his book explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition. Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a m
his book explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition. Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a m
It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in
The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulation
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
Path integrals have played a fundamental role in emphasizing the profound analogies between Quantum Field Theory (QFT), and Classical as well as Quantum Statistical Physics. Ideas coming from Statistical Physics have then led to a deeper understanding of Quantum Field Theory and open the way for a wealth of non-perturbative methods. Conversely QFT methods are become essential for the description of the phase transitions and critical phenomena beyond mean field theory. This is the point we want to illustrate here. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to an accurate determination of critical exponents of the N-vector model, and more recently of the equation of state of the 3D Ising model. The starting point is the perturbative expansion for RG functions or the effective potential to the order presently available. Perturbation theory is known to be divergent and its divergence has been related to instanton contributions. This has allowed to characterize the large order behaviour of perturbation series, an information that can be used to efficiently "sum" them. Practical summation methods based on Borel transformation and conformal mapping have been developed, leading to the most accurate results available probing field theory in a non perturbative regime. We illustrate the methods with a short discussion of the scaling equation of state of the 3D Ising model. Compared to exponents its determination involves a few additional (non-trivial) technical steps, like the use of the parametric representation, and the order-dependent mapping.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.