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Basic Definitions and Some Fundamental Properties
In this section, we discuss the mathematical description of membranes using differential geometry. Differential geometry is the branch of mathematics that studies geometrical objects in an analytical way, using differential and integral calculus (see lectures [96], [97], [98, Ch. 7]).
The development of differential geometry started in the eighteenth and nineteenth centuries with the study of curves ans surfaces mainly by Gauss, Riemann, Lobachevsky and Bolyai. The generalisation of the concept of curves and surfaces to higher dimensions is what we call manifolds.
The name manifold comes from the German Mannigfaltigkeit which is the name given by Bernhard Riemann and later translated to manifold by William Clifford.
The concept of manifold is central in physics. Polymer chains on one hand and biological membranes and graphene-based sheets on another, which are of great importance in biophysics, chemistry, condensed matter physics and material science, are respectively one-dimensional and two-dimensional manifolds. Even the space-time continuum in general relativity is a Lorentzian2 four-dimensional manifold. Higher dimensional manifolds can also be found in string and superstring theory. In statistical physics where the phase space is a symplectic manifold3. Mathematically, a D-dimensional manifold is an object that is locally flat (but not globally). Locally, aD-dimensional manifold is homeomorphic to the Euclidean space RD. Therefore, the manifold can be covered by patches parametrised in a local coordinate system fxg with = 1; : : : ;D. The choice of the coordinate system is arbitrary and depends on the needs. If two neighbouring points on the manifold are parametrised by two different coordinate systems fxg and fyg, then there is a continuous bijection between the two systems in the region where they overlap.
In this section we are only interested in smooth differentiable manifoldsMwith an inner product g on each tangent space TPM. These manifolds are called Riemannian manifolds. Manifolds do not need an embedding space. For instance, the space-time continuum is not a priori considered as embedded in a space of higher dimensions. But fluid and polymerized membranes, that interest us, do live in R3. From now on, we will only focus on manifolds embedded in a d-dimensional Euclidean space Rd, where d > D.
The normal-normal Correlation Function in the Harmonic Approximation
Below the critical temperature, the membrane is flat. But unless the membrane is at zero temperature, it is not completely flat because of thermal fluctuations. At finite temperatures, there is a competition between the energy that flattens the membrane and the entropy that bends it. This induces ripples. The normal-normal correlation function Gn(~r ) =< ~n(~r ):~n(0) > characterizes the flatness of the membrane. For a completely flat membrane the correlation is equal to 1 and for almost flat one 0 < Gn < 1.
Self-Consistent Screening Approximation (SCSA)
In the previous section, we saw that in the harmonic approximation, the flat phase cannot exists because of fluctuations. In this section, we show how this situation changes when we take into account the coupling between the out-of-plane bending and the in-plane elasticity and we recover the existence of the phase transition as in the -expansion. Nelson and Peliti [106] found that the coupling between the out-of-plane bending and in-plane phonons renormalized the bending rigidity / q with an anomalous dimension = 1. The free energy now reads: F = Z x 0.
NPRG Approach to Polymerized Membranes
To tackle the problem of the order of the phase transition as well as the behaviour in the flat phase of physical membranes (D = 2 and d = 3) we use a non-perturbative renormalization group approach. With this approach we are able to compute the critical exponents in the whole (d;D) plane and more importantly to determine the dc(D) line separating a second-order transition from a first-order one. In general the effective average action is a functional of all the invariants of the system. With O(d) rotational and T(D) translational invariance k for polymerized membranes reads in the lowest order of the derivative expansion: where u = @~r:@~r is the strain tensor, Zk corresponds to the field renormalization, U the running potential and the self-avoidance is neglected again. The tensor structure of u imposes that the potential is a function of an infinite number of invariants which are polynomials of the traces of (u)n. The tensor structure of u imposes that the effective potential depend on a infinite number of invariants which are a polynomial of the trace of (u)n. Since the dimensionDof the membrane is finite, one can show that the traces Tr u)D+i , with i 1, can be written as a combination of the trace with smaller powers of the tensor u, e.g. for D = 2 we have the relation Tr[(u)3] = 3Tr[(u)2]Tr[u] (Tr[u])3. With this simplification, the effective potential reads: U(u) =X n1;:::;nD0 an1;:::;nD (Tr [(u)])n1 : : : Tr u)DnD.
Table of contents :
Abstract
Acknowledgements
Chapter 1 Introduction
Chapter 2 Non-Perturbative Renormalization Group
2.1 A Brief Historical Introduction
2.2 Wilson Renormalization Group
2.2.1 Kadanoff’s Block Spin
2.2.2 Wilson Momentum Shell Integration
2.2.3 Polchinski and Proof of Renormalizability
2.3 Wetterich Renormalization Group
2.3.1 Effective Average Action
2.3.2 The Wetterich Equation
2.3.3 Approximations
2.3.3.1 Derivative Expansion
2.3.3.2 Field Expansion
2.3.3.3 Combination of the Derivative Expansion and the Field Expansion
2.3.3.4 Blaizot-Mendez-Wschebor Approximation
2.3.4 Optimisation and Cut-off Function Choice
2.3.5 The O(n)-model
2.3.5.1 Propagator
2.3.5.2 Definitions of the Coupling Constants
2.3.5.3 Derivation of the Flow Equations
2.3.5.4 Weak-coupling Expansion
2.3.5.5 Low-Temperature Expansion
2.3.5.6 Large-n Expansion
2.3.6 Conclusion
Appendix Chapter A Threshold Functions
Chapter 3 Membranes
3.1 Introduction
3.2 Differential Geometry of Membranes
3.2.1 Basic Definitions and Some Fundamental Properties
3.2.2 Monge Parametrization
3.3 Deformations
3.4 Long-Range Behaviour of Fluid Membranes
3.4.1 The Model
3.4.2 Fluid Membranes in Monge Parametrization
3.5 Polymerized Membranes
3.5.1 The Model
3.5.2 Mean Field Theory
3.5.3 Perturbative RG for the Crumpled-to-Flat Transition
3.5.4 The normal-normal Correlation Function in the Harmonic Approximation
3.5.5 Self-Consistent Screening Approximation (SCSA)
3.5.6 Conclusion
3.6 NPRG Approach to Polymerized Membranes
3.6.1 The propagator in Fourier Space
3.6.2 The Minimum Configuration
3.6.3 The Flow Equations of u and v and the Anomalous Dimension k
3.6.4 Derivation of the Flow Equations
3.6.4.1 The Effective Action
3.6.4.2 The Configuration
3.6.4.3 Propagator at the minimum
3.6.4.4 Flow Equations of u and v
3.6.4.5 Flow of 2
3.6.4.6 Definitions of the Coupling Constants
3.6.5 Crumpled to Flat Transition
3.6.6 Symmetry Breaking, Goldstone Bosons and Flat Phase
3.7 Conclusion
Appendix Chapter B Cayley-Hamilton Theorem and Faddeev-Leverrier Algorithm
Appendix Chapter C Derivatives of the Flow of Ueff
C.1 Derivative of the Propagator
Appendix Chapter D New Definition of the Minimum Configuration
Appendix Chapter E Threshold Functions
Chapter 4 Anisotropic Membranes
4.1 Introduction
4.2 Anisotropic Scaling Behaviour
4.3 Perturbative RG
4.4 Non-Perturbative Approach
4.4.1 The propagator
4.4.2 Flow equation of y
4.4.3 Flow equations
4.5 Physical Results
4.6 Conclusion
Appendix Chapter F Threshold Functions
Chapter 5 Lifshitz Critical Behaviour
5.1 Introduction
5.2 The Model
5.3 Anisotropic Scale Invariance
5.4 Critical Dimensions
5.5 Perturbative RG
5.5.1 Weak-Coupling -Expansion
5.5.2 Large-n Expansion
5.6 NPRG Approach
5.6.1 Lowest Order of the Derivative Expansion
5.6.2 Flow Equations
5.6.3 Upper Critical Dimension duc = 4 + m
5.6.4 Lower Critical Dimension dlc = 2 + m
5.7 Higher Order Expansion and Physical Results
5.8 Conclusion
Appendix Chapter G Threshold Functions
Chapter 6 Disordered Membranes
6.1 Introduction
6.2 Replica Formalism
6.3 The Model
6.4 Perturbative RG
6.5 NPRG
6.5.1 Effective Action
6.5.2 Propagator
6.5.3 Flow Equations
6.5.4 Conclusion
Conclusion
