Perturbation Theory for Spin Ladders Using Angular-Momentum Coupled Bases

TL;DR

This paper develops a high-order perturbation theory method for Heisenberg spin-1/2 ladders, enabling accurate energy estimates and convergence analysis, with the plaquette basis proving effective for isotropic cases.

Contribution

It introduces a novel approach to extract high-order perturbative coefficients from small clusters and compares convergence properties of rung and plaquette bases.

Findings

High-order perturbative estimates closely match DMRG results.

The plaquette basis has a larger radius of convergence than the rung basis.

The method reliably estimates ground-state energy, gap, and magnon dispersion.

Abstract

We compute bulk properties of Heisenberg spin-1/2 ladders using Rayleigh-Schr\"odinger perturbation theory in the rung and plaquette bases. We formulate a method to extract high-order perturbative coefficients in the bulk limit from solutions for relatively small finite clusters. For example, a perturbative calculation for an isotropic $2\times 12$ ladder yields an eleventh-order estimate of the ground-state energy per site that is within 0.02% of the density-matrix-renormalization-group (DMRG) value. Moreover, the method also enables a reliable estimate of the radius of convergence of the perturbative expansion. We find that for the rung basis the radius of convergence is $\lambda_c\simeq 0.8$, with $\lambda$ defining the ratio between the coupling along the chain relative to the coupling across the chain. In contrast, for the plaquette basis we estimate a radius of convergence of $\lambda_c\simeq 1.25$. Thus, we conclude that the plaquette basis offers the only currently available perturbative approach which can provide a reliable treatment of the physically interesting case of isotropic $(\lambda=1)$ spin ladders. We illustrate our methods by computing perturbative coefficients for the ground-state energy per site, the gap, and the one-magnon dispersion relation.