String order in spin liquid phases of spin ladders

TL;DR

This paper investigates two distinct Haldane-type spin liquid phases in two-leg spin ladders, demonstrating how they can be distinguished using string order parameters and exploring the coexistence of long-range string and dimer orders.

Contribution

It introduces a numerical method to differentiate topologically distinct spin liquid phases via string order parameters in spin ladders.

Findings

Two Haldane-type spin liquid phases are identified.

Different string order parameters distinguish the phases.

Long-range string and dimer orders can coexist.

Abstract

Two-leg spin ladders have a rich phase diagram if rung, diagonal and plaquette couplings are allowed for. Among the possible phases there are two Haldane-type spin liquid phases without local order parameter, which differ, however, in the topology of the short range valence bonds. We show that these phases can be distinguished numerically by two different string order parameters. We also point out that long range string- and dimer orders can coexist.