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 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.

arXiv:0704.1469Apr 11, 20076 citations

Valence Bond Spin Models: Ground States and Excitations

Élliott H. Lieb

NATO ASI series. Series B : PhysicsJan 1, 19890 citations

Loop updates for variational and projector quantum Monte Carlo simulations in the valence-bond basis

Anders W. Sandvik, Hans Gerd Evertz

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 $O({N}^{2})$ to $O(N)$ for variational calculations, where $N$ is the system size, and from $O({m}^{2})$ 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 ${M}_{s}$ 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 ${M}_{s}=0.30743(1)$ . 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.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 $O({N}^{2})$ to $O(N)$ for variational calculations, where $N$ is the system size, and from $O({m}^{2})$ 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 ${M}_{s}$ 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 ${M}_{s}=0.30743(1)$. 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.

arXiv:0807.0682Physical Review BJul 8, 2010144 citations

Towards quantum simulation of spin dynamics in frustrated systems

Ippei Nakamura, Ryuta Yamamoto, Takeshi Fukuhara

Bulletin of the American Physical SocietyMay 29, 20180 citations

Valence Bond Entanglement Entropy

Fabien Alet, Sylvain Capponi, Nicolas Laflorencie +1 more

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.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.

arXiv:cond-mat/0703027Physical Review LettersSep 12, 200765 citations

z

components and valence bonds. Compared to schemes formulated purely in the valence-bond basis, the computational effort is reduced from up to

O({N}^{2})

O(N)

for variational calculations, where

N

is the system size, and from

O({m}^{2})

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

{M}_{s}

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

{M}_{s}=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.