Series expansion methods for strongly interacting lattice by Jaan Oitmaa

By Jaan Oitmaa

Perturbation sequence growth equipment are subtle numerical instruments used to supply quantitative calculations in lots of components of theoretical physics. This publication supplies a complete consultant to using sequence enlargement tools for investigating part transitions and demanding phenomena, and lattice versions of quantum magnetism, strongly correlated electron structures and trouble-free debris. Early chapters hide the classical remedy of serious phenomena via high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The publication then discusses high-order linked-cluster perturbation expansions for quantum lattice versions, finite temperature expansions, and lattice gauge versions. additionally integrated are a number of specific examples and case reports, and an accompanying assets web site,, comprises courses for enforcing those robust numerical innovations. A priceless source for graduate scholars and postdoctoral researchers operating in condensed subject and particle physics, this e-book can also be beneficial as a reference for specialised graduate classes on sequence enlargement tools.

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7 the weights are now more complicated but still depend only on the graph topology. They can be computed in a number of (equivalent) ways, each of which can be computerized: r By direct enumeration of overlaps of disconnected bare graphs. r By a ‘finite cluster method’, whereby the partition function for the graph is computed and the weight of proper sub-graphs subtracted. 7. r By the so-called ‘complete-term method’ (Domb, 1970; Sykes and Hunter, 1974). 8 we give the graphs which contribute to order v 6 with weights and lattice constants, again for the triangular lattice.

14. 5973 15. 18 − 2v 6 + . . 16. 12 v5 v6 − 2v 6 + . . 9. Graphs and weights for star graph expansions for the inverse susceptibility χ −1 . Graph Lattice constant Weight 1. 1 1 2. 3 − 2v + 2v 2 − 2v 3 + 2v 4 − 2v 5 + 2v 6 + . . 3. 2 6v 3 − 12v 4 + 12v 5 − 6v 6 + . . 4. 3 8v 4 − 16v 5 + 16v 6 + . . 5. 6 10v 5 − 20v 6 + . . 6. 15 12v 6 + . . 7. 3 − 8v 5 + 4v 6 + . . 8. 12 − 8v 6 + . . 6 More on Ising models 47 the graphs to order v 6 , together with their weights and lattice constants for the triangular lattice.

58) and is thus not represented exactly as a [T /M, L] approximant. However we would expect good convergence as the degree of the approximant increases. 2. It is left as an exercise for the reader to show that f 1 exactly satisfies a homogeneous second-order differential equation with polynomial coefficients, and is represented exactly by an approximant [−1/1,2,2]. Turning now to test function f 2 , the task is to evaluate the series at the singular point x = 1, from a knowledge of the first 15 terms.

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