A Plaquette Basis for the Study of Heisenberg Ladders

TL;DR

This paper introduces a plaquette basis approach for studying Heisenberg ladders, enabling efficient low-energy state approximation and mapping to a simpler spin-1 ladder, with implications for understanding ladder dynamics.

Contribution

It develops a plaquette basis for Heisenberg ladders, demonstrating effective truncation and a novel mapping to a spin-1 ladder, facilitating low-energy analysis.

Findings

Over 90% of ground-state probability is captured by few basis states.

The low-energy spectrum can be reproduced with less than 1% error using a simple CORE approximation.

The approach simplifies understanding of ladder dynamics by revealing two energy scales.

Abstract

We employ a plaquette basis-generated by coupling the four spins in a $2\times2$ lattice to a well-defined total angular momentum-for the study of Heisenberg ladders with antiferromagnetic coupling. Matrix elements of the Hamiltonian in this basis are evaluated using standard techniques in angular-momentum (Racah) algebra. We show by exact diagonalization of small ($2\times4$ and $2\times6$) systems that in excess of 90% of the ground-state probability is contained in a very small number of basis states. These few basis states can be used to define a severely truncated basis which we use to approximate low-lying exact eigenstates. We show how, in this low-energy basis, the isotropic spin-1/2 Heisenberg ladder can be mapped onto an anisotropic spin-1 ladder for which the coupling along the rungs is much stronger than the coupling between the rungs. The mapping thereby generates two distinct energy scales which greatly facilitates understanding the dynamics of the original spin-1/2 ladder. Moreover, we use these insights to define an effective low-energy Hamiltonian in accordance to the newly developed COntractor REnormalization group (CORE) method. We show how a simple range-2 CORE approximation to the effective Hamiltonian to be used with our truncated basis reproduces the low-energy spectrum of the exact $2\times6$ theory at the $\alt 1%$ level.