Critical Hamiltonians with long range hopping

TL;DR

This paper investigates critical states in disordered systems with long-range hopping using real space RG, revealing non-trivial fixed points and complex scaling behaviors of participation ratios, with implications for fermionic Hamiltonians.

Contribution

Introduces a real space RG approach to analyze critical states with long-range hopping and disorder, uncovering non-trivial fixed points and complex participation ratio scaling.

Findings

Identification of non-trivial RG fixed points.

Scaling of participation ratios varies with fixed points.

RG method extended to fermionic Hamiltonians with disorder.

Abstract

Critical states are studied by a real space RG in the problem with strong diagonal disorder and long range power law hopping. The RG flow of the distribution of coupling parameters is characterized by a family of non-trivial fix points. We consider the RG flow of the distribution of participation ratios of eigenstates. Scaling of participation ratios is sensitive to the nature of the RG fix point. For some fix points, scaling of participation ratios is characterized by a distribution of exponents, rather than by a single exponent. The RG method can be generalized to treat certain fermionic Hamiltonians with disorder and long range hopping. We derive the RG for a model of interacting two-level systems. Besides couplings, in this problem the RG includes the density of states. The density of states is renormalized so that it develops a singularity near zero energy.