Classical percolation transition in the diluted two-dimensional S=1/2 Heisenberg antiferromagnet

The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground state properties of the largest connected cluster on L*L lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number Nc of spins on an infinite lattice at the percolation density are studied for Nc up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same non-zero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality d=91/48, have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site-dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.Comment: 18 pages, 21 figures (spin stiffness results added in v2

Quantum Monte Carlo studies of spinons in one-dimensional spin systems

Observing constituent particles with fractional quantum numbers in confined and deconfined states is an interesting and challenging problem in quantum many-body physics. Here we further explore a computational scheme [Y. Tang and A. W. Sandvik, Phys. Rev. Lett. {\bf 107}, 157201 (2011)] based on valence-bond quantum Monte Carlo simulations of quantum spin systems. Using several different one-dimensional models, we characterize $S=1/2$ spinon excitations using the spinon size and confinement length (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the critical region, and become ill-defined in the N\'eel state. We also verify that pairs of spinons are deconfined in these uniform spin chains but become confined upon introducing a pattern of alternating coupling strengths (dimerization) or coupling two chains (forming a ladder). In the dimerized system an individual spinon can be small when the confinement length is large---this is the case when the imposed dimerization is weak but the ground state of the corresponding uniform chain is a spontaneously formed valence-bond-solid (where the spinons are deconfined). Based on our numerical results, we argue that the situation $\lambda \ll \Lambda$ is associated with weak repulsive short-range spinon-spinon interactions. In principle both the length-scales can be individually tuned from small to infinite (with $\lambda \le \Lambda$) by varying model parameters. In the ladder system the two lengths are always similar, and this is the case also in the dimerized systems when the corresponding uniform chain is in the critical phase. In these systems the effective spinon-spinon interactions are purely attractive and there is only a single large length scale close to criticality, which is reflected in the standard spin correlations as well as in the spinon characteristics.Comment: 15 pages, 15 figure

Ground states of a frustrated quantum spin chain with long-range interactions

The ground state of a spin-1/2 Heisenberg chain with both frustration and long-range interactions is studied using Lanczos exact diagonalization. The evolution of the well known dimerization transition of the system with short-range frustrated interactions (the J1-J2 chain) is investigated in the presence of additional unfrustrated interactions decaying with distance as 1/r^a. It is shown that the continuous (infinite-order) dimerization transition develops into a first-order transition between a long-range ordered antiferromagnetic state and a state with coexisting dimerization and critical spin correlations at wave-number k=\pi/2. The relevance of the model to real systems is discussed.Comment: 4 pages, 5 figures, final published versio

Comment on ``Quantum Phase Transition of the Randomly Diluted Heisenberg Antiferromagnet on a Square Lattice''

In Phys. Rev. Lett. 84, 4204 (2000) (cond-mat/9905379), Kato et al. presented quantum Monte Carlo results indicating that the critical concentration of random non-magnetic sites in the two-dimensional antiferromagnetic Heisenberg model equals the classical percolation density; pc=0.407254. The data also suggested a surprising dependence of the critical exponents on the spin S of the magnetic sites, with a gradual approach to the classical percolation exponents as S goes to infinity. I here argue that the exponents in fact are S-independent and equal to those of classical percolation. The apparent S-dependent behavior found by Kato et al. is due to temperature effects in the simulations as well as a quantum effect that masks the true asymptotic scaling behavior for small lattices.Comment: Comment on Phys. Rev. Lett. 84, 4204 (2000), by K. Kato et al.; 1 page, 1 figur

Definitions of entanglement entropy of spin systems in the valence-bond basis

The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific hamiltonians or using importance-sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Results for the bond-based definition agrees with a previously studied definition using valence-bond wave functions (instead of the transition graphs, which involve two states). For the one dimensional Heisenberg chain, with uniform or random coupling constants, the prefactor of the logarithmic divergence with the size of the smaller subsystem agrees with exact results. For the ground state of the two-dimensional Heisenberg model (and also Neel-ordered amplitude-product states), there is a similar multiplicative violation of the area law. In contrast, the loop-based entropy obeys the area law in two dimensions, while still violating it in one dimension - both behaviors in accord with expectations for proper measures of entanglement entropy.Comment: 9 pages, 8 figures. v2: significantly expande