Exploring Contractor Renormalization: Tests on the 2-D Heisenberg Antiferromagnet and Some New Perspectives

Contractor Renormalization (CORE) is a numerical renormalization method for Hamiltonian systems that has found applications in particle and condensed matter physics. There have been few studies, however, on further understanding of what exactly it does and its convergence properties. The current work has two main objectives. First, we wish to investigate the convergence of the cluster expansion for a two-dimensional Heisenberg Antiferromagnet(HAF). This is important because the linked cluster expansion used to evaluate this formula non-perturbatively is not controlled by a small parameter. Here we present a study of three different blocking schemes which reveals some surprises and in particular, leads us to suggest a scheme for defining successive terms in the cluster expansion. Our second goal is to present some new perspectives on CORE in light of recent developments to make it accessible to more researchers, including those in Quantum Information Science. We make some comparison to entanglement-based approaches and discuss how it may be possible to improve or generalize the method.Comment: Completely revised version accepted by Phy Rev B; 13 pages with added material on entropy in COR

The Complete Characterization of Fourth-Order Symplectic Integrators with Extended-Linear Coefficients

The structure of symplectic integrators up to fourth-order can be completely and analytical understood when the factorization (split) coefficents are related linearly but with a uniform nonlinear proportional factor. The analytic form of these {\it extended-linear} symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and non-forward fourth-order algorithms with arbitrary number of operators. Most fourth-order forward integrators can now be derived analytically from this extended-linear formulation without the use of symbolic algebra.Comment: 12 pages, 2 figures, submitted to Phys. Rev. E, corrected typo

Quantum Statistical Calculations and Symplectic Corrector Algorithms

The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher order splitting will result in a higher order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Greens function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically give rises to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second order forward propagators can have fourth order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second order propagators and their minimal correctors.Comment: 9 pages, no figure, corrected typos, mostly missing right bracket

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

Horticultural therapy program for people with mental illness: A mixed-method evaluation

Horticultural therapy (HT) has long been used in the rehabilitation of people with mental illness, but many HT programs are not standardized, and there have been few evaluation studies. Aims. This study evaluated the process and outcomes of a standardized horticultural program using a mixed methodology, i.e., systematic integration (“mixing”) of quantitative and qualitative data within a study. Methods. Participants who have mental illnesses were assigned to a treatment (HT) and a comparison group (n = 41 for each group). The process and outcomes of the program, including stress and anxiety, engagement and participation, affect changes, mental well-being, and social exchange, were obtained using self-completed questionnaires, observational ratings of participants during the group, as well as through a focus group. Results. The study results supported the proposal HT is effective in increasing mental well-being, engagement, and the sense of meaningfulness and accomplishment of participants. Many participants reported a reduction in stress and anxiety in the focus group, but positive changes in affect were not fully observed during the group process or captured by quantitative measures. The participants also did not report increases in the social exchange over the HT sessions. Conclusion. The evidence supports that HT is effective in increasing mental well-being, engagement in meaningful activities, but did not result in significant affect changes during therapy, or increase social exchanges among people with mental illness

A Case Study in Macao

Funding Information: This research was funded by Fundação para a Ciência e Tecnologia, I.P., Portugal, grant number UID/AMB/04085/2020, and the APC was funded by CENSE. Funding Information: The work developed was supported by The Macao Meteorological and Geophysical Bureau (SMG). Publisher Copyright: © 2022 by the authors.Despite the levels of air pollution in Macao continuing to improve over recent years, there are still days with high-pollution episodes that cause great health concerns to the local community. Therefore, it is very important to accurately forecast air quality in Macao. Machine learning methods such as random forest (RF), gradient boosting (GB), support vector regression (SVR), and multiple linear regression (MLR) were applied to predict the levels of particulate matter (PM10 and PM2.5) concentrations in Macao. The forecast models were built and trained using the meteorological and air quality data from 2013 to 2018, and the air quality data from 2019 to 2021 were used for validation. Our results show that there is no significant difference between the performance of the four methods in predicting the air quality data for 2019 (before the COVID-19 pandemic) and 2021 (the new normal period). However, RF performed significantly better than the other methods for 2020 (amid the pandemic) with a higher coefficient of determination (R2) and lower RMSE, MAE, and BIAS. The reduced performance of the statistical MLR and other ML models was presumably due to the unprecedented low levels of PM10 and PM2.5 concentrations in 2020. Therefore, this study suggests that RF is the most reliable prediction method for pollutant concentrations, especially in the event of drastic air quality changes due to unexpected circumstances, such as a lockdown caused by a widespread infectious disease.publishersversionpublishe

Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact K\"ahler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and we show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we proved some properties of semistable Higgs sheaves.Comment: 18 pages; some typos correcte

Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation

We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically. A key contribution of this work is to show how certain double commutators in the factorization process can be simulated in practice. The method is general, applicable to the multivariable case, and systematic, with known procedures for doing fourth order factorizations. The fourth order convergence of the resulting algorithm allowed very large time steps to be used. In simulating the Brownian dynamics of 121 Yukawa particles in two dimensions, the converged result of a first order algorithm can be obtained by using time steps 50 times as large. To further demostrate the versatility of our method, we derive two new classes of fourth order algorithms for solving the simpler Kramers equation without requiring the derivative of the force. The convergence of many fourth order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure