Minimal renormalization without ϵ-expansion: Amplitude functions for O(n) symmetric systems in three dimensions below Tc

Massive field theory at fixed dimension d < 4 is combined with the minimal subtraction scheme to calculate the amplitude functions of thermodynamic quantities for the O(n) symmetric Φ⁴ model below T_c in two-loop order. Goldstone singularities arising at an intermediate stage in the calculation of O(n) symmetric quantities are shown to cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the amplitude functions in zero field for the order parameter, specific heat and helicity modulus (superfluid density) in three dimensions. We also calculate the q² part of the inverse of the wavenumber-dependent transverse susceptibility χT(q) which provides an independent check of our result for the helicity modulus. The two-loop contributions to the superfluid density and specific heat below T_c turn out to be comparable in magnitude to the one-loop contributions, indicating the necessity of higher-order calculations and Padé-Borel type resummations.     

On the ε-expansion procedure in the RNG theory of turbulence

Rescaling the equations of eliminating small scales, the physical justification for the-expansion procedure in the RNG theory of turbulence is proposed, in terms of that the inertial effects are small comparing with the viscous effects at the vicinity of the Kolmogorov dissipation wavenumber.We are grateful to Professor Chao-Hao Gu for numerous helpful suggestions. We would also like to acknowledge Professor Ke-Lin Wang and Bing-Hong Wang for many stimulating discussions of these problems.     

Analytical five-loop expressions for the renormalization group QED β-function in different renormalization schemes

We obtain analytical five-loop results for the renormalization group ??-function of Quantum Electrodynamics with the single lepton in different renormalization schemes. The theoretical consequences of the results obtained are discussed.     

Runge–Kutta methods and renormalization

Rooted trees have been used to calculate the solution of nonlinear flow equations and Runge–Kutta methods. More recently, rooted trees have helped systematizing the algebra underlying renormalization in quantum field theories. The Butcher group and B-series establish a link between these two approaches to rooted trees. On the one hand, this link allows for an alternative representation of the algebra of renormalization, leading to nonperturbative results. On the other hand, it helps to renormalize singular flow equations. The usual approach is extended here to nonlinear partial differential equations. A nonlinear Born expansion is given, and renormalization is used to partly remove the secular terms of the perturbative expansion. Received: 6 July 1999 / Published online: 10 December 1999     

ß-functions and the exact renormalization group

We relate ß-functions to the flow of relevant couplings in the exact renormalization group. The specific case of a cutoff γφ⁴ theory in four dimensions is discussed in detail. The underlying idea of convergence of the flow of effective lagrangians is developed to identify the ß-functions. A perturbative calculation of the ß-functions using the exact flow equations is sketched. The analysis may be extended to any system with a cutoff.     

The ɛ-expansion for the Hierarchical Model

In this paper, the Hierarchical Model is studied near a non-trivial fixed point of its renormalization group. Our analysis is an extension of work of Bleher and Sinai. We prove the validity of the -expansion for . We then show that the renormalization transformations around have an unstable manifold which is completely characterized by the tangent map and can be brought to normal form. We then establish relations between this result and the critical behaviour of the model in the thermodynamic limit.     

Pseudo-ε-expansion and the two-dimensional Ising model

The pseudo-ε-expansions for the coordinate of the fixed point g^*, the critical exponents, and the sextic effective coupling constant g₆ are determined for the two-dimensional Ising model on the basis of the five-loop renormalization group series. It is found that the pseudoe-expansions for the coordinate of the fixed point g^*, the inverse exponent γ^?1, and the constant g₆ possess a remarkable property, namely, the higher terms of these series are so small that reliable numerical results can be obtained without invoking Borel summation.     

The β-expansion of periodic dressed chain models

We revisit the method of calculating the $\beta$-expansion of the Helmholtz free energy of any one-dimensional (1D) Hamiltonian with invariance under space translations, presented in [O. Rojas, S.M. de Souza, M.T. Thomaz, J. Math. Phys. 43 (2002) 1390], extending this method to 1-D Hamiltonians that are invariant under translations along super-sites (sequences of $l$ sites). The method is applicable, for instance, to spin models and bosonic/fermionic versions of Hubbard models, either quantum or classical. As an example, we focus on the staggered spin-$S$ Ising model in the presence of a longitudinal magnetic field, comparing some of its thermodynamic functions to those of the standard Ising model. We show that for arbitrary values of spin ($S\in \{1,3/2,2,\dots \}$) but distinct values of the coupling constant and the magnetic field, the specific heat and the $z$-component of the staggered and usual magnetizations can be well approximated by their respective thermodynamic function of the spin-$1/2$ models in a suitable interval of temperature. These approximations are valid for the standard Ising model as well as for the staggered model, the thermodynamics of which are known exactly.     

Minimal Braid in Applied Symbolic Dynamics

Based on the minimal braid assumption, three-dimensionai periodic flows of a dynamical system are reconstructed in the case of unimodai map, and their topologicai structures are compared with those of the periodic orbits of the R6ssler system in phase space through the numerical experiment. The numerical results justify the validity of the minimai braid assumption which provides a suspension from one-dimensional symbolic dynamics in the Poincare section to the knots of three-dimensionai periodic flows.     

Perturbative renormalization of massless φ 4 4 with flow equations

Perturbative renormalizability proofs in the Wilson-Polchinski renormalization group framework, based on flow equations, were so far restricted to massive theories. Here we extend them to Euclidean massless φ ₄ ⁴ . As a by-product of the proof we obtain bounds on the singularity of the Green functions at exceptional momenta in terms of the exceptionality of the latter. These bounds seem to be new and are quite sharp.     

Tight-binding electron–phonon coupling and band renormalization in graphene

The kink structure in the quasiparticle spectrum of electrons in graphene observed at 200 me V below the Fermi level by angle-resolved photoemission spectroscopy(ARPES)was claimed to be caused by a tight-binding electron–phonon(e–ph)coupling in the previous theoretical studies.However,we numerically find that the e–ph coupling effect in this approach is too weak to account for the ARPES data.The former agreement between this approach and the ARPES data is due to an enlargement of the coupling constant by almost four times.     

The finite temperature renormalization group equation in λφ4 theory

The renormalization group equation at finite temperature is studied in the context of λφ⁴ theory. Three methods of deriving the finite temperature renormalization group equations are presented. The result shows that the effective mass becomes large as the temperature increases while the effective coupling constant becomes small.     

Can renormalization group flow end in a Big Mess?

The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan–Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental string (M?) theory. We consider various general aspects of such chaotic flows. We argue that they pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed.     

Exact renormalization group and Φ-derivable approximations

We show that the so-called Φ-derivable approximations can be combined with the exact renormalization group to provide efficient non-perturbative approximation schemes. On the one hand, the Φ-derivable approximations allow for a simple truncation of the infinite hierarchy of the renormalization group flow equations. On the other hand, the flow equations turn the non-linear equations that derive from the Φ-derivable approximations into an initial value problem, offering new practical ways to solve these equations.