Structural constraints on mobility edges in one-dimensional quasiperiodic systems

TL;DR

The paper addresses mobility edges in 1D quasiperiodic systems by introducing a structural constraint across isospectral dual Hamiltonians, derived from a Lyapunov-spectrum identity via the Thouless formula. For the bichromatic Aubry–André model, mobility-edge positions obey an energy-independent relation $ΔΓ = Γ_H - Γ_K = ln(g m / r)$, and mobility edges satisfy $F(E_c; λ_c) = 0$. On the self-dual line, the constraint yields a linear critical scaling $γ_2 ∼ |g-1|^ν$ with $ν ≈ 1$, which is confirmed numerically across parameters; in the deep localized regime both Lyapunov exponents scale as $γ_{1,2} ∼ (1/2) ln|g|$. Energy-resolved analysis shows the exponent is universal while the prefactor $A(E)$ is nonuniversal and energy-dependent, demonstrating the robustness of the constraint beyond the AA model. The results generalize to multichromatic models and point to experimental realizations in cold-atom platforms, with extensions to non-Hermitian or driven quasiperiodic systems and connections to Avila global theory.

Abstract

Mobility edges commonly arise in one-dimensional quasiperiodic systems once exact self-duality is broken, yet their origin is typically understood only at the level of individual Hamiltonians. Here we show that mobility edge positions are not independent spectral features of individual Hamiltonians, but are structurally constrained across quasiperiodic Hamiltonians related by an isospectral duality. Using a bichromatic Aubry--André model as a minimal setting, we demonstrate that this constraint is encoded in an exact identity for Lyapunov exponents derived from the Thouless formula. As a consequence, the mobility edge positions are restricted to a reduced set of energies. In the self-dual limit, these mobility edge positions coincide at a single localization--delocalization transition. This structural constraint enforces a linear critical scaling of the physical Lyapunov spectrum near the self-dual point. Numerical results confirm a critical exponent consistent with the standard Aubry--André value of $ν= 1$, while simultaneously revealing a novel, non-universal energy-dependent prefactor.

Figures (7)

• Figure 1: Classification of the parameter space in terms of isospectral duality and Lyapunov spectrum structure. Region I corresponds to the self-dual manifold. Region II represents an isospectral duality with preserved Lyapnuov spectrum dimension, where the number of positive Lyapunov exponents is identical for the dual Hamiltonians. Region III denotes the special isospectral regime, in which the Lyapunov spectrum dimension differs between the original and dual systems.
• Figure 2: Difference of Lyapunov-exponent sums $\Delta\Gamma$ as a function of the parameter $g$. (a) For $E = 1.234$ with $m = 1.2$ and $r = 0.7$, the numerical data (black) for $\Delta \Gamma$ coincide exactly with $\ln|gm/r|$ (red), as given in Eq. \ref{['eq:diff_Gamma']}. (b) For $E = 1.234$ and $m = 0.4$ at $r = 0$, the numerical results (black) agree with $\ln|gm|$ (red) in accordance with Eq. \ref{['eq:diff_Gamma_special']}. The exact agreement in both panels demonstrates that $\Delta\Gamma$ is independent of energy and is governed solely by $g$, as predicted by the theory.
• Figure 3: Phase diagram of $H$ at $r=0$ with $m=1$, with the parameter sampled over $g\in [0.1,10]$. The vertical axis represents energy parameterized by the IDOS coordinate $x = N(E)$, discretized into $N_{\rm bin}=25$ bins, to account for spectral bandwidth variations. (a) Inverse participation ratio (IPR) obtained from exact diagonalization on a lattice with $L = 3200$ sites. Low (high) IPR values indicate extended (localized) eigenstates. (b) $|F(E;g)|$ with respect to $g$, with energies obtained from diagonalization of a finite lattice with $L = 101$ sites. Values with $|F(E;g)| > 1$ are shown black. The mobility edges (white region) follow Eq. \ref{['eq:me_condition_at_r_0']}, demonstrating that they are fully determined by a Biddle--Das Sarma type $K$.
• Figure 4: Critical behavior of $F(E;g)$ near the self-dual point. The energy is parametrized by the IDOS $x = N(E)$, discretized into $N_{\rm bin}=25$ bins such that $|F(E;g)|$ is plotted as a function of $g$ for fixed values of $x$. Energies are obtained from diagonalizing a finite lattice with $L = 151$ sites. (a) $|F(E;g)|$ as a function of $g$ near $g = 1$ at $m=r =0.7$. Red crosses indicate the median over $x$, respectively. (b) Log--log plot of $\ln|F(E;g)|$ versus $\ln|g-1|$, from which a power-law scaling Eq. \ref{['eq:power_law']} is extracted. Data for $g < 1$ are omitted owing to the symmetry. Dashed lines indicate linear fits for the median (red) values. (c) Critical exponent $\nu$ as a function of $m$, evaluated for $m=0.1,\,0.3,\,0.5,\,0.7,\,0.9,\,1.1,\,1.5,\,1.7,\,2.1,\,2.7,\,9.0,$ and $81.0$.
• Figure 5: Positive Lyapunov exponents $\gamma_{1}$ and $\gamma_{2}$ of $H$ as a function of $g$ for $m =r=0.7$. The energy is parametrized by the IDOS $x = N(E)$ discretized into $N_{\rm bin}=25$ bins, such that $\gamma_1(E;g)$ is plotted as a function of $g$ for fixed values of $x$. Energies are obtained from diagonalizing a finite lattice with $L = 100$ sites. (a) $\gamma_{1}(E;g)$ and (b) $\gamma_{2}(E;g)$ shown over a wide range of $g$. The red crosses indicate the median, while solid line indicates logarithmic growth $(\ln|g|)/2$.