Scattering and transport statistics at the metal-insulator transition: A numerical study of the power-law banded random matrix model J. A. M´endez-Berm´ udez Instituto de F´ısica, Benem´ erita Universidad Aut´ onoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico Victor A. Gopar Depto de F´ısica Te´ orica, Facultad de Ciencias, and Instituto de Biocomputaci´ on y F´ısica de Sistemas Complejos (BIFI), Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain Imre Varga (Dated: October 29, 2018) We study numerically scattering and transport statistical properties of the one-dimensional An- derson model at the metal-insulator transition described by the Power-law Banded Random Matrix (PBRM) model at criticality. Within a scattering approach to electronic transport, we concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from localized to delocalized behavior in the average scattering matrix elements, the conductance probability distribution, the variance of the conductance, and the shot noise power by varying b (the effective bandwidth of the PBRM model) from small ( b ≪ 1) to large ( b > 1) values. We contrast our results with analytic random matrix theory predictions which are expected to be recovered in the limit b → ∞ . We also compare our results for the PBRM model with those for the three-dimensional (3D) Anderson model at criticality, finding that the PBRM model with b ∈ [0 . 2 , 0 . 4] reproduces well the scattering and transport properties of the 3D Anderson model. PACS numbers: 03.65.Nk, 71.30.+h, 73.23.-b Elm´ eleti Fizika Tansz´ ek, Fizikai Int´ ezet, Budapesti M˝ uszaki ´ es Gazdas´ agtudom´ anyi Egyetem, H-1521 Budapest, Hungary and Fachbereich Physik und Wissenschaftliches Zentrum f¨ ur Materialwissenschaften, Philipps Universit¨ at Marburg, D-35032 Marburg, Germany The study of systems at the Anderson metal-insulator transition (MIT) has been a subject of intensive research activity for several decades. 1–4 In particular, much in- terest has been focused on the scattering properties of critical systems by analysing the probability distribution functions of the resonance widths Γ and Wigner delay times τ W , 5–14 as well as the transmission or dimension- less conductance T . 15–26 The distribution functions of Γ and τ W have been shown to be related to the properties of the corresponding closed systems, i.e., the fractality of the eigenstates and the critical features of the MIT. On the other hand, at the MIT, the distribution of con- ductances w ( T ) has been found to be universal , i.e., size independent, but dependent on the adopted model, di- mensionality, symmetry, and even boundary conditions of the system. w ( T ) has been studied for systems in two and more dimensions with a large number of attached single-channel leads. 15–26 In fact, concerning the conduc- tance of one-dimensional (1D) systems and its statistical distribution, the regime of small number of leads has been left almost unexplored. 26–29 In the present work we study numerically several sta- tistical properties of the scattering matrix and the elec- tronic transport across disordered systems in one and three dimensions described by the Power-law Banded Random Matrix (PBRM) model at criticality and the three-dimensional (3D) Anderson model at the MIT, re- As we have mentioned above, in the present study we adopt two models, namely, the PBRM model at critical- ity and the 3D Anderson model at the MIT. The PBRM model 4,30 describes 1D samples of length L with random long-range hopping. This model is rep- resented by N × N ( N = L ) real symmetric matri- ces whose elements are statistically independent random The organization of this paper is as follows. In the next subsection we define the PBRM model, the 3D Anderson model, as well as the scattering setup. We also define the scattering quantities under investigation and provide the corresponding analytical predictions from random scattering-matrix theory (RMT) for systems with time- reversal symmetry. These analytical results will be used as a reference along the paper. In Section II we analyse the average scattering matrix elements, the conductance probability distribution, the variance of the conductance, and the shot noise power for the PBRM model as a func- tion of its effective bandwidth b . In Section III we com- pare the results of the PBRM model at criticality with the scattering and transport properties of the 3D Ander- son model. Finally, Section IV is left for conclusions. spectively. We stress that we concentrate on the case of a small number of attached leads each of them supporting one open channel. I. INTRODUCTION A. The PBRM and the 3D Anderson models arXiv:1001.2568v2 [cond-mat.dis-nn] 30 Oct 2010