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Caracciolo, Sergio; Causo, Maria Serena; Ferraro, Giovanni; Papinutto, Mauro; Pelissetto, Andrea

doi:10.1023/a:1018719226557

Cited by 15 publications

(18 citation statements)

References 40 publications

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“…Note that for η the finite-N corrections are very small. In fact, from the η 6 -values near N = 1000, we can extract numerically the 1/N-expansion 26) which agrees reasonably well with the exact expansion…”

Section: Critical Exponents Up To Six Loopssupporting

confidence: 72%

Critical Exponents from Other Expansions

Kleinert¹,

Schulte-Frohlinde²

2001

Critical Properties of Phi⁴-Theories

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It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4 -theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one-and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4.Another approach is based on high-temperature expansions for lattice models of the O(N)-symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8.Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4 -theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in Section 20.2. Sixth-Order Expansions in Three DimensionsThe three-dimensional φ 4 -theory is defined by the bare energy functionalwhere the coupling constant is normalized to obtain the most convenient perturbation expansions. The field φ B (x) is an N-dimensional vector, and the action is O(N)-symmetric in this vector space. By calculating the Feynman diagrams up to six loops, and imposing the normalization conditions (9.23), (9.24), and (9.33) we find renormalized values of mass, coupling constant, and field related to the bare input quantities by renormalization constants Z φ , Z m 2 , Z g : Sixth-Order Expansion in Three Dimensions 367The divergences are removed by analytic regularization [1]. In the literature, we find expansions for m 2 B /m 2 , g B /g, and φ B /φ in powers of g up to g 6 (six loops), the latter two having recently been extended to the power g 7 (seven loops) [2,3]. We shall first discuss in detail the variational resummation of the six-loop expansions.. The small improvements of the critical exponents brought about by the seven-loop terms will be calculated separately in Section 20.4.Introducing the reduced dimensionless coupling constantsḡ ≡ g/m andḡ B ≡ g B /m, these determine the r...

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Section: Critical Exponents Up To Six Loopssupporting

confidence: 72%

Critical Exponents from Other Expansions

Kleinert¹,

Schulte-Frohlinde²

2001

Critical Properties of Phi⁴-Theories

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“…where x = (2d) 1/2 R/R end , f (x) is a universal function, and ∆ is a correction to scaling exponent. As shown in [29], f (x) is well approximated by a phenomenological representation given first by Mc Kenzie and Moore [27,28]. A comparison with the latter is shown in Fig.…”

Section: The Thermal Correlation Lengthmentioning

confidence: 70%

Simple model for the DNA denaturation transition

2000

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We study pairs of interacting self-avoiding walks ¿omega(1), omega(2)¿ on the 3d simple cubic lattice. They have a common origin omega(1)(0)=omega 20, and are allowed to overlap only at the same monomer position along the chain: omega(1)(i) not equal omega(2)(j) for i not equal j, while omega(1)(i)=omega(2)(i) is allowed. The latter overlaps are indeed favored by an energetic gain epsilon. This is inspired by a model introduced long ago by Poland and Sheraga [J. Chem. Phys. 45, 1464 (1966)] for the denaturation transition in DNA where, however, self avoidance was not fully taken into account. For both models, there exists a temperature T(m) above which the entropic advantage to open up overcomes the energy gained by forming tightly bound two-stranded structures. Numerical simulations of our model indicate that the transition is of first order (the energy density is discontinuous), but the analog of the surface tension vanishes and the scaling laws near the transition point are exactly those of a second-order transition with crossover exponent straight phi=1. Numerical and exact analytic results show that the transition is second order in modified models where the self-avoidance is partially or completely neglected.

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“…39, without the additional refinements discussed in Ref. 42 To further speed up the dynamics, we simulated several walks simultaneously, performing a multiple Markov chain simulation ͑this technique is also known as parallel tempering or exchange Monte Carlo͒. The algorithm is very efficient as long as the attraction is not too large.…”

Section: Appendix A: Numerical Detailsmentioning

confidence: 99%

Corrections to scaling and crossover from good- to θ-solvent regimes of interacting polymers

Pelissetto

Hansen

2005

The Journal of Chemical Physics

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120

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We exploit known properties of universal ratios, involving the radius of gyration R(g), the second and third virial coefficients B(2) and B(3), and the effective pair potential between the centers of mass of self-avoiding polymer chains with nearest-neighbor attraction, as well as Monte Carlo simulations, to investigate the crossover from good- to theta-solvent regimes of polymers of finite length L. The scaling limit and finite-L corrections to scaling are investigated in the good-solvent case and close to the theta temperature. Detailed interpolation formulas are derived from Monte Carlo data and results for the Edwards two-parameter model, providing estimates of universal ratios as functions of the observable ratio A(2)=B(2)/R(g) (3) over the whole temperature range, from the theta point to the good-solvent regime. The convergence with L (L< or =8000) is found to be satisfactory under good-solvent conditions, but longer chains would be required to match theoretical predictions near the theta point, due to logarithmic corrections. A quantitative estimate of the universal ratio A(3)=B(3)/R(g) (6) as a function of temperature shows that the third virial coefficient remains positive throughout, and goes through a pronounced minimum at the theta temperature, which goes to zero as 1/ln L in the scaling limit.

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Untitled

Cited by 15 publications

References 40 publications

Critical Exponents from Other Expansions

Critical Exponents from Other Expansions

Simple model for the DNA denaturation transition

Corrections to scaling and crossover from good- to θ-solvent regimes of interacting polymers

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