A physical interpretation for the non-Hermitian Hamiltonian
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe Ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians. It is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wave function with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter the non-Hermitian is broken symmetry or even non-. An exactly solvable model is presented to demonstrate the main points of this article.
pacs:03.65.Ge, 05.30.Jp, 03.65.Nk, 03.67.Bg
In general, a non-Hermitian Hamiltonian is said to be physical when it can have an entirely real energy spectrum. Much effort has been devoted to establish a parity-time () symmetric quantum theory as a complex extension of the conventional quantum mechanics Bender 98 (); Bender 99 (); Dorey 01 (); Bender 02 (); A.M43 (); A.M (); A.M36 (); Jones () since the seminal discovery by Bender Bender 98 (). It is found that non-Hermitian Hamiltonian with simultaneous symmetry has an entirely real quantum mechanical energy spectrum and has profound theoretical and methodological implications. Reseaches and findings relevent to the spectra of the symmetric systems are presented, such as exceptional points EP (), spectral singularities for complex scattering potentials AMSS (), complex crystal and other specific models LonghiSS () have been investigated. At the same time the symmetry is also of great relevance to the technological applications based on the fact that the imaginary potential could be realized by complex index in optics Bendix (); Joglekar (); Keya (); YDChong (); LonghiLaser (). In fact, such optical potentials can be realized through a judicious inclusion of index guiding and gain/loss regions and the most interesting aspects associated with symmetric system are observed during dynamic evolution process Klaiman (); El-Ganainy (); Makris (); Musslimani ().
Thus one of the ways of extracting the physical meaning of a pseudo-Hermitian Hamiltonian with a real spectrum is to seek for its Hermitian counterparts A.M38 (); A.M391 (); A.M392 (). The metric-operator theory outlined in Ref. A.M () provides a mapping of such a pseudo-Hermitian Hamiltonian to an equivalent Hermitian Hamiltonian. Thus, most of the studies focused on the quasi-Hermitian system, or unbroken symmetric region. However, the obtained equivalent Hermitian Hamiltonian is usually quite complicated A.M (); JLPT (), involving long-range or nonlocal interactions, which is hardly realized in practice.
To anticipate these problems, alternative proposals for the connection between a pseudo-Hermitian Hamiltonian and a real physics system have been suggested in the context of scattering problems JLScat (). Central to that analysis was the recognition that the Hamiltonian may be used to depict the resonant scattering for an infinite system. It is shown that any real-energy eigenstate of certain tight-binding lattice shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice. In such a framework, further questions to ask are whether the requirements of the entireness of the real eigenvalunes and the symmetry of the non-Hermitian system are really necessary.
In this paper, we propose a physical interpretation for a general non-Hermitian Hamiltonian based on the configurations involving an arbitrary network coupled with the input and output waveguides. Relevant to our previous discussion is the interpretation of the imagiary potentials. Based on this, we make a tentative connection between a non-Hermitian system and the corresponding large Hermitian system. It is shown that for any scattering state of such a Hermitian system, the wavefunction within the center lattice always corresponds to the equal energy eigenfunction of the non-Hermitian Hamiltonian, no matter it is symmetric or not. Our formalism is generic and is not limited to the pseudo-Hermitian system.
This paper is organized as follows. Section II is the heart of this paper which presents a formulism to reduce a scattering process of a Hermitionian system to the eigen problem of the non-Hermitian system. Section III consists of two exactly solvable examples to illustrate our main idea. Section IV is the summary and discussion.
Ii Non-Hermitian reduction of a Hermitian system
A typical scattering tight-binding network is constructed by a scattering-center network and two semi-infinite chains as the input and output leads. The well-established Green function technique Datta (); YangAs (); JLTrans () can be employed to obtain the reflection and transmission coefficients for a given incoming plane wave. The corresponding wave function within the scattering center should be obtained via Bethe ansatz method. In the following we will show that this can be done by solving a finite non-Hermitian Hamiltonian. In our previous work JLScat (), a symmetric non-Hermitian Hamiltionian has been connected to a physical system in the manner that any real-energy eigenstate of a tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. The main aim of this article is to answer the question of whether such a statement still holds for broken non-Hermitian or non- lattice. In the following, we will show that a scattering state of the Hermitian system always has connection to the eigenstate of its non-Hermitian reduction. For a certain incident plane wave, the scattering problem of the whole infinite Hermitian system can be reduced to the eigen problem of a finite non-Hermitian system.
The Hamiltonian of a typical scattering tight-binding network has the form
represent the left and right waveguides and
describes an arbitrary -site network as a scattering center. Sites and are arbitrary within the network. Here , , are boson (or fermion) operators, (we denote only for the sake of simplicity) represents the potential at site . Fig. 1(a) represents a schematic scattering configuration for an arbitrary network.
For an incident plane wave incoming from waveguide with energy , the scattering wave function can be obtained by the Bethe ansatz method. The wave function has the form
Here , are the reflection and transmission coefficients. The explicit form of the Schrödinger equations for the waveguides and are
From Eq. (8), we obtain and
The explicit form of the Schrödinger equations for can be written as,
This is equivalent to the effective non-Hermitian Hamiltonian
Without losing generality, we take with . This leads to
which means that the imaginary part of the additional potentials have opposite signs, one providing gain and the other loss. This is in accordance to the conservation law of the current. It is important to stress that magnitude of the two imaginary potentials may not equal, which deviates from the general understanding of an imaginary potential.
The existence of the scattering solution of the Hermitian system ensures that there must exist at least one real solution of with eigenvalue equals to the incident energy . It possesses the identical wavefunction as that of the scattering state within the region of the scattering center. Then a scattering problem is reduced to the eigen problem of a non-Hermitian Hamiltonian. This conclusion is an extension of our previous result JLScat (). In this work, our formalism is generic: The central network is not limited to the linear geometry and the scattering is not restricted to be resonant transmission. Thus the scattering interpretation for the non-Hermitian Hamiltonian is not limited to pseudo-Hermitian system. This rigorous conclusion has important implications in both theoretical and methodological aspects.
Likewise, if we consider the inverse scattering process, i.e., taking the time-reversal operation on the above mentioned scattering process. The corresponding Bethe ansatz wave function has the form
with energy . The above conclusion still holds. Straightforward algebra shows that the corresponding non-Hermitian reduction is . In the framework of non-Hermitian quantum mechanics, takes an important role to construct a complete biorthogonal basis set, which has no physical correspondence. In the context of our approach, has the same physics as , in describing the scattering problem of the same Hermitian system.
Iii Illustrative Examples
In this section, we investigate simple exactly solvable systems to illustrate the main idea of this article. We will discuss two examples which correspond to a and a non- non-Hermitian Hamiltonian, respectively. The advangtage of these examples are that the non-Hermiltian Hamiltonians are exactly solvable.
iii.1 Exactly solvable Hamiltonian
To exemplify the previously mentioned analysis of relating the stationary states of a non-Hermitian -symmetric Hamiltonian to a scattering problem for a Hermitian one, we take the center network to be a simple network: a uniform ring system. We start with the scattering problem for a class of symmetric systems, the Hamiltonian can be written as
where we denote the connection sites as and .
The corresponding non-Hermitian Hamiltonian depends on the energy of the incident plane wave as well as the parameters and . To be concise, as an illustrative example, we would like to present the exactly solvable model, which are helpful to demonstrate our main idea. Therefore, we will focus on the following configurations:
i) , where . Here we restrict the energy of the incident plane wave since it will leads to the pure imaginary potential, thus ensures the existence of the exact solution. Straight forward algebra shows that the problem of solving the Schrodinger equation is reduced to the eigen problem of the following non-Hermitian Hamiltonian
with the imaginary potential
Obviously, this Hamiltonian depicts a -site ring with two imaginary potentials at two symmetrical sites, which is a -invariant Hamiltonian. Note that the magnitude of the imaginary potential is discrete in order to obtained the exact solutions. In Appendix A, it is shown that such lattices can be synthesized from the potential-free lattice by the intertwining operator technique generally employed in supersymmetric quantum mechanics. The eigen spectrum of consists of
and two additional levels
The eigenstates with eigenvalue can be decomposed into two sets: bonding and antibonding, with respect to the spatial reflection symmetry about the axis along the waveguides. For the scattering problem, only the bonding states are involved. It shows that there always exists a solution in to match the energy of the incident wave.
From Eqs. (22, 23), on can see that a pair of imaginary eigenvalues appear, i.e., the symmetry is broken when . In general, a non-Hermitian Hamiltonian with a broken symmetry is unacceptable because its complex energy eigenvalues make a hash of the physical interpretation. On the other hand, the symmetry breaking was observed in optics realm experimentally AGuo (). In theoretical apects, symmetry in non-Hermitian spin chain system was discussed Giorgi (). From the point of view of this article, we note that even possesses a broken symmetry, the spectrum still contains the state with the energy . It is worth mentioning that the broken symmetry does not contradict the interpretation of the non-Hermitian Hamiltonian (20). This idencates that even the symmetry is broken the non-Hermitian Hamiltonian still has physical significance.
ii) , . Here we do not restrict the energy of the incident plane wave but the magnitue of . Straight forward algebra shows that the problem of solving the Schrodinger equation is reduced to the eigen problem of the following non-Hermitian Hamiltonian
with the imaginary potential
From Appendix A, the solution of the Hamiltonian has the same form of Eqs. (22, 23) with replaced by . Here we would like to see the relation between the Hamiltonians and : Both of them come from the same model with different coupling constants (with and ) and different incident plane waves (with discrete and continuous spectra). However they have the same structure but different values of the imaginary potentials. In Appendix A, we provide the universal solution contains that of and .
Obviously, Hamiltonian is always exact symmetric. All the eigenvalues are real. Among them we can find that , one of equals to the energy of incident plane wave and thus verifies the above mentioned conclusion. Furthemore, the solution of it has the following peculiar feature: in the case of , i.e., the incident wave has wave vector , the exceptional points appear in . It is shown in Appendix A that the corresponding eigenfunctions of () and () coalesce.
According to non-Hermitian quantum mechanics, in general, has the Hermitian counterpart which possesses the same spectrum. When the potential approches , the similarity transform that connects and becomes singular. The Hamiltonian becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues , (i.e., the lack of completeness of the energy eigenstates.) Such a Hamiltonian has no Hermitian counterpart BenderPRD (). According to our analysis, one can see that even at the exceptional points EP () the coalescing eigenstates still has physical significance.
iii.2 Exactly solvable non- Hamiltonian
Now we turn to exemplify the previously mentioned analysis of relating the stationary states of a non-Hermitian non--symmetric Hamiltonian to a scattering problem for a Hermitian one. We still take the center network as a simple network: a uniform ring system with uniform coupling but none on-site real potentials. The corresponding Hamiltonian can be written as
We consider the incident plane wave with energy , where , without any restriction. Straight forward algebra shows that the problem of solving the Schrodinger equation is reduced to the eigen problem of the following non-Hermitian Hamiltonian
where the complex potentials are
We can see that, in general, Hamiltonian is not symmetric, except in some special cases. It is hardly to get the analytical solution of such a Hamiltonian in general cases. Fortunately, what we need to do is to prove that the incident energy is always one of the eigenvalues of the Hamiltonian. In fact, in single-particle basis the matrix representation of the Hamiltonian (27) satisfies
according to the derivation given in Appendix B. This result do not depend on the pseudo-Hermiticity of the Hamiltoian. In this sense, one can conclude that a non- non-Hermitian Hamiltonian still has physical significance.
In summary, we have studied the connection between a non-Hermitian system and the corresponding large Hermitian system. We propose a physical interpretation for a general non-Hermitian Hamiltonian based on the configurations involving an arbitrary network coupled with the input and output waveguides. We employed the Bethe ansatz approach to the scattering problem to show that for any scattering state of a Hermitian system, the wavefunction within the scattering center lattice always corresponds to the equal energy eigenfunction of the non-Hermitian Hamiltonian. It is important to stress that such a physical interpretation for the non-Hermitian Hamiltonian is not limited to the pseudo-Hermitian system. As an application, we examine concrete networks consisting of a ring lattice as the scattering center. Exact solutions for such types of configurations are obtained to demonstrate the results. Such results are expected to be necessary and insightful for the physical significance of the non-Hermitian Hamiltonian.
Acknowledgements.We acknowledge the support of the CNSF (Grant Nos. 10874091 and 2006CB921205).
Appendix A Construction of -Hamiltonian by Interwining operator technique
In this Appendix, we will derive the central formula for studying the eigen problem of the ring system.
a.1 Linear Transformation
First of all, the Hamiltonian can be decomposed into two independent sub-Hamiltonians
with , by using the following linear tranformation:
We will focus on the solution of the Hamiltonian . Typically, the solution can be obtained via Bethe ansatz method as shown in Ref. JLTrans (). In this Appendix, we will use the intertwining operator technique to get the solutions in order to reveal their characteristic features.
a.2 Interwining operator technique
The intertwining operator technique is generally employed in supersymmetric quantum mechanics, which provides the universal approach to creating new exactly solvable models. Recently, it is applied to discrete systems in order to construct the model which supports the desirable spectrum LonghiIOT (); LonghiDynamics ().
The critical idea of the intertwining operator technique is as the following: Consider an Hamiltonian which has the form , where and represent and matices, respectively. One can construct an new Hamiltonian () by interchanging the operators and . The spectrum of is the same as that of except for the energy level . Iterating this method results in a series of Hamiltonians , , , whose energy spectra differ from that of owing to the addition of the discrete energy levels , , , .
Our aim is to construct a -invariant Hamiltonian by adding two energy levels and (, the obtained conclusion will be extended beyond this region later) into the energy spectrum of a uniform chain system. We will show the processes of this construction explicitly. We start with the following Hamiltonian
which depicts an -site uniform chain. The spectrum of can be expressed as
On the other hand, can be written in the form
Then the Hamiltonian can be constructed in the form
which possesses an extra eigenvalue based on the spectrum .
Next step, we repeat the above procedure based on a new Hamiltonian , which is obtained from under parity operation , i.e.,
is the matix representation of mirror reflection. Note that and have identical spectra. Accordingly, can be written as the form
a.3 Eigenfunctions of
Now we turn to derive the eigen functions of . The eigenfunctions of the Hamiltonians , and are denoted by , and , respectively. The eigenfunctions of a uniform chain can be readily written as
According to the interwining operator technique of supersymmetry theory, we have
Note that the eigenfunctions are not normalized.
In the above, we restricted in the region for the purpose of obtaining as a non-Hermitian unbroken symmetric Hamiltonian with imaginary potentials at the edges in the form of Eq. (A.2). However, the obtained result can be extended beyond the region. Actually, one can simply replace by in all the expressions. Then one can obtain a Hermitian Hamiltonian with two added bound states with energy where . On the other hand, if is replaced by a complex number , one can obtain a non-Hermitian symmetric Hamiltonian in the broken phase. In this case the two added eigenstates have pure imaginary eigenvalues .
a.4 Coalescence of eigenstates
Now we investigate the eigenfunctions in the case of (). In this situation, all the eigenfunction can be written explicitly as
For odd , we have and , which means the coalescence of eigenstates. Also the norms of the above four eigenstates vanish. For even , we have the same conclusion except when . In this case, we have , which means the coalescence of the three eigenstates. Also the norms of the above three eigenstates vanish.
Appendix B Zero determinant
In this appendix we will prove the Eq. (29). Applying the linear transformation introduced in Appendix A, the -dimensional matrix can be written in a diagonal block form, i.e.,
where is -dimensional, while is -dimensional. Then we have
Consider the the -dimensional matrix , which determinant has the form
Using cofactor expansion along the first and last rows, we obtain
where is the determinant
Such kinds of determinants follow the recursion formula
which leads to
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