Plaquette order in a dimerized frustrated spin ladder

We study the effect of dimerization (due to, e.g., spin-Peierls instability) on the phase diagram of a frustrated antiferromagnetic spin-1/2 ladder, with weak transverse and diagonal rung coupling. Our analysis focuses on a one-dimensional version of the model (i.e., a single two-leg ladder) where we consider two forms of dimerization on the legs: columnar dimers (CDs) and staggered dimers (SDs). We examine in particular the regime of parameters (corresponding to an intermediate $XXZ$ anisotropy) in which the leg dimerization and the rung coupling terms are equally relevant. In both the CD and SD cases, we find that the effective field theory describing the system is a self-dual sine-Gordon model, which favors ordering and the opening of a gap to excitations. The order parameter, which reflects the interplay between the leg and rung dimerization interactions, represents a crystal of 4-spin plaquettes on which longitudinal and transverse dimers are in a coherent superposition. Depending on the leg dimerization mode, these plaquettes are closed or open, however both types spontaneously break reflection symmetry across the ladder. The closed plaquettes are stable, while the open plaquette order is relatively fragile and the corresponding gap may be tuned to zero under extreme conditions. We further find that a first-order transition occurs from the plaquette order to a valence bond crystal (VBC) of dimers on the legs. This suggests that in a higher-dimensional version of this system, this variety of distinct VBC states with comparable energies leads to the formation of domains. Effectively one-dimensional gapless spinon modes on domain boundaries may account for the experimental observation of spin-liquid behavior in a physical realization of the model.

Figure 1

Schematic representation of the exchange interactions of the NT model.

Figure 2

Two possible ordering of valence bonds on two noninteracting chains: (a) columnar dimers (CDs), (b) staggered dimers (SDs). Thick red lines represent dimers on the stronger bonds.

Figure 3

Two possible types of plaquette order on the frustrated dimerized ladder: (a) closed plaquettes corresponding to $\sigma =+$, (b) open plaquettes corresponding to $\sigma =-$. Dark purple ellipses represent strong dimers, and light purple ellipses weakened dimers. In each case, two distinct plaquette-ordered ground-state configurations emerge (corresponding to $\tau =\pm$), with spontaneously broken reflection symmetry.

Figure 4

$g\text{−}{g}_P$ phase diagram in arbitrary units for fixed $g^z=1.5$, $\sigma =-$, $K=1/\sqrt{2},\gamma =0.1,u=6.15$, and $\Lambda =\pi$. The phase boundary line denotes a first-order transition.