Universal Conductance Fluctuations in Non-integer Dimensions

We propose an Ansatz for universal conductance fluctuations in continuous dimensions from 0 up to 4. The Ansatz agrees with known formulas for integer dimensions 1, 2 and 3, both for hard wall and periodic boundary conditions. The method is based solely on the knowledge of energy spectrum and standard assumptions. We also study numerically the conductance fluctuations in 4D Anderson model, depending on system size L and disorder W. We find a small plateau with a value diverging logarithmically with increasing L. Universality gets lost just in 4D. Disordered systems usually possess the metallic regime, where Ohm's law for mean conductance g works well at least for cubic samples and the distribution of g for various realizations of the same disorder is Gaussian with constant (disorder, mean free path l e and g independent) width, called universal conductance fluctuations (UCF). Higher cummulants of g should disappear as some power of 1/g [1], though recent experiment on gold wire did not confirm this in quasi-1D [2]. Recently [3] we analysed the statistical properties of conductance on bifractal lattices [4]. It became clear, that g and var g depend (besides the spectral dimension d s) on lattice topology. Simply speaking, bifractals are no hypercubes. But we hope that by changing other parameters, say anisotropy, we can tune the systems to cubic-like. If the UCF of anisotropic bifrac-tals become those of non-integer dimensional hypercubes, other parameters of these systems may be comparable. The main goal of this work is to find a way to calculate the UCF also for non-integer dimensions. Rewriting (2) as a simple integral, we will propose its analytical continuation to real dimensions. Two (on first sight) different expressions were given for UCF in literature [5] and [6]. Let us comment this seeming ambiguity. In classical papers of Lee, Stone and Fukuyama [5]-Appendix, formulas forUCF in 3D (2D, 1D) were given as a sum of three diagrammatic terms F a , F b and F c. They can be written in terms of convolutions, e. g. (the simple one-loop " Meeron " diagram):