Parity-Time Symmetry Breaking beyond One Dimension: The Role of Degeneracy

We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere, and a group theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the T-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.


I. INTRODUCTION
Parity-Time (PT ) symmetric systems have attracted considerable interest in the past few years. These are nonhermitian systems which are invariant under the combined action of a parity and time-reversal operation. In the case of closed Hamiltonian systems the transition is from a regime of real energy eigenvalues to complex conjugate pairs of eigenvalues as the degree of non-hermiticity is increased [1][2][3]. For the case of open, scattering systems, the transition is seen in the eigenvalues of the scattering matrix, which can remain on the unit circle despite the non-hermiticity up to some threshold and then depart from it in pairs with inverse moduli [4][5][6]. In both cases the transition occurs when two eigenvalues coincide at an exceptional point (EP) which corresponds not to a degeneracy of the relevant operator but to a point at which it becomes defective (two eigenvectors coalesce), and hence is non-diagonalizable [7][8][9][10][11][12][13][14]. A major application of the theory of PT -symmetry breaking is to the wave equation of electromagnetism where the possibility of adding gain and loss in a PT -symmetric manner allows observation of many intriguing phenomena [4][5][6][15][16][17][18][19][20][21][22][23][24].
Essentially all of the work on PT -symmetry breaking has focused on one-dimensional (1D) or quasi-1D (coupled waveguide) systems. These systems can never have a high enough symmetry group to generate generic degeneracies. * Electronic address: li.ge@csi.cuny.edu In the current work we will focus on two-dimensional (2D) and three-dimensional (3D) PT -symmetric wave equations, which can have the new feature of continuous symmetries and generic degeneracies in the absence of the T -breaking non-hermitian perturbation. It will be shown that for such systems the PT -transition is absent if T is generically broken, meaning that they do not have a real spectrum even when the T -breaking is infinitesimal. However, if T is not generically broken, i.e. if some further discrete spatial symmetries are preserved, then it is possible that either the entire spectrum remains real over a finite interval (standard PT behavior) or a finite subset of the degenerate spectrum does. These scenarios are analyzed using a coupled-mode theory and generalized point groups. Our analysis also shows that other composite symmetries which can occur in higher dimension, such as RT where R represents rotation by π, can behave differently from PT and can exhibit a fully real spectrum when the corresponding PT system does not. Finally, we show that it is also possible for multimode coupling to restore the PT -symmetric phase, at finite T -breaking, if it is appropriately tuned.

II. ROLE OF DEGENERACY: A QUALITATIVE ARGUMENT
We now give a qualitative argument for the absence of PT -transition due to degeneracy, which will be quantified later using a coupled-mode theory. We focus on the case of closed, hamiltonian systems, H; analogous conclusions should hold for scattering systems with unimodularity of arXiv:1402.0428v2 [quant-ph] 13 Aug 2014 the eigenvalues replacing realness [25]. Since the hamiltonian has PT -symmetry, it is easily shown that the eigenfunctions of H have closure under the PT operation: in the symmetric "phase" each eigenfunction is mapped to itself; in the broken symmetry phase it is mapped to another in a finite set. Consider the non-degenerate case. When there is no T -breaking, then the system is hermitian and the eigenvalues, {k j }, are real. We will use a real parameter τ to denote the strength of the nonhermitian, T -breaking term in the hamiltonian. For a particular eigenvalue k j , it must move continuously as τ is increased from zero, because there are no singularities in the problem which would allow a jump (the derivative of the eigenvalue motion can be singular, e.g. at an exceptional point, but finite jumps never occur). Given this constraint, is it possible for k j to make a small step off the real axis? The answer is no, its movement must be confined to the real axis in the vicinity of τ = 0. The reason is that the PT -symmetry of H implies that under PT this eigenvalue is always mapped to its complex conjugate, which is trivially satisfied if k j remains real. But if k j moved off the real axis, then another eigenvalue would have to move off as well so as to form a complex conjugate pair with the same real part and opposite imaginary parts for this infinitesimal value of τ . But by the generic nondegeneracy assumption, all other eigenvalues are a finite distance away on the real axis and cannot possibly move enough to satisfy the symmetry as τ → 0. This explains why the PT -symmetry breaking point requires a finite τ for non-degenerate hamiltonians (and hence all hamiltonians in finite 1D systems). In both the hamiltonian case and the scattering case, pairs of eigenvalues must move a finite distance on the real axis or unit circle before they meet (at an EP) in order to undergo the transition while satisfying the exact PT -symmetry at all values of τ . Now consider a higher dimensional case with generic even degeneracy (the odd degeneracy, which can occur in 3D, will be discussed below). In this case the previous argument does not hold. As the T -breaking is turned on, it is possible for the paired eigenvalues to have a first-order splitting in τ while maintaining the required PT -symmetry, and indeed this becomes the generic case, and the PT -symmetric phase no longer occurs. In this case the degeneracy can be seen as the precursor of the PT -broken phase but no eigenfunction coalesce occurs as it would at an EP. We show rigorously some examples of this behavior below using a coupled-mode theory, and analyze the role of additional discrete symmetries and multimode coupling as means to protect or restore the PT -symmetric phase.
While the above argument is qualitative at this level, it is based on the rigorous requirement that if k j is an eigenvalue of a PT symmetric H, then so must be k * j , and the fact that the roots of a characteristic polynomial must be a continuous functional of the complex potential in the wave equation. Hence it should be straightforward to build a rigorous proof of the following statement: the existence of degeneracy at τ = 0 is a necessary condition for thresholdless PT breaking (transition for infinitesimal τ ). A second statement that also follows is: degeneracy is also a sufficient condition for a thresholdless transition if the T -breaking perturbation couples the degenerate modes of the τ = 0 system. We will not attempt rigorous mathematical proofs of these statements but will provide arguments supporting them below.

III. QUANTITATIVE STUDY OF 2D DISK: EVEN DEGENERACY
First we consider 2D circular systems of radius R, in which the eigenmodesφ( r) are determined by the Helmholtz equation We refer tok =ω/c as the eigenfrequency of the system and employ the Dirichlet boundary condition. The cavity dielectric function c ( r), gain and loss strength τ , and their spatial profile g( r) are real quantities. We adopt the convention that τ is non-negative, with which g( r) < 0 (> 0) represents gain (loss). We take a uniform c ( r) = n 2 which results in a separable system at τ = 0; each eigenmode of Eq. (1) has a well defined angular momentum m, and a second quantum number η indicates the number of intensity peaks in the radial direction. Henceforth we denote P φ as the parity operation about the φ, φ + π axis, and we choose P ≡ P 0 as the parity operation in the PT -symmetry, i.e. g(r, φ) = −g(r, −φ). Except for the m = 0 modes, the eigenmode spectrum at τ = 0 consists of degenerate pairs due to the rotational symmetry. The eigenmodes can be conveniently expressed as ϕ (p) m,η ( r) ∝ J m (nk m,η r) cos(mφ), J m (nk m,η r) sin(mφ), and we note that the corresponding quantities with and without the tilde are defined in general and at τ = 0, respectively. {k m,η } are real and the superscript p indicates even (e) or odd (o) parity about the PT -axis. The eigenmodes at τ = 0 form an orthogonal and complete basis, satisfying d rϕ j ( r)ϕ j ( r) = δ jj , in which we have used a single index j to represent {m, η, p}.
At τ = 0 the system becomes non-hermitian, and its eigenmodesφ( r) can be expanded in the hermitian basis {ϕ j ( r)}, i.e.φ( r) = j a j ϕ j ( r). For the superposition of two ϕ j 's to be in the PT -symmetric phase, i.e.φ( r) ∝ ϕ * (P r), it requires either (i) the ϕ j 's have the same parity and a j 's are in phase; or (ii) they have opposite parity and a j 's are π-out-of-phase. Condition (i) implies thatφ( r) is still real with a proper normalization, while condition (ii) requires a complexφ( r). These conditions can be easily generalized ifφ( r) contains more than two ϕ j 's.

A. Absence of PT transitions
We analyze the spectrum of disks with generically broken T symmetry, and consistent with our argument in Gain and loss strength τ Section II we find that the previously degenerate modes (at τ = 0) are now in the PT -broken phase at an arbitrarily small τ . Generically broken means that the only residual symmetry of the circle maintained in the nonhermitian system is the designated PT symmetry. One example of such a case is shown in Fig. 1, where the gain and loss profile increases linearly from φ = 0 to ∓π, retaining PT but breaking all other combinations of a discrete symmetry and T . The two pairs of eigenfrequencies shown have a dominant angular momentum m = 2, 3, respectively. For each pair, their frequencies are complex conjugates, as is demanded by PT symmetry, and their intensity profiles are mirror images of each other about the PT -axis; the one that overlaps strongly with the gain region in each pair has a positive Im[k], and it increases linearly with τ . To understand quantitatively the absence of the standard PT -phase transition at a finite τ , we develop a coupled-mode theory using the hermitian modes at τ = 0 as the basis: In this matrix equationτ ≡ τ /n 2 , G jj ≡ d rg( r)ϕ j ( r)ϕ j ( r) is the coupling between any two hermitian modes, E jj = k 2 j δ j,j is a diagonal matrix with the squared eigenvalues on the diagonal arranged in degenerate pairs; a is the column vector of the expansion coefficients in the {ϕ j ( r)} basis. Note that the coupledmode theory (2) is an exact restatement of Eq. (1) if all modes of the τ = 0 system are taken into account. Our approach here is different from the previous types of coupled-mode theory applied in PT -symmetric systems (see, for example, Ref. [15]), which consider the coupling between two eigenmodes of the subsystems, e.g. a waveguide with gain and a neighboring one with loss. We on the other hand, focus on the couplings between modes of the entire system at τ = 0 [16]. In the first part of the paper we will focus on the case in which (2) is truncated to degenerate sets of basis functions, because only for these is the effect of the the T -breaking first order in τ . Later we will use (2) to discuss multimode couplings beyond the degenerate sets.
For the degenerate pairs in the disk Eq. (2) is simply and G ee , G oo on the diagonal vanishes due to the PTsymmetry. We immediately find that which agrees nicely with the linear τ -dependence of Im[k] shown in Fig. 1. As expected from first-order perturbation theory, the rate at which the complex splitting of the eigenfrequencies increases depends on the gain and loss profile g( r), while the mixing ratio between each pair of degenerate modes does not. This can be directly seen from the 2 × 2 matrix on the left hand side of Eq. (3), whose diagonal and off-diagonal elements are the same, leading to the usual symmetric and anti-symmetric superpositions which can be easily checked to be PT -symmetric partners, satisfying (PT )φ j,± =φ j,∓ . Alternatively, we could have used the degenerate pair ϕ j,± ≡φ j,± in the hermitian basis instead of ϕ j , which has no off-diagonal elements in the perturbation, and it would have led to the same expression (4), but for the subsequent analysis it is useful to work in the parity basis.
To illustrate the crossover to the non-degenerate case where there is a PT -phase transition, we analyze Eq. (2) for two non-degenerate eigenvalues k j , k j , assuming that they are relatively close so that couplings to other eigenstates can be ignored. The two resulting eigenvalues due to the coupling of ϕ j and ϕ j are given bỹ which are necessarily real for a smallτ if k j = k j . The transition occurs whenτ becomes larger thañ It is clear thatτ TH approaches zero as k j → k j , and we recover Eq. (4) from Eq. (5) in this limit whenτ |G jj | 1. In the absence of degeneracy or quasi-degeneracy, multimode coupling will typically be important, and the PT transition strength cannot be quantitatively determined by the above expression (5). shown, and complex eigenvalues only appear above the lowest τTH 0.047. Inset: the symmetries of the system is described by the generalized dihedral group DT 2v with v = 6 (see Section III C). It includes, for example, P π 2 and R 2π 3 in addition to the PT -symmetry. Here P φ denotes the parity operation about the φ, φ + π axis, and R φ denotes clockwise rotation about the origin by φ. (b) Intensity profiles of the four broken-symmetry modes marked in (a) at τ = 0.16. The broken-symmetry modes can be non-degenerate (e.g. mode A and B), in which case they are simultaneous eigenfunctions of P π 2 and R 2π 3 ; they can also be degenerate (e.g. mode C1 and C2), in which case they can either be eigenfunctions of P π 2 or R 2π 3 but not both. Mode C1 and C2 plotted are eigenfunctions of P π 2 , and they can be linearly combined to be eigenfunctions of R 2π 3 [26].
B. Protected PT -symmetric phase due to additional discrete symmetries As noted above, the arguments for a thresholdless transition hold for generic T -breaking perturbations, and they can break down for cases in which imposition of gain and loss preserves additional discrete symmetries χ. In such cases the condition for thresholdless PT breaking for a pair of degenerate modes is that their coupling G eo must be finite. In the most extreme case, χ may decouple all degenerate pairs, i.e. all the corresponding G eo in Eq. (2) vanish, and a finite PT -transition thresholdτ TH for the entire spectrum is restored, which is due solely to the coupling between non-degenerate modes.
While the results we have obtained above follow straightforwardly from the perturbative coupled-mode approach, they have not been anticipated by earlier work; we believe this is due to the relatively simple T -breaking perturbations typically imposed in higher dimensions, which do preserve discrete symmetries. Such an example occurred in our own work [4], in which the scattering PT transitions were considered in a disk consisting of two uniform gain and loss halves. The protection of the symmetric phase for the entire spectrum at small τ can be attributed to χ ≡ P π 2 in this case. This is because the integrand of G eo for each degenerate pair is an odd function with respect to P π 2 , and thus all these G eo van- ish as a result. This observation is confirmed for more complicated g( r) satisfying the P π 2 symmetry, including the "radiation hazard" configuration shown in Fig. 2.
The situation discussed above, i.e. χ decouples all degenerate modes, is a special scenario. More generally, χ only decouples a subset of the degenerate modes, and a partial PT -breaking transition occurs as a result. One such example, the "PT -wheel" configuration, is shown in Fig. 3. It has a uniform gain and loss profile in each of the four quadrants, invariant with the P ± π 4 operations. For two degenerate modes with an even m, the integrand of the corresponding G eo is an odd function with respect to P ± π 4 . Therefore, G eo vanishes and these modes are protected from symmetry breaking at infinitesimal τ ; ϕ (e) j , ϕ (o) j need to couple to other more distant modes to break the PT -symmetry, which is the standard scenario described by Eq. (5) with a finite threshold τ TH [solid lines in Fig. 3(a)]. On the other hand, the integrand of G eo is an even function with respect to P ± π 4 for two degenerate modes with an odd m. Therefore, G eo does not vanish in this case and any pair of odd-m degenerate modes are in the PT -broken phase with infinitesimal τ [dashed lines in Fig. 3(a)]. We note that a similar partial transition in coupled waveguides was mentioned by Ctyroký in Ref. [27].

C. General symmetry analysis
The arguments and examples above support our general conjectures that degeneracy is a necessary but not sufficient condition for a thresholdless PT transition. As shown in the example of Fig. 1, when the T breaking perturbation preserves no discrete symmetries (i.e. is generic) then degeneracy is sufficient, but otherwise the situation must be analyzed separately depending on the nature of the remaining symmetries χ in the presence of T breaking. We will now do this using the generalized "point group" S ≡ {PT , χ}, where χ also includes the identity operator 1.
The examples examined above and in Ref. [27] have v angular blocks of equal area, and the corresponding S is a generalization of the dihedral group, D 2v , describing the symmetries of regular polygons of v sides. D 2v contains 2v elements, including 1, v − 1 rotations, and v reflections. We denote the relevant generalization of this group to our system as DT 2v . We see that due to the PT -symmetry, v needs to be an even number. The difference between DT 2v and D 2v is due to the effect of T breaking while PT is preserved. One finds that v elements in D 2v are no longer symmetry operators, but become so again when multiplied by T . In the simplest example, a disk with half uniform gain and loss considered in Ref. [4], S = DT 4 ≡ {1, PT , P π 2 , R π T }, where R φ denotes clockwise rotation about the origin by φ. In comparison, D 4 contains 1, P, P π 2 , and R π . Similarly, for the "PT -wheel" configuration shown in Fig. 3, and removing the 4 T -operators gives the original D 8 group. Although in these examples each angular block only contains gain or loss, this does not need to be the case. For example, the group property is not affected if we exchange gain and loss on a ring centered at the origin.
We want to find the conditions for vanishing coupling G eo between pairs of degenerate eigenfunctions, which will then lead to a protected PT transition for that set or subset of the states. Assume ϕ It is easy to see that their eigenvalues for S v can only be either 1 or -1. If S v contains T and this pair are degenerate in terms of S v , or if S v does not contain T and they are non-degenerate in terms of S v , it follows that the integrand of G eo is then an odd function with respect to S v , and G eo vanishes. P π 2 is such an operator S v for all pairs of ϕ (e) j , ϕ (o) j as indicated before, and so is R π T . The presence of either of these operators is sufficient to protect the entire spectrum from the effects of infinitesimal T breaking and restore the normal PT transition. For the generalized dihedral group DT 2v this criterion is satisfied when v is the sum of 2 and a multiple of 4, and both operators P π 2 , R π T appear (due to the closure property of a group given R π T = P π 2 PT ). Examples of this type were considered in Ref. [4] (DT 4 ) and here in Fig. 2 (DT 12 ). For the rest, i.e. DT 2v with v that is a multiple of 4 (such as DT 8 in Fig. 3), they do not include either P π 2 or R π T , and there are always some G eo that do not vanish, and an entirely real spectrum cannot be maintained at infinitesimal T -breaking. The one shown in the inset of (a) belongs to the former, and the gain and loss gradient is given by g(r, φ) = φ/π(r > R/5; 0 < φ < π), −(φ + π)/π(r > R/5; −π < φ < 0), 0(r ≤ R/5). In (a) only the two eigenfrequencies with the lowest τTH are shown. (b) shows the intensity profile of the marked mode at τ = 0.233. The intensity profile of its RT -symmetric partner is obtained by rotating (b) by π.
It is interesting to note that these two different scenarios with the generalized dihedral group can be differentiated by the presence or absence of RT -symmetry (where R ≡ R π is rotation by π and equivalent to inversion in 2D systems). It has been speculated that parity in PT -symmetric structures can be replaced by any linear symmetry operator, such as rotation or inversion, and the same spectral phase transition behavior will be observed as for PT . Here, when we take into account the role of degeneracy, we find that PT systems can have qualitatively different behavior from RT . Whereas RT always give a finite τ transition for the full spectrum, PT can give all three behaviors: finite τ , infinitesimal τ and a mixture of both.
This statement about RT generalizes beyond the DT 2v cases we have examined. We show in Fig. 4 a case similar to that of Fig. 1 but with g( r) increasing clockwise from −1 at φ = 0 to 0 at φ = −π instead. It is described by a cyclic group of order 2 (i.e. {1, RT }), and the system maintains a real spectrum until τ surpasses τ TH 0.22, because of the RT symmetry as discussed above. It is possible to apply the group theoretical analysis we have done above for DT 2v to generalized cyclic groups CT 2v (v is an integer). The original cyclic group C 2v we consider here consist of 2v rotations, which can be ordered by R π v , R 2π v , . . . , R 2π (= 1), and R π v is the generator of the group. In the generalized cyclic group, half of these rotation operators, at the odd positions in the given order, are multiplied by T , and R π v T becomes the generator of the group. For example, CT 4 = {R π 2 T , R, R 3π 2 T , 1} and CT 6 = {R π 3 T , R 2π 3 , RT , R 4π 3 , R 5π 3 T , 1}. CT 2v (v is odd) contains RT and gives a protected symmetric phase at infinitesimal T -breaking. For the rest of CT 2v (i.e. v is even), they do not include RT and a partial transition takes place, similar to that shown in Fig. 3.
We note that for all generalized point groups with T -breaking, there are always half of the elements that contain T . This observation can be proved rigorously using the closure property of a group, which holds in higher dimensions as well. This finding serves as a guideline to find other generalized point groups with T -breaking. For example, another way of generalizing the dihedral group D 2v with PT symmetry is multiplying all the v reflection operators in it by T . Since P π 2 and RT are not among the resulting elements, the spectrum always becomes (partially) complex at infinitesimal τ . Two examples of this group, denoted by MT 2v with v = 3, 4, are given in Fig. 5. They each have 2v angular blocks, instead of v as in DT 2v . We also note that we cannot simply multiply each element in a generalized point group by T to get another one. Although this procedure does not violate the guideline mentioned above, it removes 1 which is required by any group. We leave more complicated groups S derived from higher symmetry objects with imposed PT perturbations for future work. We will however discuss the case of the sphere briefly below. IV. 3D SPHERE: ODD DEGENERACY So far we have assumed pairwise degeneracy. What happens in the case of odd degeneracy? Our general arguments presented above suggest that either all degenerate eigenvalues will become complex for an infinitesimal value of τ , or an even number will be protected by additional symmetries, leaving at least some fraction of the spectrum vulnerable to infinitesimal perturbations. Here we analyze the scalar modes in a uniform sphere as an example. The hermitian eigenmodes at τ = 0 of each angular momentum l and radial quantum number η can be expressed as j η (nk l,η r)Y l,m (θ, φ), which have a (2l + 1)-fold degeneracy. Here m is the z-component of l, which we choose to be in the vertical direction, and j η , Y l,m are the spherical Bessel function and the spherical Harmonics, respectively. We define the PT -symmetry with respect to the φ = 0 plane, and for convenience we choose our basis as the parity eigenmodes about this plane, i.e. use Y l,0 (θ, φ), Y 2i ∝ sin(mφ) to denote their angular dependency. Since Y l,0 (θ, φ) is uniform in φ and thus even about the PT -plane, it couples to Y l,m . For the simplest non-trivial case with l = 1, Eq. (2) can be approximated by the following 3 × 3 form: Note that the degeneracy reduces the diagonal matrix E on right hand side of Eq.
(2) to a scalar k 2 l,η . This results in a profound influence on the multimode coupling: one of the resulting eigenfrequeices stays unchanged and real, and its wave function is given by G o0 Y (e) l,1 − G eo Y l,0 with a proper normalization. Note that this mixing is between the two hermitian modes that do not couple directly; they indirectly couple via Y (o) l,1 , which however does not appear in the mixing. Thus this mode is the analogue of a "dark state" in atomic physics, and its form does not depend on the specific values of the nonvanishing coupling G o0 , G eo . The other twok are complex conjugates, whose imaginary parts again display a linear dependence on the T -breaking: All three hermitian modes are present in the two corresponding non-hermitian modes, which resemble the "bright states" but with complex eigenvalues. We find similar results for l > 1, and we therefore conclude that in this case of odd degeneracy the standard PT -phase transition is also absent, but one has a partial PT -breaking transition, with a different fraction of the degenerate spectrum undergoing the transition for different values of l.
V. MULTIMODE COUPLING: RESTORING THE PT -SYMMETRIC PHASE AT FINITE τ Multimode coupling can be important even in the case of even degeneracy. We note that some eigenfrequencies in our 2D circular systems exhibit a more complicated dependence on τ in the PT -broken phase [see the solid lines in Fig. 6(a)]. Such nonmonotonic behaviors are signatures of multimode coupling, and in the case shown in Fig. 6(a) they are primarily due to the couplings of two degenerate m = 2 modes and the m = 0 mode closest to them in frequency, which we denote ϕ (e) 2 , ϕ (o) 2 , ϕ 0 and the corresponding eigenfrequencies k 2 , k 0 . ϕ 0 contributes a significant fraction to the broken symmetry modes, as can be seen from their intensity profiles near the origin in Fig. 6(b). The φ-dependence of these three modes is similar to the three l = 1 modes analyzed in the sphere, which leads to a vanished G e0 (not G eo ) and a similar 3 × 3 coupling matrix to that in Eq. (7). As a result, two eigenmodes are in the broken-symmetry phase with infinitesimal T -breaking and the other one stays in the symmetric phase. The existence of the latter is determined solely by the fact that these non-hermitian eigenfrequencies are given by the roots of a cubic equation with real coefficients. The latter is well known but has never found its way into non-hermitian systems as far as we know, since it is very rare to find real eigenvalues systematically in non-hermitian systems without the PTsymmetry. These PT -symmetric modes are no longer dark modes though; the bottom right diagonal element in the coupling matrix now becomes k 2 2 /k 2 0 instead of unity, and as a result, ϕ (e) 2 , ϕ (o) 2 , ϕ 0 are present in all three resulting non-hermitian modes. We further note that the "gulfs" near τ = 0.23 can develop into a restored PT -symmetric phase, if the τ -dependent discriminant of the aforementioned cubic equation becomes greater than zero. This condition can be satisfied, for example, if G o0 increases by more than 5%; at |G o0 /G eo | 3.57, this restored PTsymmetric phase is located at τ ∈ [0.152, 0.166] [see the dashed line in Fig. 6(a)].

VI. DISCUSSION AND CONCLUSIONS
Degeneracies in higher-dimensional systems are likely to be imperfect, especially for fabricated devices in pho-tonics. This will lead to a finite but small PT -transition threshold τ TH . Since it is also very difficult to probe the system near τ = 0, the results we have presented hold qualitatively as long as τ TH is lower than the minimum τ reachable. In fact, in the presence of a global or a local defect satisfying the PT -symmetry, one can tune the system across the symmetry transition threshold by controlling the amplitude of the defect. This scheme may benefit certain applications of PT -symmetry based devices, such as sensing and switching. This approach to a quasi-degenerate spectrum at τ = 0 can be regarded as coupling localized modes in momentum space; another approach is coupling two localized modes in the real space, and one example is the one-dimensional PT -symmetric random chain studied in Ref. [28]. We also note that in Ref. [29] the authors found a thresholdless condition for asymmetric modal conversion in quasi-1D waveguides with longitudinal index modulation ∆n(z) and attributed it to the spontaneous breaking of PT -symmetry. However this situation was first studied by Greenberg and Orenstein [30,31], whose analysis makes it clear that this phenomenon is not related to PT symmetry breaking in our sense.
In summary, we have shown that the standard PTtransition is absent for systems with degeneracies in the absence of T -breaking as long as the perturbations are generic, i.e. without preserving additional discrete symmetries. As a result, a real spectrum cannot be maintained even at infinitesimal strength of gain and loss. If the T -breaking also preserves additional discrete symmetries, the standard PT -transition can be restored for certain subsets of the degenerate spectrum or even its entirety. We have applied a coupled mode theory to analyze these behaviors and given a group theoretical description of the "protected" transitions, which also shows that qualitatively different behaviors can take place in PT and RT symmetric systems.