Wilson theory for 2-dimensional Ising spin systems

Niemeyer, Th.; Leeuwen, J. M. J. van

doi:10.1016/0031-8914(74)90044-5

Cited by 292 publications

(14 citation statements)

References 14 publications

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“…The procedure is often implemented in the form of block RG [21,31]. This corresponds to a factorization of the conditional probability distribution into independent contributions from equivalent (assuming translation invariance) blocks V ⊂ X :…”

Section: The Rsmi Algorithmmentioning

confidence: 99%

“…Additionally, since the RG rule is optimized iteratively, we can study the approach to the optimal solution by considering the properties of the renormalized Hamiltonian defined by the coarse-graining rule at any stage of the training. In order to do this analytically we now show how the effective Hamiltonian can be expressed by appropriate cumulant expansion [31] (though the RSMI algorithm deals with probability measure as the basic object, and at no point computes the Hamiltonian, the Hamiltonian picture is more interpretable physically). Subsequently we apply this machinery to the 1D Ising model.…”

Section: Optimality: the Measure And The Effective Hamiltonianmentioning

confidence: 99%

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Optimal Renormalization Group Transformation from Information Theory

et al. 2020

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The connections between information theory, statistical physics and quantum field theory have been the focus of renewed attention. In particular, the renormalization group (RG) has been explored from this perspective. Recently, a variational algorithm employing machine learning tools to identify the relevant degrees of freedom of a statistical system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), was proposed for real-space RG. Here we investigate analytically the RG coarse-graining procedure and the renormalized Hamiltonian, which the RSMI algorithm defines. By a combination of general arguments, exact calculations and toy models we show that the RSMI coarse-graining is optimal in a sense we define. In particular, a perfect RSMI coarse-graining generically does not increase the range of a short-ranged Hamiltonian, in any dimension. For the case of the 1D Ising model we perturbatively derive the dependence of the coefficients of the renormalized Hamiltonian on the real-space mutual information retained by a generic coarse-graining procedure. We also study the dependence of the optimal coarse-graining on the prior constraints on the number and type of coarse-grained variables. We construct toy models illustrating our findings.

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Section: The Rsmi Algorithmmentioning

confidence: 99%

Section: Optimality: the Measure And The Effective Hamiltonianmentioning

confidence: 99%

Optimal Renormalization Group Transformation from Information Theory

et al. 2020

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“…Similarly as was done in [5] we can, in the case under consideration, calculate the critical interaction KC using the results of Niemeyer and van Leeuwen [4]. From the first-order cumulant expansion they lead to the following equation for K e :…”

Section: --mentioning

confidence: 98%

“…For D large and negative f 4 For the spin f (half-integer) the result is, of course, quite different [3] and Ki2) can change only from I ? (for D + -00) to 9 E (for D + co).…”

Section: --mentioning

confidence: 99%

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Effective Hamiltonian for the Two‐Dimensional Arbitrary Spin Ising Model

Sznajd

1984

Physica Status Solidi (b)

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A method of the reduction of the generalized arbitrary spin two-dimensional Ising model to the spin-half Ising model is presented. The method is demonstrated in detail by calculating the effective interaction constants to the third order in a cumulant expansion for the triangular spin-one Ising model (the Blume-Emery-Griffiths model).Es wird eine Methode der Zuriickfiihrung des verallgemeinerten zweidimensionalen Ising-Modells fur Spins beliebiger GroDe auf das Ising-Model1 fur den Spin + dargestellt. Durch die Berechnung der effektiven Wechselwirkungskonstanten bis zur dritten Ordnung in der Cumulantenentwicklung wird die Methode fur ein dreieckiges Spin-1-Mode11 (das Blume-Emery-Griffiths-Modell) ausfuhrlich demonstriert.

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The effective‐site percolation approach in two dimensions

Pawłowski

2007

Physica Status Solidi (b)

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A new Monte Carlo cluster approach to describe the order -order and order -disorder transitions in twodimensional (2D) correlated systems is presented. It is shown that a phase transition of a physical system can be correctly described in terms of the percolation language of the effective-site approach. In contrast to the well-known bond approaches, the method proposed does not require additional assumptions as to the acceptance of the bonds. The new idea is based on the site approach to elementary ordered plaquettes, leading to the accurate coincidence of percolation and the phase transitions in 2D. Here I present the analysis of the spin-system in the Blume -Capel model and the charged-system in the atomic limit of the extended Hubbard model. In both cases the interpretation of the phase transition is made in terms of the percolation of different types of order. The new method allows precise identification of the pure and mixed phases.

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Wilson theory for 2-dimensional Ising spin systems

Cited by 292 publications

References 14 publications

Optimal Renormalization Group Transformation from Information Theory

Optimal Renormalization Group Transformation from Information Theory

Effective Hamiltonian for the Two‐Dimensional Arbitrary Spin Ising Model

The effective‐site percolation approach in two dimensions

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