Weak Quenched Disorder and Criticality: Resummation of Asymptotic(?) Series

Holovatch, Yu.; Blavatska, V.; Dudka, M.; Ferber, Christian von; Folk, R.; Yavors’kii, T.

doi:10.1142/s0217979202014760

Cited by 33 publications

(10 citation statements)

References 107 publications

(228 reference statements)

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“…This means that one probably should not consider the previous results as very reliable. In this respect, we recall the results obtained by means of the sixloop calculation in d = 3 et al [132]: N c (d = 3) = 6.4 (4) and by means of the 4 − ǫ expansion at five-loops [159]:…”

Section: Tests Of the Methods And First Resultsmentioning

confidence: 97%

“…For most systems a quantitative and, even sometimes, a qualitative description of the critical physics is either still lacking or very difficult to obtain by perturbative RG methods. This is the case, for instance, in the Potts model [2,3], in magnetic systems with disorder [4], in superconductors [5,6], in Josephson junction arrays [7], in He 3 [8,9], in smectic liquid crystals [10], in electroweak phase transitions [11,12] and in frustrated magnets like helimagnets or geometrically frustrated magnets (triangular for instance) which are our main purpose in this article (see [13] for a review).…”

Section: Introductionmentioning

confidence: 99%

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Nonperturbative renormalization-group approach to frustrated magnets

2004

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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. We 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. First, we recall the treatment of the unfrustrated -O(N ) -case with this method. This allows to introduce its technical aspects. Then, we show how this method unables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. Firstly, 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 qualitative and (semi-) quantitative agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudo-fixed point and that relies on a slowing-down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allows to explain the occurence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.

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Section: Tests Of the Methods And First Resultsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Nonperturbative renormalization-group approach to frustrated magnets

2004

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“…Several methods can be used, Padé approximants constitute one possibility. [50][51][52] However, it is generally believed that the use of a conformal mapping 53,54 is more efficient since it makes use of the convergence properties of the Borel sum. Under the assumption that all the singularities of B͑u͒ lie on the negative real axis and that the Borel-Leroy sum is analytic in the whole complex plane except for the cut extending from −1 / a to −ϱ, one can perform the change in variable,…”

Section: A Resummation Proceduresmentioning

confidence: 99%

Relevance of the fixed dimension perturbative approach to frustrated magnets in two and three dimensions

et al. 2010

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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 O͑N͒ 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 Z 2 topological defects in two-dimensional frustrated Heisenberg spins remains open.

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“…Another way to obtain the BSs, is to calculate them from the β-functions (2.3)-(2.5) fixing d at certain values. Since the RG expansions have divergent [2] nature, the special resummation techniques are needed to get convergent results [29]. The two-loop β-functions (2.3)-(2.5) β = β u i have a form of polynomials in renormalized couplings:…”

Section: Bss From Resummed β-Functionsmentioning

confidence: 99%

Marginal dimensions for multicritical phase transitions

Dudka¹,

Folk²,

Holovatch³

et al. 2012

Condens. Matter Phys.

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The field-theoretical model describing multicritical phenomena with two coupled order parameters with n || and n ⊥ components and of O(n )⊕O(n ⊥ ) symmetry is considered. Conditions for realization of different types of multicritical behaviour are studied within the field-theoretical renormalization group approach. Surfaces separating stability regions for certain types of multicritical behaviour in parametric space of order parameter dimensions and space dimension d are calculated using the two-loop renormalization group functions. Series for the order parameter marginal dimensions that control the crossover between different universality classes are extracted up to the fourth order in ε = 4 − d and to the fifth order in a pseudo-ε parameter using the known high-order perturbative expansions for isotropic and cubic models. Special attention is paid to a particular case of O(1)⊕O(2) symmetric model relevant for description of anisotropic antiferromagnets in an external magnetic field.

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Weak Quenched Disorder and Criticality: Resummation of Asymptotic(?) Series

Cited by 33 publications

References 107 publications

Nonperturbative renormalization-group approach to frustrated magnets

Nonperturbative renormalization-group approach to frustrated magnets

Relevance of the fixed dimension perturbative approach to frustrated magnets in two and three dimensions

Marginal dimensions for multicritical phase transitions

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