Quantum transport in non-Hermitian impurity array

TL;DR

This paper investigates how non-Hermitian impurities induce bound states within the band gap of a Hermitian system, leading to potential applications in quantum device fabrication through non-Hermitian impurity engineering.

Contribution

It introduces a novel approach to creating and analyzing bound states via non-Hermitian impurities in Hermitian systems, with implications for quantum transport and device design.

Findings

Bound states emerge inside the band gap with PT imaginary potential.

Bound states become localized near the exceptional point.

Impurity array exhibits Dirac probability-preserving dynamics.

Abstract

We study the formation of band gap bound states induced by a non-Hermitian impurity embedded in a Hermitian system. We show that a pair of bound states emerges inside the band gap when a parity-time ($\mathcal{PT}$) imaginary potential is added in a strongly coupled bilayer lattices and the bound states become strongly localized when the system approaches to the exceptional point (EP). As a direct consequence of such $\mathcal{PT}$ impurity-induced bound states, an impurity array can be constructed and protected by energy gap. The effective Hamiltonian of the impurity array is non-Hermitian Su-Schrieffer-Heeger (SSH) type and hosts Dirac probability-preserving dynamics. We demonstrate the conclusion by numerical simulations for the quantum transport of wave packet in right-angle bends waveguide and $Y$-beam splitter. Our finding provides alternative way to fabricate quantum device by non-Hermitian impurity.

Equations

$$H$$

$$H_0$$

$$H_T$$

$$H_{\mathrm{Chain}}$$

$$|{\psi }_{\mathrm{B}}⟩$$

$$E_{\mathrm{B}}=-2\kappa \cosh {\beta }^{+}+\Delta =2\kappa \cosh {\beta }^{-}-\Delta .$$

$$\left\{\begin{array}[]{c}\cosh\beta^{-}+\cosh\beta^{+}=\Delta/\kappa\\ \sinh\beta^{+}\sinh\beta^{-}=T^{2}/(2\kappa)^{2}\end{array}\right..$$

$$|{\psi }_{\mathrm{B}}^{\mathrm{EP}}⟩=C\sum _{j=-\infty }^{\infty }\left(e^{i\pi j}{e}^{-\beta |j|}{a}_{j,+}^{†}+i{e}^{-\beta |j|}{a}_{j,-}^{†}\right)|\mathrm{vac}⟩,$$

$$H=H_0+H_T.$$

$$H_0$$

$$H_{\lambda }$$

$$H_{12}$$

$$H_T=i\sum _{\lambda =1}^2\sum _{j,l}(-1{)}^{\lambda +j+l}{\gamma }_{jl}{\alpha }_{j,l,\lambda }^{†}{\alpha }_{j,l,\lambda }.$$

$$a_{j,l,\pm }^{†}=\frac{1}{\sqrt{2}}\left({\alpha }_{j,l,1}^{†}\pm {\alpha }_{j,l,2}^{†}\right),$$

$$H_0$$

$$H_T=i\sum _{j,l}(-1{)}^{j+l+1}{\gamma }_{j,l}\left(a_{j,l,+}^{†}{a}_{j,l,-}+H.c.\right),$$

$$H=\left(\begin{array}[]{cc}0&D\\ D^{{\dagger}}&0\end{array}\right),$$

$$\mathcal{H}=H+i\gamma {\sigma }_z\otimes I_N,$$

$$|{\varphi }_n⟩=\left({\sigma }_z\otimes I_N\right)|{\varphi }_{-n}⟩.$$

$${ϵ}_{\pm n}=\pm {\left({\epsilon }_n^2-{\gamma }^2\right)}^{1/2}$$

$$|{\phi }_{\pm n}⟩=M_{\pm n}|{\varphi }_{\pm n}⟩$$

$$M_{\pm n}=\left(\begin{array}[]{cc}a_{\pm n}I_{N}&0\\ 0&I_{N}\end{array}\right),$$

$$⟨{\phi }_m|{\phi }_n⟩={\delta }_{mn},$$

$$H_{\mathrm{Ladd}}$$

$$|\psi \left(0\right)⟩={\Omega }^{-1/2}\sum _{\lambda =1,2}\sum _{n=1}^N{e}^{-{\alpha }^2{\left(n-N_{\mathrm{c}}\right)}^2/2}{e}^{i{k}_{\mathrm{c}}n}{\alpha }_{n,\lambda }^{†}|\mathrm{vac}⟩,$$

$$|\psi \left(t\right)⟩=e^{-iHt}|\psi \left(0\right)⟩,$$

$$p(i,j,t)=\sum _{\lambda =1,2}{\left|⟨i,j,\lambda |\psi \left(t\right)⟩\right|}^2.$$

$$P(i,j)=\frac{p(i,j,t{)}_{\max }}{{\left[{\sum }_{t}p(i,j,t)\right]}_{\max }}\sum _{t}p(i,j,t),$$

$$|{\psi }_{\mathrm{B}}⟩=\sum _{\sigma =\pm }\sum _{j=-\infty }^{\infty }{C}^{\sigma }{\left(-\sigma \right)}^{|j|}{e}^{-{\beta }^{\sigma }|j|}{a}_{j,\sigma }^{†}|\mathrm{vac}⟩,$$

$$E_{\mathrm{B}}$$

$$\left\{\begin{array}[]{c}\left(\Delta-2e^{-\beta^{+}}\kappa-E_{\mathrm{B}}\right)C^{+}+iTC^{-}=0\\ iTC^{+}+\left(-\Delta+2e^{-\beta^{-}}\kappa-E_{\mathrm{B}}\right)C^{-}=0\end{array}\right.,$$

$$\left|\begin{array}[]{cc}\left(\Delta-2e^{-\beta^{+}}\kappa-E_{\mathrm{B}}\right)&iT\\ iT&\left(-\Delta+2e^{-\beta^{-}}\kappa-E_{\mathrm{B}}\right)\end{array}\right|=0.$$

Full text

I Introduction

Throughout physics, stable or equilibrium phenomena can be understood by bound state, ranging from quantum to classical objects. In quantum physics, a bound state is a localized state of a particle subject to a real valued potential, which may be the result of the presence of another particles. The concept of bound state is ubiquitous in numerous branches of physics, such as optics and condensed matter. In experiment, engineering bound state can be generated by artificial defects in photonic crystals and an array of defects, which are known as coupled-resonator optical waveguides providing almost lossless guiding, bending of wave packetJoannopoulos ; John ; Yablonovitch ; Skorobogatiy ; Engelen ; Hughes ; Kuramochi ; Thomas ; Mazoyerprl ; Mazoyeroe ; Faolain . By introducing artificial defects, various photonic crystal devices can be realized for the applications in a wide variety of fields. Nowadays a complex potential is not forbidden since non-Hermitian quantum mechanics has emerged as a versatile platform for fabricating functional devices in non-Hermitian regime. The main mechanism is based on the existence of imaginary potential which has been investigated theoretically Bender ; JPA2 ; Ali ; Znojil ; Jones ; OL07 ; PRL08a ; PRL08b ; Joglekar10 ; Joglekar11 ; YDChong ; HJingPRL2014 and realized in experiment AGuo ; CERuter ; Wan ; Sun ; LFeng ; BPeng ; LChang ; LFengScience ; HodaeiScience ; NC2015 as an ideal building block of non-Hermitian systems. A non-Hermitian term has a distinguishing feature that differs from a Hermitian one can be exemplified by a simple two level system. An extra pseudo-Hermitian non-Hermitian term always shrinks the level spacing, while a nontrivial Hermitian perturbation always leads to the repulsion of the two levels. Presumably, mid-gap levels can be generated by adding non-Hermitian impurity on a Hermitian gapped system.

In this paper, we study the possibility of a quantum channel generated by an array of non-Hermitian defects. Based on the Bethe ansatz solution we show that a pair of bound states emerges inside the band gap when a parity-time ($\mathcal{PT}$) imaginary potential is added in a strongly coupled bilayer. Such bound states are protected by energy gap, especially as the bound-state energy tends to zero when the system approaches to the exceptional point (EP). Consequently, an impurity array can be constructed in the mid-gap and provides a low-loss waveguide since other Hermitian degrees of freedom are adiabatically eliminated when we consider the dynamics in the impurity array. In addition, we show that the effective Hamiltonian of the impurity array is equivalent to a non-Hermitian Su-Schrieffer-Heeger (SSH) system and obeys chiral-time ($\mathcal{CT}$) symmetry. This ensures quasi orthogonality of the mid-gap modes in the framework of Dirac inner product, and therefore the Dirac probability-preserving dynamics in the waveguide. Therefore, although the waveguide is engineered by non-Hermitian impurities, it acts as a conditional Hermitian device. We demonstrate the conclusion by numerical simulations for the quantum transport of wave packet in right-angle bends waveguide and $Y$-beam splitter. Our finding provides alternative way to fabricate quantum device by non-Hermitian impurity.

This paper is organized as follows. In Sec. II, we present the main ideas of non-Hermitian-impurity induced waveguide. In Sec. III, we provide a concrete example to illustrate our theory. Sec. IV demonstrates the dynamics of waveguide in the concrete system. Finally, our conclusion is given in Sec. V.

II Formalism

Consider a bilayer system (Fig. 1(a)), composed of two identical lattice but with opposite on-site energies. The interlayer tunneling is non-Hermitian and sparse, which is referred to as non-Hermitian impurity. The Hamiltonian is given by

[TABLE]

where $\sigma$ corresponds to the $+$ and $-$ layers, and $\left\{iT_{j}\right\}$ is a set of imaginary intralayer hopping parameters. Fig. 1(a) is the schematic illustration of the model. We consider the region with $\Delta\gg\left|\kappa_{ij}\right|$, which ensures the energy gap of order $\Delta$ between two ($+$/$-$) enargy bands for the system $H_{0}$. In the following, we show that $H_{T}$ may induce local states within the energy gap.

In the absence of $H_{T}$, the states of $H_{0}$ are extended states spreading the probability over the two layers. In the limit case with  $\left|T_{j}\right|\sim\Delta\gg\left|\kappa_{ij}\right|$, each non-Hermitian tunneling $iT_{j}$ may induce two isolated energy levels around the midgap, which have the form $\pm\sqrt{\Delta^{2}-\left(T_{j}\right)^{2}}$. It is responsible for the non-Hermiticity of the impurity, since a real tunneling cannot form the isolated levels within the energy gap, but beyond the two bands. Fig. 1(b) is the band structures for $H_{0}$ and $H$. We note that the eigenvalues of $H$ are real or imaginary, i.e., the EPs only appear at zero energy. Considering the case with a single nonzero $T_{j}$, there are two isolated levels within the gap and such two bound states coalesce to a single state $(1/\sqrt{2})\left(a_{j,-}^{{\dagger}}+ia_{j,+}^{{\dagger}}\right)\left|\mathrm{vac}\right\rangle$ at the EP when $\Delta=\left|T_{j}\right|$. If there are many such nonzero tunneling $T_{j}$ within the unbroken region and the corresponding bound states overlap with each other, sub-bands within the gap will form and a non-Hermitian-impurity induced waveguide is achieved for a array of impurity.

To demonstrate the main idea, we consider an example model, which is consist of two infinite chains with a single non-Hermitian tunneling. The corresponding Hamiltonian has the form

[TABLE]

where $\sigma=+$ or $-$ is the index that respectively labels the position in the top or bottom chains, and $j$ is the in-chain site index. Parameters $\kappa$ and $T$ of this model are intra- and interchain hopping strengths. The schematic illustration is shown in Fig. 2(a). Bethe ansatz method shows that there are two bound states around the center with the eigenvector (see Appendix 1)

[TABLE]

Here $C$ is the normalization coefficient, which is determined in the context of Dirac or biorthogonal inner products. Profile of the bound state is shown in Fig. 2(b). The corresponding eigenenergy is

[TABLE]

where $\beta^{\pm}$ are positive real numbers and fulfill the equations

[TABLE]

We note that two bound states coalesce to a single one

[TABLE]

at EP with $\beta=\ln\left[\Delta/\left(2\kappa\right)+\sqrt{\Delta^{2}/\left(4\kappa^{2}\right)-1}\right]$ and $T=T_{\mathrm{c}}=\sqrt{\Delta^{2}-(2\kappa)^{2}}$. When $T>T_{\mathrm{c}}$, the bound-state energy becomes an imaginary number. This exact solution indicates that stable bound states can be formed by the non-Hermitian impurity. In the case of multi impurities, the above exact solution is still applicable when the distance between two neighboring impurities is sufficiently large. Nevertheless, the overlap of the wave functions occur and extended states form within the region of the impurity array. Fig. 2(c) and (d) are plots of several typical bound states for two and five imaginary impurities. For $n$ impurities, the system supports $2n$ bound states confined the region of the array. Consequently, the mid-gap sub-bands may form for large $n$.

This result has many implications. (i) Similar to the Hermitian regime, imaginary impurities can induce local bound states with real energies. (ii) In contrast to a Hermitian impurity, energy levels of imaginary impurity can be protected by band gap and coalesce at EP. (iii) Deliberately designed impurity array can take the role of waveguide. It allows a variety of non-Hermitian models with various of geometries and wide range of parameters to be candidates of waveguides. In the next section, we will show its application in an example.

III Bilayer square lattice

The bound states induced by non-Hermitian impurities can be employed to construct sub-bands in the energy gap, which constitute the channel for specific transport of particle, as a waveguide in a Hermitian systems. In the Hermitian realm, the waveguides of most kinds of discrete systems have been well studied. It is usually be done by distroying the translational symmetry. In the present work, our strategy is doing the same thing but by adding non-Hermitian terms. In the following, we will present an example that implements a 1D waveguide over a square lattice along any desired path.

We consider a bilayer square lattice model which is shown in Fig. 3(a). The Hamiltonian can be written as the Hermitian part $H_{0}$ and non-Hermitian part $H_{T}$,

[TABLE]

The corresponding Hermitian Hamiltonian has the form

[TABLE]

where $\lambda=1$ or $2$ is the index that respectively labels the position in the top or bottom layers, and $(j,l)$ is the in-plane site index. Parameters $\kappa$ and $\Delta$ of this model are intra and interlayer hopping strengths. And the distribution of imaginary potentials is given as the form

[TABLE]

By taking the linear transformations

[TABLE]

the Hamiltonian Eq. (8) can be written as

[TABLE]

We note that the bond (antibond) state of a rung can only be transited to the bond (antibond) state next to it with hopping strength $\kappa$. Therefore it can be decomposed into two independent single layer square lattices with on-site potentials $\Delta$ and $-\Delta$, respectively.

Accordingly, the non-Hermitian term reads

[TABLE]

which takes the role of interlayer imaginary tunneling. Obviously, the present model is a concrete example of the system depicted in Eq. (1). The obtained result is applicable to a wide kind of systems.

IV Dynamics in waveguide

Before proceeding, we present a general non-Hermitian model which exhibits Dirac-probability preserving dynamics. Considering a chiral symmetric system, the Hamiltonian can be written in the block off-diagonal form Ryu

[TABLE]

where $D$ as an arbitrary $N\times N$ matrix. The basis in $H$ can be a complete set of site states for a bipartite lattice. Based on $H$ as the original Hermitian Hamiltonian, a non-Hermitian Hamiltonian $\mathcal{H}$ is generated as the form

[TABLE]

where $\sigma_{z}$ is the Pauli matrix, and $I_{N}$ denotes the $N\times N$ identity matrix. The chiral symmetry of $H$ ensures its eigenvalues and eigenvectors have the following properties: (i) The eigenvalues are always in pairs, i.e., the spectrum has the form $\left\{\varepsilon_{n},\varepsilon_{-n}\right\}$ with $\varepsilon_{-n}=-\varepsilon_{n}$. (ii) The corresponding eigenvector $\left\{\left|\phi_{n}\right\rangle,\left|\phi_{-n}\right\rangle\right\}$ obeys

[TABLE]

It turns out that (see Appendix 2) the spectrum of $\mathcal{H}$ has the form $\left\{\epsilon_{n},\epsilon_{-n}\right\}$ with

[TABLE]

and the eigenvector of $\left|\varphi_{\pm n}\right\rangle$ can be mapped directly from $\left|\phi_{\pm n}\right\rangle$

[TABLE]

with the mapping matrix

[TABLE]

where $a_{\pm n}=[\epsilon_{\pm n}+i\gamma]/\varepsilon_{\pm n}$ is a complex number with unit modulus, $\left|a_{\pm n}\right|=1$ for real $\epsilon_{\pm n}$.

Obviously, $M_{\pm n}$ is unitary matrix when the spectrum is fully real. Remarkably, it can be proved that eigenvector set $\left\{\left|\varphi_{\pm n}\right\rangle\right\}$ obeys quasi-orthonormal relation under the Dirac inner product, i.e.,

[TABLE]

for $mn>0$ and $\epsilon_{m}\epsilon_{n}\neq 0$. It indicates that the non-Hermitian system acts as a Hermitian one when only one of subspaces with positive or negative spectrum is concerned. A direct conclusion is that, the evolved state in one of the subspace of $\mathcal{H}$ maintains the preservation of Dirac probability, exhibiting the Hermitian dynamic behavior.

Based on the above analysis, the bilayer system is a candidate for waveguide allowing quantum transport with probability preserving. We take the waveguide array by a collection of sites $\left\{\left(j,l\right)\right\}$ and setting $\gamma_{j,l}=\gamma$, but zero for the rest of sites. In this paper, the set of sites $\left\{\left(j,l\right)\right\}$ is selected under the rule: it forms a non-Hermitian ladder system, described by the Hamiltonian

[TABLE]

The scheme is schematically illustrated in Fig. 3(b). There are numerous configurations for the path to connect any two distant locations. When the parameters are in the range $\gamma\sim\Delta\gg\left|\kappa\right|$, the sub-system $H_{\mathrm{Ladd}}$ decouples from the the bilayer system. The dynamics in the waveguide array is governed by the Hamiltonian $H_{\mathrm{Ladd}}$. In certain parameter region, $H_{\mathrm{Ladd}}$ possesses fully real spectrum and obeys the Hermitian dynamics since it has the form of Eq. (14).

The system $H_{\mathrm{Ladd}}$ can be regarded as an extended non-Hermitian SSH chain with long-range hopping term (see Appendix 3). It turns out that such a model share the same dynamic behaviors with the simplest non-Hermitian SSH chain within certain parameter region (strong dimerization limit), which has been studied in the previous works HWH ; ZKL1 ; ZKL2 . It is expected that the dynamics of a non-Hermtian SSH chain emerges in the bilayer squrare lattice with pre-engineered imaginary impurities.

To see waveguide dynamics in the bilayer square lattice system, we perform the numerical simulations. The initial state is taken as the bilayer Gaussian wave packet, which has the form

[TABLE]

where $n$ is the site index along the waveguide array, $N_{\mathrm{c}}$ is the center of Gaussian wave packet, $k_{\mathrm{c}}$ is the central momentum, and $\Omega=2\sqrt{\pi}/\alpha$ is the Dirac normalization factor. According to previous work HWH ; ZKL1 ; ZKL2 , the wave packet should propagte along the array without spreading approximately, if the sub-bands are sufficiently separated from the bilayer bands. The evolved state has the form

[TABLE]

which can be computed by exact diagonalization.

The probability distribution at position $(i,j)$ at time $t$ is defined by the sum of the Dirac probabilities of the top and bottom layers

[TABLE]

where $\left|i,j,\lambda\right\rangle$ denotes the position state. To demonstrate the efficiency of the waveguide, we define the function

[TABLE]

to record the trace of the wave packet. The simulation is performed for the systems far from EP, i.e., the positive and negative waveguide bands are well separated, and two kinds of waveguide configurations: (i) a single path waveguide with two right-angle bends, and (ii) a beam splitter with two right-angle bends. Their schematics are shown in Fig. 4(a) and Fig. 5(a). The initial state is a Gaussian wave packet with central momentum $k_{\mathrm{c}}=-\pi/2$. Since it only relates to a single sub-band in large gap limit, it is expected that the time evolution exhibits a probability preserving behavior approximately. For case (i), the numerical simulation results indicates that the wave packet propagate along the designed waveguide array path as expected. For the (ii), it shows that the wave packet split into two parts after passing through the joint. In both cases, the wave packet is confined within the defects well and transmitted efficiently around the corners. These numerical results demonstrate and verify our theory for quantum transport in non-Hermitian impurity array.

V Conclusion and Discussion

In summary, we have studied the formation of band gap bound states induced by a non-Hermitian impurity embedded in a Hermitian system. We have shown that a pair of bound states emerges inside the band gap when a $\mathcal{PT}$ imaginary potential is added in a strongly coupled bilayer and the bound states become strongly localized when the system approaches to the exceptional point. Inspired by this, we construct an impurity array which can be described by a non-Hermitian SSH type effective Hamiltonian which possesses $\mathcal{CT}$ symmetry. It is the first time to establish a theory for non-Hermitian impurity-induced waveguide, which paves the way for the non-Hermitian device design. As a supplementary material (Appendix 2) for our theory, we first provide a rigorous proof for the features of a $\mathcal{CT}$-symmetric system, including the reality spectrum and quasi orthogonality of Dirac inner product. We would like to point out that such two conclusions are independent of other requirements for the original Hermitian system, such as translational symmetry. Thus a variety of non-Hermitian models with various of geometries is allowed. The numerical simulations for the quantum transport of wave packet in right-angle bends waveguide and $Y$-beam splitter have demonstrated this point.

Appendix

In this appendix we present the Bethe ansatz solution for the Hamiltonian from Eq. (2) and properties of the non-Hermitian ladder system.

1 Bound states induced by non-Hermitian impurity

For a single-tunneling non-Hermitian Hamiltonian (2), the bound-state Bethe ansatz wave function has the form

[TABLE]

where $\beta^{\sigma}$ is a positive real number, indicating the strength of localization around the non-Hermitian tunneling $iT$. The Schrödinger equation $H_{\mathrm{Chain}}\left|\psi_{\mathrm{B}}\right\rangle=E_{\mathrm{B}}\left|\psi_{\mathrm{B}}\right\rangle$ with bound-state energy $E_{\mathrm{B}}$ gives

[TABLE]

at $\left|j\right|\geqslant 1$, and

[TABLE]

at $j=0$. The existence of bound-state solution requires

[TABLE]

Then evanescent coefficient $\beta^{\sigma}$ can be determined by

[TABLE]

And Eq. (A3) leads to

[TABLE]

Then the wave function can be written as

[TABLE]

where the normalization coefficient is neglected since it should be valued in the frameworks of Dirac or biothonormal inner product in practice. We note that Eq. (A5) is symmetric under the operation $\beta^{+}\leftrightarrows\beta^{-}$. Thus, if $(\beta^{+},\beta^{-})=(x,y)$ is a solution with the eigenvalue $E_{\mathrm{B}}$, then (\beta^{+},\beta^{-})=(y,x)\corresponds to another solution with eigenvalue $-E_{\mathrm{B}}$. Accordingly, when take $T=T_{\mathrm{c}}=\sqrt{\Delta^{2}-(2\kappa)^{2}}$, we have $\beta_{c}=\beta^{+}=\beta^{-}$ with

[TABLE]

and two bound states coalesce to a single state

[TABLE]

with the eigenenergy $E_{\mathrm{B}}^{\mathrm{EP}}=0$, indicating the occurrence of EP. We note that $\left|\psi_{\mathrm{B}}^{\mathrm{EP}}\right\rangle$ has the identical Dirac probability distributions on the two chains.

In strong localization limit with $e^{2\beta^{\sigma}}\gg 1$, we have the approximate solutions

[TABLE]

which are still in agreement with the symmetry of Eq. (A5).

2 Dirac probability preservation

Consider a Hermitian Hamiltonian with chiral symmetry,

[TABLE]

where $D$ as an arbitrary $N\times N$ matrix. The Schrödinger equation is

[TABLE]

and obey

[TABLE]

due to the chiral symmetry

[TABLE]

A non-Hermitian Hamiltonian $\mathcal{H}$ can be generated as the form

[TABLE]

where $\sigma_{z}$ is the Pauli matrix, and $I_{N}$ denotes the $N\times N$ identity matrix. We note that non-Hermitian term breaks the chiral symmetry, but $\mathcal{H}$ has $\mathcal{CT}$ symmetry. The corresponding Schrödinger equation is

[TABLE]

and obey

[TABLE]

with the mapping matrix

[TABLE]

where $a_{\pm n}=[\epsilon_{\pm n}+i\gamma]/\varepsilon_{\pm n}$ fulfills $\left|a_{\pm n}\right|=1$ for real $\epsilon_{\pm n}$ and is pure imaginary for imaginary $\epsilon_{\pm n}$. For real $\epsilon_{\pm n}$, the factor $a_{\pm n}$ can be written in the form of $a_{\pm n}=e^{i\theta_{n}}$, with $\theta_{n}=\arctan\left(\gamma/\epsilon_{\pm n}\right)$. This can be shown as following.

Actually, from

[TABLE]

we have $H\left|\varphi_{\pm n}\right\rangle=\left(HM_{\pm n}\right)\left|\phi_{\pm n}\right\rangle$, and

[TABLE]

Therefore, from

[TABLE]

and $a_{\pm n}=\left(\epsilon_{\pm n}+i\gamma\right)/\varepsilon_{\pm n}$, $\epsilon_{\pm n}=\pm\left(\varepsilon_{n}^{2}-\gamma^{2}\right)^{1/2}$, we have $\varepsilon_{\pm n}\left(a_{\pm n}\right)^{-1}+i\gamma=\varepsilon_{\pm n}a_{\pm n}-i\gamma=\epsilon_{\pm n}$, which leads to

[TABLE]

The mapping relation between vectors $\left|\phi_{n}\right\rangle$ and $\left|\varphi_{n}\right\rangle$ can result in many interesting and useful applications. In general, set $\left\{\left|\phi_{n}\right\rangle\right\}$ obeys orthogonal relation in the framework of Dirac inner product, while set $\left\{\left|\varphi_{n}\right\rangle\right\}$ does not due to the non-Hermiticity of $\mathcal{H}$. We will show that $\left\{\left|\varphi_{n}\right\rangle\right\}$ still obeys Dirac orthogonal relation within one of the subspace ($n>0$ or $n<0$).

We start with the eigenvector of Hermitian Hamiltonian $H$, which has the form

[TABLE]

where $A_{n}$ and $B_{n}$ are two $N\times 1$ vectors, representing the wave function of sublattice $\mathrm{A}$ and $\mathrm{B}$. $\left\{\left|\phi_{n}\right\rangle\right\}$ fulfill orthogonal normalization condition

[TABLE]

The Schrödinger equation of $\left|\phi_{n}\right\rangle$ has the form

[TABLE]

or explicitly

[TABLE]

Multiplying by $A_{m}^{{\dagger}}$ or $B_{m}^{{\dagger}}$, respectively, we have

[TABLE]

Considering the conjugation of above Schrödinger equation (A38)

[TABLE]

Eq. (A39) together with Eq. (A40) gives

[TABLE]

and consequently leads to

[TABLE]

In the case of $mn>0$ and $\varepsilon_{n}\varepsilon_{m}\neq 0$, we have

[TABLE]

Together with the orthogonal normalization condition Eq. (A36), we obtain

[TABLE]

which means that eigenvector $\left|\phi_{n}\right\rangle$ has the same Dirac probability in sublattice $\mathrm{A}$ and $\mathrm{B}$.

This factor leads to an important conclusion for vector $\left\{\left|\varphi_{n}\right\rangle\right\}$: different eigenstates in one of the subspace ($mn>0$) of $\mathcal{H}$ are still orthogonal. In fact, the mapping matrix gives

[TABLE]

In addition, for full real spectrum we have

[TABLE]

since $M_{n}$ is unitary, i.e., $\left|a_{n}\right|=$ $\left|e^{i\theta_{n}}\right|=1$.

As an application, we will show that for any initial state in one of the subspace ($n>0$ or $n<0$) of the non-Hermitian system $\mathcal{H},$ the time evolution preserves the Dirac probability. Considering an initial state in one of the subspace with the form $\left|\psi(t=0)\right\rangle=\sum_{n=1}^{N}C_{n}\left|\varphi_{n}\right\rangle$ with $\sum_{n=1}^{N}C_{n}^{\ast}C_{n}=1$, the evolved state can be written as

[TABLE]

In the condition of full real spectrum, the Dirac probability is

[TABLE]

which maintains the preservation of Dirac probability, exhibiting the Hermitian dynamic behavior.

3 Non-Hermitian ladder and SSH models

In the condition of $\Delta\gg\kappa$, (strong dimerization limit) the non-Hermitian ladder described by the Hamiltonian in Eq. (20)

[TABLE]

is equivalent to a non-Hermitian SSH model

[TABLE]

Actually, the core matrix of $H_{\mathrm{Ladd}}$ is

[TABLE]

For $H_{\mathrm{SSH}}$, we have the core matrix

[TABLE]

which has the same spectrum with

[TABLE]

In the case of strong dimerization limit $\Delta^{2}\gg\kappa^{2}$, we have

[TABLE]

or

[TABLE]

The spectra of ladder model and SSH model with several $\gamma$ are shown in Fig. 6(a) and (b), respectively. It indicate that two spectra are almost identical.

Acknowledgements.

This work was supported by National Natural Science Foundation of China (under Grant No. 11874225).