Surface critical behaviour at<b><i>m</i></b>-axial Lifshitz points: continuum models, boundary conditions and two-loop renormalization group results

Diehl, H. W.; Rutkevich, S. B.; Gerwinski, A.

doi:10.1088/0305-4470/36/16/101

Cited by 8 publications

(8 citation statements)

References 19 publications

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“…In this paper we have extended previous field-theoretic work on boundary critical behaviour at m-axial LPs [15,16,17,22] by studying the critical behaviour at the ordinary transition of a semi-infinite system that is bounded by a surface perpendicular to an α-direction. This geometry was the one considered in the earliest investigations [20,21] of boundary critical behaviour at LPs.…”

Section: Discussionmentioning

confidence: 94%

“…As a consequence of the different scaling behaviour of distances along α-and β-directions, it depends on whether the orientation of the surface is parallel or perpendicular which boundary contributions L 1 are potentially infrared relevant below the upper critical dimension d * (m) = 4 + m/2 and hence must be included in the action. The problem of constructing semi-infinite extensions of the models with bulk density (4) for d = d * (m) − ǫ that are "minimal" in the sense that all irrelevant and marginal boundary contributions not compatible with the presumed O(n) and Euclidean ‡ symmetries are discarded was considered for the case of parallel surface orientation in references [15,16], where it was found that a contribution ∝ α (∂ α φ) 2 had to be included in the corresponding surface density L 1 ≡ L 1 , in addition to the one in equation ( 3). The so-obtained semi-infinite model, defined by equation ( 4) in conjunction with the boundary density…”

Section: Introductionmentioning

confidence: 99%

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Boundary critical behaviour atm-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Diehl¹,

Shpot²,

Прудников³

2006

J. Phys. A: Math. Gen.

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Semi-infinite d-dimensional systems with an m-axial bulk Lifshitz point are considered whose (d − 1)-dimensional surface hyper-plane is oriented perpendicular to one of the m modulation axes. An n-component φ 4 field theory describing the bulk and boundary critical behaviour when (i) the Hamiltonian can be taken to have O(n) symmetry and (ii) spatial anisotropies breaking its Euclidean symmetry in the m-dimensional coordinate subspace of potential modulation directions may be ignored is investigated.The long-distance behaviour at the ordinary surface transition is mapped onto a field theory with the boundary conditions that both the order parameter φ and its normal derivative ∂nφ vanish at the surface plane. The boundary-operator expansion is utilized to study the short-distance behaviour of φ near the surface. Its leading contribution is found to be controlled by the boundary operator ∂ 2 n φ. The field theory is renormalized for dimensions d below the upper critical dimension d * (m) = 4 + m/2, with a corresponding surface source term ∝ ∂ 2 n φ added. The anomalous dimension of this boundary operator is computed to first order in ǫ = d * − d. The result is used in conjunction with scaling laws to estimate the value of the single independent surface critical exponent β (ord,⊥) L1 for d = 3. Our estimate for the case m = n = 1 of a uniaxial Lifshitz point in Ising systems is in reasonable agreement with published Monte Carlo results.

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Section: Discussionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Boundary critical behaviour atm-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Diehl¹,

Shpot²,

Прудников³

2006

J. Phys. A: Math. Gen.

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“…The discovery of the existence of this kind of multicritical points in 1975 led to numerous theoretical studies of their bulk critical properties. With the exception of an early attempt by Gumbs [114], surface critical phenomena at a bulk Lifshitz point have only been studied very recently [115,116,117,118,119,120]. Most of the results have been obtained for the semi-infinite ANNNI model.…”

Section: Surface Critical Behaviour Near a Lifshitz Pointmentioning

confidence: 99%

“…Very recently Diehl and coworkers [119,120] analysed the surface critical behaviour at bulk Lifshitz points using renormalization group methods. They thereby considered general m-axial Lifshitz points where the wave vector instability takes place in an mdimensional subspace of the d-dimensional space.…”

Section: Surface Critical Behaviour Near a Lifshitz Pointmentioning

confidence: 99%

Critical phenomena at perfect and non-perfect surfaces

Pleimling¹

2004

J. Phys. A: Math. Gen.

100

135

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Abstract. In the past perfect surfaces have been shown to yield a local critical behaviour that differs from the bulk critical behaviour. On the other hand surface defects, whether they are of natural origin or created artificially, are known to modify local quantities. It is therefore important to clarify whether these defects are relevant or irrelevant for the surface critical behaviour.The purpose of this review is two-fold. In the first part we summarise some of the important results on surface criticality at perfect surfaces. Special attention is thereby paid to new developments as for example the study of surface critical behaviour in systems with competing interactions or of surface critical dynamics. In the second part the effect of surface defects (presence of edges, steps, quenched randomness, lines of adatoms, regular geometric patterns) on local critical behaviour in semi-infinite systems and in thin films is discussed in detail. Whereas most of the defects commonly encountered are shown to be irrelevant, some notable exceptions are highlighted. It is shown furthermore that under certain circumstances non-universal local critical behaviour may be observed at surfaces.

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“…(2,0) R is ill-defined, and since the renormalization parts of Γ (2,0) appear as di-vergent subgraphs of other vertex functions, his "renormalized" theory quite generally has this deficiency. The fact that the counterterms he employs are insufficient to subtract all primitive q-dependent divergences of Γ (2,0) (q) implies that uv singular pieces of nonlocal form produced by higher-order graphs containing the subgraph 7,8 , such as , will not get canceled by the subtractions provided by the counterterms to two-loop order.…”

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confidence: 99%

Comment on “Renormalization-group picture of the Lifshitz critical behavior”

Diehl

Shpot

2003

Phys. Rev. B

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We show that the recent renormalization-group analysis of Lifshitz critical behavior presented by Leite ͓Phys. Rev. B 67, 104415 ͑2003͔͒ suffers from a number of severe deficiencies. In particular, we show that his approach does not give an ultraviolet finite renormalized theory, is plagued by inconsistencies, misses the existence of a nontrivial anisotropy exponent 1/2, and therefore yields incorrect hyperscaling relations. His ⑀-expansion results to order ⑀ 2 for the critical exponents of m-axial Lifshitz points are incorrect in both the anisotropic (0ϽmϽd) and isotropic (mϭd) cases. The inherent inconsistencies and the lack of a sound basis of the approach makes its results unacceptable even if they are interpreted in the sense of approximations.

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Surface critical behaviour atm-axial Lifshitz points: continuum models, boundary conditions and two-loop renormalization group results

Cited by 8 publications

References 19 publications

Boundary critical behaviour atm-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Boundary critical behaviour atm-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Critical phenomena at perfect and non-perfect surfaces

Comment on “Renormalization-group picture of the Lifshitz critical behavior”

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