A Monte Carlo method for quantum spin systems is formulated in the valence-bond basis. The nonorthogonality allows for an efficient importance-sampled projection of the ground state out of an arbitrary state. The method provides access to resonating valence-bond physics, enables a direct estimator for the singlet-triplet gap, and extends the class of models that can be studied without negative-sign problems. As a demonstration, the valence-bond distribution in the ground state of the 2D Heisenberg antiferromagnet is calculated. Generalizations of the method to fermion systems are also discussed.
We discuss a projector Monte Carlo method for quantum spin models formulated in the valence bond basis, using the S=1/2 Heisenberg antiferromagnet as an example. Its singlet ground state can be projected out of an arbitrary basis state as the trial state, but a more rapid convergence can be obtained using a good variational state. As an alternative to first carrying out a time consuming variational Monte Carlo calculation, we show that a very good trial state can be generated in an iterative fashion in the course of the simulation itself. We also show how the properties of the valence bond basis enable calculations of quantities that are difficult to obtain with the standard basis of Sz eigenstates. In particular, we discuss quantities involving finite-momentum states in the triplet sector, such as the dispersion relation and the spectral weight of the lowest triplet.
Every singlet state of a quantum spin-
1/2 system can be decomposed into a linear combination of valence bond basis states. The range of valence bonds within this linear combination as well as the correlations between them can reveal the nature of the singlet state and are key ingredients in variational calculations. In this work, we study the bipartite valence bond distributions and their correlations within the ground state of the Heisenberg antiferromagnet on bipartite lattices. In terms of field theory, this problem can be mapped to correlation functions near a boundary. In dimension $d\ensuremath{\ge}2$, a nonlinear $\ensuremath{\sigma}$ model analysis reveals that at long distances the probability distribution
P(r) of valence bond lengths decays as ${|r|}^{\ensuremath{-}d\ensuremath{-}1}$ and that valence bonds are uncorrelated. By a bosonization analysis, we also obtain $P(\mathbf{r})\ensuremath{\propto}{|r|}^{\ensuremath{-}d\ensuremath{-}1}$ in
d=1 despite the different mechanism. On the other hand, we find that correlations between valence bonds are important even at large distances in
d=1, in stark contrast to $d\ensuremath{\ge}2$. The analytical results are confirmed by high-precision quantum Monte Carlo simulations in
d=1, 2, and 3. We develop a single-projection loop variant of the valence bond projection algorithm, which is well designed to compute valence bond probabilities and for which we provide algorithmic details.
I define a set of wave functions for
SU(N) lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988)], in which the singlets are extended over
N-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with
N-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break
SU(N) and exhibit
N-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of
SU(N). I analyze this transition using a classical mean field approach. For
N>2, the ordered state is selected via an ``order by disorder'' mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.
The bond operator representation of quantum
S=21 spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.
The theory of the resonating valence bond (RVB) for the antiferromagnetic Heisenberg model of spin 1/2 proposed by Anderson is generalized by taking account of singlet bonds connecting distant spins, and is solved exactly for the finite lattice. It is shown that the RVB wave function is S tot (total spin quantum number)=0, and that replacing a part of singlet bonds by triplet bonds, eigenfunctions with S tot =1, 2, … can be constructed. These are the new representation of eigenfunctions of the Heisenberg Hamiltonian.