Quantum phase transitions between the magnetically ordered and disordered states are studied for the two-dimensional antiferromagnetic quantum spin systems with ladder, plaquette, and mixed-spin structures. Starting with properly chosen singlet-cluster configurations, we perform the series expansion for the staggered magnetic susceptibility. The phase boundary is determined by applying the Dlog and biased Padé approximants to the staggered susceptibility thus obtained. The resulting phase diagram allows us to discuss the quantum phase transitions quantitatively, which agrees fairly well with the quantum Monte Carlo results for several cases previously studied.
The spin-1/2 quantum transverse Ising model, defined on a ladder structure, with nearest-neighbor and four-spin interaction on a plaquette, was studied by using exact diagonalization on finite ladders together with finite-size-scaling procedures. The quantum phase transition between the ferromagnetic and paramagnetic phases has then been obtained by extrapolating the data to the thermodynamic limit. The critical transverse field decreases as the antiferromagnetic four-spin interaction increases and reaches a multicritical point. However, the exact diagonalization approach was not able to capture the essence of the dimer phase beyond the multicritical transition.
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.