Scaling behavior of crystalline membranes: An ε-expansion approach

Mauri, Achille; Katsnelson, M. I.

doi:10.1016/j.nuclphysb.2020.115040

Cited by 22 publications

(39 citation statements)

References 32 publications

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“…In that respect, an essential feature of the novel fixed point P c found in [41] is that its coordinates near D uc = 4 differ only from those of the vanishing-temperature fixed point P 5 by terms of order 2 , with = 4 − D, strongly suggesting that P c could also be identified within a perturbative expansion up to this order. This is the reason why, in this Letter, we investigate quenched disordered membranes at two loops in the vicinity of the upper critical dimension, extending both the one-loop computation of Morse et al performed 30 years ago [24,25] at the next order and the recent two-loop computation of Coquand et al [44] (see also [45]) on disorder-free membranes to the disordered case. We derive the RG equations, analyze them, and provide the critical quantities, notably the anomalous dimension η, at order 2 .…”

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confidence: 82%

Wrinkling transition in quenched disordered membranes at two loops

Coquand

Mouhanna

2021

Phys. Rev. E

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We investigate the flat phase of quenched disordered polymerized membranes by means of a two-loop, weak-coupling computation performed near their upper critical dimension D uc = 4, generalizing the one-loop computation of Morse et al. [D. C. Morse et al., Phys. Rev. A 45, R2151 (1992); D. C. Morse and T. C. Lubensky, Phys. Rev. A 46, 1751 (1992)]. Our work confirms the existence of the finite-temperature, finite-disorder wrinkling transition, which has been recently identified by Coquand et al. [O. Coquand et al., Phys. Rev. E 97, 030102(R) ( 2018)] using a nonperturbative renormalization group approach. We also point out ambiguities in the two-loop computation that prevent the exact identification of the properties of the novel fixed point associated with the wrinkling transition, which very likely requires a three-loop order approach.

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confidence: 82%

Wrinkling transition in quenched disordered membranes at two loops

Coquand

Mouhanna

2021

Phys. Rev. E

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“…(6.7) 10 The next order correction should also contain a pole corresponding to the second radial excitation. 11 For instance consider a very recent application to elastic membranes [44]. Although similar notation is used for similar quantities elsewhere in this paper, throughout this section the functions g, v, Π, J should be understood as being functions within the SCSA, which is expressed diagrammatically in Fig 11:…”

Section: λ As a Hubbard-stratonovich Fieldmentioning

confidence: 99%

Lessons from $O(N)$ models in one dimension

Schubring

2021

Preprint

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Various topics related to the O(N ) model in one spacetime dimension (i.e. ordinary quantum mechanics) are considered. The focus is on a pedagogical presentation of quantum field theory methods in a simpler context where many exact results are available, but certain subtleties are discussed which may be of interest to active researchers in higher dimensional field theories as well.Large N methods are introduced in the context of the zero-dimensional path integral and the connection to Stirling's series is shown. The entire spectrum of the O(N ) model, which includes the familiar l(l + 1) eigenvalues of the quantum rotor as a special case, is found both diagrammatically through large N methods and by using Ward identities. The large N methods are already exact at O N −1 and the O N −2 corrections are explicitly shown to vanish. Peculiarities of gauge theories in d = 1 are discussed in the context of the CP N −1 sigma model, and the spectrum of a more general squashed sphere sigma model is found. The precise connection between the O(N ) model and the linear sigma model with a φ 4 interaction is discussed. A valid form of the self-consistent screening approximation (SCSA) applicable to O(N ) models with a hard constraint is presented. The point is made that at least in d = 1 the SCSA may do worse than simply truncating the large N expansion to O N −1 even for small N . In both the supersymmetric and nonsupersymmetric versions of the O(N ) model, naive equations of motion relating vacuum expectation values are shown to be corrected by regularization-dependent finite corrections arising from contact terms associated to the equation of constraint.

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“…graphs in a larged c expansion, leads to quantitatively small corrections to universal quantities for D = 2, d c = 1 [51], which supports the accuracy of the method. Recently, SCSA predictions have been compared with exact analytical calculations of η in second-order large-d c [76] and ε-expansions [17,18]. In Ref.…”

Section: Self-consistent Screening Approximationmentioning

confidence: 99%

“…In Ref. [17], it was shown that the SCSA equations are exact at O(ε 2 ) within a nonstandard dimensional continuation of the theory to arbitrary D. A more general two-loop theory was developed in Ref. [18], where a larger space of theories was considered.…”

Section: Self-consistent Screening Approximationmentioning

confidence: 99%

“…As a result of strong nonlinear coupling between bending and shear deformations, thermal fluctuations in the flat phase present anomalous scale invariance characterized by universal noninteger exponents. In the long-wavelength limit, the scale-dependent effective compression and shear moduli are driven to zero as power laws of the wave vector q, while the effective bending rigidity diverges as κ (q) ≈ q −η [6][7][8]10,11,17,18]. This anomalous infrared behavior sets in at a characteristic Ginzburg scale q * ≈ 3TY/(16πκ 2 ), where κ, Y , and T are, respectively, the bare bending rigidity, Young modulus, and temperature [3,19].…”

Section: Introductionmentioning

confidence: 99%

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Thermal ripples in bilayer graphene

Mauri¹,

Soriano²,

Katsnelson³

2020

Phys. Rev. B

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We study thermal fluctuations of freestanding bilayer graphene subject to vanishing external tension. Within a phenomenological theory, the system is described as a stack of two continuum crystalline membranes, characterized by finite elastic moduli and a nonzero bending rigidity. A nonlinear rotationally invariant model guided by elasticity theory is developed to describe interlayer interactions. After neglection of in-plane phonon nonlinearities and anharmonic interactions involving interlayer shear and compression modes, an effective theory for soft flexural fluctuations of the bilayer is constructed. The resulting model, neglecting anisotropic interactions, has the same form of a well-known effective theory for out-of-plane fluctuations in a single-layer membrane, but with a strongly wave-vector-dependent bare bending rigidity. Focusing on AB-stacked bilayer graphene, parameters governing interlayer interactions in the theory are derived by first-principles calculations. Statistical-mechanical properties of interacting flexural fluctuations are then calculated by a numerical iterative solution of field-theory integral equations within the self-consistent screening approximation. The bare bending rigidity in the considered model exhibits a crossover between a long-wavelength regime governed by in-plane elastic stress and a short wavelength region controlled by monolayer curvature stiffness. Interactions between flexural fluctuations drive a further crossover between a harmonic and a strong-coupling regime, characterized by anomalous scale invariance. The overlap and interplay between these two crossover behaviors is analyzed at varying temperatures.

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Scaling behavior of crystalline membranes: An ε-expansion approach

Cited by 22 publications

References 32 publications

Wrinkling transition in quenched disordered membranes at two loops

Wrinkling transition in quenched disordered membranes at two loops

Lessons from $O(N)$ models in one dimension

Thermal ripples in bilayer graphene

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