Nonorientability-induced $\mathcal{PT}$ phase transition in ladder lattices

We study parity-time ($\mathcal{PT}$) phase transitions in the energy spectra of ladder lattices caused by the interplay between nonorientability and non-Hermitian $\mathcal{PT}$ symmetry. The energy spectra show level crossings in circular ladder lattices with increasing on-site energy gain-loss because of the orientability of a normal strip. However, the energy levels show $\mathcal{PT}$ phase transitions instead of the avoided level crossings of a Hermitian situation in $\mathcal{PT}$-symmetric Möbius ladder lattices due to the nonorientability of a Möbius strip. The latter effectively presents a perturbation that would cause avoided level crossing in a Hermitian situation, but leads, in the presence of $\mathcal{PT}$ symmetry, to locked real energy parts and conjugate values of the imaginary parts. In order to understand the level crossings of $\mathcal{PT}$-symmetric phases, we generalize the rotational transformation using a complex rotation angle. We also study the modification of resonant tunneling induced by a sharply twisted interface in $\mathcal{PT}$-symmetric ladder lattices. Finally, we find that perfect transmissions at the zero energy are recovered at the exceptional points of the $\mathcal{PT}$-symmetric system due to the self-orthogonal states.

Figure 1

(a) A circular ladder lattice (CLL) and (b) a Möbius ladder lattice (MLL). The blue box represents the unit cell of the CLL, which has two sites with intra-unit cell hopping strength $d$ and inter-unit cell hopping strength $t$.

Figure 2

(a) Real and (c) imaginary parts of the eigenenergies as a function of $\gamma$ in a $\mathcal{PT}$-symmetric CLL with 100 unit cells when $d=t=1$. (b) Real and (d) imaginary parts of the selected eigenenergies. (e) Real and (g) imaginary parts of the eigenenergies as a function of $\gamma$ in a $\mathcal{PT}$-symmetric MLL with 100 unit cells when $d=t=1$. (f) Real and (h) imaginary parts of the selected eigenenergies. See also Figs. 3 and 4 below.

Figure 3

(a) Real and (c) imaginary parts of the eigenenergies in a $\mathcal{PT}$-symmetric CLL showing level crossings of real eigenenergies of a non-Hermitian system in the $\mathcal{PT}$-unbroken phase. (b) ${\left|\alpha \right|}^2$ of two pairs of degenerate eigenstates and (d) $|{\alpha }_{\theta }{|}^2$ of two pairs of degenerate eigenstates.

Figure 4

Same as Fig. 3 but for the MLL. (a) Real and (c) imaginary parts of the eigenenergies in a $\mathcal{PT}$-symmetric MLL. (b) ${\left|\alpha \right|}^2$ of four eigenstates and (d) $|{\alpha }_{\theta }{|}^2$ of four eigenstates. The formation of a pair of EPs sandwiching a region of broken $\mathcal{PT}$ phase is clearly visible.

Figure 5

(a) A ladder lattice with two leads. The unit cell has two sites, $a$ and $b$, with hopping strengths $d$ (dashed lines) and $t$ (solid lines) between the sites. The coupling strength between sites and leads is ${\gamma }_{u\left(d\right)}^{i\left(o\right)}$ (long-dashed lines). (b) A twisted-ladder lattice with two leads. (c) Detangled Fano lattices from a ladder lattice with symmetric contacts corresponding to (a). The hopping strength between $f_n$ and $p_n$ is ${\epsilon }_{-}=({\epsilon }_u-{\epsilon }_d)/2$.

Figure 6

(a) Eigenenergies as a function of imaginary antisymmetric on-site potential $\gamma$ in a ladder lattice with 100 unit cells. (b) Transmission probability for the ladder lattice in which both $a$ and $b$ sites of the end unit cells are connected to the input and output leads. Yellow, red, and black colors denote the highest, middle, and lowest transmission probabilities, respectively. (c) Transmission probability as a function of $\gamma$ when $\mathrm{Re}\left(E\right)=0$. (d) Eigenenergies as a function of $\gamma$ in a ladder lattice with 100 unit cells and one twisted hopping. (e) Transmission probability for the twisted-ladder lattice in which both $a$ and $b$ sites of the end unit cells are connected to the input and output leads. (f) Transmission probability as a function of $\gamma$ when $\mathrm{Re}\left(E\right)=0$. The dark regions in (e) appear larger than in (b) reflecting the presence of evanescent modes corresponding to imaginary eigenenergies in the broken $\mathcal{PT}$ phase.