Real space renormalization group approach to the 2d antiferromagnetic Heisenberg model

TL;DR

This paper applies a real space renormalization group method to analyze the low energy behavior of the 2D antiferromagnetic Heisenberg model, focusing on spin sectors and ground state energies as system size increases.

Contribution

It introduces a recursive approach for the interaction matrices that explicitly conserves total spin, providing insights into the ground state energies and relevant couplings in the model.

Findings

Ground state energies for different spins approach each other with increasing system size.

The renormalization procedure generates relevant couplings via spin transition matrix elements.

Conservation of total spin is crucial in the renormalization process.

Abstract

The low energy behaviour of the 2d antiferromagnetic Heisenberg model is studied in the sector with total spins $S=0,1,2$ by means of a renormalization group procedure, which generates a recursion formula for the interaction matrix $\Delta_S^{(n+1)}$ of 4 neighbouring "$n$ clusters" of size $2^n\times 2^n$, $n=1,2,3,...$ from the corresponding quantities $\Delta_S^{(n)}$. Conservation of total spin $S$ is implemented explicitly and plays an important role. It is shown, how the ground state energies $E_S^{(n+1)}$, $S=0,1,2$ approach each other for increasing $n$, i.e. system size. The most relevant couplings in the interaction matrices are generated by the transitions $<S',m';n+1|S_q^*|S,m;n+1>$ between the ground states $|S,m;n+1>$ ($m=-S,...,S$) on an $(n+1)$-cluster of size $2^{n+1}\times 2^{n+1}$, mediated by the staggered spin operator $S_q^*$