The Role of Boundary Conditions in the Real-Space Renormalization Group

TL;DR

This paper demonstrates that the effectiveness of real-space RG methods in 1D models depends on boundary conditions, introducing a new approach that accurately captures ground and excited states for free boundary conditions.

Contribution

The authors show boundary conditions critically influence RG success and develop a new analytical block RG method for free BCs that yields exact ground states and correct excitation energies.

Findings

Exact ground state for free boundary conditions

Correct $1/N^2$-law for first excited state energy

Reconstruction method for excited state wave-functions

Abstract

We show that the failure of the real-space RG method in the 1D tight-binding model is not intrinsic to the method as considered so far but depends on the choice of boundary conditions. For fixed BC's the failure does happen. For free BC's we present a new analytical block RG-method which gives the exact ground state of the model and the correct $1/N^2$-law for the energy of the first excited state in the large $N$(size)-limit. We also give a reconstruction method for the wave-functions of the excited states.