控制物理结晶膜的平坦相的长波长行为的反常指数在应用于1/d_C（d_C是余维）中的二级展开的自洽屏蔽近似（SCSA）内计算，扩展了Le Doussal和Radzihovsky的开创性工作[Phys. Rev. Lett. 69，1209（1992）]。特别地，发现弯曲刚度在长波长极限下代数硬化，指数η =0.789..用于提取弹性软化指数eta_u=0.422.并且粗糙度指数zeta=0.605.证明了标度关系eta_u=2- 2 eta在SCSA中适用于所有阶。此外，将SCSA应用到1/dC展开式中是必要的，因为在两个气泡水平下没有找到自洽方程的解，这是朴素的二阶展开式。令人惊讶的是，尽管物理膜的膨胀参数为1/d_C=1，但应用于二级膨胀的SCSA仅与一级膨胀略有偏离，仅使zeta增加0.016。这支持了物理结晶膜的SCSA的高质量，并改善了与这些系统的实验和数值模拟的比较。SCSA的预测适用于泊松比的一阶扩展被证明是准确的所有订单。
英文摘要：
The anomalous exponents governing the long wavelength behavior of the flat phase of physical crystalline membranes are calculated within a self-consistent screening approximation (SCSA) applied to second order expansion in 1/d_C (d_C is the co-dimension), extending the seminal work of Le Doussal and Radzihovsky [Phys. Rev. Lett. 69, 1209 (1992)]. In particular, the bending rigidity is found to harden algebraically in the long wavelength limit with an exponent eta=0.789..., which is used to extract the elasticity softening exponent eta_u=0.422..., and the roughness exponent zeta=0.605.... The scaling relation eta_u=2-2eta is proven to hold to all orders in SCSA. Further, applying the SCSA to an expansion in 1/d_C, is found to be essential, as no solution to the self-consistent equations is found in a two bubble level, which is the naive second order expansion. Surprisingly, even though the expansion parameter for physical membrane is 1/d_C=1, the SCSA applied to second order expansion deviates only slightly from the first order, increasing zeta by mere 0.016. This supports the high quality of the SCSA for physical crystalline membranes, as well as improves the comparison to experiments and numerical simulations of these systems. The prediction of SCSA applied to first order expansion for the Poisson ratio is shown to be exact to all orders.
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）
