Observation of scale-free localized states induced by non-Hermitian defects

Wave localization is a fundamental phenomenon that appears universally in both natural materials and artificial structures and plays a crucial role in understanding the various physical properties of a system. Usually, a localized state has an exponential profile with a localization length independent of the system size. Here, we experimentally demonstrate a new class of localized states called scale-free localized states, which has an unfixed localization length scaling linearly with the system size. Using circuit lattices, we observe that a non-Hermitian defect added to a Hermitian lattice induces an extensive number of states with scale-free localization. Furthermore, we demonstrate that, in a lattice with a parity-time-symmetric non-Hermitian defect, the scale-free localization emerges because of spontaneous parity-time symmetry breaking. Our results uncover a type of localized states and extend the study of defect physics to the non-Hermitian regime.

Figure 1

(a) Upper panel: Tight-binding model of a 1D chain with nearest-neighbor coupling $t$ and a non-Hermitian defect $-i\gamma$ at the middle of the chain (site $K+1$). Lower panel: Schematic diagram of the designed circuit realizing the tight-binding model. $C$, $R$, and $L$ denote capacitors, resistors, and inductors, respectively. (b) The real part of eigenvalues of a finite circuit chain with the fixed size $N=2K+1=101$, plotted as a function of frequency for fixed $C=22.7\mathrm{nF}$, $L=1.18\mu \mathrm{H},$ and $R=3.58\mathrm{\Omega }$. (c) Spatial distributions of all eigenstates at $0.9\mathrm{MHz}$ (the upper panel) and $1.02\mathrm{MHz}$ (the lower panel), as indicated by the yellow and brown dotted lines in (b). The inset in the upper panel shows the spatial distributions of eigenstates without the conventional LS. The colors denote the real part of eigenvalues. (d) Rescaled spatial distributions of conventional LSs (the upper panel) and scale-free LSs with the smallest localization lengths (the lower panel) in systems with different sizes. The horizontal axis is normalized by the system size $N$ and each eigenstate is normalized by its maximum value.

Figure 2

(a) Photo of the fabricated circuit. The zoomed-in image shows 13 nodes where the red one denotes the lossy node and white boxes highlight switches composed of two-pin headers. (b) The real part of measured eigenvalues of a finite circuit chain with $N=21$. (c) Spatial distributions of all measured eigenstates at $0.9\mathrm{MHz}$ (the upper panel) and $1.02\mathrm{MHz}$ (the lower panel), as indicated by the yellow and brown dotted lines in (b). The colors denote the real part of eigenvalues. (d) Rescaled spatial distributions of measured eigenstates with the smallest localization length at 0.9 MHz (the upper panel) and 1.02 MHz (the lower panel) in systems with different system sizes. Note that the horizontal axis is normalized by the system size $N$ and each eigenstate is normalized by its maximum value. (e) Plots of localization length against $N$ (the upper panel) and the IPR against $1/N$ (the lower panel) of measured conventional LSs (red dots) and scale-free LSs (blue dots) shown in (d). The black dotted lines denote linear fits. (f) Defect strength dependence of localization length. The plots show the IPR against frequencies of conventional LSs (the upper panel) and scale-free LSs (the lower panel) with the smallest localization lengths in a circuit chain with the fixed size $N=21$. Note that the increased frequency is equivalent to decreased defect strength.

Figure 3

(a) Upper panel: Tight-binding model of a 1D finite chain with nearest–neighbor coupling $t$ and two non-Hermitian defect sites ($i\gamma$ and $-i\gamma$) respecting PT symmetry. Lower panel: Schematic diagram of the designed circuit realizing the tight-binding model. (b) Photo of the fabricated circuit. The zoomed-in image shows 14 nodes where the red and orange ones denote the two lossy nodes and the white boxes highlight switches composed of two-pin headers. (c) The real part of eigenvalues of a finite circuit chain with the fixed size $N=2K=102$, plotted as a function of frequency. The black arrow denotes the PT phase transition point. (d) Spatial distributions of all numerical eigenstates at $1.5\mathrm{MHz}$ (the upper panel) and $0.68\mathrm{MHz}$ (the lower panel, with two conventional LSs excluded), as indicated by the yellow and brown dotted lines in (c). The colors denote the real part of eigenvalues. (e) Spatial distributions of two measured eigenstates ($N=46)$ at $1.5\mathrm{MHz}$ (the upper panel) and $0.68\mathrm{MHz}$ (the lower panel), which correspond to an extended state and a scale-free LS, respectively. (f) Upper panel: Rescaled spatial distributions of the measured scale-free LSs with the largest IPR values in systems with different sizes at 0.68 MHz. Note that the horizontal axis is normalized by the system size $N$ and each eigenstate is normalized by its maximum value. Lower panel: Plot of the IPR of the measured scale-free LSs shown in the upper panel against $1/N$. The black dotted line denotes the linear fit.