The stochastic series expansion (SSE) quantum Monte Carlo method with directed loops is very efficient for spin and boson systems. The Heisenberg model and its generalizations, such as the
JQ2 model, are extensively simulated via this method. When introducing magnetic field in these models, the SSE method always combines the field with the diagonal part of the Heisenberg interactions (
SizSjz) and take them as the new diagonal operators. In general, this treatment is reasonable. However, when studying Hamiltonians, which have other interactions or even do not contain the Heisenberg interactions, this general treatment will not be efficient or even not work. We suggest that when doing directed-loop simulations, the magnetic field can be put into other interactions. This treatment, in some cases, improves the simulation efficiency. Using the
JQ2 model with magnetic field as an example, we here demonstrate this SSE method. Such treatment significantly improves the efficiency when the
Q2 interactions are large. The autocorrelations are reduced a lot compared to the previous approach. In addition, we argue that we can divide the magnetic field into two parts and combine them with both the
J and
Q operators respectively. This treatment also improves the simulation efficiency. The underlying mechanism is that these two SSE methods can utilize the main part or even all part of operators in operator products to do the directed-loop updates. Such idea can also be applied to other models with magnetic field and it will speed up the simulations.
In this paper we develop the stochastic cluster series expansion (SCSE) which is a (cluster) variant of the stochastic series expansion (SSE) method. For certain systems with longer-range interactions the SCSE is considerably more efficient than the standard implementation of the SSE, at low temperatures. As an application of this method we calculated the
T=0 conductance for a linear Heisenberg chain with a (diagonal) next-nearest-neighbor interaction.
The Stochastic Series Expansion (SSE) quantum Monte Carlo method with loop updates is among the most powerful approaches for quantum spin and boson systems. In the standard formulation, the magnetic field term is routinely absorbed into the Heisenberg interactions---a strategy that has proven highly efficient across a wide range of field strengths. In this work, we propose a more general SSE framework in which the magnetic field operator is treated as an independent operator class, enabling it to participate in Monte Carlo updates on an equal footing with all other operators. Importantly, the field operator and other interaction operators can transform into one another during the update process. We demonstrate this algorithm using the two-dimensional antiferromagnetic Heisenberg model in a magnetic field as a concrete example. The simulation results agree with those of the conventional algorithm, confirming the correctness of the new formulation. A comparison of the integrated autocorrelation time shows that the new algorithm is competitive. This flexibility not only facilitates measurements of observables associated with operators but also offers potential computational advantages for a broader class of problems.
The Stochastic Series Expansion (SSE) technique is a quantum Monte Carlo method that is especially efficient for many quantum spin systems and boson models. It was the first generic method free from the discretization errors affecting previous path integral based approaches. These lecture notes give a brief overview of the SSE method and its applications. In the introductory section, the representation of quantum statistical mechanics by the power series expansion of
e−βH will be compared with path integrals in discrete and continuous imaginary time. Extensions of the SSE approach to ground state projection and quantum annealing in imaginary time will also be briefly discussed. The later sections introduce efficient sampling schemes (loop and cluster updates) that have been developed for many classes of models. A summary of generic forms of estimators for important observables are also given. Applications are discussed in the last section.
The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.
We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the
S=1 antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.