Exact minimal effective amounts of three 1D continuous functions and their use in Anderson transitions

Abstract

A conceptual quantity—the minimal effective amount of a quantum state ϕ(rj) in d-dimensional systems, defined by N∗=∑j=1Nmin{N|ϕ(rj)|2,1} , is newly proposed, where system sizes N=Ld . The effective dimension d IR can be calculated by N∗=h∗(L)LdIR , where h∗(L) does not change faster than any nonzero power. However, the nature of h∗(L) is unknown priori in any given model, but is at the same time very important for its numerical analysis. Hence, analytical results can provide insights on h∗(L) in more complex situations. In this paper, we get exact results of 1D continuous sine functions, exponential decay functions and power-law decay functions. They are used to distinguish extended and localized phases in the 1D uniform potential model, Anderson model and HMP (hopping rates modulated by a power-law function) model.