Scaling exponent beyond the semi-localized phase

Determine whether the scaling exponent y governing the momentum-space localization-profile parameter R\sim L^y satisfies y=2a-3 beyond the critical disorder W_c(a), particularly for a<2 at weak disorder.

Background

The paper analyzes the semi-localized momentum-space ground state of the one-dimensional Anderson model with deterministic long-range hopping t(r)\sim r{-a} in the weak-disorder regime 1<a<3/2. It estimates a critical disorder W_c(a) separating the semi-localized and extended phases, but does not resolve the structure of the phase beyond W_c(a).

Exact-diagonalization data suggest that the disorder-averaged momentum-space intensity decays as exp[-(m/R)x], with R\sim Ly. The authors conjecture y=2a-3; if valid, this would imply a non-ergodic extended phase with fractal dimension y<1 for a<2 and a genuinely ergodic momentum-space phase only for a>2 at weak disorder.

References

What happens beyond W_{c}(a) is currently a discussive issue. A limited information that can be obtained from the exact diagonalization numerics shows that: \begin{equation} \langle |\psi(k)|{2}\rangle \sim {\rm exp[-(m/R){x}]}, \end{equation} where $m=k L/\pi$ and $R\sim L{y}$ is an extensive parameter that diverges in the thermodynamic limit. However, for $a<2$ at weak disorder $y<1$. At the same time at $a=3/2$ the exponent $y$ must be zero. So, we conjecture that \begin{equation} y=2a-3. \end{equation}

— Semi-localized ground state in a 1D system with long-range hopping  (2608.25806 - Bahovadinov et al., 26 Aug 2026) in Conclusion and discussion