A minimal approach for the local statistical properties of a one-dimensional disordered wire

TL;DR

This paper introduces a minimal zero-dimensional quantum model to analyze local statistical properties of a one-dimensional disordered wire, simplifying complex calculations by using independent matrix elements in a novel representation.

Contribution

It constructs a minimal quantum system representation for the disordered wire, connecting spatial properties to a non-unitary, infinite-dimensional U(1,1) representation, and compares it to existing methods.

Findings

The model captures local statistical properties effectively.

The quantum system is a non-unitary, infinite-dimensional U(1,1) representation.

The approach is shown to be minimal compared to supersymmetry and Berezinskii techniques.

Abstract

We consider a one-dimensional wire in gaussian random potential. By treating the spatial direction as imaginary time, we construct a `minimal' zero-dimensional quantum system such that the local statistical properties of the wire are given as products of statistically independent matrix elements of the evolution operator of the system. The space of states of this quantum system is found to be a particular non-unitary, infinite dimensional representation of the pseudo-unitary group, U(1,1). We show that our construction is minimal in a well defined sense, and compare it to the supersymmetry and Berezinskii techniques.