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Abstract
We study the topological features in a trimerized lattice of parity-time symmetry with comparable nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings as well as a Peierls phase. Eigen energies of four edge states in two bandgaps, of topological origin verified by the quantized total Zak phase, are surprisingly independent of the NNN coupling and the Peierls phase. Topological regions with respect to the intercell NN coupling, as the intracell NN coupling is fixed, can be extended with reinforced localization strengths for one pair of edge states but reduced with weakened localization strengths for the other pair of edge states, by increasing the NNN coupling. The partial overlapping between extended and reduced topological regions promises then a two-step phase transition of ‘zero – two – four’ edge states, viable to be periodically modulated by the Peierls phase.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Significant advances have been made recently in investigating the topological phases and the band structures of various one-dimensional (1D) crystals, among which those described by the Aubry-André-Harper (AAH) model of different features have attracted particular attentions [1–11]. As a classical topological model, the AAH model was initially proposed to study the localization transition for certain quasi-periodic systems [5–7], and typically exhibit two topological phases: the nontrivial one featured by an energy spectrum containing four edge states and the trivial one featured by an energy spectrum containing zero edge states, in two bandgaps. This model can also be mapped into a two-dimensional (2D) Hall effect system with topological edge states, and may give rise to a topological quasicrystal when the incommensurate modulation is considered [12–14]. We further note that it is viable to realize exponential localization, surface soliton, and superfluid-to-insulator phase transition in various optical lattices described by this model [15–17]. It is thus clear that, due to its wide applicability to optical, optomechanical, and other physical systems, the AAH model provides indeed a versatile platform for exploring the topological states of matterr [4,18–20] and quantum many-body localization [21–24].
Realistic topological systems inevitably interact with nearby environments so that the dissipative effects have to be considered, usually featured as complex potentials in relevant Hamiltonians [25,26]. Such non-Hermitian systems have been well explored to generate exotic phenomena not available yet in the Hermitian systems, especially in regard of parity-time ($\mathcal{PT}$) symmetry [27–29] that requires balanced gain and loss, e.g., in the optical settings [30–34]. In particular, dissipative effects of gain and/or loss have been examined in various AAH models [35–40], whose topological features can be somehow altered but are robust against the non-Hermiticity. On the other hand, a synthetic gauge field may be introduced into topological systems to attain an alternative freedom of dynamic modulation - the generated Peierls phase, yielding thus complex hopping rates [41–44]. The realization of synthetic gauge fields in optical resonators and waveguides are attracting more interests [45–51], but little endeavors have been made to explore them in combination with other effects like $\mathcal{PT}$ symmetry, e.g., in the phase control of topological edge states for achieving stronger photonic localizations and richer topological phases.
Since Haldane and Raghu transferred topological phases from electronics to photonics [52], photonic topological insulators have triggered extensive research interests [53–61], providing thus a perspective for robustly manipulating photon transport against system impurities. Now it is known that photonic topological insulators offer the possibility of eliminating backscattering losses, which then enable largely improved efficiencies of integrated photonic devices and optical communication systems [53–55]. Meanwhile, Floquet quantum states in optically driven semiconductors have been examined to reveal topological non-equilibrium effects with impacts on optoelectronic applications [56–58]. Optically driven semiconductors have also been studied in the strong coupling regime to control polariton lasing or condensates [59–61]. While most topological structures own fixed optical properties, recent advances show that topological photonic crystals may become tunable with all-optical free-carrier excitation allowing for fast refractive-index modulations [61]. These significant researches can be well extended as synthetic gauge fields are further introduced to manipulate topological effects.
In this paper, we investigate a trimerized optical lattice composed of $\mathcal{PT}$-symmetric triangle unit cells in the presence of balanced gain and loss. Then we realize an extended AAH model including the intracell next-nearest-neighbor (NNN) coupling and the Peierls phase in addition to the intracell and intercell nearest-neighbor (NN) couplings. The specially introduced NNN coupling and Peierls phase are found to have no effects on the complex eigen energies of two upper and two lower edge states, whose topological origin can be inferred from the total Zak phase of values ‘0’, ‘1’, and ‘2’ corresponding to zero, two, and four edge states in order. Modulating the NNN coupling and/or the Peierls phase, however, it is viable to extend the topological regions for one pair of edge states with reinforced localization strengths of photon distributions and meanwhile reduce the topological regions for the other pair of edge states with weakened localization strengths of photon distributions. This indicates a strong correlation of periodic and inverse modulations on two upper and two lower edge states in terms of topological region sizes and photonic localization strengths.
2. $\mathcal{PT}$-symmetric trimerized lattice
We start by considering in Fig. 1 a trimerized 1D optical lattice [62,63] with $\mathcal{N}$ triangle unit cells, which may be composed of waveguides [64] or resonators [65]. The unit cells exhibit real couplings: the intracell NN coupling $g_{1}$, the intracell NNN coupling $\nu$, and the intercell NN coupling $g_{2}$. Balanced gain and loss are introduced into each unit cell described by $\mathcal{PT}$-symmetric potentials: $i\gamma$ on the gain sites $A$, $0$ on the neutral sites $B$, and $-i\gamma$ on the loss sites $C$. A synthetic gauge field is also applied to generate the Peierls phase $\varphi$ for each unit cell. These considerations yield an extended $\mathcal{PT}$-symmetric AAH model, in which $\varphi$ is meaningful only if $\nu \ne 0$. Consequently, we can write down the Hamiltonian
Fig. 1. A $\mathcal{PT}$-symmetric trimerized lattice with $\mathcal{N}$ unit cells. The gain sites $A$ (green), neutral sites $B$ (white), and loss sites $C$ (red) are coupled by the intracell constants $g_1$ and $\nu$ as well as the intercell constant $g_2$, with the Peierls phase $\varphi$ generated by a synthetic gauge field.
Using $\hat {a}_{n}$, $\hat {b}_{n}$, and $\hat {c}_{n}$ to denote the annihilation operators on sites $A$, $B$, and $C$ of the $n$th unit cell, it is easy to attain from $H_{r}$ the following dynamic equations
Our AAH lattice is equivalent to a more general non-Hermitian one, whose sites $A$, $B$, and $C$ in each unit cell have potentials $0$, $-i\gamma$, and $-2i\gamma$, respectively [45]. Then an experimental implementation of our proposal in coupled waveguides requires to attain the desired losses by introducing microscopic scattering points with suitable concentrations [45] and realize the synthetic gauge field by spiralling the parallel waveguides with a common helix radius [46]. In this case, the three intracell and intercell couplings become $g_{1}e^{i\varphi _{1}}$, $g_{2}e^{i\varphi _{2}}$, and $\nu e^{i\varphi _{3}}$ instead, which can be replaced by $g_{1}$, $g_{2}$, and $\nu e^{i(2\varphi _{1}+\varphi _{3})}$ after a simple gauge transformation like $\hat {a}_{n}\rightarrow \hat {a}_{n}e^{i2(n-1)\varphi _{1}+(n-1)\varphi _{2}}$, $\hat {b}_{n}\rightarrow \hat {b}_{n}e^{i(2n-1)\varphi _{1}+(n-1)\varphi _{2}}$, and $\hat {c}_{n}\rightarrow \hat {c} _{n}e^{i2n\varphi _{1}+(n-1)\varphi _{2}}$. Our proposal may also be realized in coupled resonators, in which balanced gain and loss are much easier to be introduced [66] and the synthetic gauge field can be realized in more flexible ways [47–49].
3. Effects of NNN coupling on topological edge states
In the case of $\nu =0$, Figs. 2($a_1$) and 2($a_2$) show that the real and imaginary energy spectra are symmetric with respect to both $E_{r}=0$ and $\theta =\pi$; the three bands of real eigenvalues coalesce in two small regions centered at $\theta =0.5\pi$ and $\theta =1.5\pi$ of widths determined by $\gamma$; four edge states (green and brown) exist inside the two gaps of real bands but extrude into the coalescence region [62]. In the presence of NNN coupling with $\nu =0.2$ but absence of Peierls phase with $\varphi =0$, however, Figs. 2($b_1$) and 2($b_2$) show that the real and imaginary energy spectra are no longer symmetric with respect to $E_{r}=0$ though still symmetric with respect to $\theta =\pi$. It is of special interest that the upper (lower) bandgap centered at $\theta =\pi$ becomes narrower (wider), which then results in a reduction (expansion) of the topological region containing two upper (lower) edge states. Thus, we expect a controlled transition from zero edge states to four edge states through two edge states as $\theta$ is varied around $0.5\pi$ or $1.5\pi$. In the case of $\nu =0.4$, Figs. 2($c_1$) and 2($c_2$) show that a further reduction (expansion) of the upper (lower) bandgap results in a much wider transition region where only two lower edge states can be found.
Fig. 2. Real ($a_{1}$, $b_{1}$, $c_{1}$) and imaginary ($a_{2}$, $b_{2}$, $c_{2}$) energy spectra of $\mathcal{PT}$-symmetric Hamiltonian $H_{r}$. Solid green and dashed brown lines denote four edge states. Relevant parameters are given in the main text except $\varphi =0$ and $\nu =0$ (first row); $\nu =0.2$ (second row); $\nu =0.4$ (third row).
Taking $\theta =\pi$ as an example, we check photon distributions for the two upper edge states with Re$(E_{r})\simeq 1.0$ and Im$(E_{r})=\pm 0.1$ in the left columns of Fig. 3, while for the two lower edge states with Re$(E_{r})\simeq -1.0$ and Im$(E_{r})=\pm 0.1$ in the right columns of Fig. 3. This is done in order to understand how the four edge states are affected by the NNN coupling constant $\nu$ in terms of their localization behaviors. It is clear that the four edge states have identical localization behaviors in the case of $\nu =0$ because their spectra of real and imaginary energies are symmetric with respect to $E_{r}=0$ and $\theta =\pi$. In the case of $\nu =0.2$, however, we find that the two upper edge states extend more toward the lattice center, while the two lower edge states are more localized at the lattice boundaries. As the NNN coupling constant is increased to $\nu =0.4$, localization behaviors of the two lower and upper edge states are further reinforced and become roughly vanishing, respectively.
Fig. 3. Photon distributions for upper ($a_{1}$, $b_{1}$, $c_{1}$) and lower ($a_{2}$, $b_{2}$, $c_{2}$) edge states attained with $\theta =\pi$. Squares, circles, and triangles denote $|a_{n}|^{2}$, $|b_{n}|^{2}$, and $|c_{n}|^{2}$ in order. Relevant parameters are the same as in Fig. 2.
In order to reveal the topological origin of these edge states, we transform Hamiltonian $H_{r}$ into
Fig. 4. Real energy bands in the momentum space attained with the same parameters as in Fig. 2 except $\nu =0.2$ and $\theta =0.3\pi$ ($a$); $0.5\pi$ ($b$); $0.7\pi$ ($c$); $1.0\pi$ ($d$).
The topological origin of these edge states can be verified by examining the total Zak phase [67]
Here $\delta _\pm =-i\gamma /2\pm g^{\prime }_{1}$ with $g_{1}^{\prime }=\sqrt {g_{1}^{2}-\gamma ^{2}/4}$ refer to the effective on-site potentials while
It is then not difficult to derive from $H_{m}^{\prime }$ the left and right eigenvectors corresponding to $E_{m,\alpha }^{\prime }$
Fig. 5. Total Zak phase against $\theta$ attained with $\nu =0$ (black solid); $\nu =0.2$ (red dashed); $\nu =0.4$ (blue dotted). Other parameters are the same as in Fig. 2.
4. Effects of Peierls phase on topological edge states
Now we try to find the analytical solutions determining the topological region sizes $\Delta \theta _{\pm }$ of the upper ($+$) and lower ($-$) edge states in the limit of $\gamma <2g_{1}$ with $\nu \ne 0$ and $\varphi \ne 0$. In addition, we will quantify $\Delta \theta _{\pm }$, together with the corresponding localization strengths, as a function of $\varphi$ via numerical calculations. This is important to understand how these edge states of topological origin can be modulated by the Peierls phase $\varphi$.
Without the loss of generality, we just consider the two edge states at the left boundary. They can be observed when, ($i$) $c_{n}=0$ so that their energies are independent of $g_{2}(\theta )$ and ($ii$) $|a_{n+1}|<|a_{n}|$ so that their profiles decay toward the lattice center, are simultaneously satisfied. With the first requirement $c_{n}=0$, it is straightforward to attain the eigen equations of $H_{r}$
Next, we define the inverse participation ratio (IPR) for the upper and lower edge states [70,71] as
It is clear that the lower and upper edge states will exhibit alternately stronger localization behaviors as the Peierls phase changes from $\varphi =0$ to $\varphi =2\pi$.
Numerical calculations based on Eq. (11) and Eq. (12) show in Fig. 6 that both region sizes $\Delta \theta _{\pm }$ and IPRs $I_{\pm }$ are $\varphi$-independent (see the red lines) in the absence of NNN coupling, but become sensitive to $\varphi$ with a period $2\pi$ (see the blue and green lines) in the presence of NNN coupling. To be more specific, the upper and lower edge states have an inverse dependence on $\varphi$ so that the increase (decrease) of $\Delta \theta _{+}$ or $I_{+}$ must be accompanied by the decrease (increase) of $\Delta \theta _{-}$ or $I_{-}$, indicating a strong correlation of the phase modulations on $\Delta \theta _{\pm }$ and $I_{\pm }$. It is of particular interest that we can predict, by setting $\nu =-\gamma \sin \varphi \mp 2g_{1}^{\prime } \cos \varphi$ in Eq. (11), intersections of the blue and red lines ($\varphi =0.566\pi$ and $\varphi =1.5\pi$ for $\nu =0.2$; $\varphi =0.595\pi$ and $\varphi =1.468\pi$ for $\nu =0.4$) as well as intersections of the green and red lines ($\varphi =0.434\pi$ and $\varphi =1.5\pi$ for $\nu =0.2$; $\varphi =0.405\pi$ and $\varphi =1.532\pi$ for $\nu =0.4$). The blue and green lines, however, always intersect at $\varphi =0.5\pi$ and $\varphi =1.5\pi$ (independent of $\nu$) as predicted by setting $\cos \varphi =0$ in Eq. (13).
Fig. 6. Region sizes $\Delta \theta _{\pm }$ and IPRs $I_{\pm }$ of upper (blue lines) and lower (green lines) edge states against Peierls phase $\varphi$ attained with $\gamma =0.2$ and $\nu =0.2$ ($a$, $c$); $\nu =0.4$ ($b$, $d$). Red lines show corresponding results attained with $\nu =0$ as a reference. Other parameters are the same as in Fig. 2.
Finally, we show in Fig. 7 numerical results on region sizes $\Delta \theta _{\pm }$ and IPRs $I_{\pm }$ for two even larger values of the NNN coupling: $\nu =0.6$ and $\nu =0.8$. We find on one hand that no valid values of $\Delta \theta _{\pm }$ and $I_{\pm }$ exist for certain values of $\varphi$, indicating thus the vanishing of both edge states, as $\nu$ is so large that Eq. (11) cannot be satisfied for arbitrary values of $\varphi$. To be more concrete, the upper (lower) edge state exists for $0.266<\varphi /\pi <0.908$ and $1.156<\varphi /\pi <1.798$ ($0.092<\varphi /\pi <0.734$ and $1.202<\varphi /\pi <1.844$) in the case of $\nu =0.6$, while for $0.407<\varphi /\pi <0.867$ and $1.197<\varphi /\pi <1.656$ ($0.133<\varphi /\pi <0.593$ and $1.344<\varphi /\pi <1.803$) in the case of $\nu =0.8$. On the other hand, localization strengths in terms of $I_{\pm }$ can be increased (reduced) in larger (smaller) topological regions in terms of $\Delta \theta _{\pm }$, until they reach the saturation values $I_{\pm }^{max}\simeq 0.40$ and $I_{\pm }^{min}\simeq 0.02$ together with $\Delta \theta _{\pm }^{max}/\pi =2.0$ and $\Delta \theta _{\pm }^{min}/\pi =0.0$ when the NNN coupling is increased to be $\nu \geq 0.5$. It is therefore the interplay of Peierls phase $\varphi$ and NNN coupling $\nu$ that determines region sizes and localization strengths of both edge states.
Fig. 7. Region sizes $\Delta \theta _{\pm }$ and IPRs $I_{\pm }$ of upper (blue lines) and lower (green lines) edge states against Peierls phase $\varphi$ attained with $\gamma =0.2$ and $\nu =0.6$ ($a$, $c$); $\nu =0.8$ ($b$, $d$). Red lines show corresponding results attained with $\nu =0$ as a reference. Other parameters are the same as in Fig. 2.
5. Conclusions
In summary, we have studied the topological features in a $\mathcal{PT}$-symmetric trimerized lattice, of coupled waveguides or resonators, with two specially introduced parameters, i.e., NNN coupling $\nu$ and Peierls phase $\varphi$. Analytical and numerical results show that $\nu$ and $\varphi$ can jointly result in the correlated, periodic, and inverse modulations on two pairs of upper and lower edge states of topological origin verified by the quantized total Zak phase. This promises, in particular, a well controlled manipulation on topological regions of edge states and localization strengths of photon distributions. As our PT-symmetric trimerized lattice refers to optical resonators, the balanced dissipation and gain constants reflect as usual lifetimes of optically excited states [61], with which the two edge states of energies $E_r=i\gamma /2\pm g_{1}^{\prime }$ ($E_r=-i \gamma /2 \pm g_{1}^{\prime }$) will increase (decrease) at the rate $e^{\gamma t/2}$ ($e^{-\gamma t/2}$). Our results should be instructive for further studies on topological features in similar $\mathcal{PT}$-symmetric or general non-Hermitian systems with aperiodic or disordered structures [72,73].
Funding
Ministry of Science and Technology of the People's Republic of China (No. 2021YFE0193500); National Natural Science Foundation of China (No. 11674049, No. 11704064, No. 11861131001, No. 12074061).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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