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.
We show how efficient loop updates, originally developed for Monte Carlo simulations of quantum spin systems at finite temperature, can be combined with a ground-state projector scheme and variational calculations in the valence-bond basis. The methods are formulated in a combined space of spin
z components and valence bonds. Compared to schemes formulated purely in the valence-bond basis, the computational effort is reduced from up to
O(N2) to
O(N) for variational calculations, where
N is the system size, and from
O(m2) to
O(m) for projector simulations, where
m⪢N is the projection power. These improvements enable access to ground states of significantly larger lattices than previously. We demonstrate the efficiency of the approach by calculating the sublattice magnetization
Ms of the two-dimensional Heisenberg model to high precision, using systems with up to $256\ifmmode\times\else\texttimes\fi{}256$ spins. Extrapolating the results to the thermodynamic limit gives
Ms=0.30743(1). We also discuss optimized variational amplitude-product states, which were used as trial states in the projector simulations, and compare results of projecting different types of trial states.
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.
A Quantum Monte Carlo method of calculating operator expectation values for the ground state of the nearest-neighbor Heisenberg model on large square lattices is presented, along with some comparisons of the results obtained from this method and alternative methods for 2D lattices. In the simulations the ground state is projected out from an arbitrary state and sampled in a valence\nbond basis spanning the spin singlet subspace. The projection method is then extended to be used on a simplified Hubbard model at half-filling, which is\naptly called the T0-less Hubbard model, for arbitrary relative strengths of the electron hopping and electron-electron repulsion energy parameters. Results for calculated ground state energies in 1D are presented. These calculations are performed using a spin-charge separated basis, in which the valence bonds\nare generalized to also carry charge.
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.
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.
We develop a generalization of the singlet sector valence-bond basis projection algorithm of Sandvik, Beach, and Evertz (Sandvik 2005 Phys. Rev. Lett. 95 207203; Beach and Sandvik 2006 Nucl. Phys. B 750 142; Sandvik and Evertz 2010 Phys. Rev. B 82 024407) to cases in which the ground state of an antiferromagnetic Hamiltonian has total spin S tot = 1/2 in a finite-size system. We explain how various ground state expectation values may be calculated by using generalizations of the estimators developed in the singlet case, and illustrate the power of the method by calculating the ground state spin texture and bond energies in an L × L Heisenberg antiferromagnet with L odd and free boundaries.