Exact Mobility Edges in a Disorder-Free Dimerized Stark Lattice with Effective Unbounded Hopping

TL;DR

This work tackles the existence of exact mobility edges in a disorder-free 1D system by engineering a dimerized (SSH-like) lattice with a sublattice-selective linear Stark potential, which produces unbounded, site-dependent hopping and breaks the assumptions of standard no-go theorems. Through a sublattice-based gauge transformation and momentum-space analysis, the authors map the problem to a real, unbounded Jacobi matrix and identify an explicit mobility edge at $|E|=t_2$, separating a continuum of extended states for $|E|<t_2$ from two localized branches above the edge—the unbounded Wannier-Stark ladder and an anomalous bounded branch accumulating at the ME. They provide large-scale numerical evidence (up to $L\sim 10^9$) showing $ ext{IPR}\sim L^{-1}$ in the extended region and a localized behavior for the two above-edge branches, confirming the thermodynamic persistence of the extended phase. An experimental route using photonic frequency synthetic dimensions in Creutz-ladder-like ring-resonator setups is proposed, with a detailed stability analysis indicating robustness against detuning and photon loss in realistic devices, paving the way for observing exact MEs in disorder-free systems. Overall, the work establishes a concrete mechanism to realize and observe delocalization in the presence of unbounded potentials, expanding the landscape of mobility-edge physics beyond traditional disordered settings.

Abstract

We propose a disorder-free one-dimensional single-particle Hamiltonian hosting an exact mobility edge (ME), placing the system outside the assumptions of no-go theorems regarding unbounded potentials. By applying a linear Stark potential selectively to one sublattice of a dimerized chain, we generate an effective Hamiltonian with unbounded, staggered hopping amplitudes. The unbounded nature of the hopping places the model outside the scope of the Simon-Spencer theorem, while the staggered scaling allows it to evade broader constraints on Jacobi matrices. We analytically derive the bulk spectrum in reciprocal space, identifying a sharp ME where the energy magnitude equals the inter-cell hopping strength. This edge separates a continuum of extended states from two distinct localized branches: a standard unbounded Wannier-Stark ladder and an anomalous bounded branch accumulating at the ME. The existence of extended states is supported by finite-size scaling of the inverse participation ratio up to system sizes $L \sim 10^9$. Furthermore, we propose an experimental realization using photonic frequency synthetic dimensions. Our numerical results indicate that the ME is robust against potential experimental imperfections, including frequency detuning errors and photon loss, establishing a practical path for observing MEs in disorder-free systems.

Figures (6)

• Figure 1: (a) Schematic of the original lattice with linear Stark potential $V_n = -F n$ on the B-sublattice. (b) Effective nearest-neighbor hopping lattice after unitary rotation. (c) Sorted energy spectrum for $L=2001$ ($N=1000$), $t_1=1$, $t_2=1.2$, and $F=0.6$ under open boundary conditions. Colors denote the extended continuum (red), unbounded Wannier-Stark ladder (blue), and bounded anomalous branch (green). Inset: Low-energy zoom and the theoretical dispersion $E \approx t_2 \sin k$ (dashed). (d) Fractal dimension ($D_2$) phase diagram in the ($t_2, E$) plane (truncated at $|E| < 6 t_1$). Dashed lines indicate the analytical mobility edge $|E|=t_2$. Other parameters match those in (c).
• Figure 2: Theoretical mechanism. (a) Graphical solution of the quantization condition. Black dots mark eigenenergies $E_m^2$ at the intersections of the constraint $f(E^2)=E^2-t_1^2-t_2^2$ (black line) and mode curves $y_m(E^2)$ (colored lines). Vertical dashed line indicates the threshold $t_2^2$. (b) Reciprocal space wavefunction amplitudes (log scale). Solid lines show the extended state ($E \approx 0.5 t_2$) for A (red, left axis) and B (green, right axis) sublattices. Dashed lines (numerical) and markers (theoretical) plot the localized state ($m=50$). Vertical dotted lines mark the resonant momenta $k_c=\pm \arcsin (E/t_2)$. Other parameters match those in Fig. 1 [$L=2001$ ($N=1000$), $t_1=1$, $t_2=1.2$, and $F=0.6$].
• Figure 3: Characterization of localization. (a) Fractal dimension ($D_2$) versus energy ($L=20001$, $N=10000$, truncated at $|E| < 6t_1$). Black dashed lines mark the analytical mobility edges at $|E|=t_2$. (b) Finite-size scaling of inverse participation ratio (IPR) versus system size $L$ for representative states: extended continuum ($E \approx 0.5t_2$), unbounded branch ($m=50$), and bounded branch ($m=-10$). Dashed line indicates the scaling $\text{IPR} \sim L^{-1.00}$. (c, d) Real-space probability density $|\psi_n|^2$ on A and B sublattices (log scale, truncated at $10^{-7}$) shows the spatial distribution for the representative states ($L=2001, N=1000$). Other parameters match those in Fig. 1 ($t_1=1$, $t_2=1.2$, and $F=0.6$).
• Figure 4: Experimental proposal and robustness analysis. (a) Schematic of the proposed photonic platform. Two ring resonators (A and B) are coupled via a tunable Mach-Zehnder Interferometer (MZI). RF electrodes on the rings generate intra-resonator hopping ($\pm it_2/2$), while the MZI generates inter-resonator couplings ($t_1, t_2/2$). A DC bias on resonator B detunes its FSR to tune the Stark potential gradient $F$. (b) Robustness of the fractal dimension ($D_2$) against leakage potential strength $\epsilon$ on the A-sublattice with $\epsilon/F \in\{0.001, 0.01, 0.1\}$ (system size $L=201, N=100$). The energy is truncated by $|E| < 10$. (c, d) Simulated steady-state output spectra with leakage potential $\epsilon=0.01F$ and different loss rates $\kappa/t_1\in\{0, 0.01, 0.05\}$ from resonator A (c) and resonator B (d) under injection at the center mode of resonator A ($E=0$). The spectra are normalized to the maximum intensity at port A. Other parameters match those in Fig. 1 ($t_1=1$, $t_2=1.2$, and $F=0.6$).
• Figure E1: Maximum relative energy error compared to standard scipy solvers (calculated as the maximum over 10 independent runs per point) for $L\approx10^3$ to $10^6$. Plots include a $10^{-20}$ offset. (b) Residual $R = ||(H-E)\psi||/||\psi||$ versus system size $L$ for $L\approx 10^3$ to $10^9$.