Dualities in Spin Ladders

TL;DR

This paper explores dualities and phase relationships in Heisenberg spin ladders using discrete modular transformations, revealing invariances and mappings between different quantum phases such as RVB, Haldane, and mixed phases.

Contribution

It introduces a set of discrete modular transformations to analyze phase relationships in Heisenberg ladders, uncovering invariances and mappings between distinct quantum phases.

Findings

RVB phase is invariant under $S__$ transformation.

Haldane phase is invariant under $U__$ transformation.

A mixed phase maps to RVB and Haldane phases under different transformations.

Abstract

We introduce a set of discrete modular transformations $T_\ell,U_\ell$ and $S_\ell$ in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the 2 legged ladder we show that the $RVB$ phase is invariant under the $S_\ell$ transformation, while the Haldane phase is invariant under $U_\ell$. These two phases are related by $T_\ell$. Moreover there is a "mixed" phase, that is invariant under $T_\ell$, and which under $U_\ell$ becomes the RVB phase, while under $S_\ell$ becomes the Haldane phase. For odd ladders there exists only the $T_\ell$ transformation which, for strong coupling, maps the effective antiferromagnetic spin 1/2 chain into the spin 3/2 chain.