UNIVERSITY OF SOUTHAMPTON 1 0 0 2 The g u Local Potential Approximation A 3 of the 1 v 8 Renormalization Group 1 0 8 0 1 0 / h t - by p e h : v i X Christopher Simon Francis Harvey-Fros r a A thesis submitted for the degree of Doctor of Philosophy Department of Physics and Astronomy March 1999 This thesis was submitted for examination in March 1999. It does not neces- sarily represent the final form of the thesis as deposited in the University after examination. Dedicated to my family UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF SCIENCE PHYSICS Doctor of Philosophy The Local Potential Approximation of the Renormalization Group Christopher Simon Francis Harvey-Fros We introduce Wilson’s, or Polchinski’s, exact renormalization group, and review the Local Potential Approximation as applied to scalar field theory. Focusing on the Polchinski flow equation, standard methods are investigated, and by choos- ing restrictions to some sub-manifold of coupling constant space we arrive at a very promising variational approximation method. Within the Local Potential Approximation, we construct a function, C, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows and is stationary only at fixed points - where it ‘counts degrees of freedom’, i.e. is extensive, counting one for each Gaussian scalar. In the latter part of the thesis, the Local Potential Approximation is used to derive a non-trivial Polchinski flow equation to include Fermi fields. Our flow equation does not support chirally invariant solutions and does not reproduce the features associated with the corresponding invariant theories. We solve both forafinitenumber ofcomponents, N, andwithinthelargeN limit. TheLegendre flow equation provides a comparison with exact results in the large N limit. In this limit, it is solved to yield both chirally invariant and non-invariant solutions. Contents Preface ix Acknowledgements x 1 Introduction 1 1.1 Motivation and Structure . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Lattice Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Kadanoff blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 Fixed points and Perturbations . . . . . . . . . . . . . . . . . . . 12 1.9 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 i 1.10 β-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.11 Anomalous dimension . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.12 The Momentum expansion . . . . . . . . . . . . . . . . . . . . . . 18 1.13 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Scalar field theory 21 2.1 The Wilsonian effective action . . . . . . . . . . . . . . . . . . . . 22 2.2 The Polchinski flow equation . . . . . . . . . . . . . . . . . . . . . 24 2.3 The Legendre flow equation . . . . . . . . . . . . . . . . . . . . . 26 2.4 Application of the Local Potential Approximation . . . . . . . . . 29 2.5 Flow and β-functions . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Shooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8 Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9 The Wilson-Fisher fixed point by Variation . . . . . . . . . . . . . 41 2.10 The large N limit of the LPA . . . . . . . . . . . . . . . . . . . . 42 2.11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Appendices 48 3 Zamalodchikov’s C-function 53 ii 3.1 The conformal group . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Special properties in two dimensions . . . . . . . . . . . . . . . . 55 3.3 The stress tensor as the generator of scaling . . . . . . . . . . . . 57 3.4 Conformal weights . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 The operator product expansion . . . . . . . . . . . . . . . . . . . 61 3.6 The Virasoro algebra . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.7 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.8 Zamalodchikov’s C-theorem . . . . . . . . . . . . . . . . . . . . . 66 3.9 M as a metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ij 3.10 A C-function representation of the LPA . . . . . . . . . . . . . . 70 3.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.12 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Appendices 76 4 Fermionic field theory 79 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Conditions on Fermionic potentials . . . . . . . . . . . . . . . . . 81 4.3 The Legendre flow equation . . . . . . . . . . . . . . . . . . . . . 83 4.4 The exact large N limit . . . . . . . . . . . . . . . . . . . . . . . 86 iii 4.5 Solutions in the large N limit . . . . . . . . . . . . . . . . . . . . 90 4.6 The Polchinski flow equation . . . . . . . . . . . . . . . . . . . . . 94 4.7 Application of the Local Potential Approximation . . . . . . . . . 96 4.8 The large N limit of the LPA . . . . . . . . . . . . . . . . . . . . 99 4.9 Finite N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Appendices 106 5 Concluding remarks 107 5.1 The Local Potential Approximation . . . . . . . . . . . . . . . . . 107 5.2 Remaining issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Alternative methods . . . . . . . . . . . . . . . . . . . . . . . . . 109 Bibliography 110 iv List of Figures 1.1 Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Magnetisation as a function of temperature for the Ising model with zero external magnetic field . . . . . . . . . . . . . . . . . . 6 1.3 Kadanoff Blocking . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 A two dimensional projection of the flow near a fixed point . . . . 14 2.1 A schematic representation of the momentum cutoffs used . . . . 22 2.2 An ultra-violet cutoff function . . . . . . . . . . . . . . . . . . . . 23 2.3 An infra-red cutoff function . . . . . . . . . . . . . . . . . . . . . 27 2.4 The flow equations for the vertices of the generating functional W 29 Λ 2.5 The β-function for the mass term of three dimensional one com- ponent Z -invariant scalar field theory . . . . . . . . . . . . . . . 33 2 2.6 Shooting: φ v s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 c 2.7 The Wilson-Fisher fixed point . . . . . . . . . . . . . . . . . . . . 37 v 2.8 A plot of s v D for the Wilson-Fisher and two subleading fixed ∗ points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 The simplest polynomial variational approximation to the Wilson- Fisher fixed point (dashed line) compared to the exact solution (full line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.10 N-dependence of the two leading contributions to the four point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.11 Time varying potential in reparameterised functional . . . . . . . 51 3.1 Evaluation of the commutator (3.38) on the conformal plane . . . 61 3.2 Evaluation of the commutator (3.48) on the conformal plane . . . 65 4.1 SolutiontothelargeN limit of theLegendre flowequationinthree dimensions with cutoff introduced in a chirally symmetric manner 92 4.2 Region of (α, f ) space allowed by the boundedness condition (4.55) 93 1 4.3 Ultra-violet and Infra-red cutoff functions . . . . . . . . . . . . . 95 4.4 Solution to the large N limit of the LPA Polchinski flow equation in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Non-divergent solution to the large N limit of the LPA Polchinski flow equation in three dimensions . . . . . . . . . . . . . . . . . . 101 4.6 z versus V(0) for the Fermionic Polchinski flow equation at N = 4 c and D = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 vi