Critical exponents predicted by grouping of Feynman diagrams in 4 model (original) (raw)
Related papers
Precise determination of critical exponents and equation of state by field theory methods
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
An investigation of the spatial fluctuations and their manifestations in the vicinity of the quantum critical point within the framework of the renormalized phi4\phi^{4}phi4 theory is proposed. Relevant features are reported through the Ginzburg-Landau-Wilson (GLW)-based calculations, combined with an efficient non perturbative technique. Both the dimension and size, but also microscopic details of the system, leading to critical behavior, and strongly deviating from the classical mean-field approach far from the thermodynamic limit, are taken into account. Further, the important role that harmonic and anharmonic fluctuations and finite-size effects can play in the determination of the characteristic properties of corresponding various systems, involving phase transitions and critical phenomena, is then discussed in detail with emphasis on the qualitative validity of the analysis
Surface critical exponents using the renormalization group ε-expansion
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 B, 1985
The renormalization-group recursion relations are solved for the effective Hamiltonians relative to phase transitions with four-component order parameters. For this value of n there are 22 types of Hamiltonians which can be classified into two categories according to the action of their normalizer Gz on the corresponding parameter space (Gz is the symmetry group leaving globally invariant this space). In the first case, G~generates a finite number of isolated fixed points whose characteristics can be deduced from the detailed investigation of five Hamiltonians only. In the second category, for which G~is a continuous group, there are, in addition to isolated fixed points, continuous manifolds of physically equivalent fixed points (the dimension of the manifolds is either one or three). In the search for a stable fixed point, the continuous manifolds can be ignored, while the isolated points are related to the five former Hamiltonians. For n =4, it is necessary to solve the recursion relations to two-loop order. The only possible stable ones among the fixed points then arise from a splitting of points which coincide, to one-loop order, with the isotropic fixed point. Extending, to twoloop order, a result recently established to the preceding order, we show that if a stable fixed point exists, it is unique. For n =4, the stable fixed point has one of three possible symmetries: diicosahedral, hypercubic, or dicylindrical. Despite this anisotropy of the critical fluctuations, the exponents associated with any of the stable fixed points are identical to order e to the "isotropic" exponents corresponding to n =4. The cubic point is destabilized by any operator of symmetry lower than cubic. The dicylindrical one remains stable with respect to certain anisotropies of lower symmetry. We examine the available experimental data in light of the preceding theoretical results concerning the critical behavior and the thermodynamic order of the transitions. On the other hand, we establish two general symmetry conditions relative to the stable fixed points determined by the renormalization-group equations. The first one specifies group theoretically, for each value of n, the possible symmetries G; of the stable fixed points. The second one formulates a necessary condition for the occurrence of an anisotropic stable fixed point: The normalizer G~of the considered parameter space must fulfill the condition G~L:G;* for one, at least, of the former G;* groups. These rules are shown to be very restrictive for n =4: On the basis of symmetry the lack of a stable fixed point can be asserted for 10 Hamiltonians out of 22, without solving the fixed-point equations.
Critical exponent η in 2D O(N )-symmetric ϕ 4 -model up to 6 loops
Critical exponent η (Fisher exponent) in O(N)-symmetric ϕ 4-model was calculated using renor-malization group approach in the space of fixed dimension D = 2 up to 6 loops. The calculation of the renormalization constants was performed with the use of R-operation and specific values for diagrams were calculated in Feynman representation using sector decomposition method. Presented approach allows easy automation and generalization for the case of complex symmetries. Also a summation of the perturbation series was obtained by Borel transformation with conformal mapping. The contribution of the 6-th term of the series led to the increase of the Fisher exponent in O(1) model up to 8%.
Critical Exponents for Long-Range Interactions
Physical Review Letters, 1972
Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's 'effective dimension'. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component lsing (and more generally Griffiths-Simon class) spin variables. In particular, it is established that even in dimensions d < 4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as Jx = 1/Ixl a+~ with 0 < a~< d/2, then the exponents ~, 6, ? and A 4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with try> 2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.
Critical Exponents of the Superfluid–Bose-Glass Transition in Three Dimensions
Physical Review Letters, 2014
Recent experimental and numerical studies of the critical-temperature exponent φ for the superfluid-Bose glass universality in three-dimensional systems report strong violations of the key quantum critical relation, φ = νz, where z and ν are the dynamic and correlation length exponents, respectively, and question the fundamental concepts underlying quantum critical phenomena. Using Monte Carlo simulations of the disordered Bose-Hubbard model, we demonstrate that previous work on the superfluid-to-normal fluid transition-temperature dependence on chemical potential (or magnetic field, in spin systems), Tc ∝ (µ − µc) φ , was misinterpreting transient behavior on approach to the fluctuation region with the genuine critical law. When the model parameters are modified to have a broad quantum critical region, simulations of both quantum and classical models reveal that the φ = νz law [with φ = 2.7(2), z = 3, and ν = 0.88(5)] holds true, resolving the φ-exponent "crisis". PACS numbers: 67.85.Hj, 67.85.-d,64.70.Tg