Electron in a tangled chain: multifractality at the small-world critical point

Abstract

We study a simple model of conducting polymers in which a single electron propagates through a randomly tangled chain. The model has the geometry of a small-world network, with a small density $p$ of crossings of the chain acting as shortcuts for the electron. We use numerical diagonalisation and simple analytical arguments to discuss the density of states, inverse participation ratios and wave functions. We suggest that there is a critical point at $p=0$ and demonstrate finite-size scaling of the energy and wave functions at the lower band edge. The wave functions are multifractal. The critical exponent of the correlation length is consistent with criticality due to the small-world effect, as distinct from the previously discussed, dimensionality-driven Anderson transition.

7 pages, 6 figures, REVTEX4; Phys. Rev. B, accepted; minor changes

References in corpus (6)

• Statistical mechanics of complex networks
• Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law
• Equilibrium shapes of flat knots
• Quantum and classical diffusion in small-world networks
• Netons: Vibrations of Complex Networks
• Small-Worlds, Mazes and Random Walks

Cited by in corpus (6)

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• Complex Quantum Networks: a Topical Review
• From compact localized states to many-body scars in the random quantum comb
• Discrete nonlinear Schrödinger equation in complex networks
• Symmetry Violation of Quantum Multifractality: Gaussian fluctuations versus Algebraic Localization
• Electronic transmission in bent quantum wires