Bulk and boundary critical behavior at Lifshitz points

Diehl, H. W.

doi:10.1007/bf02704584

Cited by 10 publications

(17 citation statements)

References 79 publications

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“…Since z now is an α-direction, the classical equation of motion is of fourth rather than of second order in ∂ z . In conformity with this we find two (instead of one) mesoscopic BC at either boundary plane, namely [25,26]…”

Section: Perpendicular Slab Orientationsupporting

confidence: 88%

“…In the case of FBC, different boundary densities L j (x) and L ⊥ j (x) (dictated by relevance/irrelevance considerations) must be chosen to define appropriate minimal models for slabs with parallel or perpendicular orientation. Work on semi-infinite systems § [22,23,24,25,26] suggests the choices…”

Section: Modelsmentioning

confidence: 99%

“…However, when considering FBC, we shall restrict ourselves in two ways: We assume that the BC that result in the large length-scale limit (i) do not break the O(n) symmetry and (ii) are associated with the respective most stable renormalization-group (RG) fixed point. For parallel orientation this simply means that Dirichlet BC φ = 0 hold asymptotically [22,23,24,25]. In the case of perpendicular orientation, two BC hold on either ‡ Note that since ξα and ξ β are bulk correlation lengths, they diverge at the bulk critical point.…”

Section: Introductionmentioning

confidence: 99%

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Fluctuation-induced forces in strongly anisotropic critical systems

2010

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Strongly anisotropic critical systems are considered in a ddimensional film geometry.Such systems involve two (or more) distinct correlation lengths ξ β and ξα that scale as nontrivial powers of each other, i.e. ξα ∼ ξ θ β with anisotropy index θ = 1. Thus two fundamental orientations, perpendicular (⊥) and parallel ( ), for which the surface normal is oriented along an α-and β-direction, respectively, must be distinguished. The confinement of critical fluctuations caused by the film's boundary planes is shown to induce effective forces F C that decay as F C ∝ −(∂/∂L)∆ ⊥, L −ζ ⊥, as the film thickness L becomes large, where the proportionality constants involve nonuniversal amplitudes. The decay exponents ζ ⊥, and the Casimir amplitudes ∆ ⊥, are universal but depend on the type of orientation. To corroborate these findings, n-vector models with an m-axial bulk Lifshitz point are investigated by means of RG methods below the upper critical dimension d * (m) = 4 + m/2 under periodic boundary conditions (PBC) and free BC of an asymptotic form pertaining to the respective ordinary surface transitions. The exponents ζ ⊥, are determined, and explicit results to one-or two-loop order are presented for several Casimir amplitudes ∆ BC ⊥, . The large-n limits of the Casimir amplitudes ∆ BC /n for periodic and Dirichlet BC are shown to be proportional to their critical-point analogues at dimension d−m/2. The limiting values ∆ PBC ,⊥,∞ = limn→∞ ∆ PBC ,⊥ /n are determined exactly for the uniaxial cases (d, m) = (3, 1) under periodic BC. Unlike ∆ PBC ,∞ , the amplitude ∆ PBC ⊥,∞ is positive, so that the corresponding Casimir force is repulsive.

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Section: Perpendicular Slab Orientationsupporting

confidence: 88%

Section: Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fluctuation-induced forces in strongly anisotropic critical systems

2010

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“…However, the difference θ − 1/2 appears to be small. Field-theory estimates based on the ǫ expansion gave θ(3, 1, 1) ≃ 0.47 [39,46], and Monte Carlo simulation data seem to be consistent with values 0.48 θ 1/2 [42,37].…”

Section: O(n) Model For Critical Behavior At M-axial Lifshitz Pointsmentioning

confidence: 75%

On conjectured local generalizations of anisotropic scale invariance and their implications

2011

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“…The original BDS result was derived using the technique of Mellin-Barnes contour integral representations 10 . The result (6)- (7) or its immediate generalizations, expressed in terms of generalized hypergeometric functions of two variables, have been reproduced several times by different authors using different means 11,12,13,14 . All of them either contained symmetrizations like (2) or comprised some hidden symmetries in apparently non-symmetric expressions.…”

Section: The Results Of Berends Davydychev and Smirnovmentioning

confidence: 91%

A massive Feynman integral and some reduction relations for Appell functions

Shpot

2007

Journal of Mathematical Physics

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We consider finite iterated generalized harmonic sums weighted by the binomial 2k k in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic , generalized harmonic, and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for N → ∞ and the iterated integrals at x = 1 lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit N → ∞ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to N ∈ C. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as, e.g., for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into bi-nomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums. C 2014 AIP Publishing LLC. [http://dx.

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Bulk and boundary critical behavior at Lifshitz points

Cited by 10 publications

References 79 publications

Fluctuation-induced forces in strongly anisotropic critical systems

Fluctuation-induced forces in strongly anisotropic critical systems

On conjectured local generalizations of anisotropic scale invariance and their implications

A massive Feynman integral and some reduction relations for Appell functions

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