We study the scaling behavior of two-dimensional (2D) crystalline membranes in the flat phase by a renormalization group (RG) method and an $ε$-expansion. Generalization of the problem to non-integer dimensions, necessary to control the $ε$-expansion, is achieved by dimensional continuation of a well-known effective theory describing out-of-plane fluctuations coupled to phonon-mediated interactions via a scalar composite field, equivalent for small deformations to the local Gaussian curvature. The effective theory, which will be referred to as Gaussian curvature interaction (GCI) model, is equivalent to theories of elastic $D$-dimensional manifolds fluctuating in a $(D + d_{c})$-dimensional embedding space in the physical case $D = 2$ for arbitrary $d_{c}$. For $D\neq 2$, instead, the GCI model is not equivalent to a direct dimensional continuation of elastic membrane theory and it defines an alternative generalization to generic internal dimension $D$. We calculate explicitly RG functions at two-loop order and determine the exponent $η$ characterizing the long-wavelength scaling of correlation functions to order $ε^{2}$ in an $ε=(4-D)$-expansion. The self-consistent screening approximation (SCSA) for the GCI model is shown to be exact to O($ε^{2}$). For $d_{c} = 1$, the O($ε^{2}$) correction is suppressed by a small numerical prefactor. As a result, despite the large value of $ε= 2$, extrapolation of the first and second order results to $D = 2$ leads to very close numbers, $η= 0.8$ and $η\simeq 0.795$. The calculated exponent values are close to earlier reference results obtained by non-perturbative RG, the SCSA and numerical simulations. These indications suggest that a perturbative analysis of the GCI model could provide an useful framework for accurate quantitative predictions of the scaling exponent even at $D = 2$.
Reply to 'Comment on "Thermodynamics of quantum crystalline membranes"'
在这篇文章中，我们回复了E.I.Kats和V.V.Lebedev [arXiv：1407.4298]对我们最近的工作“量子结晶膜的热力学”[Phys. Rev. B 89，224307（2014）]的评论。Kats和Lebedev质疑我们工作中提出的计算的有效性，特别是使用德拜动量作为理论的紫外调节器。我们处理和反驳Kats和Lebedev对我们工作的批评。
Critical Ripples and Dirac Fermions in Crystalline Membranes
Universiteit Leiden（莱顿大学）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Universidad Técnica Federico Santa María（费德里科·Santa María 技术大学）
