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Exceptional Points of Non-hermitian Operators

Exceptional points associated with non-hermitian operators, i.e. operators being non-hermitian for real parameter values, are investigated. The specific characteristics of the eigenfunctions at the exceptional point are worked out. Within the domain of real parameters the exceptional points are the points where eigenvalues switch from real to complex values. These and other results are exemplified by a classical problem leading to exceptional points of a non-hermitian matrix.

There are a number of phenomena, where the physical effect of an EP has been at least indirectly observed. Laser induced ionisation states of atoms [9] are a clear manifestation of an EP even though in [9], it has not been analysed as such. A recent theoretical paper [10] shows that, for a suitable choice of parameters associated with an EP, the only acoustic modes in an absorptive medium are circular polarized waves with one specific orientation for a given EP. Similarly in optics, experimental observations in absorptive media [11] reveal the existence of handedness since the stable mode of light propagation is either a left or a right circular polarized wave for appropriately chosen parameters. This has been interpreted in [12] in terms of EPs. A particular resonant behavior of atom waves in crystals of light [13] has been interpreted [14] in terms of EPs. While absorption is essential in all cases, some situations clearly point to a chiral behavior of an EP. In fact, the wave function at the EP has been shown to have definitive chiral character [15] and this has been experimentally confirmed recently [16].
In a previous experiment [17], EPs have been investigated in a flat microwave cavity. Major findings have been the confirmation of a forth order branch point of the coalescing wave functions and -depending on the path in the complex λ-plane -level avoidance associated with width crossing or level crossing with width avoidance. These results are the consequence of the topological structure of Riemann sheets at a branch point [4]. The experiment thus showed that this topology is a physical reality.
In the following we carry the analysis further in that we investigate the EPs of H 0 + λH 1 when H 0 or H 1 or both are no longer hermitian. As a consequence, even for real values of λ, the problem H 0 + λH 1 is no longer hermitian. This lack of hermiticity is different in nature from that discussed above where dissipation -like for instance in the optical model in nuclear physics [18] -makes H 0 + λH 1 non-hermitian for complex λ. There is a great variety of problems in the literature where either the perturbation or the full Hamiltonian is non-hermitian. Boson mapping [19], effective interactions [20] and the the random phase approximation (RPA) in many body theory [21] yield non-hermitian operators. More recently a wider class of non-hermitian Hamiltonians has been proposed to address specific symmetries [22] or transitional points in specific delocalisation models [23]. These suggestions have led to a further thorough study [24] of non-hermitian operators.
The present study is motivated by the classical problem of two coupled damped oscillators. It gives rise to non-hermitian matrices in a natural way. The problem is stated in the following section. The ensuing general treatment of section three yields new insights and special features regarding level repulsion. It is shown that the change from a complex to a pure real spectrum of a (real) non-hermitian matrix under variation of the (real) parameter λ is due to the occurrence of a real EP. As expected, the coalescence at the EP of two complex eigenvalues into one real eigenvalue (which then bifurcates in two real eigenvalues) yields only one eigenfunction in contrast to the usual two for a genuine degeneracy. A typical example is the instability point of the RPA. In addition, the pattern of level repulsion is distinctly different from that of a hermitian problem: the levels approach each other in the form of a cusp and not in a smooth way as is the case for hermitian H 0 and H 1 . These general findings are illustrated in section four where the example of section two is resumed.
We stress that the present paper focusses upon EPs and not on the study of non-hermitian operators as such. A summary and suggestion is given in the last section.

II. TWO COUPLED DAMPED OSCILLATORS
As a first illustration we consider a simple classical case of two damped coupled oscillators in one dimension. Denoting by p 1 , p 2 , q 1 , q 2 the momenta and spatial coordinates of two point particles of equal mass the equations of motion read for the driven system d dt where ω j − ik j , j = 1, 2 are essentially the damped frequencies without coupling and f and g are the coupling spring constant and damping of the coupling, respectively. The driving force is assumed to be oscillatory with one single frequency and acting on each particle with amplitude c j . Here we are interested only in the stationary solution being the solution of the inhomogeneous equation which reads Resonances occur for the real values ω of the complex solutions of the secular equation det |iω − M| = 0 (4) and EPs occur for the complex values ω where is fulfilled simultaneously together with Eq.(4). We choose the parameter f as the second variable needed to enforce the simultaneous solution of Eqs. (4) and (5) and keep the other parameters of M fixed, but of course any other preference -like choosing g -would be just as good and not alter the essential results. Thus we encounter the problem of finding the EPs of the matrix problem with and Note that M 0 and M 1 are not symmetric. Before we turn to explicit solutions and characteristics of Eqs. (4) and (5) we first address the general problem of EPs of non-hermitian matrices.

III. GENERAL NON-HERMITIAN CASE
Like for the hermitian operators the behavior around an EP can be described locally by a 2 × 2 matrix. The reduction of an N dimensional to a two dimensional problem is given below. We always consider a situation where the unperturbed problem, denoted by H 0 , or h 0 for the two dimensional case, is assumed to be diagonal. At first we discuss the two dimensional case and assume that the Jordan decomposition of the non-hermitian perturbation h 1 , i.e. h 1 = SJS −1 , yields a diagonal matrix J. We thus consider Note that for h 1 to be symmetric we would have φ 1 = φ 2 , i.e. S would be orthogonal. For convenience we have exploited the freedom to use normalized column vectors in S. The two eigenvalues of h are given by with the discriminant The two levels coalesce when D = 0 that is for In terms of Eq.(9) the parameters are The spectrum is complex between the EPs at λ + EP = 7.3 . . . and λ − EP = 3.4 . . ..
Note that even when all parameters are real the two EPs can now occur on the real axis. It happens when the signs of φ 1 and φ 2 are different. The implication is that the spectrum is no longer real when λ lies between λ − EP and λ + EP . In Fig.1 we display a typical case of a spectrum of that nature 1 . We recall that at the EPs, where the real spectrum ends or begins, only one eigenfunction exists of the two by two matrix problem; its precise form is given below. Here we stress a general property of a matrix at an EP: the Jordan decomposition of h(λ EP ) = T JT −1 yields a non-diagonal matrix J given by the standard form 1 in Fig.1 of [22] the spectra E k (N ) exhibit manifestations of real EPs in the variable N .
whereas for all points λ = λ EP the matrix J λ is diagonal and reads We recall that T is orthogonal (or unitary) only if h(λ) is hermitian. While it is well known that a non-hermitian operator can have a non-real spectrum, the deviation from the hermitian case has, for the real part of the spectrum, distinct consequences for the shape of level repulsions; this is exemplified in the following section where typical results of the two oscillators introduced in the previous section are presented.
Yet the local behavior at the EP is basically the same as for the hermitian case. It is clear from Eqs. (11) and (12) that the two eigenvalues are connected at the square root branch points situated at λ = λ ± EP just as in the hermitian case. The difference arises in the eigenfunction at the EP. Recall that the coalescence of two eigenvalues at the EP is not to be confused with a true degeneracy in that there is only one eigenfunction at the EP. At λ ± EP this single and unique eigenfunction (up to a possible common factor) turns now out to be We note that, in contrast to the symmetric case, the left hand eigenfunction at the EP (or the eigenfunction of the adjoint problem) is now different and reads As a result, the relation still prevails just as in the symmetric case. The basis vectors |j , j = 1, 2 refer to the eigenstates of h 0 . We only mention the special case where either φ 1 or φ 2 assume the value 0 or π/2: in contrast to the hermitian case the confluence of the two EPs does not invoke a true degeneracy with two independent eigenvectors even though it gives rise -for real parameters -to a real level crossing; also Eq. (18) is upheld at such points.
To summarize: for real matrix elements and different signs of the angles, the spectrum is complex between the two real EPs. Regarding the wave function the quotient of the amplitudes of the two coalescing wave functions deviates from that of the hermitian case. The genuinely complex superposition of the eigenfunction at the EP remains, however. For real angles φ j (of equal sign), the fixed phase difference of ±π/2 between the basis states at λ ± EP occurs just as in the symmetric case, but the ratio of the modulus of the amplitudes is in general not equal to unity. In addition, as the angles may be complex, not only the ratio of the modulus but also the phase difference can be different from the hermitian case. The important point is, however, that Eq.(16) describes, up to a common factor, the only possible eigenmode at the EP.
The reduction locally of an N -dimensional problem to the appropriate effective two-dimensional problem around an EP is, mutatis mutandis, achieved along the same lines as for hermitian operators [15]. Owing to their vanishing norm the two coalescing eigenfunctions dominate the complete set of all N normalized eigenfunctions in the immediate vicinity of an EP. The expansion of the N -dimensional vector in terms of the complete bi-orthogonal set , contains virtually only two terms for λ → λ EP . In fact we may write up to a common factor; the two non-zero positions are given by the values k, k + 1 for which |χ k (λ) and |χ k+1 (λ) coalesce. From Eqs. (19,20) the effective two dimensions for any |ψ(λ) becomes obvious within a small neighbourhood of λ EP .
We do not discuss cases where the Jordan decompositions of h 0 or h 1 or both do not yield diagonal but block matrices as this does not affect the local behavior at an EP. This should not be confused with the fact that in all cases an EP of the full problem H 0 + λH 1 (or h 0 + λh 1 ) is characterized by a non-diagonal matrix J (see Eq. (14)) of its Jordan decomposition.

IV. EXAMPLES
While there are various physical reasons to consider non-hermitian operators, we here focus on the simple mechanical model introduced in section two. Note that the model can be easily translated into a corresponding electronic setting using two coupled R-L-C circuits. EPs can always be found for some complex values of the pair (ω, f ), but a complex value of the spring constant f does not appear physical. This is in contrast to quantum mechanical cases discussed previously [4] where dissipation is often described by an effective complex interaction. In the classical model we therefore introduce the damping term of the coupling denoting its strength by the real constant g. For given values of ω j and k j we determine g such that an EP occurs at a real value of f . The associated two coalescing energies are then complex describing a damped oscillation being sustained by the driving force. The particular model reduces the resultant of Eqs. (4) and (5) to a polynomial of fifth order in f which is readily solved. The symmetry of the model implies that an EP at the pair (ω EP , f EP ) is always associated with an EP at (−ω * EP , −f EP ); we focus our attention on positive f , i.e. a repulsive spring, and the physical requirement ℑω EP < 0 implying proper damping. Obviously, EPs can occur only if either ω 1 = ω 2 or k 1 = k 2 or both as otherwise a genuine degeneracy is found for f = g = 0.
To get a good understanding for the EPs we first turn our attention to the behavior of the eigenvalues of iM in Eq.(2) as functions of f and g; the eigenvalues are the solutions of Eq.(4). In Fig.2 the real and imaginary parts of two levels coalescing at an EP very close to the real g-axis are plotted versus g using for f a real fixed value chosen such that an EP occurs in the vicinity of g = 0.00075. The parameters are the same as in Fig.2 The level repulsion of the real parts and the expected crossing [4] of the imaginary parts are distinctly different in shape from the usual appearance for hermitian matrices. The cusp originates from the plain square root behavior of the singularity, i.e. the difference between the two levels is controlled by ∼ √ λ − λ EP (in Fig.2  λ ≡ g); this is in contrast to the two complex conjugate EPs occurring in the hermitian case where this difference is controlled by ∼ (λ − λ EP )(λ − λ * EP ) = (λ − ℜλ EP ) 2 + (ℑλ EP ) 2 and hence produces a smooth approach. For illustration a typical shape of the latter is drawn schematically in the inset. The deviation from a hermitian case is even more dramatic when the two levels are plotted against f for a fixed g = 0.00075 as illustrated in Fig.3. Yet, the pattern is understood by the same mechanism being a square root branch cut running along the real f -axis and having a branch point at f ≈ 1.
As we deal with a classical system we now turn to the behavior of the complex amplitudes q 1 (ω) and q 2 (ω) of Eq.(3). The overall oscillatory time behavior is of no interest, we rather concentrate on the modulus and the phase difference. In general these complex amplitudes depend on the amplitudes c j of the driving force by Eq. (3). However, as discussed in the previous section, at the EP there is only one mode possible given by Eq.(16) up to a global constant factor. In other words, at the EP the ratio of the amplitudes of the two coalescing modes is given by i cot φ 1 / cot φ 2 which is a function of only M 0 and M 1 and is independent of a driving force, i.e. of the c j . At close distance to ω EP the correct value for the ratio must therefore approximately be attained. This is well demonstrated in Fig.4 where the moduli and phases are plotted against the driving frequency for the same parameters as in the previous figures giving rise to an EP at ω EP = 10.05 − 0.15 i. The top drawings are the moduli of the amplitudes with the left one referring to c 1 = i, c 2 = 1 and the right one to c 1 = −i, c 2 = 1; below are the respective phases. The former choice (left column in Fig.4) is driving the two masses with equal strength but with a leading phase of π/2 for the first mass. From the drawing we see that q 1 (ω)/q 2 (ω) ≈ +i through the whole resonance. This value is almost equal to the exact value being q 1 (ω EP )/q 2 (ω EP ) = 0.0049 + i 1.000 . . . (see (21) below). If, however, the 'incorrect' input is enforced like the lagging phase (right column), there is more variation in the response. Yet, at the resonance the phase difference still is +π/2, i.e opposite to the driving force and in line with the mode at the EP, even though the ratio of the moduli is quite different from unity. If the driving force is getting closer to ω EP , i.e. if a slightly damped excitation is used, the ratio does approach the exact value irrespective of the values c j . In fact, after some slightly tedious but straightforward algebra the result is obtained yielding the numerical value indicated above for the parameters considered. While this result is obtainable analytically for the particular case of Eq.(6), in general one has to resort to the two dimensional reduction by numerical means and then use Eqs. (9), (10) and (16) to find the amplitude ratio.

V. SUMMARY
The physical relevance of EPs and their observability has been presented in the introduction. Consideration of a simple mechanical problem leads to non-hermitian matrices and the study of the associated EPs has produced new general insights. The parameters for which a real spectrum switches to complex values is clearly related to the occurrence of EPs on the real axis. The instability point of the RPA is just one case in point. In addition, the specific shape of the spectrum can be quite different from the one encountered for symmetric matrices. On the other hand, the topological structure, i.e. the Riemann sheet structure of the energy surfaces is independent of whether H 0 and/or H 1 are hermitian or not. The eigenfunctions at the EP have a structure similar to the symmetric case except for the value of the ratio of the two relevant states. This changes from ±i for the symmetric case to ±i cot φ 1 / cot φ 2 for the nonhermitian case. Note also, that this ratio is different for the left hand eigenfunction at the EP.
The universal significance of the EPs is once more underlined by the particular example from classical physics. While the general features of EPs for non-hermitian H 0 and H 1 have been presented in section three, we believe that the particular results of section four can be experimentally confirmed, results whose analogues have so far been implemented in sophisticated microwave cavities [16,17], optical systems [11] and atomic spectra [9].