1. Introduction

Non-Hermitian physics has attracted much attention because it combines the topological insulator field and condensed matter physics [1–9]. In previous work, the phase transition between trivial and nontrivial in topological systems has been widely studied [2–5]. Generally, the phase transition can be characterized by topological invariants, which means that the topology phase remains unchanged until topological invariants undergo a sudden change. Based on the definition and calculation of bulk topological invariants under the periodic boundary condition (PBC), the presence of edge states can be accurately predicted under the open boundary condition (OBC), which is the famous bulk-boundary correspondence (BBC) [7,10]. Non-Hermitian physics brings novel unprecedented phenomena and applications, which can describe open systems [11], electronic systems with interactions [12], and quantum systems with gain or loss [13]. The non-Hermitian systems usually exhibit a complex energy spectrum and can be realized by introducing gain or loss terms [14], nonreciprocal hoppings [8,9], and nonlinear terms [15]. However, when the system satisfies the parity-time ($PT$) symmetry [14,16] or pseudo-Hermiticity [17], it has the same purely real energy band structure as the Hermitian systems. Furthermore, non-Hermitian systems exhibit unique topological features exotic to the Hermitian case, including half-integer winding [18], non-Hermitian skin effect (NHSE) [19,20], topological semimetal phase [21], and Weyl exceptional rings [22].

The NHSE is a striking feature of non-Hermitian systems with nonreciprocal hopping, which refers to topologically protected edge states under OBC that will be hidden in the bulk states. Unlike the Hermitian case, non-Hermitian systems are very sensitive to boundary conditions, which makes the traditional BBC invalid. Previous studies have proposed several methods to restore the BBC relation and determine topological properties. For instance, one can compute the so-called generalized Brillouin zone (GBZ) by the non-Bloch band theory [8,9,23,24], or using the modified PBC [25] and the biorthogonal polarization [26,27] to directly reflect OBC properties. In addition, the NHSE has been observed in different physical platforms, including the acoustic ring resonators [28,29], coupled resonant optical waveguides [30], topoelectrical circuits [9,31], and cold atom system [32,33].

Very recently, researchers have also shown that non-Bloch $PT$ symmetry is the general way for realizing $PT$ symmetry in the presence of NHSE [34]. As to the experimental realization, the non-Bloch $PT$ phase transition is realizable in optical systems [35]. When the non-Hermiticity is below (above) a critical value, it causes the Hamiltonian to be a real (complex) spectrum. The non-Bloch $PT$ symmetry Hamiltonian typically has two phases, the $PT$-exact phase and the $PT$-broken one, respectively. The transition points between these phases are associated with the exceptional points (EPs), where the eigenstates coalesce and the Hamiltonian becomes nondiagonalizable (defective) [36,37]. The EPs often lead to inspiring novel designs and promising applications, such as unidirectional invisibility and enhanced transmission [38,39]. Moreover, as we know that the degenerate zero-energy edge states in $PT$ symmetry systems with non-Hermitian Hamiltonians are driven to undergo a topological transition from a trivial $PT$ symmetry edge state to a topological broken $PT$ one, this process that needs to cross the EPs. Therefore, a crucial question arises: Can skin states caused by the NHSE possess the eigenvalues and the corresponding eigenvectors coalescence? If so, what type is this EP associated with the non-Bloch $PT$ symmetry? We aim to answer these questions and explore novel topological features. Here, we present a simple and feasible method to simulate the non-Hermitian Su-Schrieffer-Heeger (SSH) model, where the nonreciprocal intracell hopping can be realized by tuning the total phases of three-mode systems consisting of two optical modes and an auxiliary mode. Our work systematically investigates the non-Hermitian properties of the current system. We find that the bulk states experience energy spectrum degeneracy and eigenvector coalescence at the non-Bloch $PT$ phase transition point, corresponding to the higher-order EPs. Further, to gain a comprehensive understanding of the localization behaviors among eigenstates and the non-Hermitian skin phase transition, we analyze and calculate the directional mean inverse participation ratio and GBZ. Our work contributes to a better understanding of the impact of nonreciprocal hopping on non-Hermitian optical systems.

2. Model and Hamiltonian

We consider a cyclic three-mode optical system consisting of two linear optical modes $a$ and $b$ coupled with one auxiliary mode $c$, as shown in Fig. 1. The total Hamiltonian of the system is given by ($\hbar = 1$)

The quantum Langevin equations of operators is obtained as

The nonreciprocal hopping between modes $a$ and $b$ can be realized by adiabatically eliminating the auxiliary mode $c$, i.e., $\dot {c} = 0$. Under the adiabatical condition $\kappa _{c} \gg \max \{\kappa _{a}=\kappa _{b}, J_{ac}=J_{bc}, J\}$, we obtain

Then the effective dynamical equations of $a$ and $b$ are given by

Thus, the effective Hamiltonian of the system can be written as

 

Fig. 1. Schematic diagram of 1D non-Hermitian SSH model. The dotted rectangle represents the unit cell. The intercell hopping is reciprocal with tunnel coupling strength $G$. The intracell hopping between $a$ and $b$ with strengths $J_{R}=J+J'\cos \theta$ and $J_{L}=J-J'\cos \theta$ holds the nonreciprocal forms, which can be implemented by using a three modes optical system.

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To extend the cyclic three-mode optical system into a lattice model, we arrange the unit cells in a periodic array and connect them through the photon tunnel coupling between optical modes $b$ and $a$, with a hopping strength of $G$ [42]. Thus, the current system is equivalent to a 1D non-Hermitian SSH lattice model with nonreciprocal hopping. Each unit cell contains two sub-lattices. Then, the Hamiltonian of the system can be expressed as

One can see that the Hamiltonian possesses the chiral symmetry with the relationship of $\sigma _{z} H(k)\sigma _{z} = -H(k)$, where $\sigma _{x}$, $\sigma _{y}, \sigma _{z}$ are the Pauli operators. The energy eigenvalues are $E_{\pm }(k)=\pm \sqrt {d^{2}_{x}(k)+d^{2}_{y}(k)}$, and the positions of band gap closing is a vital component of the topological information, which requires $J=G \pm J'\cos \theta ~(k=\pi )$ or $J=-G \pm J'\cos \theta ~(k=0)$. Figure 2 shows that this information does not appear to be preserved properly under different boundary conditions. Clearly, the energy spectrum of the Hamiltonian is gapless under OBC, while the spectrum of $H(k)$ is gapped. This indicates that the breakdown of traditional BBC is an undeniable fact in non-Hermitian systems caused by the NHSE.

 

Fig. 2. Comparison of energy spectrum of the system as a function of $\theta$ with length $N=20$ (unit cell) under OBC and PBC, respectively. Other parameters are taken as $J = 0.4$ and $J' = 0.5$.

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We know that the NHSE is a classic feature of the non-Hermitian system with nonreciprocal hopping, which leads to that the bulk states tend to be localized at the boundary of the system under OBC. To characterize the left or right localization properties of the eigenstates, we can use the directional inverse participation ratio (dIPR) defined as [43,44]

3. Results and discussions

3.1 Zero-energy edge states and winding numbers

We first consider constructing a diagonal matrix $S$ with matrix elements $\{1,r,r,r^{2},r^{2},\ldots,r^{L-1},$ $r^{L-1},r^{L}\}$ with $r=\sqrt {|(J+J'\cos \theta )/(J-J'\cos \theta )|}$. Due to similarity transformation $\widetilde {H}=S^{-1}HS$ always keeps the eigenvalues unchanged, the Hamiltonian of the sysytem under OBC becomes

Applying the Fourier transformations, the Hamiltonian of the system in momentum space can be written as

We investigate the energy band structure of the 1D non-Hermitian SSH model with nonreciprocal hopping. As shown in Fig. 3, we plot the dIPR as a function of $\theta$ for each real and imaginary parts of the energy level under OBC with $J'=0.5$ and $N = 20$. We find that the red regions represent all the eigenstates localized at the right end of the system with dIPR > 0. Painted in blue are the regions of dIPR < 0, corresponding to all the eigenstates localized at the left end of the system. The parameter $\theta$ in the green regions leads to the appearance of extended states with dIPR closed to 0. Due to the inherent symmetry of the cosine modulation in nonreciprocal hopping, the energy band structure holds mirror symmetric for $\theta$ = 0. Therefore, the whole system undergoes the energy gap opening to closure and reopening to reclosure with the variation of $\theta$. It is well known that the topologically distinguishable nontrivial and trivial phases can be distinguished by the presence or absence of zero-energy edge states under OBC. As shown in Fig. 3(a), when $J=0.2$, we can determine that the energy gap closes at $\theta =\pm 0.44\pi$ and $\theta =\pm 0.56\pi$ by Eq. (14). The eigenstates coalesce at these points, referred to as EPs (red solid dots). In addition, the zero-energy edge states of the system have geometric multiplicity 1 and algebraic multiplicity 2. It is indicated that the current nonreciprocal system holds the more plentiful zero-energy edge states. As $J$ increases, the region of existence for zero-energy edge states gradually decreases, accompanied by a narrowing region of the imaginary energy spectrum. Until $J$ increases to 0.8, the energy levels are purely real in Fig. 3(b). The current system holds the Hermitian SSH model for $|J'\cos \theta |<|J|$, which leads to a purely real eigenenergy through the whole parameter region $\theta$. However, the non-Hermitian system leads to the appearance of a complex energy spectrum when the nonreciprocal hopping $|J'\cos \theta |>|J|$. Namely, in Fig. 3(a) the imaginary part can only appear in the region of $\theta \in [-\pi,-0.63\pi ] \cup [-0.37\pi,0.37\pi ]\cup [0.63\pi,\pi ]$. One can see that the eigenenergies of the current system undergo a real-complex phase transition process under OBC. The critical point of the phase transition can be obtained by the equation $|J'\cos \theta |=|J|$ as we show later, corresponding to a non-Bloch $PT$ phase transition.

 

Fig. 3. Energy spectrum of the system as a function of $\theta$ with (a) $J = 0.2$ and (b) $J = 0.8$. The colorbar indicates the dIPR value of eigenstates. The brown and pale blue regions represent the $PT$-exact and the $PT$-broken phases, respectively. Other parameters are taken as $J'$ = 0.5 and $N$ = 20.

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To further understand the zero-energy edge states shown in Fig. 3, we present the state distributions of the system with $\theta =-\pi, 0$, as shown in Fig. 4. We find that the zero-energy edge states and the bulk states are both localized at the one end of the system for $J=0.2$ and $0.8$, as shown in Figs. 4(a) and 4(b). Note that the state distributions of the system show mirror symmetry distributions about $\theta$ = 0. Especially, when $J$ = 0, there exist doubly degenerate zero-energy edge states in black lines localized at the two ends of the system, and the bulk states of the system are extended, as shown in Fig. 4(c).

 

Fig. 4. State distributions of the system with (a) $J$ = 0.2, (b) $J$ = 0.8, and (c) $J$ = 0. The blue, red, and green lines represent the probability distribution of the left edge states, right edge states, and bulk states, respectively. Other parameters are taken as $J'$ = 0.5 and $N$ = 20.

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Next, we proceed to a theoretical description of the band topology and demonstrate the validity of the BBC relationship in the current non-Hermitian system. In non-Bloch band theory, the Bloch phase factor $e^{ik}$ needs to be replaced by $\beta$ ($\beta =re^{ik}$), where the GBZ $C_{\beta }$ is instead of the traditional BZ. $r$ deviates from 1 due to the NHSE, corresponding to the degree of amplification with $|r|>1$ or attenuation with $|r|<1$ of the wave function. By this means, we can rewrite the Hamiltonian $H(k)$ as $H(\beta )$, which can be expressed as

The GBZ is a closed trajectory on the complex plane determined by $|\beta _{i}(E)| = |\beta _{i+1}(E)|$, where $\beta _{i}(E)$ and $\beta _{i+1}(E)$ are the middle two among all solutions of the eigenvalue equation $|H(\beta ) - E| = 0$ sorted by their moduli. Hence, the GBZ can be described by

To better explore the topological properties between trivial phase and nontrivial phase, the topological invariant needs to be calculated. Based on the GBZ approach, the critical "$\mathcal {Q}$ matrix" can be constructed from the eigenstates of $H(\beta )$, the non-Bloch winding number $W$ is defined as [8]

 

Fig. 5. (a) Non-Bloch winding number as a function of $\theta$. Parameter values are taken as $J=0.2$ and $J'=0.5$. (b) phase diagram of non-Bloch winding number on the $\theta -J$ plane with $J'$ = 0.5. The lattice length is $N=20$.

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3.2 Non-Bloch $\mathcal {PT}$ symmetry and its breaking

In this subsection, we investigate the non-Bloch $PT$ transition and discuss the reasons for the emergence of high-order EPs. Based on the non-Bloch $PT$ symmetry theory [34,35], the $PT$-exact phase of the system holds the purely real eigenenergies, while the presence of nonzero imaginary parts in the energy spectrum corresponding to $PT$-broken phase.

As shown in Fig. 3(a), the brown regions correspond to the system owning the $PT$-exact phase, whereas the pale blue regions correspond to the system holding the $PT$-broken phase. We find that the existence of EPs is related to the degeneration of band structure for the zero-energy edge states and bulk states of the system. According to the real part of the energy spectrum, one can see that there are three linearly independent eigenvectors at $\theta = \pm 0.37\pi$ and $\pm 0.63\pi$ ($|J'\cos \theta | = |J|$). Meanwhile, the bulk states in the upper (lower) energy band coalesce into a single energy with $E = 2J$ or $E = -2J$, indicated by the black empty dots. It is easy to verify that the edge states with $E = 0$ have an algebraic (geometric) multiplicity 2 (1). The bulk states with $E = \pm 2J$ have algebraic multiplicities scaling with the system size, whereas the geometric multiplicities remain 1. Therefore, the degeneration of the energy band corresponds to the higher-order EPs. On the other hand, from the imaginary part of the energy spectrum, one can see that the phase transition between the $PT$-broken and $PT$-exact regions must go through the high-order EPs.

To demonstrate the non-Bloch $PT$ phase transition mentioned above, we adopt the saddle point theorem [34], and the coalescence of saddle point energies can serve as a criterion for identifying the non-Bloch $PT$ phase transition, where the point fulfilling $\partial _{\beta }H(\beta )$ = 0 is referred to the saddle point. In this theory, solve the equations $f(\beta, E)\equiv \mathrm {det}[H(\beta )-E]=0$ and $\partial _{\beta }f(\beta, E)=0$ to get all information about saddle points. The ends of the OBC bulk spectrum always correspond the saddle points. As shown in Fig. 6, we present the complex energy plane of the systems under OBC with $J = 0.2$, $J' = 0.5$, and $N = 20$. In Fig. 6(a), the saddle points $S_{1}$ and $S_{2}$ ($S_{3}$ and $S_{4}$) are located at the end of the lower (upper) energy band with $\theta =\arccos 0.3$. The energy spectrum of the system holds purely real eigenvalues, where the red arrows indicate whether the saddle points tend to coalesce or separate. With an increase of $\theta$, the two pairs of saddle points move toward the center of the energy band, accompanied by the continuous deformation of the energy bands. Until these saddle points coalesce into single points corresponding to the higher-order EPs at $\theta =\arccos 0.4$, as shown in Fig. 6(b). Further increase of $\theta$, the two coalescent saddle points separate each other, accompanied by the appearance of the complex eigenenergy spectrum, as shown in Fig. 6(c). Analyzed results show that the non-Bloch $PT$ phase transition can be well characterized by the motion of saddle points, which is accompanied by a real-complex phase transition of the eigenenergy spectrum of the system. The bulk states coalesce to form higher-order EPs at the critical point of the non-Bloch $PT$ phase transition. Further, we show the phase boundary by using the ratio of the complex eigenenergies defined as [45]

 

Fig. 6. Evolution of the saddle points on the complex energy plan for different values of $\theta$ under OBC. The red arrows denote direction of motion of the saddle points. The parameters are taken as $J = 0.2$, $J' = 0.5$, and $N = 20$.

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Fig. 7. Ratio of the complex eigenenergies $f_{\rm Im}$ as a function of $\theta$ and $J$. The brown and pale blue regions denote the $PT$-exact and the $PT$-broken phases, respectively. The Parameter are taken as $J'$ = 0.5 and $N$ = 20.

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3.3 Non-Hermitian skin phase transition

To investigate the non-Hermitian skin phase transition of the current nonreciprocal system, we numerically calculate the dMIPR and GBZ $\beta$. From the variation of the dIPR of the energy spectrum shown in Fig. 3 and the distributions of all eigenstates shown in Fig. 4, one can see that the bulk states exhibit the clear signatures of the NHSE. Interestingly, the NHSE is excluded since the left and right hopping amplitudes are $|J'\cos \theta |$, though their signs are opposite, as shown in Fig. 4(d).

To further explore the properties of the NHSE, we plot the phase diagram for the dMIPR of the system as a function of $\theta$ and $J$ with $J'=0.5$ and $N=20$, as shown in Fig. 8(a). We find that the phase diagram will be divided into three regions. The brown and purple regions represent the left (dMIPR < 0) and right (dMIPR > 0) localization phases, respectively. The white lines with $\theta =\pm 0.5\pi$ correspond to the phase boundary. For $-\pi <\theta <-0.5\pi$ and $0.5\pi <\theta <\pi$, the skin states of the system represent the left localization as $J_{L}>J_{R}$. In the region of $-0.5\pi <\theta <0.5\pi$, the system exhibits the right localization of skin states as $J_{L}<J_{R}$. At $\theta =-0.5\pi$ and $\theta =0.5\pi$, the nonreciprocal hopping terms become zero, implying the disappearance of the NHSE, which leads to the extended bulk states. The results indicate that the NHSE is caused by nonreciprocal hopping and only depends on the sign of the nonreciprocal term, in which the eigenstates tend to localize toward the direction with stronger hoppings strength. To further illustrate the skin phase transition of the system, we adopt the method of the non-Bloch band theory, which suggests that the bulk states will be localized at the right (left) end of the system when $|\beta |>1~(|\beta |<1)$, while $|\beta |=1$ implying that the bulk states are extended. Figure 8(b) shows that the left (right) skin states in the blue (red) regions are determined by $|\beta |$. When $|\beta |>1$, leading to rightward amplification of wave propagation, the skin states show the right localization under OBC. When $|\beta |<1$, the bulk states will always be localized at the left end with the leftward amplification of wave propagation. For $|\beta |=1$, the system restores the delocalization, which leads to the extended bulk states. The GBZ shows consistency with the results of dMIPR and simultaneously elucidates the skin phase transition of the current nonreciprocal system.

 

Fig. 8. (a) Phase diagram of dMIPR as a function of $\theta$ and $J$. (b) The left and right skin states on the $\theta -J$ plane by GBZ. The parameters are taken as $J'$ = 0.5 and $N$ = 20.

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4. Conclusion

In conclusion, we have proposed a feasible scheme to realize a 1D modulated nonreciprocal SSH model using cyclic three-mode optical systems. We have investigated the topological properties by analyzing the energy spectrum and winding numbers of the current non-Hermitian system. Notably, the eigenenergies of the system undergo a real-complex transition with the change of nonreciprocal hopping, which is closely related to the non-Bloch $PT$ symmetry. We can describe the non-Bloch $PT$ phase transition through the evolution of saddle points on the complex energy plane and the ratio of complex eigenenergies. Our results reveal that the higher-order EPs caused by the non-Hermitian skin effect correspond to the critical point of the non-Bloch $PT$ symmetry. Moreover, we have studied the non-Hermitian skin phase transition by analyzing and calculating the dMIPR and GBZ. Our work provides an alternative way to investigate the novel topological and non-Hermitian effects in nonreciprocal optical systems.