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arXiv:1012.3739 · Condensed Matter Physics

Analysis of the 3d massive renormalization group perturbative expansions: a delicate case

Delamotte, M. Dudka, Yurij V. Holovatch, Mouhanna

Published Jan 1, 2010• 21 citations
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Abstract

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.

DOI
10.5488/cmp.13.43703

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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.
We show that the critical behavior of two- and three-dimensional frustrated magnets cannot reliably be described from the known five- and six-loop perturbative renormalization-group results. Our conclusions are based on a careful reanalysis of the resummed perturbative series obtained within the zero-momentum massive scheme. In three dimensions, the critical exponents for XY and Heisenberg spins display strong dependences on the parameters of the resummation procedure and on the loop order. This behavior strongly suggests that the fixed points found are in fact spurious. In two dimensions, we find, as in the O(N)O(N)O(N) case, that there is apparent convergence of the critical exponents but toward erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by Z2{Z}_{2}Z2​ topological defects in two-dimensional frustrated Heisenberg spins remains open.
The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Pad\'e-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for N> 3,N>~3,N> 3, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when N=0N=0N=0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.
The critical behavior of a model describing phase transitions in cubic and tetragonal anti-ferromagnets with 2N-component (N>1) real order parameters as well as the structural transition in NbO 2 crystal is studied within the field-theoretical renormalization-group (RG) approach in three and (4-∊)-dimensions. Perturbative expansions for RG functions are calculated up to three-loop order and resummed, in 3D, by means of the generalized Padé–Borel procedure which is shown to preserve the specific symmetry properties of the model. It is found that a stable fixed point does exist in the three-dimensional RG flow diagram for N>1, in accordance with predictions obtained earlier within the ∊-expansion. Fixed-point coordinates and critical-exponent values are presented for physically interesting cases N=2 and N=3. In both cases critical exponents are found to be numerically close to those of the 3DXY model. The analysis of the results given by the ∊-expansion and by the RG approach in three dimensions is performed resulting in a conclusion that the latter provides much more accurate numerical estimates.
We estimate, using a large-scale Monte Carlo simulation, the critical exponents of the antiferromagnetic Heisenberg model on a stacked triangular lattice. We obtain the following estimates: $γ/ν= 2.011 \pm .014 $, $ν= .585 \pm .009 $. These results contradict a perturbative 2+ε2+ε2+ε Renormalization Group calculation that points to Wilson-Fisher O(4) behaviour. While these results may be coherent with 4−ε4-ε4−ε results from Landau-Ginzburg analysis, they show the existence of an unexpectedly rich structure of the Renormalization Group flow as a function of the dimensionality and the number of components of the order parameter.
The critical properties of the antiferromagnetic Heisenberg model on the three-dimensional stacked-triangular lattice are studied by means of a large-scale Monte Carlo simulation in order to get insight into the controversial issue of the criticality of the noncollinear magnets with the $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$ symmetry. The maximum size studied is 3843{384}^{3}3843, considerably larger than the sizes studied by the previous numerical works on the model. Availability of such large-size data enables us to examine the detailed critical properties including the effect of corrections to the leading scaling. Strong numerical evidence of the continuous nature of the transition is obtained. Our data indicate the existence of significant corrections to the leading scaling. Careful analysis by taking account of the possible corrections yields critical exponents estimates, $\ensuremath{\alpha}=0.44(3), \ensuremath{\beta}=0.26(2), \ensuremath{\gamma}=1.03(5), \ensuremath{\nu}=0.52(1), \ensuremath{\eta}=0.02(5)$, and the chirality exponents, ${\ensuremath{\beta}}_{\ensuremath{\kappa}}=0.40(3)$ and ${\ensuremath{\gamma}}_{\ensuremath{\kappa}}=0.77(6)$, supporting the existence of the O(3)O(3)O(3) chiral [or $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$] universality class governed by a ``chiral'' fixed point. We also obtain an indication that the underlying fixed point is of the focus type, characterized by the complex-valued correction-to-scaling exponent, $\ensuremath{\omega}=0.{1}_{\ensuremath{-}0.05}^{+0.4}+i\phantom{\rule{4pt}{0ex}}0.{7}_{\ensuremath{-}0.4}^{+0.1}$. The focus-like nature of the chiral fixed point accompanied by the spiral-like renormalization-group (RG) flow is likely to be the origin of the apparently complicated critical behavior. The results are compared and discussed in conjunction with the results of other numerical simulations, several distinct types of RG calculations including the higher-order perturbative massive and massless RG calculations and the nonperturbative functional RG calculation, and the conformal-bootstrap program.
This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems and argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach---the effective average action method---that we have used to investigate the physics of frustrated magnets and show how it enables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. First, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudofixed point and that relies on a slowing down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allow us to explain the occurrence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.