The Matrix Product Approach to Quantum Spin Ladders

TL;DR

This paper develops a rotationally invariant matrix product approach for quantum spin ladders, analyzing ground states, correlations, and topological order for various spin magnitudes and coupling configurations.

Contribution

It introduces a new rotationally invariant matrix product formulation applicable to spin ladders, including novel analysis of topological order and AKLT states with 9-j symbols.

Findings

Long-range topological order is broken by interchain coupling.

Correlation lengths are finite and computed for different ladders.

Duality properties are numerically evidenced in spin 1/2 ladders.

Abstract

We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to 2 legged spin ladders with spins 1/2, 1 and 3/2 and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder We compute ground state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the 3 different non ferromagnetic spin 1/2 ladders. We show that the long range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally for ladders with spin equal or greater than 3/2 we define a class of AKLT states whose matrix product coefficients are given by 9-j symbols.