Multifractality as a sensing resource: a D2 figure of merit for Rosenzweig-Porter systems, and a lattice platform

Abstract

Two established results have never been composed. The fidelity susceptibility of a quantum state to a Hamiltonian parameter equals one quarter of its quantum Fisher information, F_Q = 4·chi_f; and an exact calculation in the Rosenzweig-Porter (RP) ensemble shows that eigenfunction fidelity susceptibility is enhanced in the fractal non-ergodic-extended phase (Skvortsov, Amini and Kravtsov, PRB 106, 054208, 2022). Composed, they make the eigenvector fractal dimension D2 — until now a diagnostic of localization physics — an operational metrological figure of merit: along the RP fractal phase, at fixed distance from criticality and for field-like (local) couplings to the sensed parameter, the finite-size enhancement exponent of the quantum Fisher information over the ergodic reference is 1 − D2. We confirm this in-house under preregistered criteria on the generalized-RP ensemble with D2 measured from inverse-participation-ratio scaling (not assumed): enhancement = 1 − D2 within 0.15 at 4/4 interior points of the phase (deviations 0.009–0.078; correlation 0.996). Two further preregistered experiments fix the criterion's boundaries: geometric fractality alone — a disorder-free Sierpinski-gasket lattice with certified eigenpairs and degenerate-multiplet treatment — shows no effect (enhancement −0.12; the RP formula does not extend), while critical power-law banded random matrices extend the law at all three tested bandwidths (deviations 0.016–0.064). The criterion is therefore: multifractal eigenstates of random ensembles, sensed through local couplings — randomness, not fractal geometry, is the missing ingredient, and an operator-structure scope law (diagonal inherits; short-range inherits once its range fits the fractal support; collective and dense couplings are D2-blind) gives the design rule. On the platform side we construct — because it cannot be cited — the minimal model that carries the physics onto cold atoms: an interacting tilted+disordered lattice in which the tilt is only the sensed parameter (the bare Wannier-Stark ladder is integrable, so sourcing RP statistics from gravity would be immediately refutable) and the dense RP-like spectrum comes from interactions and disorder. A preregistered coexistence test on the interacting t–V chain finds a textbook ergodic-to-localized crossover whose typical susceptibility peaks at the crossover's ergodic edge — not mid-crossover — with the predicted heavy x^−2 tail (Hill index 1.96–2.04) in the chaotic regime and estimator-dependent rare-event gains inside the crossover. The platform statement is scoped accordingly, and one instrument caveat is stated up front: this attaches to lattice/Bloch gravimetry, not to free-fall light-pulse interferometry. Tier-2 throughout: a shared random-matrix description with preregistered numerical evidence, not a hardware claim.