Exponentially Fragile $\mathcal{P}\mathcal{T}$ Symmetry in Lattices with Localized Eigenmodes

We study the effect of localized modes in lattices of size $N$ with parity-time ($\mathcal{P}\mathcal{T}$) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting $\delta \sim {exp}^{-N/\xi }$ where $\xi$ is their localization length. The level “evolution” with respect to the $\mathcal{P}\mathcal{T}$ breaking parameter $\gamma$ shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous $\mathcal{P}\mathcal{T}$ symmetry breaking occurs at ${\gamma }_{\mathcal{P}\mathcal{T}}\sim \min \{\delta \}$, thus resulting in an exponentially narrow exact $\mathcal{P}\mathcal{T}$ phase. As $N/\xi$ decreases, it becomes more robust with ${\gamma }_{\mathcal{P}\mathcal{T}}\sim 1/N^2$ and the distribution $\mathcal{P}({\gamma }_{\mathcal{P}\mathcal{T}})$ changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.

Figure 1

Pairs of exponentially localized modes in a 1D chain with $\mathcal{P}\mathcal{T}$-symmetric disorder [(a)–(c)] and surface states in a $\mathcal{P}\mathcal{T}$-symmetric periodic chain (d) yielding the smallest energy splittings ${\delta }_1$ for $\beta =3$. For $\gamma <{\gamma }_{\mathcal{P}\mathcal{T}}$ (b) the two eigenfunctions [blue (dark gray) and red (medium gray); the imaginary part is shown in the inset] are complex but $\mathcal{P}\mathcal{T}$ symmetric and their absolute values remains equal and symmetric [coinciding on the black line in (a)]. For $\gamma >{\gamma }_{\mathcal{P}\mathcal{T}}$ (c) the eigenfunctions (magenta and cyan) are no longer $\mathcal{P}\mathcal{T}$ symmetric and shift their weight towards separate sides of the chain (a). (d) Surface states [blue (dark gray) and green (light gray)] showing exponential localization. The red (medium gray) dashed [solid] lines in (a) [(d)] are guides to the eye, pointing out the exponential localization.

Figure 2

(a) Bifurcations for the dipole moment $D$ and for the imaginary part $\mathrm{Im}E$ of energy levels for a $\mathcal{P}\mathcal{T}$-symmetric 1D chain with $\gamma =\mathrm{const}$ and ${\beta }_n$ given by a box distribution [$-\frac{\beta }{2}$; $\frac{\beta }{2}$] for $N/\xi \gg 1$. The first two bifurcations (corresponding to splittings ${\delta }_1$ and ${\delta }_2$) are shown. The square-root behavior at the branching point (see text) is indicated with dashed cyan lines (on top of the $\mathrm{Im}E$ lines). Inset: a linear relation between ${\delta }_1$ and ${\gamma }_{\mathcal{P}\mathcal{T}}$ is evident for almost 10 orders of magnitude. (b) Distribution $\mathcal{P}(x)$ of $x\equiv (\log {\delta }_1-⟨\log {\delta }_1⟩)/\sigma$ ($\sigma$ is the standard deviation) for various disorder strengths $\beta$. In the limit of large $\beta$ the distribution becomes log-normal. Inset: The distribution $\mathcal{P}(s)$ of $s\equiv \log ({\delta }_2)-\log ({\delta }_1)$ reported in a semilogarithmic plot. A Poisson distribution is approached as $\beta$ increases.

Figure 3

(a) Same as in Fig. 2a but now for “weak” disorder $N/\xi \ll 1$. The same bifurcation scenario is observed. (b) The distribution $\mathcal{P}(\tilde{\Delta })$ of the rescaled minimal energy split $\tilde{\Delta }=\Delta \delta /\sigma$ ($\sigma$ is the standard deviation), for various disordered strengths, all of them being in the weak localized regime. The distribution has an upper cutoff at $\tilde{\Delta }=0$. A Gaussian distribution is shown also by a red dashed line. Inset: The same data in a semilogarithmic manner vs ${\tilde{\Delta }}^2$.