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arXiv:hep-lat/9703017 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

2n-point renormalized coupling constants in the three-dimensional Ising model:Estimates by high temperature series to order Β17

P. Butera, M. Comi

Published Jun 1, 1997• 40 citations
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Abstract

We compute the 2n-point renormalized coupling constants in the symmetric phase of the three-dimensional (3D) Ising model on the simple cubic lattice in terms of the high temperature expansions O(${\mathrm{\ensuremath{\mathrm{B}}}}^{17}$) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3D scalar field at the infrared fixed point or, equivalently, in the critical equation of state of the 3D Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations, as well as previous high temperature expansions.

DOI
10.1103/physreve.55.6391

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We compute the 2n-point renormalized coupling constants in the symmetric phase of the three-dimensional (3D) Ising model on the simple cubic lattice in terms of the high temperature expansions O(\ensuremathB17{\mathrm{\ensuremath{\mathrm{B}}}}^{17}\ensuremathB17) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3D scalar field at the infrared fixed point or, equivalently, in the critical equation of state of the 3D Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations, as well as previous high temperature expansions.
25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining gamma=1.2373(2), nu=0.63012(16), alpha=0.1096(5), eta=0.036 39(15), beta=0.326 53(10), and delta=4.78 93(8). Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate Delta=0.52(3) for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
Using a finite-size scaling method combined with extensive Monte Carlo measurements of high statistics, we calculate the four-, six- and eight-point renormalized coupling constants defined at zero momentum in the symmetric phase of the three-dimensional Ising system. The results of the 2D Ising system that are directly measured are also reported. Our value of the four-point coupling constant for the 3D system agrees very well with the available estimates from other methods. Our values of the six- and eight-point coupling constants are significantly different from those obtained by other methods for the 3D system, although they agree reasonably well in the 2D system.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
Higher{order vertices at zero external momenta for the scalar eld theory, describing the critical behaviour of the Ising model, are studied within the eld{theoretical renormalization group (RG) approach in three dimensions.Dimensionless six{point g6 and eight{point g8 eective coupling constants are calculated in the three{loop approximation.Their numerical values, universal at criticality, are estimated by means of the Pad e and Pad e{Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal xed{point value known from six{loop RG calculations.The values of g 6 obtained are compared with their analogs resulting from the {expansion, Monte Carlo simulations, the Wegner{Houghton equations and the linked cluster expansion series.The eld{theoretical estimates for g 6 are shown to be in a good agreement with each other, diering considerably from the values given by other methods.