Checkerboard decomposition

The path-integral formulation of a quantum systems goes back to [37], and forms the basis of most QMC algorithms. Instead of following the historical route and discussing the Trotter-Suzuki (checkerboard) decomposition [38,39] for path integrals with discrete time steps At we will directly describe the continuous-time formulation used in modern codes. 

In ID, the fermion sign problem can actually be eliminated completely. In lattice models of electronic systems, this is accomplished by a trick called the checkerboard decomposition. This method is best illustrated using noninteracting... 

To eliminate the fermion sign problem, the checkerboard decomposition follows an unconventional method to obtain the path integral. First, the single-particle Hamiltonian is decomposed into two pieces hj = /i /t , where /t is given by the... 

The checkerboard decomposition can actually be thought of as a special case of the blocking scheme described in Section 3.2. By blocking together exchange and nonexchange paths in the checkerboard decomposition, the fermion sign... 

Using the checkerboard decomposition, the stability of the distorted Peierls phase has been examined. It was found that for spin-1 electrons, the distorted phase is always robust, even in the presence of nuclear tunneling of the C atoms, which tends to increase the quantum dispersion and destablize the distorted phase. 

Like the checkerboard decomposition in ID, the auxiliary field method in 2D can be viewed as an application of the general blocking strategy described in Section 3.2. Although the auxiliary field method sums over all the exchanges, it does... 

The fifth and final chapter, on Parallel Force Field Evaluation, takes account of the fact that the bulk of CPU time spent in MD simulations is required for evaluation of the force field. In the first paper, BOARD and his coworkers present a comparison of the performance of various parallel implementations of Ewald and multipole summations together with recommendations for their application. The second paper, by Phillips et AL., addresses the special problems associated with the design of parallel MD programs. Conflicting issues that shape the design of such codes are identified and the use of features such as multiple threads and message-driven execution is described. The final paper, by Okunbor Murty, compares three force decomposition techniques (the checkerboard partitioning method. 

Here we use the infinite-checkerboard coordinates R (t) of the particles that initially are in the primary (v = 0) cell. If we make the decomposition... 

The composition profile in depth direction was investigated using a 3D model, whereby a checkerboard stmcture was observed in the early stages of decomposition, but this decayed as the intrinsic value of R(f) in the bulk domain increased. The strip patterns were confined to the surface domain in the later stages of phase decomposition. A consideration of the isotropic elastic energy also caused a slower evolution in the depth direction. 

The numerical simulation was consistent with experimental data with regards to influences of substrate pattern width, polymer weight ratio and polymer molecular weight of PAA. Both, experimental and simulated results showed that the intrinsic characteristic length should be compatible with the substrate pattern size for a reflned phase separation pattern, according to the substrate pattern. Simulations also showed that the increase in polymer molecular weight can increase the speed of morphologic evolution. As the film thickness is very small the checkerboard structure of spinodal decomposition cannot be seen in these results, but they are present. 

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