1(f), we need to analysis the eigenvector of EP in the EHS [45,46]. The eigenvector at the EP can be formally written as . Note that the linear crossing point here are still EPs due to at these points. This is the consequence of coalescence of EPs with the same chirality [45]. The EDPs, which labelled by blue and red stars, have opposite chirality. However, the linear crossing at in Fig. 1(f) is not an EP becasue the EPs with opposite chirality (blue and red lines in Fig. 1(f)) coalesce at the nodal line.
The topological properties of EDPs can be also confirmed through calculation the change of geometric phase before and after the transition point. When , we choose the yellow loop 1 and green loop 3 as shown in Figs. 1(c) and 3(a), and the values of along these two loops are shown by blue circles and red crosses in Fig. 3(f), respectively. The represents the polar angle according to the center of the two loops, and the arrows in Fig. 1(c) show the directions. We can see that the change of geometric phase on loop 1 and 3 is ; while the change of geometric phase on loop 2 and 4 is .
When the exceptional tours expand, the loop 1 and 2 will touch at , while the loop 3 and 4 will touch at , as shown in Fig. 3(b). When It is known that Dirac cone in band structure can lead to many interesting properties, such as zero-index medium [47], Klein tunneling [48], etc. For exceptional points, there are also some potential of application have be proposed, such as realizing unidirectional waveguides [18,19], and lasing [20][21][22] . Usually, the reflection or transmission spectra is widely used in sensors [49,50], in this section, we will show the EDPs can also be used as sensors by measuring the refection spectrum, which is quite sensitive to the change of refractive index in the surrounding.
In the following, we first investigate the transmission and reflection properties of the PCs to confirm the existence of EDPs. We design a finite PC with 25 unit cells shown in Fig. 5(a). The refractive index of environment medium on the left and right side are and respectively. The whole system is PT-symmetric when and PTsymmetry broken when . We first study transport properties near the EHS when . Figures 5(b) and 5(c) show the calculated reflections from left side and right side in the log scale, respectively. For comparsion, we plot two boundaries of EHS in space from Fig. 1(e) by black dashed rings. It can be seen that is enhanced on the outer ring and reduced on the inner ring. On the contrary, is reduced on the outer ring and enhanced on the inner ring. Correspondingly, the unitary transmission is obtained from both sides which is protected by reciprocal symmetry, and this phenomenon is referred as anisotropic transmission resonance (ATR) [51]. With increase of , the inner ring will shrink and the outer ring will expand. At a particular , the outer ring touches the boundary and the inner ring shrinks to one point, which corresponds to the EDPs defined in Fig. 3(b).
The EDPs found in the scattering calculation here happen at around , but the EDPs obtained from band structure calculations is at . The small deviation is due to the finite size effect.
To further confirm EDPs, we plot the calculated on the axis and the axis near EDP-1, as shown in Figs. 6(a) and 6(b). We see two reflection dips near the EDP-1 in both axes because EDP-1 is a vertical exceptional cone (Fig. 3(d)). We label the distance between these two dips as and . Near the EDP-1, two dips are close to each other and this makes a narrow plateau between them. When approaches the g 0 0.2698 critical value , and will approach zero, as shown by blue dots and red circles in Fig. 4(c). The inset in Fig. 6(c) gives the dependence of on and , which is linearly fitted with slope 1/2. This slope further confirms the Hamiltonian in Eq. (3). In Fig. 7(a), we also plot the reflection of the PCs with q=0. When changing the value of (marked by different color), we can clearly see the two dips fuse and then disappear. The two dips coalesce between .
Similar phenomena are observed near EDP-2 as shown in Figs. 6(d) and 6(e) for the change of reflection on the axis and axis. Since EDP-2 is a horizontal Dirac cone, so only the axis intersect with the dashed line of ATR and the axis q=1 does not intersect with dashed line before reaching . Therefore, we find two reflection dips on the p=1 axis, and no reflections dip on the q=1 axis. When approach the critical value , the value of will approach zero as shown in Fig. 6(f). The inset in Fig. 6(f) gives the dependence of on , which is linearly fitted with slope 1/2. This also confirms the Hamiltonian in Eq. (3) of main text.
Since small difference between environment index on two sides will break the PTsymmetry, so the EHS here can be used as optical sensing [23][24][25]43]. We consider the change of reflection plateau induced by the perturbation of environment index of PC near the EDP-1. For the system in Fig. 5(a), we firstly consider the environment index on both sides changed with the same value . Figure 7(b) shows reflections on the axis q=0 in the log scale for different index change . From the inset of Fig. 7(b), we  Fig. 7(c). Compared with the results in Fig. 4(d), the change of reflection plateau under different is distinguished. The dependence of reflection at EDP-1 for these two cases are given in Fig. 7(d) with red dots (with PT symmetry) and blue crosses (without PT symmetry). This shows that the reflection is quite sensitive to the environment perturbation which breaks PT-symmetry of the system.

Discussion
Previously, the synthetic dimension must be a degree of freedom, which can be used to simulate the behavior of a spatial dimension, for example, the 4D quantum hall effect [2,[52][53][54]. However, the concept of the synthetic dimension has been extended very recently to the scenario that an independent systematic parameter can also be regarded as a 'synthetic dimension'[1, 10], which is not necessary to mimic a real spatial dimension. We can introduce the synthetic dimension by using the parameter to either construct a lattice or tune the system. Similarly, we have introduced two geometric parameter p, q and the non-Hermitian term in our work. Although the cannot behave like a spatial dimension, it actually is a new dimension (independent of the other three dimensions). And we can exploit the physics in the Non-Hermitian 4D parameter space.
In Hermitian system, when the topological phase transition occurs, the bands will always have degenerate points, for example, the Dirac cone behaves as the phase transition point. However, the degenerate point of exceptional surface has not been discussed before.
Here we have observed the exceptional cone in the 4D parameter space firstly, which behaves topological phase transition of the exceptional surface. The exceptional cone is a linearly degenerate point of the exceptional surface. This is quite different from the previous case, that the linear crossing point of exceptional lines is always not an EP, because the two EPs with opposite chirality coalesce at this point, as shown in Fig. 1(f) [45,46].

Conclusion
Through introducing two parameters (p, q), we construct 4D parameter space in a 1D PC with a complex unit cell. Based on the symmetry of parameter space, we obtain exceptional cone in the EHS. Through calculation of the geometric Phase, we prove such degeneracy points are the consequence of topological properties of EPs. We