We use quantum Monte Carlo (stochastic series expansion) and finite-size scaling to study the quantum critical points of two
S=1∕2 Heisenberg antiferromagnets in two dimensions: a bilayer and a Kondo-lattice-like system (incomplete bilayer), each with intraplane and interplane couplings
J and ${J}_{\ensuremath{\perp}}$. We discuss the ground-state finite-size scaling properties of three different quantities---the Binder moment ratio, the spin stiffness, and the long-wavelength magnetic susceptibility---which we use to extract the critical value of the coupling ratio $g={J}_{\ensuremath{\perp}}∕J$. The individual estimates of
gc are consistent provided that subleading finite-size corrections are properly taken into account. For both models, we find that the spin stiffness has the smallest subleading finite-size corrections; in the case of the incomplete bilayer we find that the first subleading correction vanishes or is extremely small. In agreement with predictions, we find that at the critical point the Binder ratio has a universal value and the product of the spin stiffness and the long-wavelength susceptibility scales as
1∕L2 with a universal prefactor. Our results for the critical coupling ratios are
gc=2.5220(1) (full bilayer) and
gc=1.3888(1) (incomplete bilayer), which represent improvements of more than an order of magnitude over the previous best estimates. For the correlation length exponent we obtain $\ensuremath{\nu}=0.7106(9)$, consistent with the expected three-dimensional Heisenberg universality class.
A relation between geometric and gauge symmetries for the Heisenberg model of magnet is pointed out. The space of quantum states of the magnet exhibits the structure of a bundle, with the base consisting of nodes of the crystal, and the typical fiber spanned on the set of the single-node spin projections. The geometric symmetry group acts on the base, the gauge group — on the typical fiber, and all combined operations form the wreath product. In particular, all global gauge transformations yield the direct product group — a subgroup of the wreath product.
Abstract A quantum spin model of a delta-chain with ferro- and antiferromagnetic exchange interactions has been analyzed. Special attention has been paid to the analysis of magnetic properties at the critical point that separates the ferro- and ferrimagnetic phases of the ground state. The ground state is macroscopically degenerate in this case, and the entropy at zero temperature is finite. The gap in the excitation spectrum is exponentially small, and the spectrum itself has a multiscale structure that determined the low-temperature thermodynamics. The results obtained are used to analyze the magnetic properties of recently synthesized molecules with a record-breaking spin value of the ground state.