Phase transitions and spectral singularities in a class of one-dimensional parity-time-symmetric complex potentials

TL;DR

The paper addresses how spectral singularities (SSs) govern phase transitions in a 1D PT-symmetric non-Hermitian potential with two tunable parameters. It analyzes the class $V(x) = -W(x)^2 - 2 g W(x) - i W'(x)$ with $W(x) = A sech(x)$, derives SS conditions via a transfer-matrix approach, and presents an exact bound-state solution for $A = 1/(1 - 4 g^2)$ with $E = -16 g^4/(1 - 4 g^2)^2$. A universal counting law is established: the number of SSs satisfies $N_{ss} = n+1$ for $A$ in the interval $[n+ frac{1}{2}, n+ frac{3}{2})$, enabling deterministic placement of SSs by tuning $A$. The results connect SSs with continuous-spectrum transitions and bound-state formation, offering a tunable framework for engineering non-Hermitian scattering in optical and quantum devices and outlining open questions regarding generalization to broader potentials.

Abstract

We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time ($\mathcal{PT}$) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.

Figures (3)

• Figure 1: (a) Imaginary part of the energy eigenvalues, $\operatorname{Im}(E)$, as a function of the parameter $g$ for $A=3/2$. (b) Real parts of the complex eigenvalues between the two phase transitions points $g=g_{c,1}=-0.928$ and $g=g_{c,2}=-0.365$, indicated by the two vertical lines in both (a) and (b). (c)-(f) Energy spectra in the complex plane for $A=3/2$ at specific values of $g$: (c) $g=-1$, (d) $g=-3/5$, (e) $g=-2/5$, and (f) $g=-1/2\sqrt{3}=-0.289$.
• Figure 2: Reflection coefficients $R^{r}(k)$ (solid lines) and $R^{l}(k)$ (dashed lines), and transmission coefficient $T(k)$ (dotted lines) as functions of (a) $g$ ( $A=3/2$ and $k=1.019$), and (b) $k$ ($A=3/2$ and $g=-0.929$).
• Figure 3: (a) Critical values $g_c$ associated with the phase transition from the continuous spectrum as a function of $A$. (b) Number of SSs, $N_{ss}$, as a function of the parameter $A$.