The fate of quantum wavefunction multifractality under perturbation

Ref. : R. Dubertrand, I. García-Mata, B. Georgeot, O. Giraud, G. Lemarié, and J. Martin, Two scenarios for quantum multifractality breakdown, Phys. Rev. Lett. 112, 234101 (2014).

The concept of fractal geometry was introduced by Mandelbrot in the seventies, to describe a range of phenomena characterized by the fact that a certain quantity has a similar distribution at all scales. This notion has proven very useful in the study of many areas of science, including fluid mechanics, biology, economy or geophysics. Multifractals are characterized by the existence of a whole range of fractal dimensions. The application of these ideas to quantum mechanics is much more recent, and the experimental observation of multifractal wavefunctions remains very difficult; in particular, how multifractality can survive under experimental conditions is a challenging question. In a paper recently published in Physical Review Letters, together with collegues from Toulouse, Liège and Mar del Plata, we have investigated theoretically how imperfections unavoidably present in experimental realizations will affect multifractality of quantum wave functions. Multifractality is destroyed under a sufficiently large perturbation in essentially two different ways: either multifractality survives below a certain scale of the quantum fluctuations (in which case one can compensate a small enough perturbation by using high resolution to resolve very small scales),  or multifractality is destroyed at all scales at a similar rate. These results should help interpret or predict experimental results in a real setting.

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