Exact Skin Critical Phase and Configurable Fractal Wavefunctions via Imaginary Gauge Phase Imprint in Non-Hermitian Lattices

Abstract

The generation of complex states like multifractal critical states has been an outstanding challenge in both classical and quantum physics. Here we propose a general framework, termed the imaginary gauge phase imprint, allowing to engineer rigorous wavefunctions in any-dimensional non-Hermitian lattices. Using this method, we uncover a novel phase with exact critical wavefunctions in one (and two) dimension, dubbed the skin critical phase (SCP). Unlike conventional critical phases with overall uniform density distributions and non-Hermitian skin effect with eigenstate accumulation at open boundaries, the SCP is marked by a macroscopically multifractal distribution with all critical eigenstates sharing an identical profile and always accumulating at specific bulk interfaces under periodic boundary condition, which become topology-dependent boundary or interface skin modes under open boundary condition. We also show the ballistic dynamics in the SCP, in contrast to the diffusive behaviour in conventional critical phases. Moreover, we validate our method by imprinting configurable wavefunctions in higher dimensions, including complex fractal states with Sierpinski-carpet and Koch-snowflake profiles in non-fractal lattices and Moire states in non-Moire lattices. Our work not only offers fresh insights into fractal phenomena and critical phases, but also provides a rigorous paradigm for wave manipulations in engineered non-Hermitian systems.

Figures (8)

• Figure 1: Imprinting exact multifractal critical states. (a) 1D critical state $|\psi_n|$ with the self-similar structure under imaginary phase $X_n$. (b) 2D critical state $|\psi_{n,m}|$ and (c) 3D critical state $|\psi_{n,m,p}|$ with Sierpiński-carpet fractal profiles, under imaginary phases $S_{n,m}$ and $V_{n,m,p}$, respectively. (d) Large-size scaling to extract the fractal dimension $D_f=0.605,1.939,2.884$ for critical states in (a-c), with system size $N=L^d>2.5\times10^{9}$ sites.
• Figure 2: SCP in 1D. (a) Phase diagram on the $W$-$h$ plane. The system exhibits the SCP below the diagonal (grey dashed line) and extended phase in the striped region above. In the SCP, all eigenstates are skinned to bulk interfaces (Int) under the PBC for any $\omega$; while under the OBC, they become interface, left (L) or right (R) boundary skin modes for $\omega=0,1,-1$, respectively. The orange and blue shading regions has $\omega=\pm1$ for $\bar{g}<0$ and $\bar{g}>0$, separating by exact boundaries as blue and red dashed lines with $\omega=\bar{g}=0$. (b) $\omega$ and $D_f$ as functions of $W$ with $h=1.25$. (c) Wavefunction distributions $|\psi_n|$ under the PBC and OBC for $\omega=0$ ($W=2$ and $h=1$) and $\omega=1$ ($W=1.5$ and $h=1$). (d) Corresponding $X_n$ (blue lines) and $X_n-\bar{g}n$ (red line) with quasiperiodic fluctuations and interfaces (green dash lines). (e) Expansion dynamics with time-evolved density profiles $\log P(n,t)$ in the SCP ($\bar{g}=0$ and $\bar{g}<0$) and the extended (Ext) phase from top to bottom panels). (f) Corresponding time-evolved second moment $M_2(t)$ with ballistic scaling $\delta=1$. The simulated $M_2(t)$ at the critical point of the AA model with diffusive scaling $\delta=0.5$ is shown for comparison. (g) Summary of properties of the CCP and SCP.
• Figure 3: Configurable exact wavefunctions. In (a,b,c), the top panels show the profiles of the imaginary gauge fields $g_{n,m}$, and bottom panels show the Sierpiński-carpet (a), Koch-snowflake (b), and Moiré (c) profiles of the imprinted wavefunctions $|\psi_{n,m}|$ in square lattices. The OBC and PBC are set in (a) and (b,c), respectively.
• Figure S1: Blue and red lines denote the exact topological boundaries from Eq.(\ref{['eq16']}), separated by the dashed line $W = h$, with numerically obtained critical exponents $\nu = z = 1$ along both boundaries. (b-c) Scaling of $\Lambda^{-1}$ and $\Delta E_{im}$ for the red boundary at $(W=2,h=0)$ and $(W=2,h=1)$. (d-e) Scaling for the blue boundary at $(W=0,h=1)$ and $(W=1.25,h=1)$. $L=987$ for (b-e).
• Figure S2: (a) Schematic of the 2D Hatano-Nelson model with a quasiperiodic imaginary gauge field. (b) Phase diagram of eigenstate distributions under OBC. The red point denotes the 2D skin critical state, the green line corresponds to the line-boundary skin critical mode, and the bule regions represent corner skin mode. (c) Spatial distribution of the total imaginary phase $X_n + X_m$ for $W_x = W_y = 2$. Inset: Diagonal elements of total imaginary phase with interfaces marked by dashed lines. (d) Spatial distribution of 2D skin critical state. Inset: Analytical versus numerical wave function distribution along the diagonal. (e) Spatial distribution of line-boundary skin critical mode. (f) Representative corner skin mode localized at the upper-right lattice site.