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Register a new userDuality symmetry, strong coupling expansion and universal critical amplitudes in two-dimensional Phi^{4} field models
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Authors
Giancarlo Jug
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We show that the exact beta-function beta(g) in the continuous 2D gPhi^{4} model possesses the Kramers-Wannier duality symmetry. The duality symmetry transformation tilde{g}=d(g) such that beta(d(g))=d(g)beta(g) is constructed and the approximate values of g^{*} computed from the duality equation d(g^{*})=g^{*} are shown to agree with the available numerical results. The calculation of the beta-function beta(g) for the 2D scalar gPhi^{4} field theory based on the strong coupling expansion is developed and the expansion of beta(g) in powers of g^{-1} is obtained up to order g^{-8}. The numerical values calculated for the renormalized coupling constant g_{+}^{*} are in reasonable good agreement with the best modern estimates recently obtained from the high-temperature series expansion and with those known from the perturbative four-loop renormalization-group calculations. The application of Cardys theorem for calculating the renormalized isothermal coupling constant g_{c} of the 2D Ising model and the related universal critical amplitudes is also discussed.
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The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of ${mathbb R}^d$. Our aim is to sort out which ones of the previously published partly contradictory $epsilon$-expansion results to second order in $epsilon=4+frac{m}{2}-d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+frac{m}{2}-epsilon$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents $eta_{l2}$ and $eta_{l4}$ and of the wave-vector exponent $beta_q$ to order $epsilon^2$ are reduced to single integrals, which for general m=1,...,d-1 can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For m=2 and m=6, we confirm the results of Sak and Grest [Phys. Rev. B {bf 17}, 3602 (1978)] and Mergulh{~a}o and Carneiros recent field-theory analysis [Phys. Rev. B {bf 59},13954 (1999)].
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The massive field-theory approach for studying critical behavior in fixed space dimensions $d<4$ is extended to systems with surfaces.This enables one to study surface critical behavior directly in dimensions $d<4$ without having to resort to the $epsilon$ expansion. The approach is elaborated for the representative case of the semi-infinite $|bbox{phi}|^4$ $n$-vector model with a boundary term ${1/2} c_0int_{partial V}bbox{phi}^2$ in the action. To make the theory uv finite in bulk dimensions $3le d<4$, a renormalization of the surface enhancement $c_0$ is required in addition to the standard mass renormalization. Adequate normalization conditions for the renormalized theory are given. This theory involves two mass parameter: the usual bulk `mass (inverse correlation length) $m$, and the renormalized surface enhancement $c$. Thus the surface renormalization factors depend on the renormalized coupling constant $u$ and the ratio $c/m$. The special and ordinary surface transitions correspond to the limits $mto 0$ with $c/mto 0$ and $c/mtoinfty$, respectively. It is shown that the surface-enhancement renormalization turns into an additive renormalization in the limit $c/mtoinfty$. The renormalization factors and exponent functions with $c/m=0$ and $c/m=infty$ that are needed to determine the surface critical exponents of the special and ordinary transitions are calculated to two-loop order. The associated series expansions are analyzed by Pade-Borel summation techniques. The resulting numerical estimates for the surface critical exponents are in good agreement with recent Monte Carlo simulations. This also holds for the surface crossover exponent $Phi$.
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