Low-dimensional optical chirality in complex potentials

Chirality is a universal feature in nature, as observed in fermion interactions and DNA helicity. Much attention has been given to chiral interactions of light, not only regarding its physical interpretation but also focusing on intriguing phenomena in excitation, absorption, refraction, and topological phase. Although recent progress in metamaterials has spurred artificial engineering of chirality, most approaches are founded on the same principle of the mixing of electric and magnetic responses. Here we propose nonmagnetic chiral interactions of light based on low-dimensional eigensystems. Exploiting the mixing of amplifying and decaying electric modes in a complex material, the low-dimensionality in polarization space having a chiral eigenstate is realized, in contrast to 2-dimensional eigensystems in previous approaches. The existence of optical spin black hole from low-dimensional chirality is predicted, and singular interactions between chiral waves are confirmed experimentally in parity-time-symmetric metamaterials.


INTRODUCTION
Complex potentials that violate the Hermitian condition have been treated restrictively to describe nonequilibrium processes [1]. However, since the pioneering work of Bender [2], it has been widely accepted that the condition of parity-time (PT) symmetry allows real eigenvalues, even in complex potentials with non-Hermitian Hamiltonians. The concept of PT symmetry has opened a pathway for handling complex potentials, overcoming traditional Hermitian restrictions and stimulating the field of complex quantum mechanics [3]. In contrast to Hermitian potentials, PT-symmetric potentials support regimes in which some eigenvalues are complex [2]. Accordingly, the phase of eigenvalues can be divided into a real and a complex regime, with a border line called the exceptional point (EP) [2,4,5] marking the onset of PT symmetry breaking.
Substantial researches have focused on optical analogues of PTsymmetric dynamics [4][5][6][7][8][9][10][11][12]. Based on the equivalence of the Schrodinger and paraxial-wave equations, the classical simulation of complex quantum mechanics has been tested [11]. Exotic behaviors of light have also been implemented in PT-symmetric potentials: asymmetric modal conversion [8,13,14], abnormal beams [4,12], unidirectional invisibility [15], and PT-symmetric resonances [6,7,10]. These phenomena originate from complex eigenstates near the EP, in relation to their skewness [8,14] and unidirectional modal conversion [4,13]. Although there has been an effort to observe PT-symmetric dynamics in polarization space [16] as well, the previous work has focused only on the observation of PT-symmetric phase transition of eigenstates around the EP, lacking the in-depth investigation in the context of chiral 'light-matter interactions': handedness-dependent optical phenomena through the interactions with two-dimensional (2D) or three-dimensional (3D) chiral structures. For example, because the PT-symmetric metamaterial in ref. [16] was considered as a meta-'surface' between air and substrate, the studies for the evolution of propagating waves along PT-symmetric 'bulk' materials, for the increase of singular dynamics at the EP, and for the incidences with arbitrary polarizations have been neglected.
In this paper, generalizing light interactions with PT-symmetric electrical materials, we investigate a new class of 2D chiral 'bulk materials' with complex potentials, which possess the low-dimensional (1D) eigensystem and can be applied to exhibit the design of arbitrary singular polarization state on the Poincare sphere. In contrast to previous approaches [17][18][19][20][21][22][23][24] for chiral optical materials, we derive chiral interactions from the mixing of amplifying and decaying electric responses, not from the mixing of electric and magnetic responses. We demonstrate that the dimensionality of PT-symmetric chiral system is reduced to one at the EP, leading to a perfectly singular modal helix. This provides a pathway toward chiral light-matter interactions fundamentally distinct from conventional optical chirality [17][18][19][20]23,24], all of them based upon 2-dimensional eigensystems. Unique properties arising from the low-dimensionality are experimentally demonstrated in high-index metamaterials. We also show the convergence of arbitrarily-polarized incidences to a single chiral eigenstate without any reflections, realizing an optical spin black hole.

CHIRAL INTERACTIONS IN PT-SYMMETRICALLY POLARIZED MATERIAL
To examine the role of chiral eigenstates in PT-symmetric material, here we focus on the 'chiral interaction' with singularity, extending the discussion in ref. [16] which focused on the eigenstate itself. Firstly, we study the modal transfer between CP modes through propagation. Utilizing the eigenstates and eigenvalues from Eq. (1) and employing the CP bases of vR, L = (1/2) 1/2 •(1, ±i) T , the transfer relation between the incident Einc = (ERI, ELI)CP T and transmitted field Etrn = (ERT, ELT)CP T is written as Etrn = MPTEinc using only structural (propagation distance d) and material (εr0, εi0, and εκ0) parameters (Supplementary Note 3).
A closer investigation of the transfer matrix MPT in Supplementary Note 3 provides a straightforward understanding of chiral interactions in PT-symmetric potentials. First, the inequality between off-diagonal terms (|tR→L| > |tL→R|) leads to an asymmetric modal conversion between the right-and left-CP (RCP and LCP) modes. Because the selfevolutions of CP modes are identical (tR→R = tL→L), the chiral response of the system is governed by the intermodal chirality CIM as which is obtained in Supplementary Note 3. Note that the intermodal chirality CIM is solely determined by the ratio of εi0 and εκ0, directly related to the degree of PT symmetry, i.e., λPT = (εκ0 2 -εi0 2 ) 1/2 . Accordingly, at the EP (εi0 = εκ0), a one-way chiral conversion CIM  ∞ from the RCP to LCP mode is achieved (Fig. 1i), as expected from the reduction of the eigensystem to a 1-dimensional LCP eigenstate (Fig. 1f). Before and after the EP, CIM decreases (Fig. 1i) due to two elliptically-polarized eigenstates (Figs 1e,1g). Note that this low-dimensional PT-chiral system (tR→R = tL→L and tR→L >> tL→R ~ 0) is fundamentally distinct from conventional chirality based on a 2dimensional Hermitian eigensystem of circular birefringence (tR→R ≠ tL→L) and zero intermodal coupling (tL→R = tR→L = 0) [18,20,22,23,25].
When the intermodal chirality CIM defines the unique origin of lowdimensional chirality in PT-symmetric material, the strength of the chiral conversion CCS (Fig. 1j) is determined by the competition between the intermodal transfer and the self-evolution (CCS R→L = |tR→L / tR→R|, CCS L→R = |tL→R / tL→L|, see Supplementary Note 4 for details). In agreement with the observations made for CIM, CCS R→L is always larger than CCS L→R , resulting in LCP-favored chiral conversion (Fig. 1j). Near the EP (see Supplementary Note 4, CCS R→L ~ 2πLeff•(εi0/εr0) and CCS L→R ~ 0), the chiral conversion becomes unidirectional, and its strength is solely determined by the material and structural parameters: εi0/εr0 and effective interaction length Leff = neff•d/Λ0 where neff is the effective index of the structure. We note that while the magnitude of the gain and loss εi0 contributes both to intermodal chirality CIM and chiral conversion CCS, the achievement of large neff, e.g., with unnaturally highindex metamaterials [26], enables strong chiral conversion (or large CCS) in the 'bulk', overcoming the material restriction on ±εi0. Figures 2a and 2b show the chirality of the transmitted wave for RCP and LCP incidences. While εi0/εr0 determines the regime of chiral transfers (oscillatory for εi0/εr0 < 1, and LCP-favored for εi0/εr0 ≥ 1), Leff describes the effect of the interaction length of PT-symmetric chiral materials. For large Leff, strong LCP chirality from the one-way chiral conversion is apparent near the EP, which emphasizes the role of the singularity. This chiral singularity forms the optical spin black hole (the south pole of the Poincaré sphere, Fig. 2c), to where all the states of polarization (SOP) converge (Supplementary Note 5 for details, and Supplementary Movie 1 for the spin black hole behavior at the EP), differentiating the analysis based on low-dimensional chirality from conventional chirality [17][18][19][20][21][22][23][24] or the analysis of the 'eigenstate' singularity in PT-symmetric metasurfaces [16]. Because the eigenstate of the polarization singularity can be designed systematically (e.g. see Supplementary Note 6 for a low-dimensional eigenstate of linear polarization (LP)), the optical spin black hole can be achieved in the entire polarization space of the Poincaré sphere, by combining the imaginary potential for linear and circular bases. We also reveal another unique property of low-dimensional chirality in Supplementary Note 7, chirality reversal for LP incidences (Supplementary Movie 2 for LP incidence), which is impossible for the optical activity in conventional chiral materials [27].

GIANT CHIRAL CONVERSION IN THE RESONANT STRUCTURE
To achieve a large chiral conversion within a compact footprint, a resonant structure for the effective increase of interaction length Leff = εr0 1/2 •d/Λ0 can be considered (Fig. 3a). The resonator is composed of PT-symmetric anisotropic material at the EP (length d = 834 nm, same parameters as those of Fig. 2), sandwiched between two metallic mirrors (of thickness δ). Here, S-matrix analysis [28] is utilized to calculate the frequency-dependent transmission, reflection, and field distributions inside the resonator. In detail, while the fields of the background air and mirrors are expressed as the linear combinations of y-and z-orthogonal bases, the field inside the PT-symmetric material is expressed by the nonorthogonal bases of veig1, 2, as E( , including forward (+) and backward (-) components. From the continuity condition of the electric field across the boundary (E and ∂xE), we derive the S-matrix relation of (EOy + , EOz + , EIy -, EIz -) T = S•(EIy + , EIz + , EOy -, EOz -) T . For mirrors with thicknesses of δ = 40, 50, 60, or 70 nm, the obtained Q values of the resonators are 620, 1500, 3600, and 8200, respectively. Figures 3b and 3c show the S-matrix calculated power ratio of LCP over RCP of a transmitted and reflected wave for the forward y-linear polarized incidence (EIy + = 1). Also shown in Fig. 3d are the S-matrixcalculated wave evolutions at the on-resonance condition of the 3/2 wavelength. Enhanced by the chiral standing wave in the resonator, a giant LCP-favored chirality of the transmitted wave is observed (Fig. 3b, IL/IR = 20 dB within Leff = 1.4 at Q = 8200; to compare, for the non-resonant structure, [IL/IR = 0.08 dB, Leff = 1.4] and [IL/IR = 20 dB, Leff = 1450]). It is also notable that, in contrast to the non-resonant structure where the reflection is absent, pure chiral reflection (RCP-only) results from the backward-propagating RCP waves inside the resonator (Fig.  3c).

LOW-DIMENSIONAL CHIRAL METAMATERIAL
We now investigate the experimental realization of the PTsymmetric chiral medium, focusing on the observation of the one-way chiral conversion CIM as a clear and direct evidence of the lowdimensionality. To achieve the complex anisotropic permittivity of Eq.
(1) with isotropic media, we employ the platform of metamaterials; which enables the use of local material parameter from the subwavelength structure and the implementation of gaugetransformed [29] PT-symmetry in a passive manner (Supplementary Note 2) through the designer permittivity. Figure 4 shows the realization of low-dimensional chiral metamaterial, achieved by transplanting the ideal point-wise anisotropic permittivity (Fig. 4a) into the subwavelength structure (see Appendix B and C for the fabrication and THz measurement). We emphasize that the chiral conversion CCS R→L = |tR→L / tR→R| ~ 2πLeff•(εi0/εr0) is directly proportional to neff = (μrεr) 1/2 . Considering that most of capacitive metamaterials (including split-ring resonators in [16]) have strong diamagnetic behavior (μr << 1) [30] hindering the realization high effective index, we employ the I-shaped metamaterial which provides an ultrahigh permittivity [26] (εr >> 1) and suppressed magnetic moments (μr ~ 1) [30]; achieving strong and purely electrical light-matter interaction in the THz regime. We also note that the multilayer extension can be obtained for I-shaped structures [26], and the operation condition of high effective index and purely electrical response can be satisfied in the optical regime as well, for example by utilizing hyperbolic metamaterials [31].
By changing θ (the coupling εκ0) in the fabricated sample, now we measure the θEP, where the one-way chiral conversion of singularity occurs with εκ0 = |Im[εy-εz]|. The experimentally measured intermodal chirality CIM(θ,ω) is shown each for dielectric (Fig. 5a) and metallic (Fig.  5b) state realization, in good agreement with the COMSOL simulation (Figs 5c and 5d, respectively). In both samples, θEP of singularity in (θ,ω) space, satisfying the one-way chiral conversion for the sensitive EP are observed (cross point of black dotted lines). For the dielectric state metamaterial with propagating waves inside, the large CIM = 17.4 dB is observed at θEP = 2.0°, and CIM = 16.4 dB is observed at θEP = 1.6° in the design in the metallic state with evanescent waves inside the metamaterial. Despite the different propagating features of the two regimes, the EP design derives the chiral interaction of light in terms of 'one-way chiral conversion' for both regimes, confirming the role of low-dimensionality with a singular chiral eigenstate. Note that the separated y and z local modes that are highly-concentrated within the gaps are well-converted to a single planewave-like beam due to the deep-subwavelength scale of the structure, conserving the pure spin angular momentum of light without additional orbital angular momentum. Dotted lines represent the condition of EPs in spectral and θ domains. All simulated results were obtained using COMSOL Multiphysics.

LOW-DIMENSIONAL MODAL HELIX IN GUIDED-WAVE STRUCTURES
While ref. [16] has been focused on the singular 'scattering' from PTsymmetric meta-'surfaces' which can be analyzed through the scattering matrix, the impact of 'eigenstates' defined the system Hamiltonian is critical for propagating waves along the 'bulk'. Extending the discussion to the guided-wave and the optical frequency, we propose a modal helix in an optical waveguide platform utilizing isotropic materials. The point-wise permittivity (Fig. 4a) is transplanted into the complex-strip waveguide as a passive form (Fig.  6a), where the lossy Ti layer (grey region, thickness tTi) under a lossless Si-strip waveguide imposes the selective decay of the z mode, which is well-separated from the y mode (Fig. 6b, εy for low-loss and εz for highloss). The structural parameters are designed to satisfy Re[εy] = Re [εz], and the coupling εyz is achieved with the deviation Δ, which breaks the orthogonality between the y and z polarized modes. It is worth mentioning that the sign change of εκ0 (=εyz) can also be controlled by the mirror offset of -Δ for the deterministic control of the handedness (Fig. 1c). Therefore, the chirality of the proposed modal helix has directionality from the sign reversal of Δ for the backward (-x) view, which is absent in the structural helicity [18,23]. Figures 6c and 6d show the COMSOL-calculated modal chirality and the difference between eigenvalues as a function of the structural parameters (tTi and Δ). Because the control of the Ti layer alters the complex part of ε, the two structural parameters provide three degrees of freedom (Re[ε], Im[ε], and εκ0), resulting in the single EP in the 2D parameter space (tTi = 19 nm, Δ = 91 nm, IL/IR = 21 dB, and 18 dB for modes 1 and 2). The finite modal chirality (~20 dB) originates in the separated intensity profiles of the y and z modes (Fig. 6b), resulting in the non-uniform local chirality (82 dB maximum, Fig. 6e). Therefore, the chiral guided-wave includes the additional orbital angular momentum from the varying wave-front, in contrast to the case of the subwavelength structure (Fig. 4), which supports a planewave. With an experimentally accessible geometry (IL/IR ≥ 10 dB in Δ = 80~100 nm) and the coalescence of eigenmodes (Fig. 6d), the complex-strip waveguide will be an ideal building block for chiral guided-wave devices and for the utilization of active materials such as GaInAsP. It is also worth mentioning that our approach based on the guided-wave platform can derive the low-dimensional chirality through the simple spatial displacement of the lossy dielectric waveguide, without the use of molecular designs for low-loss and high-loss components [16].  Δ=0). c shows the modal chirality by IL/IR as a function of Δ and tTi. d. The absolute value of the difference between eigenvalues as a function of Δ and tTi. The intensity profile and the local chirality (IL(y,z) / IR(y,z)) at the EP are shown in e. All results were obtained using COMSOL Multiphysics with an optical wavelength of Λ0=1500 nm. L11=190 nm, L12=300 nm, L21=620 nm, and L22=190 nm.

CONCLUSION
We proposed and investigated a new class of optical chiral interactions based on complex potentials, without any bi-anisotropic mixing of electric and magnetic dipoles [27]. Based on the mixing of amplifying and decaying responses and on the presence of the lowdimensional eigenstate at the EP, exotic chiral behaviors of one-way CP convergence are observed, which cannot be observed in conventional chiral or gyrotropic materials. The reduced dimensionality also enables reflectionless CP generation following the dynamics of optical spin black holes, which is impossible in conventional approaches based on Hermitian elements. We also emphasize that our result, supporting the saturation of linear polarizations to a single circular polarization (Supplementary Note 7), is distinct from conventional chirality, which maintains optical activity for linear polarizations. We demonstrated the physics of low-dimensional chirality by using ultrahigh index polar metamaterial [26] with two design strategies: utilizing propagating and evanescent waves. The results prove the existence of EP and the one-way chiral conversion in the spectral regime, and the manipulation of the angular property is achieved. As an application, a chiral waveguide using isotropic materials is also proposed, which can be achieved by transplanting the point-wise anisotropic permittivity of an ideal complex-potential. Compared to previous results for polarization tunability based on detuned dipoles [32], temporal retardation [33], and singular scattering [16], our study focuses on designing target eigenstates in a chiral form, also revealing the exotic phenomenon of optical spin black hole.
Our new findings of low-dimensional chirality will pave a route toward active chiral devices [25], such as on-chip guided-wave devices for chiral lasers, amplifiers, absorbers, and switches [34] as well as complex chiral metamaterials [9] and topological phases. From the evident correlation between complex potentials and optical chirality, we can also imagine the eigenstate with nontrivial optical spins in 'non-PT-symmetric' complex potentials, by utilizing supersymmetry technique [35,36] or the inverse design of an eigenstate in disordered media [37]. Based on the general framework of non-Hermitian physics, we note that our work can be further extended using different polarization bases (Supplementary Note 6) to enable SOP collection for the arbitrary designer polarization.

A. Density of optical chirality for complex eigenstates
For the time-harmonic field of E = E0·e iωt and B = B0·e iωt , the timevarying representation of the optical chirality density [21,22] From the definition of χ and the condition of weak coupling in the PT-symmetric system (εi0 ~ εκ0 << εr0), the chirality density of each eigenstate before and after the EP is now expressed as x e ) Im( 2 where exp(2•Im[β1, 2]x) = |E0| 2 ≡ Ue represents the amplifying and decaying electric field intensities after the EP (Fig. 1b, Ue = 1 before the EP). Herein, we adopt χ1, 2/Ue to express the energy-normalized chirality of the eigenstates. Note that χ1 ~ χ2 from the condition of weak coupling (line and symbol of Fig. 1c).

B. Fabrication process of THz chiral polar metamaterials
Serving as a flexible and vertically symmetric environment of a metamaterial, a polyimide solution (PI-2610, HD MicroSystems) was spin-coated (1 μm) onto a bare Si substrate and converted into a fully aromatic and insoluble polyimide (baked at 180°C for 30 min and cured at 350°C). A negative photoresist (AZnLOF2035, AZ Electronic Materials) was spin-coated and patterned using photolithography. Then, Au (100 nm) was evaporated on the Cr (10 nm) adhesion layer and patterned as crossed 'I'-shaped array structures via the lift-off process. Repeating the polyimide coating and curing (1 μm), singlelayered metamaterials were fabricated by peeling off the metamaterial layers from the substrate.

C. THz-TDS system for the measurement of intermodal chirality C IM
To generate a broadband THz source, a Ti:sapphire femtosecond oscillator was used (Mai-Tai, Spectra-physics, 80 MHz repetition rate, 100 fs pulse width, 800 nm central wavelength, and 1 W average power). The pulsed laser beam was focused onto a GaAs terahertz emitter (Tera-SED, Gigaoptics). The emitted THz wave was then focused onto the samples using a 2 mm spot diameter. The propagating THz radiation was detected through electro-optical sampling using a nonlinear ZnTe crystal. The THz-TDS system has a usable bandwidth of 0.1-2.6 THz and a signal-to-noise ratio greater than 10,000:1.

A. Broken symmetry in the real part of permittivity
To introduce the real-part imperfection (Re[εy] ≠ Re[εz]), we change Supplementary Eq. (1), including the real part difference Δεr0, as where η1, 2 is the new normalization factor for each eigenmode satisfying |veig1, 2| 2 = 1. Due to the complex form inside the square root of Supplementary Eq. (7), perfect coalescence does not occur if Δεr0 ≠ 0, as shown in Supplementary Figs 1a and 1b. From Supplementary Eq. (8), we can also calculate the modal chirality χ1, 2 (see Appendix A), which shows the effect of the imperfection on the modal chirality ( Supplementary Fig. 1c). From those results, we can determine the boundary of the tolerance for the modal chirality ( Supplementary Fig. 1d).

Supplementary Note 3. Transfer between RCP and LCP modes in the PT-symmetric chiral material
For an incident wave impinging upon the PT-symmetric material (Einc = Aeig1·veig1 + Aeig2·veig2), the transmitted wave at x = d is expressed as where veig1 and veig2 are complex eigenmodes in Supplementary Eq.
The magnitude of the chirality in the intermodal transfer between CP modes is quantified by CIM as Eq. (2) in the main manuscript.

Supplementary Note 4. Strength of chiral conversion C CS
After the EP (εi0 ≥ εκ0), CCS R→L and CCS L→R can be expressed as (15) When εi0 ~ εκ0, CCS R→L ~ 2πLeff•(εi0/εr0) and CCS L→R ~ 0 near the EP, showing unidirectional chiral conversion (Fig. 1j in where S0 is the radius of the Poincaré sphere, and (S1, S2, S3) is the Cartesian coordinate of the SOP on the sphere. To describe the general tendency of the SOP at the EP, we assume the 400 randomly polarized incidences (Ex = a + bi and Ey = c + di for the incidence where a, b, c, and d are real numbers with the uniform random distribution in [-1,1]) on the PT-symmetric optical potential for Fig. 2c in the main manuscript.

Supplementary Note 6. Low-dimensional linear polarization
Because the formulation of Supplementary Eq. (1) is based on the general framework for two-level PT-symmetric potentials [2,4,5], our analysis of the low-dimensional polarization can be extended beyond the case of chirality treated in the main manuscript. For example, instead of mixing amplifying y-LP and decaying z-LP modes, consider the mixing of amplifying RCP and decaying LCP modes, which is possible with the recent development of active chiral materials [6,7] and circular dichroism [8,9]. The eigenmodes of the PT-symmetric material are then expressed as veig1, 2  Supplementary Figure 3 shows the corresponding profiles of the eigenpolarizations for points d-h. For the Hermitian case (point d), the eigenmodes are linearly polarized (LP) due to the even and odd couplings of the RCP and LCP modes. When εi0 increases (0 < εi0 < εκ0, point e), the eigenmodes begin to converge. At the EP (εi0 = εκ0, point f), two LP eigenmodes have coalesced, and the reduction to a 1-dimensional LP basis is evident. After the EP (εi0 > εκ0, points g and h), each eigenmode is saturated to a CP mode (RCP, amplifying, and LCP, decaying).
Because a low-dimensional eigenstate can be designed deliberately in the form both of CP (main manuscript, poles on the Poincaré sphere) and LP modes ( Supplementary Fig. 3, the equator on the Poincaré sphere), all of the states on the Poincaré sphere can be designed as the low-dimensional EP, by combining the linear loss and circular dichroism. Therefore, the dynamics of optical spin black hole can be achieved on the entire Poincaré sphere.

Supplementary Figure 3. Spatial evolutions of eigenmodes in the low-dimensional LP material corresponding to points d-h
At the exceptional point f, the complex eigenmode has the singular form of an LP state (y-z). εr0 = 12.25, and εκ0 = εr0/10 3 > 0.

Supplementary Note 7. Chirality reversal for LP incidences
Focusing on the vicinity of the EP, chirality reversal can be achieved for the SOP of linear polarization (LP) incidences. For the LP incidence Einc = (ERI, ELI)CP T = (1/2) 1/2 •E0•(e -iθ , e iθ )CP T of an arbitrary polarized angle θ, the transmitted field Etrn at the EP is obtained from the transfer matrix MPT as E E From Supplementary Eq. (19), it is clear that LCP transmission can be controlled by the polarized angle θ of LP and CCS R→L |EP ( Supplementary Fig. 4a), whereas RCP transmission is invariant (see Supplementary Movie 2 for the SOP evolutions by the degree of PT symmetry with LP incidences; θ = 0, π/2, π, and 3π/2). In the strong chiral conversion regime of CCS R→L |EP > 2 (outside the dotted circle in Supplementary Fig. 4a), the transmitted wave is always left-handed (|ELT| > |ERT|) for all angles of θ. In contrast, in the regime of CCS R→L |EP ≤ 2, chirality reversal to a right-handed output is permitted for input angles of cos(2θ) < -CCS R→L |EP/2. A pure RCP transmission can also be achieved in the special case of CCS R→L |EP = 1 and z-LP (θ = 90 o , on the solid circle in Supplementary Fig. 4a), which is counterintuitive regarding the singular existence of the LCP modal helix. This paradoxical result arises from the unidirectional (RL) intermodal transfer, which leads to a completely destructive interference for the LCP mode only (e iθ + CCS R→L |EP•e -iθ ). Note that the observed phenomena of CP interference and chirality reversal are absent in conventional chiral materials [10] which are based on uncoupled LCP and RCP modes (tL→R = tR→L = 0), and thus, they maintain the LP state with natural optical rotation during propagation, a.k.a. optical activity. In practice, the regime of chirality reversal can be controlled by changing Leff or εκ0 (Supplementary Fig. 4b) based on the definition of CCS R→L |EP (=2πLeff•(εκ0/εr0)).

Supplementary Figure 4. Chirality reversal for LP incidences a.
The ratio of IL/IR at the EP for LP incidences is shown in a as a function of CCS R→L |EP and the polarized angle θ of LP and in b as a function of coupling permittivity (εκ0/εr0). The black dotted arrows denote the points at which CCS R→L |EP = 1. All the results are based on the transfer matrix method. εr0=6.5 and Leff=10 3 for a.

Supplementary Note 8. Realization of PT-symmetric permittivity in metamaterial platforms
To transplant the point-wise anisotropic permittivity of a PTsymmetric chiral material into the structure composed of isotropic materials, we first consider a metamaterial platform in a THz regime. As a unit element, we adopt the I-shaped patch ( Supplementary Fig. 5a) for each polarization, which induces the enhanced light-matter interaction through the deep subwavelength (~λ0/200) gap structure [11]. By crossing the yand z-unit elements ( Supplementary Fig. 5b), we obtain the metamaterial, which supports the well-known electrical response of the Lorentz model both for y-and z-polarizations, as where εpoly is the permittivity of the polyimide (εpoly = 3.238 -0.144i), ω0(y, z) is the characteristic frequency, γ(y, z) is the damping coefficient, and ωp(y, z) is the plasma frequency of the metamaterial for y-and z-polarizations. Because the near-field intensity distribution is widely separated for each polarization ( Supplementary Figs 5c and 5d), it is worth mentioning that the Lorentz permittivity curve of each polarization can be detuned independently to realize the required anisotropic metamaterial.  Supplementary Fig. 6. Also corresponding to the Supplementary Eq. (20), it is necessary to change the characteristic frequency of ω0 for the spectral shift ( Supplementary Fig. 6a), while the magnitude of the permittivity is mainly tuned with the plasma frequency ωp ( Supplementary Fig.  6b compared to 6c). Figure 6. The effect of changing the physical quantities (ω0, ωp, and γ) in the Lorentz model The permittivities ε(ω) are shown for the changes of a. the characteristic frequency ω0 (black: ω00, red: 1.2•ω00, blue: 0.8•ω00), b. the plasma frequency ωp (black: ωp0, red: 1.2•ωp0, blue: 0.8•ωp0), and c. the damping coefficient γ (black: γ0, red: 1.2•γ0, blue: 0.8•γ0). ω00 = 1.27THz, ωp0 = 8.2THz, and γ0 = 0.11THz. Solid (dotted) lines denote the Re[ε] (Im[ε]).

Supplementary
By considering the results of Supplementary Fig. 6, now we investigate the role of the structural parameters (a, g, L, and w of Supplementary Fig. 5a) for the design of PT-symmetric permittivity relating to the physical quantities (ω0, ωp, and γ). To satisfy the condition of Re[εy] = Re [εz] and Im[εz] < Im[εy] < 0 with two Lorentz curves in the dielectric regime, it is necessary to introduce the frequency shift between these curves by designing different characteristic frequencies of ω01 ≠ ω02 for I-shaped patches 1 and 2 (1, 2 represent y or z). To observe the EP with the spectral stability, we consider only the cases of the broadband design of EP (black circles in Supplementary Figs 7a,7b), neglecting cases that are too sensitive (green circles in Supplementary Figs  7a,7b). If I-shaped patch 1 in the high-frequency regime (ω01) supports a lower plasma frequency than that of I-shaped patch 2 (ωp1 < ωp2, Supplementary Fig. 7a), the condition with broad bandwidth is satisfied at the metallic state (Fig. 4e in the main manuscript), which supports an evanescent mode inside the PTsymmetric metamaterial. Meanwhile, if the design satisfying ω01 > ω02 and ωp1 > ωp2 is applied simultaneously (Supplementary Fig.  7b), the propagating mode can be utilized to observe the lowdimensionality (Fig. 4c in the main manuscript).
As shown in Supplementary Figs 7c-7d, increasing the parameters related to the 'length' of the patch (L and a) increases the plasma frequency ωp due to the large number of participating electrons (note that ωp 2 ~ Ne 2 /(mε0), where N is the density of electrons), while the characteristic frequency decreases due to the weak restoring force. Similarly, decreasing gap g ( Supplementary  Fig. 7e) increases the attractive force, yielding the reversed response of ω0↓ and ωp↑. Therefore, the strategy for lowdimensional evanescent waves ( Supplementary Fig. 7a) can be achieved with the manipulation of L, a, or g. However, by widening the patches (Supplementary Fig. 7f), we can increase the participating electrons (ωp↑) and the restoring force (ω0↑) simultaneously, enabling the case of Supplementary Fig. 7b, with low-dimensional propagating waves. Now, by changing the tilted angle θ (Supplementary Fig. 5a), we can obtain the lowdimensional chiral dynamics at the EP by obtaining the coupling between the y-and z-polarizations for both cases. Figure 7. The effect of changing the structural parameters The strategy for satisfying the condition of PT symmetry with a. ω01 > ω02 and ωp1 < ωp2 (low-dimensional evanescent waves) and b. ω01 > ω02 and ωp1 > ωp2 (low-dimensional propagating waves). Black (or green) circles denote the condition of low-dimensional EP with broad (or narrow) bandwidths. The COMSOL-calculated permittivities ε(ω) are shown for the changes of c. length L, d. arm length a, e. gap thickness g, and f. width w. The unchanged parameters are L = 30 μm, a = 20 μm, g = 1.5 μm, and w = 5 μm.