We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to ${L}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but they seem to not diverge as the dimerization decreases and the system approaches the N\'eel-VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.
We introduce for SU(2) quantum spin systems the valence bond entanglement entropy as a counting of valence bond spin singlets shared by two subsystems. For a large class of antiferromagnetic systems, it can be calculated in all dimensions with quantum Monte Carlo simulations in the valence bond basis. We show numerically that this quantity displays all features of the von Neumann entanglement entropy for several one-dimensional systems. For two-dimensional Heisenberg models, we find a strict area law for a valence bond solid state and multiplicative logarithmic corrections for the Néel phase.
In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.
We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence-bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high-temperature series expansions, and zero-temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field N\'eel state. Second, the corner logs do not agree with the known results for noninteracting Boson modes. And, third, even the finite-temperature mutual information shows an anomalous behavior as
T goes to zero, suggesting that the $T\ensuremath{\rightarrow}0$ and $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in
d>1, which deserves further attention.
Using a modified spin-wave theory which artificially restores zero sublattice magnetization on finite lattices, we investigate the entanglement properties of the N\'eel ordered ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on the square lattice. Different kinds of subsystem geometries are studied, either corner-free (line, strip) or with sharp corners (square). Contributions from the
nG=2 Nambu-Goldstone modes give additive logarithmic corrections with a prefactor
nG/2 independent of the R\'enyi index. On the other hand, $\ensuremath{\pi}/2$ corners lead to additional (negative) logarithmic corrections with a prefactor
lqc which does depend on both
nG and the R\'enyi index
q, in good agreement with scalar field theory predictions. By varying the second neighbor coupling
J2 we also explore universality across the N\'eel ordered side of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnet, from the frustrated side
0<J2/J1<1/2 where the area law term is maximal, to the strongly ferromagnetic regime $\ensuremath{-}{J}_{2}/{J}_{1}\ensuremath{\gg}1$ with a purely logarithmic growth
Sq=2nGlnN, thus recovering the mean-field limit for a subsystem of
N sites. Finally, a universal subleading constant term ${\ensuremath{\gamma}}_{q}^{\mathrm{ord}}$ is extracted in the case of strip subsystems, and a direct relation is found (in the large-
S limit) with the same constant extracted from free lattice systems. The singular limit of vanishing aspect ratios is also explored, where we identify for ${\ensuremath{\gamma}}_{q}^{\text{ord}}$ a regular part and a singular component, explaining the discrepancy of the linear scaling term for fixed width vs fixed aspect ratio subsystems.
We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.