Statistical Field Theory and Applications: An Introduction for (and by) Amateurs. −−−−−−•−−−−−− by Denis BERNARD LPT-ENS & CNRS January 25, 2019 Preliminary version : 2018..... Chapter 3 was written in part by Jesper Jacobsen Comments are welcome 1 Contents 1 Introduction: what are we aiming at describing ? 5 2 Brownian motions, random paths and stochastic processes 10 2.1 Random walks and random paths . 10 2.2 Scaling limits . 11 2.3 Brownian motions and path integrals . 15 2.4 The 2D Brownian motion . 17 2.5 Brownian motions and stochastic differential equations . 18 2.6 Path integral representation for SDEs . 22 2.7 The small noise limit and the Arrhenius law . 27 2.8 Exercises . 30 3 Statistical lattice models 34 3.1 Examples of statistical lattice models . 34 3.2 Transfer matrices . 36 3.3 The 2D Ising model . 38 3.4 Lattice field theory . 45 3.5 Partition functions and path integrals . 45 3.6 The classical/quantum correspondance . 49 3.7 Exercises . 52 4 Critical systems and mean field theory. 56 4.1 Critical systems: phenomenology . 56 4.2 The Ising mean field theory . 57 4.3 Critical exponents and universality . 59 4.4 Landau theory . 61 4.5 Landau-Ginzburg theory . 62 4.6 Upper and lower critical dimensions . 65 4.7 Deviation from mean field theory . 67 4.8 Symmetry breaking and Goldstone modes . 70 4.9 Exercises . 72 5 Statistical field theory: free theory 74 5.1 Classical field theory: basics . 74 5.2 Euclidean free field theories . 75 5.3 Gaussian field theories . 77 5.4 Green functions . 81 5.5 Products and composite operators . 83 5.6 Quantization of free field theories . 85 5.7 Exercises . 90 6 Statistical field theory: interactions. 93 6.1 Preliminaries . 93 6.2 Symmetries and Ward identities . 94 6.3 Generating functions . 98 2 6.4 Perturbation theory and Feynman rules . 102 6.5 Diagrammatics . 107 6.6 One-loop effective action and potential . 109 6.7 The O(N) vector models . 113 6.8 Exercises . 115 7 Conformal field theory: basics 118 7.1 The group of conformal transformations . 118 7.2 Conformal invariance in field theory . 120 7.3 Fields and operator product expansions . 124 7.4 Massless gaussian free field in 2D . 126 7.5 2D conformal field theory . 132 7.6 Operator formalism in 2D . 133 7.7 Exercises . 134 8 The renormalisation group 138 8.1 Block spins and RG transformations . 138 8.2 Momentum RG transformations in field theory . 141 8.3 RG fixed points and universality . 143 8.4 Scaling functions and critical exponents . 145 8.5 Corrections to scaling and finite size effects . 147 8.6 Field transformations . 148 8.7 The perturbative renormalization group . 151 8.8 The Wilson-Fisher fixed point . 154 8.9 Scaling limits and renormalized theories . 156 8.10 Perturbatively renormalized φ4 theory . 161 8.11 Exercises . 166 9 Miscellaneous applications 169 9.1 The XY model . 169 9.2 Self avoiding walks . 173 9.3 Et un... 178 9.4 Et deux... 178 9.5 Et trois... 178 9.6 Zero... 178 10 Fermionic techniques 179 10.1 Fermions and Grassmannian variables . 179 10.2 The 2D Ising model again . 179 10.3 Dirac fermions . 179 10.4 Chiral anomaly . 179 10.5 Fermionisation-Bosonisation . 179 10.6 Exercises . 179 11 Stochastic field theories and dynamics 179 3 12 Exercise corrections 179 4 1 Introduction: what are we aiming at describing ? Statistical Field Theory is an important subject in theoretical physics, with a wide range of interdisciplinary applications ranging from statistical physics or condensed matter physics to higher energy physics or random geometry, which has undergone many progresses in the recent years. It is of course intimately linked to Quantum Field Theory. Statistical Field Theory aims at dealing with the behavior of systems (classical or quantum) with a large {actually infinitely large{ number of interacting degrees of freedom. These systems have very interesting and peculiar behaviors: they have different phases with different characters, they manifest phase transitions, their behaviors are dominated by collective modes and/or refined geometrical patterns, etc. Their understanding and analysis make contact with very elegant mathematical structures (say probability theory, representation theory, geometry) and with remarkable concepts, notably the renormalization group which is nowadays a cornerstone of Physics and its ramification. Statistical Field Theory aims at an understanding of those behaviors on the basis of a few physical principles. This is particularly true for its application to critical phase transitions through their universality property. These are characterized by sharp transitions in the physical properties of statistical systems controlled by external parameters. They are induced by collec- tive phenomena which involve large fluctuations over long distances without scale separation. Statistical field theory provides tools to deal with many nested degrees of freedom, with large fluctuations, over a cascade of scales. A sample of Samples of 2D self-avoiding walks, a Brownian curve in 2D. alias polymers. Understanding random patterns is at the core of the comprehension of many physical phe- nomena or mathematical structures, and the Brownian motion is a historical example of such structures. Although the relevant geometries can be as simple as gentle curved or surfaces, many relevant patterns are however not well-described by an integer dimension but have a frac- tal character, at least over some length scale. For instance, the singular behavior of second order phase transitions are compatible with fractal surfaces separating phases at the critical point. In general, the deeper our understanding of these geometries, the more complete our predictions can be. See Figure. Polymers provide simple examples of such fractal geometry going beyond the simple model of Brownian motions. These can be modeled by self avoiding random walks which may be viewed as the paths drawn by a random walker on a square lattice constrained not to visit twice any site 5 of the lattice. This constraint mimics the self-repulsion of the polymers. They undergo a phase transition when the temperature is varied: at low temperature the polymers is compactly curled up while at large temperature it extends macroscopically and resembles a fluctuating smooth curve. At the critical point, the polymers possess macroscopic fractal shapes which nowadays can be described using statistical field theory tools. See Figure. Percolation is another geometrical example manifesting phase transition and generating ran- dom fractal structures. Imagine that we randomly color the cells of a honeycomb lattice black or white. The rule is that each cell has a probability p of being white and 1 − p of being black. If p is small, most of the cells will be black, with a few small islands of white cells. As p is increased, these white clusters grow larger until a critical value pc at which there is a non-zero probability that one of these clusters spans the whole domain, no matter how large it is. Per- colation is important as a model for random inhomogeneous systems, for instance if the black clusters represent untapped oil pockets it is much easier to extract the oil if they percolate. See Figure. O A small sample of definition (left). A critical percolating clusters (right). Critical phase transitions (2nd order phase transitions) of statistical systems are the main actors in statistical field theory. There is a large variety of physical systems exhibiting second order phase transitions. Standard examples are the para-to-ferro magnetic transition in mag- netic materials, the superfluidity transition in quantum fluids, the superconductivity transition of certain metallic materials at low temperature, etc. Second order phase transitions have a universality property in the sense that the types of phase transitions fall into a relatively small number of categories, known as universality classes, which all behave similarly. Close to the tran- sition there are singularities in the thermodynamical functions and, in parallel, large anomalous fluctuations and power law behavior on the correlation functions. These singular behaviors, characterized by scaling exponents of thermodynamical functions, are understood within sta- tistical field theory. The size of the fluctuations is characterized by a length, the correlation length ξ, which is the typical extension of the domain over which the degrees of freedom are correlated. At the critical point, fluctuations are all over the scale and as a consequence this length diverges. On one hand, this has dramatic consequences because all degrees of freedom are then coupled, at any scale, making the analysis difficult (to say the least), but one the other hand, it renders a continuous description with statistical field theory possible and it is at the origin of the universality property. 6 Magnetization versus temperature Heat capacity versus temperature in the 2D Ising model. in the 2D Ising model. The archetypical statistical model of phase transition is the Ising model for magnetic transi- tion. The Ising degrees of freedom are simple spin variables si, whose values are either + or −, defined on the sites i of a lattice. The statistical weight of a given spin configuration is propor- tional to the Boltzmann weight e−E[fsg]=T , with T the temperature and with interaction energy P E[fsg] = −J i;j sisj where the sum is restricted to the neighbour spins on the lattice. For J > 0, the configuration with aligned spins are the most probable at low temperature. There is a phase transition at a certain critical temperature Tc. At T > Tc, spins have a tendency not to be aligned and.