The stochastic series expansion (SSE) method is a quantum Monte Carlo (QMC) technique working directly in the imaginary time continuum and thus avoiding "Trotter discretization" errors. Using a nonlocal "operator-loop update," it allows one to treat large quantum mechanical systems of many thousand sites. In this paper we first give a comprehensive review on SSE and present benchmark calculations of SSE scaling behavior with system size and inverse temperature, and compare it to the loop algorithm, whose scaling is known to be one of the best of all QMC methods. Finally we introduce an efficient algorithm to measure Green's functions and thus dynamical properties within SSE.
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.
A cluster update (the ``operator loop'') is developed within the framework of a numerically exact quantum Monte Carlo method based on the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ (stochastic series expansion). The method is generally applicable to a wide class of lattice Hamiltonians for which the expansion is positive definite. For some important models the operator-loop algorithm is more efficient than loop updates previously developed for ``worldline'' simulations. The method is here tested on a two-dimensional anisotropic Heisenberg antiferromagnet in a magnetic field.
Combined with numerical analytic continuation techniques, quantum Monte Carlo (QMC) methods enable the extraction of real-frequency dynamical properties from imaginary-time correlation functions. However, the efficient computation of imaginary-time correlation functions by QMC simulations can (depending on the particular model used) be challenging, particularly for operators that are off-diagonal in the computational basis. In this work, we present an efficient and general algorithm within the stochastic series expansion (SSE) framework for evaluating imaginary-time correlation functions of both diagonal and off-diagonal operators. The algorithm builds on a discrete imaginary-time slicing of the SSE operator string, which provides correlation functions on a grid of well-defined imaginary-time points with no discretization error. For off-diagonal operators, we derive estimators that integrate directly into the existing SSE directed-loop or cluster updating schemes, introducing only minimal computational overhead. We benchmark the method on the one-dimensional transverse-field Ising model (sampling with cluster updates) and XXZ spin chain (using directed-loop sampling), demonstrating excellent agreement (with only statistical errors) with exact diagonalization of small systems. We also study larger systems to demonstrate efficiency.
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) 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.