Asymmetric spin-1/2 two-leg ladders

TL;DR

This paper investigates the properties of asymmetric spin-1/2 two-leg ladders with ferromagnetic rungs, revealing how the spin gap behaves under different coupling regimes and proposing an effective low-energy spin-1 model.

Contribution

It introduces a detailed analysis of the spin gap and correlation functions in asymmetric ladders, including numerical results and a new low-energy effective theory for large blocks.

Findings

In the large J_perp limit, the gap matches the Haldane gap for spin-1 chains.

Two universality classes for the gap behavior at small rung coupling are identified.

The string order parameter behaves differently in weak and strong asymmetry regimes.

Abstract

We consider asymmetric spin-1/2 two-leg ladders with non-equal antiferromagnetic (AF) couplings J_|| and \kappa J_|| along legs (\kappa <= 1) and ferromagnetic rung coupling, J_\perp. This model is characterized by a gap \Delta in the spectrum of spin excitations. We show that in the large J_\perp limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at \kappa=1, admits a linear scaling for the spin gap \Delta ~ J_\perp. The second class appears for a strong asymmetry of the coupling along legs, \kappa J_|| << J_\perp << J_|| and is characterized by two energy scales: the exponentially small spin gap \Delta ~ J_\perp \exp(-J_|| / J_\perp), and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange ~ J_\perp^2 /J_|| . We report numerical results obtained by exact diagonalization, density matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetry. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.