Non-divergent representation of a non-Hermitian operator near the exceptional point with application to a quantum Lorentz gas

We propose a non-singular representation for a non-Hermitian operator even if the parameter space contains exceptional points (EPs), at which the operator cannot be diagonalized and the usual spectral representation ceases to exist. Our representation has a generalized Jordan block form and is written in terms of extended pseudo-eigenstates. Our method is free from divergence in the spectral representation at EPs, at which multiple eigenvalues and eigenvectors coalesce and the eigenvectors cannot be normalized. Our representation improves the accuracy of numerical calculations of physical quantities near EPs. We also ﬁnd that our method is applicable to various problems related to EPs in the parameter space of non-Hermitian operators. We demonstrate the usefulness of our representation by investigating Boltzmann’s collision operator in a 1D quantum Lorentz gas in the weak-coupling approximation.


Introduction
The importance of non-Hermitian operator has been recognized in many areas of physics in recent years, both on an applied level and on a fundamental level.Such non-Hermitian operators commonly appear, for example, when we discuss irreversible processes in open systems [1][2][3][4][5], as well as PT (parity-time) symmetric systems [6][7][8][9].It is well known that Liouville operators in describing open quantum systems take a non-Hermitian form after a partial trace of the environment is carried out [10].
Among many characteristic properties of the non-Hermitian operators, the appearance of exceptional points (EPs) in parameter space is a surprising result that has been studied in many contexts in recent years [3-5, 9, 11-27].The EP is a singular point in the parameter space at which two eigenstates coalesce.As a result the non-Hermitian operator is not diagonalizable at this point.Instead, the operator can be reduced to the Jordan block form with the aid of a pseudo-eigenstate for a pair of coalesced eigenstates [21,28].This singularity manifests itself in that the normalization constant of the eigenstate diverges at the EP [5].
Due to the divergence in the eigenstate representation, we encounter difficulties when we investigate physical properties of the system near the EPs.This causes problems, for example, when we evaluate numerical values of physical quantities on the basis of the eigenstate expansion, because each contribution from the eigenstates becomes divergent.
In this paper, we propose a non-divergent representation of the non-Hermitian operator which is applicable in all regions in the parameter space.In this representation, we can remove the difficulty in the numerical calculation for the eigenstate representation.We obtain this representation by introducing a generalization of the pseudo-eigenstate that appears at the EPs for the Jordan block calculation.
We introduce this non-divergent representation using Boltzmann's collision operator for a one-dimensional (1D) perfect quantum Lorentz gas [29,30] as a working example.The Lorentz gas gives us one of the simplest examples of the non-Hermitian operators that has the EPs in the eigenvalue problem of the Liouvillian.To our knowledge, this is the first time that the EP problem has been studied at the level of the Liouvillian.We stress, however, that this representation is applicable to a wider class of non-Hermitian operators having EPs.
The structure of the paper is as follows: In section II we briefly summarize the framework of the complex spectral representation of the Liouvillian and apply it to the 1D quantum Lorentz gas.Then we show that the effective Liouvillian thus obtained is equivalent to the collision operator of the Boltzmann equation in a certain domain of the wavenumber on which we focus.In section III we solve the eigenvalue problem of the collision operator.Then we show the appearance of the EPs in the parameter space of the wavenumber.We also present an important physical consequence of the EPs in the collision operator, namely it gives rises to the telegraph equation, instead of the diffusion equation that one often encounters in the kinetic theory.In section IV we derive a standard Jordan block structure at the EPs in terms of the pseudo-eigenstate representation of the collision operator.In section V we introduce an extension of the pseudo-eigenstate to obtain a representation that remains nondivergent in all regions of the parameter space.In section VI we discuss the time evolution of the distribution function using both the eigenstate representation and the extended pseudoeigenstate representation.In section VII we numerically estimate the time evolution of the system near the EP, and demonstrate that our non-divergent representation significantly improves the accuracy of numerical calculations as compared with the calculation using the conventional eigenstate representation.In section VIII we summarize our results and present concluding remarks.In appendix A we discuss the generalization of the extended pseudo-eigenstate representation to a general 2 non-Hermitian operator and to the multiple coalescence of an arbitrary number N of eigenstates at a so-called EPN [17].

General Formalism
We consider a quantum system described by a Hamiltonian H.The time evolution of the system is governed by the Liouville-von Neumann equation for the density matrix ρ(t), 2/17 Here L H is the Liouville-von Neumann operator (Liouvillan in short) which is defined by the commutation relation with the Hamiltonian of the system H, In the Liouville space, the eigenvalue problem of the Liouvillian for each correlation subspace [31], which is specified by the index ν, is given by where the double bra-ket vectors stand for vectors in the Liouville space and the index α specifies the eigenstate in the correlation subspace denoted by ν.We denote the righteigenstate as and the left-eigenstate as F (ν) α |.We solve the eigenvalue problem by using the well-known Brillouin-Wigner-Feshbach formalism [31] with projection operators P (ν) and Q (ν) which satisfy By applying these projection operators on (3), the eigenvalue equation of the Liouvillian takes the form where is the effective Liouvillian and the second term is the self-frequency part.In the eigenvalue problem of the Hamiltonian, a similar expression to ( 6) is called the effective Hamiltonian, and the second term in the case is called the self-energy operator [32].The effective Liouvillian is also called the collision operator which is of central importance in the kinetic theory in non-equilibrium statistical mechanics [31].One can see from its eigenvalue equation ( 5) that the collision operator has the same eigenvalues as those of the Liouvillian.Moreover, the eigenvalue equation is non-linear since the collision operator itself depends on the eigenvalue.It is well-known for an unstable quantum system with a continuous spectrum that the effective Hamiltonian becomes a non-Hermitian operator due to the resonance singularity in the self-energy.Similarly, the collision operator also becomes a non-Hermitian operator in the Liouville space in the thermodynamic limit.As a result, the collision operator has eigenstates with complex eigenvalues, which are called resonance states.For the collision operator, the imaginary part of the complex eigenvalue gives a transport coefficient of the system.
In terms of the right-and left-eigenstates of the collision operator Ψ (ν) (z), the eigenstates of the Liouvillian L H are, respectively, expressed by with the creation of correlation operator and the destruction of correlation operator which are non-diagonal transitions between the P (ν) subspace and the Q (ν) subspace [31]. 3/17 2.2.Application to Weakly-coupled One-dimensional Quantum Perfect Lorentz Gas We apply the general formalism to a weakly-coupled one-dimensional (1D) quantum perfect Lorentz gas.The Lorentz gas consists of one light-mass particle (the test particle) with mass m and N heavy particles with mass M .The Hamiltonian of the system is given by where g is the coupling constant and the interaction is assumed to be a short-range repulsive force.In this paper, we shall consider the weak-coupling regime (g ≪ 1).We also restrict our interest to the case m/M → 0, which is called the perfect Lorentz gas [29].We suppose that the system is enclosed in a large 1D box of volume L with the periodic boundary condition.Hence, the interaction potential is expanded in the Fourier series as where q's are integer multiples of 2π/L.In this paper, we shall consider the thermodynamic limit, where n is the concentration of heavy particles.In this limit, the wavenumber and the momentum become continuous variables.Hence we shall replace a summation with an integration and a Kronecker delta with a Dirac δ-function as at an appropriate stage.
In this paper we investigate the time evolution of the reduced density matrix for the test particle, which is defined as where Tr hev.denotes a partial trace over the heavy particles.This procedure is equivalent to that used for quantum master equations for open quantum systems.[10] We assume that the initial condition of the system is given by where ρ eq hev. is the Maxwell distribution of the heavy particles with temperature T , where k B is the Boltzmann constant.In the thermodynamic limit the time evolution of the density matrix associated with the heavy particle is negligible since its deviation from ρ eq hev.
is proportional to 1/L in this limit, as can be easily shown.

4/17
To discuss the space and momentum dependence of the distribution of the particles in parallel with classical mechanics, it is convenient to introduce the Wigner distribution function: which is a quantum analog of the phase space distribution function [31].Here the notation {X j } represents a set of variables for the N heavy particles and where the single bra-ket vectors stand for vectors in the wave function space and the double bra-ket vectors stand for vectors in the Liouville space [31].Here the "wavenumbers" and the "momenta" in the Wigner representation are defined as and the Wigner basis is defined by a dyad of two eigenstates of H 0 as |k, {k j }; P, We represent a linear operator A in the wave function space as a ket-vector |A in the Liouville space.The inner product of the bra-and ket-vectors is then defined by where B † is the Hermitian conjugate of a liner operator B. As a result, it is easy to show that the Wigner basis vectors are normalized with respect to the box normalization condition k, k,{k j };P,{P j }|k ′ ,{k δ Kr (k j − k ′ j )δ Kr (P j − P ′ j ).
(21) We define the projection operator that acts on the distribution function (16) as and it satisfies Using the double bra-ket notation, the projection operator is written as In the weak-coupling situation, the collision operator (6) can be approximated up to the second order in g as with Note that, in the expression (25), the first-order term in g vanishes according to the condition (23).

5/17
In this paper, we shall study a situation where the wavenumber k satisfies where Here, γ P is the momentum relaxation rate of the test particle, which is evaluated using Fermi's golden rule.We consider the case where the interaction range which we denote d is much shorter than l P , Hence, a typical value of q appearing in ( 10) is much larger than k in (25), For this case with the weak coupling g ≪ 1, we can approximate the collision operator as where +i0 means that the collision operator Ψ 2 (z) is evaluated on the real axis approaching from the upper half-plane to ensure the time evolution is properly oriented to the future t > 0 [31].
Let us now introduce the reduced collision operator acting on the reduced density matrix of the test particle (13) as In the thermodynamic limit, the matrix element of this operator in the Wigner representation is given by (33) with the reduced state of the test particle that is normalized by the δ-function in the continuous spectrum limit.Here, the operator ∂ q/2 P is a displacement operator defined by where exp [a∂/∂P ] acts on a function of P as exp [a∂/∂P ]f (P ) = f (P + a).Furthermore, we have ignored the k which appears in the denominator in (33) as compared with q (see ( 30)).Note that the expression (33) does not depend on the temperature of the heavy particles T .This is because in the limit of the perfect Lorentz gas m/M → 0 there is no energy transfer between the test particle and the heavy particles.
Performing the q integration in (33), a matrix element of the collision operator ψ (k) is expressed as Hence, it has non-vanishing matrix elements only between the states |k; P )) and |k; −P )).Physically, this is because there are only forward and backward scattering in this 1D system.

6/17
Therefore, in terms of this basis, the collision operator is represented by the 2 × 2 matrix where where µ and µ ′ take values P or −P , a is the non-dimensionalized wavenumber defined by and Î is the unit matrix of size 2.
In terms of the collision operator, the time evolution equation for the reduced density matrix for the test particle is given by where and the collision operator ψ (k) is given by (33).This is equivalent to the Boltzmann equation for the perfect quantum Lorentz gas [30,33], for which the first term in the square bracket in ( 33) is called the flow term, and the second term is called Boltzmann's collision term.

Eigenstates of the collision operator
Let us denote the right-and left-eigenstates of the collision operator (38) as |φ α and φα |, respectively, i.e., ψ(a) where the double bra-ket vectors stand for vectors in the Liouville space (see Eq.( 17)).The eigenvalue equation of the collision operator is det ψ(a) Then, we have and where we have explicitly indicated the parameter a in the eigenstates.Note that we have not normalized the eigenvectors (45) considering the fact that they cannot be normalized for a = ±1.The inner products of these right-and left-eigenstates are given by φ± (a)|φ In (44), each of the two eigenvalues is associated with one of the two values taken by the square root function, and the assignment is fixed with the following definition, We also impose the following condition to the relative phase of the components of the vectors (45), since the expression has ambiguity because the values taken by the two-valued square root functions are not fixed.With the additional condition (48) each eigenvector is determined up to an overall sign.We show the a-dependence of the real part and the imaginary part of the eigenvalues in Fig. 1.In the figures, the blue and the red lines represent the eigenvalues z+ and z− , respectively, and the purple lines represent that these two lines are overlapping.
As a function of the parameter a, the eigenvalues (44) have branch point singularities at At these points, they degenerate as At these degeneracy points, the eigenstates (45) also "degenerate" in the sense that these two eigenstates collapse into a single eigenstate as

(51b)
Since there is only one eigenstate at the degeneracy points (49), the collision operator ( 38) is non-diagonalizable at these points.This collapse of eigenstates does not take place at the usual degeneracy point in a Hermitian operator, where a degenerate eigenvalue is shared by two distinct eigenstates [5].For this reason, such a degeneracy point in the present case is often called a non-Hermitian degeneracy point [12] to emphasize that it appears only in non-Hermitian operators.It is also known as an exceptional point (EP) [11].
Note that the norm of the eigenstates vanishes at the EPs.Except at the EPs, the eigenstates (45) are normalizable as Then, they satisfy the following bi-orthonormality and bi-completeness relations for a = ±1, where α and α ′ take the values "+" or "−".Before discussing the Jordan block structure at the EPs, let us discuss a physical consequence of the EPs in the 1D Lorentz gas.We show that the existence of the EPs leads to the telegraph equation.In terms of the original variables the characteristic equation ( 43) is written as where v P ≡ |P |/m.The inverse Fourier-Laplace transformation of (55) leads to the telegraph equation In other words, Eq.( 55) is the same as the characteristic equation of the telegraph equation with regard to the X-and t-dependence as exp[i(kX − zt)].Hence our Boltzmann equation ( 40) is equivalent to the telegraph equation [30,34] with regard to the dependence of the Wigner function on X and t.
We now show that the telegraph equation reduces to the diffusion equation in long-time behavior.To see this we observe Fig. 1B.All decaying modes except for eigenstates with small pure imaginary eigenvalues have vanished in a long time region.For the remaining modes, |z| ≪ γ P , and the first term of (55) is much smaller than the second term.Hence, the characteristic equation (55) reduces to Then, inverse Fourier-Laplace transformation of (57) leads to the diffusion equation, It is remarkable that Boltzmann's equation for the 1D perfect quantum Lorentz gas is equivalent to the telegraph equation in their time development in the real space, because 9/17 the telegraph equation represent a prototypical behavior of systems with an EP with respect to a parameter.In such systems the time development changes from an over-damped one to a damped oscillation and shows a critical damping behavior just at the EP.
Moreover, the analytic solution of the initial value problem for the telegraph equation is known [34].Thus an analytic expression for the time evolution of Boltzmann's equation for the 1D quantum perfect Lorentz gas model is also available to us.We will discuss elsewhere physical behaviors of the system employing this solution of Boltzmann's equation.

Jordan block representation of the collision operator and the pseudo-eigenstate
In this section, we summarize the well known Jordan block structure and its relation to the pseudo-eigenstate [21,28] in order to prepare for the introduction of the extended pseudoeigenstate in the next section.The right pseudo-eigenstate, denoted by |φ ′ ±1 , is defined through the following relation, which is called the Jordan chain relation [21].Similarly, the left pseudo-eigenstate φ′ We then have These pseudo-eigenstates satisfy the following bi-orthonomality relations and the bi-completeness relation In terms of this basis, the collision operator is represented by the standard Jordan block, As a result, the collision operator ψ(±1) is represented by the Jordan block form by introducing the pseudo-eigenstate at the EPs.However, the representation does not remove the divergent behavior of eigenstates (52) near the EPs since it is only applicable at the EPs.In the next section, we introduce a representation that removes the divergence by extending the pseudo-eigenstate for a = ±1.

The extended pseudo-eigenstate representation
So far, we have shown that the collision operator has the EPs (49) at which the operator cannot be diagonalized.Instead, the collision operator can be reduced to a Jordan block form at these points.In this section, we introduce a representation which does not have any singularity at the non-Hermitian degeneracy points by extending the concept of the pseudo-eigenstate representation for a = ±1.
Similarly, we can introduce an left extended pseudo-eigenstate for z± , denoted by φ′ We impose the normalization conditions for the right and left extended pseudo-eigenstates, Then, we have right and left pseudo-eigenstates In terms of either basis set, the collision operator is represented by the Jordan block-like matrix for arbitrary values of a as Note that this matrix differs from the Jordan block (63) because both eigenvalues z+ and z− appear on the diagonal.By taking the limit a → 1 or a → −1 for (70), we recover the Jordan block representation (63) just at the EPs.The extended pseudo-eigenstates (67) also reduce to the usual pseudo-eigenstates (60) in this limit.Let us emphasize that the extended pseudo-eigenstates (64), (65) and expressions (66), (68), ( 69) and (70) are not model dependent and are always possible for any 2 × 2 non-Hermitian operator having EPs.In Appendix we will present the extended pseudo-eigenstate representation for more general 2 × 2 non-Hermitian operators.There, we will also present the extended pseudo-eigenstate representation for the case where multiple coalescence occurs at an EP.
11/17 6.Two different descriptions of time evolution in terms of the eigenstate representation and the extended pseudo-eigenstate representation In the previous section, we obtained two different representations of the collision operator: one is the eigenstate representation which is not normalizable at the exceptional points a = ±1, the other is the extended pseudo-eigenstate representation that is normalizable at the exceptional points.As one might expect, the usual representation in terms of the eigenstates leads to a serious difficulty in the vicinity of the exceptional points, while we have no such difficulty with the normalizable representation in terms of the extended pseudo-eigenstates.
Let us now compare the expressions in Eqs.(72) and (75).Each matrix element in the eigenstate expansion (72) diverges at the EPs z+ = z− .Hence, the expression (72) following from the eigenstate expansion generally leads to serious difficulty when we consider in the vicinity of the EPs.On the other hand, we have no such difficulty with the expression (75) since each matrix element is well defined even at the EPs.
In Fig. 2(A), we present the result using the eigenstate representation (72).We evaluate this by calculating the time evolution of each component separately in this representation, and sum up the results of all components.As we can see, each component of (72) diverges at the EP.Therefore, numerical behavior with the eigenstate representation near the EP is an artifact, and the result is not reliable when the parameter a approaches the value at the EP.
In Fig. 2(B), we present the corresponding result in terms of our extended pseudo-eigenstate representation given in Eq.( 75), where each component has a well-defined value at the EP.As expected, there is no singularity in the vicinity of the EPs.Actually the curve in Fig. 2 The true behavior does not have any singularity as in the case of Fig. 2(B), because the original equation does not have any singularity at the EP.Hence, we believe that the extended pseudo-eigenstate representation gives the correct result.

Summary and concluding remarks
We have introduced a non-divergent representation of non-Hermitian operators that remains finite even at the EPs in a parameter space.The representation has been obtained by extending the pseudo-eigenstates to the entire parameter space.
We have applied this representation to the collision operator of the Boltzmann equation for the 1D perfect quantum Lorentz gas.Then we have shown that this representation removes the difficulty resulting from the divergence of the normalization constant at the EPs in the usual eigenstate expansion.Indeed, we have demonstrated a dramatic improvement in the accuracy of the numerical evaluation of the time evolution of the distribution function in terms of our representation.
In recent years, multiple coalescence where an arbitrary number N of eigenstates coalesce at a single EP, called EPN , has also been studied [17].In appendix A we also show that our extended pseudo-eigenstate representation can be generalized to the EPN s.
The EPs are non-Hermitian degeneracy points.It is well-known that degeneracy in a Hermitian operator leads to the Berry phase effect [35] as reported in many experiments, see e.g.[36].Recently, it is clarified that the Berry phase-like effect plays an important role in the study of quantum pumping processes based on quantum kinetic equations (quantum master equations) [37][38][39][40][41]. Similar to the degenerate Hermitian operator, it is interesting to investigate the effects coming from the degeneracy in non-Hermitian operators and the phase change of the eigenstates around the EPs.Indeed, one can find many theoretical [3][4][5][12][13][14][15][16][17][18][19][20][21][22] and experimental papers [9,[23][24][25][26][27] on this subject.We will discuss the implication of our extended pseudo-eigenstate representation on the phase of the eigenstates in the vicinity of the EPs elsewhere.

Fig. 1
Fig. 1 Spectrum of the Boltzmann collision operator.(A) is the Imaginary part of the spectrum and (B) is its real part.In each figure, the blue line represents eigenvalue z+ and the red line represents eigenvalue z− .The purple lines represent that these two lines are overlapping.

Fig. 2
Fig.2Comparison of the numerical result of Im[f k;P ( t)] calculated by (A) the eigenstate representation with (B) the extended pseudo-eigenstate representation.The calculations are performed with a precision of eight significant digits.
with different values of a, but we cannot distinguish them in the resolution drawn in this figure.