Trotter dimensions

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

Thus this kind of troublesome check has to be repeated until the intersection of the set of ordered pairs of the linear extensions coincide with that of the poset. The lowest number of linear extensions -written as ordered pairs as shown above- whose intersection is the actual poset (together with its transitive relations), is its dimension. Following the explanation above one would have to check 14 T 3/2 intersections, just to verify that the dimension equals 2. If such pair of ordered sets, derived from any two linear extensions is found, one has found a "realizer" of the poset (Trotter 1991). 

Note, it is not a good policy to derive the dimension by finding explicitly the realizers. Here five useful theorems are taken from the literature (Trotter 1991) ... 

Trotter WT (1991) Combinatorics and Partially Ordered Sets Dimension Theoiy John Hopkins Series in the Mathematical Science. The J Hopkins University Press Baltimore... 

Mathematicians have termed a set of elements in a poset that are all mutually incomparable an anti-chain. (See chapter by Briiggemann and Carlsen, p. 61 and for more detailed mathematics and definitions see Combinatorics and Partially Ordered Sets Dimension Theory by Trotter (Trotter 1992)). If we consider all anti-chains that contain a partition [A] as an element, the complexity of [a] is the number of elements in those antichains (i.e. the cardinality or size of the anti-chains) that have the maximum number of elements, maximum anti-chains. Clearly, this concept can be generalized to any poset, though, as we have seen, the case of the YDL is of particular interest and relevance to physics and chemistry. 

Another technique which has gained prominence in recent years is the Quantum Monte Carlo (QMC) technique. This technique maps a d-dimensional quantum model onto a d - - 1 dimensional classical model via a Trotter decomposition of the partition function or the ground state projection operator [56, 57]. The quantum model is then studied by performing a Monte Carlo sampling procedure on the classical model in higher dimension. For fermions, the mapping of the interacting quantum model system to the classical system could... 

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