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arXiv:cond-mat/9705086 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Two-point correlation function of three-dimensional O(N) models: The critical limit and anisotropy

Massimo Campostrini B, Andrea Pelissetto C, Paolo Rossi, Ettore Vicari

Published Jan 1, 1998• 74 citations
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Abstract

In three-dimensional $O(N)$ models, we investigate the low-momentum behavior of the two-point Green's function $G(x)$ in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotationally invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice $N$-vector model, and $1/N$ expansion, field-theoretical methods within the ${\ensuremath{\varphi}}^{4}$ continuum formulation. Non-Gaussian corrections to the universal low-momentum behavior of $G(x)$ are evaluated, and found to be very small. In nonrotationally invariant physical systems with $O(N)$-invariant interactions, the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent $\ensuremath{\rho}$, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At $N=\ensuremath{\infty}$ one finds $\ensuremath{\rho}=2$. We show that, for all values of $N>~0$, $\ensuremath{\rho}\ensuremath{\simeq}2$.

DOI
10.1103/physreve.57.184

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In three-dimensional O(N)O(N)O(N) models, we investigate the low-momentum behavior of the two-point Green's function G(x)G(x)G(x) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotationally invariant fixed point. Several approaches are exploited, such as strong-coupling expansion of lattice NNN-vector model, and 1/N1/N1/N expansion, field-theoretical methods within the \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 continuum formulation. Non-Gaussian corrections to the universal low-momentum behavior of G(x)G(x)G(x) are evaluated, and found to be very small. In nonrotationally invariant physical systems with O(N)O(N)O(N)-invariant interactions, the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent \ensuremathρ\ensuremath{\rho}\ensuremathρ, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞ one finds \ensuremathρ=2\ensuremath{\rho}=2\ensuremathρ=2. We show that, for all values of N> 0N>~0N> 0, \ensuremathρ\ensuremath≃2\ensuremath{\rho}\ensuremath{\simeq}2\ensuremathρ\ensuremath≃2.
In three-dimensional O( N ) models, we investigate the low-momentum behavior of the two-point Green's function G ( x ) in the critical region of the symmetric phase. We consider physical systems whose criticality is characterized by a rotational-invariant fixed point. In non-rotational invariant physical systems with O( N )-invariant interactions, the vanishing of space-anisotropy approaching the rotational-invariant fixed point is described by a critical exponent ρ, which is universal and is related to the leading irrelevant operator breaking rotational invariance. At N = ∞ one finds ρ = 2. We show that, for all values of N ⩾ 0, ρ ≃ 2. Non-Gaussian corrections to the universal low-momentum behavior of G ( x ) are evaluated, and found to be very small.
The critical behavior of two-dimensional (2D) O(N) \ensuremath{\sigma} models with N\ensuremath{\le}2 on square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function G(x), calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For N2 the critical behavior is of power-law type, and the exponents \ensuremath{\gamma} and \ensuremath{\nu} extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At N=2, i.e., for the 2D XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent \ensuremath{\sigma}=1/2, and with universality. The value \ensuremath{\sigma}=1/2 is confirmed within an uncertainty of few percent. The prediction \ensuremath{\eta}=1/4 is also roughly verified. For various values of N\ensuremath{\le}2, we determine some ratios of amplitudes concerning the two-point function G(x) in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) in the critical region is essentially Gaussian at all values of N\ensuremath{\le}2. Exact results for the long-distance behavior of G(x) when N=1 (Ising model in the strong-coupling phase) confirm this statement. \textcopyright{} 1996 The American Physical Society.
We study the O(2) model with Z_{4}-symmetric perturbations within the framework of the nonperturbative renormalization group (RG) for spatial dimensionality d=2 and 3. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In d=3 the system is controlled by the XY, Ising, and low-T fixed points in the presence of a dangerously irrelevant anisotropy coupling λ. In d=2 the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed points, giving rise to nonuniversal critical behavior, and an isolated Ising fixed point. In addition to inducing crossover behavior in universal properties, the presence of the Ising fixed point yields a generic, abrupt change of critical temperature at a specific value of the anisotropy field.
We investigate the controversial issue of the existence of universality classes describing critical phenomena in three-dimensional systems characterized by a matrix order parameter with symmetry $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)$ and symmetry-breaking pattern $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)\ensuremath{\rightarrow}\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N\ensuremath{-}2)$. Physical realizations of these systems are, for example, frustrated spin models with noncollinear order. Starting from the field-theoretical Landau-Ginzburg-Wilson Hamiltonian, we consider the massless critical theory and the minimal-subtraction scheme without ϵϵϵ expansion. The three-dimensional analysis of the corresponding five-loop series shows the existence of a stable fixed point for N=2N=2N=2 and N=3N=3N=3, confirming recent field-theoretical results based on a six-loop expansion in the alternative zero-momentum renormalization scheme defined in the massive disordered phase. In addition, we report numerical Monte Carlo simulations of a class of three-dimensional $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$-symmetric lattice models. The results provide further support to the existence of the $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$ universality class predicted by the field-theoretical analyses.
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.
We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O(n) symmetric anisotropic phi;{4} lattice model with periodic boundary conditions in a d -dimensional hypercubic geometry above, at, and below Tc. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. The anisotropy parameters are observable by scattering experiments at Tc. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension d for 2<d<4 is employed. In contrast to the epsilon=4-d expansion, the fixed- d finite-size approach keeps the exponential form of the order-parameter distribution function unexpanded. For the case of cubic symmetry and for n=1 , our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at Tc by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. The epsilon expansion result is in less good agreement. Below Tc, a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the nonmonotonic dependence of the Binder cumulant on the ferromagnetic next-nearest-neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large- L behavior at T++Tc violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the phi4 theory, on the cutoff procedure, and on subleading long-range interactions.