Motivated by recent Monte Carlo simulations of H\"oglund and Sandvik (arXiv:0808.0408), we study edge response in square lattice quantum antiferromagnets. We use the O(3) nonlinear $\ensuremath{\sigma}$ model to compute the decay asymptotics of the staggered magnetization, energy density, and local magnetic susceptibility away from the edge. We find that the total edge susceptibility is negative and diverges logarithmically as the temperature $T\ensuremath{\rightarrow}0$. We confirm the predictions of the continuum theory by performing a
1/S expansion of the microscopic Heisenberg model with the edge. We propose a qualitative explanation of the edge dimerization seen in Monte Carlo simulations by a theory of valence-bond-solid correlations in the N\'eel state. We also discuss the extension of the latter theory to the response of a single nonmagnetic impurity, and its connection to the theory of the deconfined critical point.
We study the low-energy dynamics of
S=1/2 antiferromagnetic Heisenberg clusters constructed by diluting a square lattice at vacancy concentration
p at and below the percolation threshold ${p}^{\ensuremath{\ast}}\ensuremath{\approx}0.407$. The finite-size scaling behavior of the average excitation gap, $⟨\ensuremath{\Delta}⟩\ensuremath{\sim}{L}^{\ensuremath{-}z}$, where
L is the cluster length, is obtained using quantum Monte Carlo results for an upper bound ${\ensuremath{\Delta}}^{\ensuremath{\ast}}$ to $\ensuremath{\Delta}$, derived from sum rules. At the percolation threshold, we obtain a dynamic exponent $z=3.6\ifmmode\pm\else\textpm\fi{}0.1\ensuremath{\approx}2{D}_{f}$ for clusters with singlet
(S=0) ground state. Here
Df=91/48 is the fractal dimensionality of the percolating cluster. We argue that this large dynamic exponent---roughly twice that expected for quantum-rotor excitations---is a consequence of weakly interacting localized effective magnetic moments, which form due to local sublattice imbalance. This picture is supported by an extremal-value analysis of local spectral gaps, which delivers an exponent relation (between
z and two exponents characterizing the local-gap distribution) reproduced by our simulation data. However, the average $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ over all clusters, which have mostly ground-state spin
S>0, scales with a smaller exponent than for the
S=0 clusters alone; $z\ensuremath{\approx}1.5{D}_{f}$. Lanczos exact diagonalization for small clusters show that typically, $S\ensuremath{\rightarrow}S\ensuremath{-}1$ in the lowest-energy excitations while the dominant spectral weight originates from $S\ensuremath{\rightarrow}S+1$ excitations. Thus, the scaling of $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ for clusters with ground state
S>0 does not reflect the lowest-energy excitations but the higher $S\ensuremath{\rightarrow}S+1$ excitations. This result can be understood within a valence bond picture. To further explore the scenario of localized moments, we introduce a classical dimer-monomer aggregation model to study the distribution of nearest-neighbor sites forming dimers (which are the objects used in mapping to the quantum-rotor model) and unpaired spins (monomers). The monomers are localized and, thus, effective magnetic moments should form in the spin system. We also study the lowest triplet excitation of
S=0 clusters using quantum Monte Carlo calculations in the valence bond basis. The triplet is concentrated at some of the classical monomer regions, confirming the mechanism of moment formation. The number of spins (and moment regions) affected by the excitation scales as a nontrivial power of the cluster size. For a dimer-diluted bilayer Heisenberg model with weak interlayer coupling (where the system remains N\'eel ordered), there is no sublattice imbalance. In this case we find $z\ensuremath{\approx}{D}_{f}$, consistent with quantum-rotor excitations. For a single layer at $p<{p}^{\ensuremath{\ast}}$ we find $z\ensuremath{\approx}2=D$, which indicates that the weakly interacting localized moment mechanism is valid only exactly at the percolation point. There is a crossover behavior close to the percolation point.
Dilution effects on spin-1/2 quantum Heisenberg antiferromagnets with a non-magnetic spin-gapped ground state are studied by means of the quantum Monte Carlo simulation. In the site-diluted system, an antiferromagnetic long-range order (AF-LRO) is induced at an infinitesimal concentration of dilution due to an effective coupling ̃Jmn between induced magnetic moments. In the bond-diluted case, on the other hand, the AF-LRO is not induced up to a certain concentration of dilution due to singlet pairs reformed by an AF coupling ̃J af between the edge spins of the diluted bond through the two-dimensional shortest paths. The competition between ̃Jmn and ̃J af yields peculiar phenomena in the bond-diluted system.
We study a variational wave function for the ground state of the two-dimensional
S=1∕2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes
h(x,y) for valence bonds connecting spins separated by
(x,y) lattice spacings. In contrast to previous studies, in which a functional form for
h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to $32\ifmmode\times\else\texttimes\fi{}32$ spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes
h(r) for valence bonds of long length
r decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.
We are considering the interplay between several non-magnetic impurities in the spin-(1/2) Heisenberg antiferromagnet in chains, ladders and planes by introducing static vacancies in numerical quantum Monte Carlo simulations. The effective potential between two and more impurities is accurately determined, which gives a direct measure of the quantum correlations in the systems. Large effective interaction potentials are an indication of strong quantum correlations in the system and reflect the detailed nature of the valence bond ground states. In two-dimensions (2D) the interactions are smaller, but can still be analyzed in terms of valence bonds.
We are considering the interplay between several non-magnetic impurities in the spin-(1/2) Heisenberg antiferromagnet in chains, ladders and planes by introducing static vacancies in numerical quantum Monte Carlo simulations. The effective potential between two and more impurities is accurately determined, which gives a direct measure of the quantum correlations in the systems. Large effective interaction potentials are an indication of strong quantum correlations in the system and reflect the detailed nature of the valence bond ground states. In two-dimensions (2D) the interactions are smaller, but can still be analyzed in terms of valence bonds.