Subleading critical exponents from the renormalisation group

1994

We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with the local polynomial approximation (LPA), and discuss the advantages and shortcomings of these methods, and compare our results with the literature. In particular, convergence of the LPA is discussed in some detail. We finally integrate the flows numerically and find a c-function which determines these flows to be gradient in this approximation. * Talk presented

Arxiv preprint hep-th/ …, 1996

λϕ 4 theory at finite temperature suffers from infrared divergences near the temperature at which the symmetry is restored. These divergences are handled using renormalization group methods. Flow equations which use a fiducial mass as flow parameter are well adapted to predicting the non-trivial critical exponents whose presence is reflected in these divergences. Using a fiducial temperature as flow parameter, we predict the critical temperature, at which the mass vanishes, in terms of the zero-temperature mass and coupling. We find some universal amplitude ratios which connect the broken and symmetric phases of the theory which agree well with those of the three-dimensional Ising model obtained from high-and low-temperature series expansions.

Condensed Matter Physics, 2010

The effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed. Three models are considered: the O(N ) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice. We consider all models at fixed d = 3 and analyze the resummation procedures currently used to compute the critical exponents. We first show that, for the O(N ) model, the resummation does not eliminate all non-physical (spurious) fixed points (FPs). Then the dependence of spurious as well as of the Wilson-Fisher FPs on the resummation parameters is carefully studied. The critical exponents at the Wilson-Fisher FP show a weak dependence on the resummation parameters. On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters. For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters. It appears to be spurious, as expected. As for the frustrated models, there are two cases depending on the value of the number of spin components. When N is greater than a critical value Nc, the stable FP shows common characteristic with the Wilson-Fisher FP. On the contrary, for N < Nc, the results obtained at the stable FP are similar to those obtained at the spurious FPs of the O(N ) and cubic models. We conclude from this analysis that the stable FP found for N < Nc in frustrated models is spurious. Since Nc > 3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.

Physics Letters B, 1994

The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found: the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.

1998

The desirability of evaluating the effective potential in field theories near a phase transition has been recognized in a number of different areas. We show that recent Monte Carlo simulations for the probability distribution for the order parameter in an equilibrium Ising system, when combined with low-order renormalization group results for an ordinary ϕ^4 system, can be used to extract the effective potential. All scaling features are included in the process.

Annals of Physics, 1981

A Ginzburg-Landau-Wilson (GLW) model extended to include surface energy terms is used to discuss the surface critical behaviour of a system of interacting spins. A straightforward extension of the bulk renormalization group transformation is worked out to first order in E = 4d (where d is the number of spatial dimensions). In contrast to previous work, the coefficients of the quadratic (r) and quartic (u) terms in the GLW free energy are allowed to depend on the distance from the surface. To lowest order in E, we show that the fixed point u*(z) is given by the usual bulk value. The RG recursion formula we obtain has a line of possible fixed points r*(z) associated with both the "special" and "ordinary" transitions. The fact that the GLW model can only describe systems with short-range interactions allows one to select the correct fixed point, in which r*(z) is given by its bulk value up to atomic distances from the surface. We discuss the ordinary and special transitions in a unified way. The correlation functions involving surface spins are worked out and the critical exponents I,, and I]~ are evaluated to O(E). Our results agree with those of Lubensky and Rubin (ordinary transition) and Bray and Moore (special transition), who used less transparent methods of carrying out the RG transformation.

Physical Review Letters, 2002

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two-and threedimensional Ising models for a variety of renormalization group transformations.

Universality and Renormalization, 2007

In computing quantum effects, it is necessary to perform a sum over all intermediate states consistent with prescribed initial and final states. Divergences arising in the course of evaluating this sum forces one to "renormalize" parameters characterizing the system. An ambiguity inherent in this rescaling is parameterized by a dimensionful parameter µ 2 which serves to set a scale for the process. Requiring that the explicit and implicit dependence of a physical quantity on µ 2 conspire to cancel leads to the so-called "renormalization group" equation [1-10]. It has proved possible to extract a lot of useful information from this equation; we will enumerate a number of these in this report.

Physics Reports, 2001

Renormalization group, and in particular its Quantum Field Theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized φ 4 3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N-vector model [1,2] and of the equation of state of the 3D Ising model [3]. These results are among the most precise available probing field theory in a nonperturbative regime. Precise calculations first require enough terms of the perturbative expansion. However perturbation series are known to be divergent. The divergence has been characterized by relating it to instanton contributions. The information about large order behaviour of perturbation series has then allowed to develop efficient "summation" techniques, based on Borel transformation and conformal mapping [4]. We first discuss exponents and describe our recent results [2]. Compared to exponents, the determination of the scaling equation of state of the 3D Ising model involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order dependent mapping method. From the knowledge of the equation of state a number of ratio of critical amplitudes can also be derived. Finally we emphasize that few physical quantities which are predicted by renormalization group to be universal have been determined precisely, and much work remains to be done. Considering the steady increase in the available computer

Physical Review E

We revisit the approach to the lower critical dimension d lc in the Ising-like ϕ 4 theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d 2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d lc. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d → d lc with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d lc and for the behavior of the critical temperature as d → d lc , which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.