Topological triple phase transition in non-Hermitian quasicrystals with complex asymmetric hopping

The triple phase transitions or simultaneous transitions of three different phases, namely, topological, parity-time $\left(\mathcal{P}\mathcal{T}\right)$ symmetry breaking, and metal-insulator transitions, are observed in an extension of the $\mathcal{P}\mathcal{T}$ symmetric non-Hermitian Aubry-André-Harper model. In this model, besides the non-Hermitian complex quasiperiodic on-site potential, non-Hermiticity is also included in the nearest-neighbor hopping terms. Moreover, the nearest-neighbor hopping terms are also quasiperiodic. The presence of two non-Hermitian parameters, one from the on-site potential and another one from the hopping part, ensures $\mathcal{P}\mathcal{T}$ symmetry transition in the system. In addition, tuning these two non-Hermitian parameters, we identify a parameters regime, where we observe the triple phase transition. Following some recent studies, an electrical circuit based experimental realization of this model is also discussed.

Figure 1

Typical nature of the eigenstates corresponding to the Hamiltonian given in Eq. (1) with the OBC (top) and the PBC (bottom) are compared for the number of lattice sites $L=233$. (a),(b) The top row shows three randomly picked eigenstates under the OBC that are all localized at some lattice site for both $h=0.4$ and $h=1.2$, respectively. (c),(d) The same eigenstates are presented in the bottom figure with the PBC, where all the eigenstates are extended for $h=0.4$, but are localized at different lattice sites for $h=1.2$.

Figure 2

The phase diagrams are presented for the topological, $\mathcal{P}\mathcal{T}$ symmetry, and the MI transitions as the function of the parameters $\gamma$ and $h$. (a) The topological phases are characterized by the calculation of the winding number $w$ given in Eq. (2). The colors from black to white represent $w=0$ to 1 values. The fluctuations in the winding number are less for $\gamma <0.1$ (refer to color bar). In this region, the phase transition occurs at $h_c=0.68$ as marked in the figure. (b) The largest value of imaginary parts of the energy eigenvalues are presented to observe the $\mathcal{P}\mathcal{T}$ symmetry transition. The different color palettes separate the real and complex regions of the energies (refer to color bar). The $\mathcal{P}\mathcal{T}$ symmetry is preserved in the regime $\gamma <0.5$. A common transition point is observed at $h_c=0.68$ for $\gamma <0.1$. (c) The maximum values of the inverse participation ratio are shown to track the MI transitions. The MI transitions are observed for the whole range of $\gamma$, but at different values of the parameter $h$. For $\gamma <0.3$, the MI transition occurs at $h_c=0.68$. The vertical yellow lines in all the figures are describing the parameter regime $\gamma <0.1$, where the triple phase transition is possible to observe. The horizontal cyan line at $h=h_c=0.68$ marks the triple phase transition points.

Figure 3

The phase argument of $\text{det}(H-E_B)$ versus $\theta$ is shown for $h=0.6$ (left) and $h=0.76$ (right) for $\gamma =0.05$ and the number of lattice sites $L=233$. As $\theta$ increases from 0 to $2\pi$, the spectral trajectory does not encircle the base energy (left). Effectively, the spectrum of $H$ does not wind. However, for $h=0.76$, the spectral trajectory encircles the base energy once and the corresponding winding number is 1.

Figure 4

The figure shows the eigenspectra, MI transition, PT transition, and topological transition by varying the parameter $h$, for $V=t=1.0$, and $\gamma =0.05$. We observe three simultaneous phase transitions. Panel (a) displays the real and imaginary parts of eigenvalues (on the complex plane) and IPRs (in color scale) under the PBC for six typical values of $h$. With an increase in the value of $h$, the eigenstates tend to get localized, which is shown by color. We observe that the eigenspectra form a loop for $h>0.68$, encircling the origin of the complex energy plane, indicating a topologically nontrivial phase with nonzero windings. Also, we observe that the eigenvalues are real for $h<0.68$, but the $\mathcal{P}\mathcal{T}$ symmetry tends to break completely for $h\ge 0.68$, and thus we observe complex eigenvalues. Panel (b) shows the real parts of the eigenenergies of the Hamiltonian under PBC, where the overall energy becomes on average constant, for $h<h_c$. The bottom part of the panel displays the imaginary parts of the eigenenergies with increasing $h$. The color coding is done according to the IPR, where the red-dashed line corresponds to the phase boundary. Panel (c) shows the winding number versus the complex phase shift $h$. The winding number makes a transition from 0 to 1 as we vary $h<h_c$ to $h>h_c$. Panel (d) represents the largest value of the imaginary part of energy versus the complex phase shift $h$. For $h<h_c$, the system changes to the unbroken $\mathcal{P}\mathcal{T}$ phase, where the spectrum becomes real. For $h>h_c$, there is a broken $\mathcal{P}\mathcal{T}$ phase owing to the appearance of complex eigenvalues. Panel (e) represents the maximum value of IPRs contrasted to the complex phase shift $h$. For $h<h_c$, we observe the delocalized phase, marked by a zero IPR value. For $h>h_c$, all eigenstates become exponentially localized, which is marked by an increase in the IPR value. The length of the lattice is chosen to be $L=233$ for all the panels.