Renormalization Group Series Research Articles

A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

Abstract We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.

Universal effective couplings of the three-dimensional n-vector model and field theory

We calculate the universal ratios R2k of renormalized coupling constants g2k entering the critical equation of state for the generalized Heisenberg (three-dimensional n-vector) model. Renormalization group (RG) expansions of R8 and R10 for arbitrary n are found in the four-loop and three-loop approximations respectively. Universal octic coupling R8⁎ is estimated for physical values of spin dimensionality n=0,1,2,3 and for n=4,...64 to get an idea about asymptotic behavior of R8⁎. Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-ε expansion approach. Regarding R10 our calculations show that three-loop RG and pseudo-ε expansions possess big and rapidly growing coefficients for physical values of n what prevents getting fair numerical estimates.

Effective potential of the three-dimensional Ising model: The pseudo-ϵ expansion study

The ratios R2k of renormalized coupling constants g2k that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar λϕ4 field theory (3D Ising model) within the pseudo-ϵ expansion approach. Pseudo-ϵ expansions for the critical values of g6, g8, g10, R6=g6/g42, R8=g8/g43 and R10=g10/g44 originating from the five-loop renormalization group (RG) series are derived. Pseudo-ϵ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Padé approximants yields proper numerical results. Use of Padé–Borel–Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values R6⁎=1.6488 and R6⁎=1.6490 which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-ϵ expansions is less favorable. Nevertheless, the conform-Borel resummation gives R8⁎=0.868, the number being close to the lattice estimate R8⁎=0.871 and compatible with the result of 3D RG analysis R8⁎=0.857. Pseudo-ϵ expansions for R10⁎ and g10⁎ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.

Phase transitions in two dimensions and multiloop renormalization group expansions

We discuss using the field theory renormalization group (RG) to study the critical behavior of twodimensional (2D) models. We write the RG functions of the 2D λϕ4 Euclidean n-vector theory up to five-loop terms, give numerical estimates obtained from these series by Pade-Borel-Leroy resummation, and compare them with their exact counterparts known for n = 1, 0,−1. From the RG series, we then derive pseudo-e-expansions for the Wilson fixed point location g*, critical exponents, and the universal ratio R6 = g6/g2, where g6 is the effective sextic coupling constant. We show that the obtained expansions are “friendler” than the original RG series: the higher-order coefficients of the pseudo-e-expansions for g*, R6, and γ−1 turn out to be considerably smaller than their RG analogues. This allows resumming the pseudo-e-expansions using simple Pade approximants without the Borel-Leroy transformation. Moreover, we find that the numerical estimates obtained using the pseudo-e-expansions for g* and γ−1 are closer to the known exact values than those obtained from the five-loop RG series using the Pade-Borel-Leroy resummation.

Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation

The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.

Critical behavior of two-dimensional cubic andMNmodels in the five-loop renormalization group approximation

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization-group (RG) approach. The beta functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N=2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta functions closer to each another. For the cubic model with N\geq 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M) models ($MN$ models with N=0) the results are compatible with a stable pure fixed point for M\geq1. For the MN model with M,N\geq2 all the non-perturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.

Five-loop renormalization-group expansions for the three-dimensionaln-vector cubic model and critical exponents for impure Ising systems

The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pade-Borel-Leroy resummation under n = 0. These exponents are found to be: \gamma = 1.325 +/- 0.003, \eta = 0.025 +/- 0.01, \nu = 0.671 +/- 0.005, \alpha = - 0.0125 +/- 0.008, \beta = 0.344 +/- 0.006. For the correction-to-scaling exponent, the less accurate estimate \omega = 0.32 +/- 0.06 is obtained.

Universal critical coupling constants for the three-dimensional n-vector model from field theory.

The field-theoretical renormalization group (RG) approach in three dimensions is used to estimate the universal critical values of renormalized coupling constants g6 and g8 for the O(n)-symmetric model. The RG series for g6 and g8 are calculated in the four- and three-loop approximations, respectively, and then resummed by means of the Padé-Borel-Leroy technique. Under the optimal value of the shift parameter b providing the fastest convergence of the iteration procedure, numerical estimates for g*6 are obtained with an accuracy no worse than 0.3%. The RG expansion for g8 demonstrates a stronger divergence, and results in considerably cruder numerical estimates.

Critical exponents for a three-dimensional O(n)-symmetric model with n>3.

Critical exponents for the 3D O(n)-symmetric model with n > 3 are estimated on the base of six-loop renormalization-group (RG) expansions. A simple Pade-Borel technique is used for the resummation of the RG series and the Pade approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location g_c and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for g_c and critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n-expansion. The same value n = 28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.

Deposition of SixC1−x: films by reactive r.f. sputtering

The ratios R2k of renormalized coupling constants g2k that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar λϕ4 field theory (3D Ising model) within the pseudo-ϵ expansion approach. Pseudo-ϵ expansions for the critical values of g6, g8, g10, R6=g6/g42, R8=g8/g43 and R10=g10/g44 originating from the five-loop renormalization group (RG) series are derived. Pseudo-ϵ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Padé approximants yields proper numerical results. Use of Padé–Borel–Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values R6⁎=1.6488 and R6⁎=1.6490 which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-ϵ expansions is less favorable. Nevertheless, the conform-Borel resummation gives R8⁎=0.868, the number being close to the lattice estimate R8⁎=0.871 and compatible with the result of 3D RG analysis R8⁎=0.857. Pseudo-ϵ expansions for R10⁎ and g10⁎ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.

Comments on renormalization group series

In previous articles by Nauenberg and by Niemeijer and Ruijgrok, the renormalization group approach has been used to obtain series expansions for the free energies of certain one- dimensional spin systems. We investigate here the question of whether this method provides a fundamentally new way of approximating the largest eigenvalue of a transfer matrix or kernel. For the models mentioned above, this is not the case. The partial sums of the series are essentially equivalent to the free energies of a sequence of finite models of increasing size.