Correlated volumes for extended wavefunctions on a random-regular graph

• URL: http://arxiv.org/abs/2311.07690v2
• Date: Tue, 9 Apr 2024 09:36:22 GMT
• Title: Correlated volumes for extended wavefunctions on a random-regular graph
• Authors: Manuel Pino, Jose E. Roman,
• Abstract summary: We analyze the ergodic properties of a metallic wavefunction for the Anderson model in a disordered random-regular graph with branching number $k=2. We extract their corresponding fractal dimensions $D_q$ in the thermodynamic limit together with correlated volumes $N_q$ that control finite-size effects.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We analyze the ergodic properties of a metallic wavefunction for the Anderson model in a disordered random-regular graph with branching number $k=2.$ A few q-moments $I_q$ associated with the zero energy eigenvector are numerically computed up to sizes $N=4\times 10^6.$ We extract their corresponding fractal dimensions $D_q$ in the thermodynamic limit together with correlated volumes $N_q$ that control finite-size effects. At intermediate values of disorder $W,$ we obtain ergodicity $D_q=1$ for $q=1,2$ and correlation volumes that increase fast upon approaching the Anderson transition $\log(\log(N_q))\sim W.$ We then focus on the extraction of the volume $N_0$ associated with the typical value of the wavefunction $e^{<\log|\psi|^2>},$ which follows a similar tendency as the ones for $N_1$ or $N_2.$ Its value at intermediate disorders is close, but smaller, to the so-called ergodic volume previously found via the super-symmetric formalism and belief propagator algorithms. None of the computed correlated volumes shows a tendency to diverge up to disorders $W\approx 15$, specifically none with exponent $\nu=1/2$. Deeper in the metal, we characterize the crossover to system sizes much smaller than the first correlated volume $N_1\gg N.$ Once this crossover has taken place, we obtain evidence of a scaling in which the derivative of the first fractal dimension $D_1$ behaves critically with an exponent $\nu=1.$

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