71 research outputs found
Scale invariance implies conformal invariance for the three-dimensional Ising model
Using Wilson renormalization group, we show that if no integrated vector
operator of scaling dimension exists, then scale invariance implies
conformal invariance. By using the Lebowitz inequalities, we prove that this
necessary condition is fulfilled in all dimensions for the Ising universality
class. This shows, in particular, that scale invariance implies conformal
invariance for the three-dimensional Ising model.Comment: Phys. Rev. E 93, 012144 (2016
Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence
In this paper, we present theoretical results on the statistical properties
of stationary, homogeneous and isotropic turbulence in incompressible flows in
three dimensions. Within the framework of the Non-Perturbative Renormalization
Group, we derive a closed renormalization flow equation for a generic -point
correlation (and response) function for large wave-numbers with respect to the
inverse integral scale. The closure is obtained from a controlled expansion and
relies on extended symmetries of the Navier-Stokes field theory. It yields the
exact leading behavior of the flow equation at large wave-numbers ,
and for arbitrary time differences in the stationary state. Furthermore,
we obtain the form of the general solution of the corresponding fixed point
equation, which yields the analytical form of the leading wave-number and time
dependence of -point correlation functions, for large wave-numbers and both
for small and in the limit . At small , the leading
contribution at large wave-number is logarithmically equivalent to , where is a nonuniversal
constant, the integral scale and the mean energy injection
rate. For the 2-point function, the dependence is known to originate
from the sweeping effect. The derived formula embodies the generalization of
the effect of sweeping to point correlation functions. At large wave-number
and large , we show that the dependence in the leading order
contribution crosses over to a dependence. The expression of the
correlation functions in this regime was not derived before, even for the
2-point function. Both predictions can be tested in direct numerical
simulations and in experiments.Comment: 23 pages, minor typos correcte
Spatiotemporal velocity-velocity correlation function in fully developed turbulence
Turbulence is an ubiquitous phenomenon in natural and industrial flows. Since
the celebrated work of Kolmogorov in 1941, understanding the statistical
properties of fully developed turbulence has remained a major quest. In
particular, deriving the properties of turbulent flows from a mesoscopic
description, that is from Navier-Stokes equation, has eluded most theoretical
attempts. Here, we provide a theoretical prediction for the {\it space and
time} dependent velocity-velocity correlation function of homogeneous and
isotropic turbulence from the field theory associated to Navier-Stokes equation
with stochastic forcing. This prediction is the analytical fixed-point solution
of Non-Perturbative Renormalisation Group flow equations, which are exact in a
certain large wave-number limit. This solution is compared to two-point
two-times correlation functions computed in direct numerical simulations. We
obtain a remarkable agreement both in the inertial and in the dissipative
ranges.Comment: 8 pages, 4 figures, improved versio
Non-perturbative renormalisation group for the Kardar-Parisi-Zhang equation: general framework and first applications
We present an analytical method, rooted in the non-perturbative
renormalization group, that allows one to calculate the critical exponents and
the correlation and response functions of the Kardar-Parisi-Zhang (KPZ) growth
equation in all its different regimes, including the strong-coupling one. We
analyze the symmetries of the KPZ problem and derive an approximation scheme
that satisfies the linearly realized ones. We implement this scheme at the
minimal order in the response field, and show that it yields a complete,
qualitatively correct phase diagram in all dimensions, with reasonable values
for the critical exponents in physical dimensions. We also compute in one
dimension the full (momentum and frequency dependent) correlation function, and
the associated universal scaling functions. We find an excellent quantitative
agreement with the exact results from Praehofer and Spohn (J. Stat. Phys. 115
(2004)). We emphasize that all these results, which can be systematically
improved, are obtained with sole input the bare action and its symmetries,
without further assumptions on the existence of scaling or on the form of the
scaling function.Comment: 21 pages, 6 figures, revised version, including the correction of an
inconsistency and accordingly updated figures 5 and 6 and table 2, as
published in an Erratum (see Ref. below). The results are improve
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