Abstract One of the fundamental problems of statistical mechanics and its quantum field theory formulation is the characterization of the order parameters and the computation of their correlation functions. Besides the intrinsic interest of this problem, the correlation functions are the key quantities in the determination of the universal ratios of the renormalization group and therefore they can have direct experimental confirmation (see Chapter 8). In general, the computation of correlation functions is a difficult task, usually achieved with partial success through perturbative methods.
Abstract One of the fundamental problems of statistical mechanics and its quantum field theory formulation is the characterization of the order parameters and the computation of their correlation functions. Besides the intrinsic interest of this problem, the correlation functions are the key quantities in the determination of the universal ratios of the renormalization group and therefore they can have direct experimental confirmation (see Chapter 8). In general, the computation of correlation functions is a difficult task, usually achieved with partial success through perturbative methods.
Since the form of PN is the same as that occurring in the statistical mechanics of the classical gas (replace f 2 ( r ij ) by exp[- V ( r ij /kT ]), we can use the same integration techniques that are used in the classical problem. W. L. McMillan, 1965 Summary Building on the random walk methods developed in the previous chapter, we show how to compute properties of many-body trial wavefunctions using a random walk. This method, called variational Monte Carlo, is the simplest stochastic quantum many-body technique. Whereas mean-field methods are usually limited to single determinants, variational Monte Carlo can treat any correlated trial function, as long as its values are computable. We discuss how to optimize such trial wavefunctions, how to compute their momentum distribution, how to use non-local pseudopotentials, how to compute excited states, and how to correct for the finite size of the simulation cell. Deterministic quantum methods have difficulties. For example, the Hartree–Fock method assumes the wavefunction is a single Slater determinant, neglecting correlation. If one expands as a sum of determinants, it is very difficult to have the results size-consistent since the number of determinants needed will grow exponentially with the system size. As we have seen, the DMFT method introduced in Ch. 16 assumes locality. In Ch. 6 we discussed general properties of many-body wavefunctions. Using Monte Carlo methods, we can directly incorporate correlation into a wavefunction, without having to make any further approximations other than the form of the correlation factors. In many cases the energy and other properties are very close to the exact results. Some of the usual restrictions on the form of the many-body wavefunction are not an issue in variational Monte Carlo. The most important generalization of the HF wavefunction is to put correlation directly into the wavefunction via the “Jastrow” factor. At next order, one can use the “backflow” wavefunction, in which correlation is also built into the determinant. The variational Monte Carlo method (VMC) was first used by McMillan [44] to calculate the ground-state properties of superfluid 4 He. One of the key problems at that time was whether the observed superfluid properties were a consequence of Bose condensation.
In the successful framework of the Renormalization Group (RG) approach to Quantum Field Theories (QFT), establishing general constraints on the RG flow has become a central problem. This is formally addressed by the monotonicity theorems, such as Zamolodchikov’s c-theorem in d=2, the a-theorem in d=4, and the F-theorem in d=3. Essentially, these theorems state that if a RG trajectory leads from a UV fixed point to a IR one, each described by a Conformal Field Theory (CFT), then a suitable positive quantity decreases. In both even and odd dimensions, this quantity can be extracted, at fixed points, from the free energy of the CFT on a sphere, leading to the conjecture of a generalized F-theorem valid in continuous dimension d. In this thesis we investigate this framework in a multicritical multiscalar theory with a generic sextic interaction for N massless scalar fields below its upper critical dimension d=3. This model captures, among many others, the critical features of the Tricritical Ising universality class. A first step is to establish whether the RG flow admits a gradient structure, i.e. that there exists a monotonic function A whose gradient yields the beta function up to a positive-definite metric in the space of couplings. This is one of the strongest realizations of monotonicity, as it guarantees the irreversibility of the RG flow and gives an intrinsic geometric structure to the theory space. Given the known beta function up to next-to-leading order, we solve the gradient flow equation perturbatively, finding a parametric family of solutions for the monotonic A-function together with the metric at and below the upper critical dimension, so that the RG flow admits a gradient structure at this order. As this procedure leaves some parameters undetermined, to turn it into a physical statement these functions must be matched to suitable observables of the CFTs describing the fixed points. We therefore compute the CFT candidates suggested by the generalized F-theorem, namely the sphere free energy of the CFT and the Zamolodchikov metric. Specializing to the O(N)-symmetric fixed point, we consistently match the RG and CFT results, fixing all the free parameters in a unique way. This matching is the central result of this work: it establishes that the RG flow admits a gradient structure perturbatively, with F as the monotonic function and Zamolodchikov's metric as the metric on the theory space, recovered from the geometric construction for a specific and unique choice of the free parameters. Importantly, this result provides a nontrivial, perturbative confirmation of the F-theorem in its strongest form, at and below the upper critical dimension, for renormalizable three-dimensional multiscalar theories.
Abstract At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations that can exactly solved in many cases of physical interest. Chapter 19 covers general properties of form factors, Faddeev–Zamolodchikov algebra, symmetric polynomials, kinematical and bound state poles, the operator space and kernel functions, the stress-energy tensor and vacuum expectation values and the Ising model in a magnetic field.
Experimental determinations of correlation functions typically involve events of differing particle number. In order to interpret such data, it is important to relate these measurements to fixed-number correlations. We exhibit the total correlation function in terms of fixed n correlations and fluctuations with respect to the average. We note that moments, constructed as integrals of the appropriate correlation functions, can be dominated in the case of broad distributions by cumulant moments constructed from single-particle density fluctuations.
This dissertation involves the renormalization group theory and its application to two systems exhibiting renormalization symmetry. Both the real space and the field theoretic formulations of the renormalization group theory are discussed. A continuum of real space renormalization transformations is used to study the critical behavior of the two-dimensional Ising model. We find that there exists an optimal transformation that produces sharply more accurate critical properties that the others. We establish a set of conditions that characterize the optimal transformation. These conditions suggest criteria that could enable one to control a given real space renormalization approximation. Field theory renormalization is used to exploit the renormalization symmetry in the excluded volume problem. We develop a self-contained formalism that incorporates the concepts and methods of field theory renormalization directly into the language of the excluded volume problem. This formalism avoids the Lagrangian model isomorphism, thereby providing a simpler and more natural framework for underst and ing the excluded volume problem from first principles. This formalism is used to investigate the spatial correlations in a large self-repelling ring polymer. In particular, we find that the ratio of the radius of gyration of a ring polymer to that of a linear polymer having the same number of monomers is a universal quantity whose value we calculate to first order in (epsilon) (TBOND) 4 - (dimension of space). In addition, we study the mutual excluded volume characterizing a system of two chains with different lengths. The renormalization of the polymer perturbation theory through two-loop order enables us to calculate the universal value of the interpenetration function to second order in (epsilon). Our renormalization group results are in very good agreement with the results obtained from numerical and experimental studies of rings and chains in three dimensional space. We also discuss the connection between real space and field theory renormalization. We outline the basic conceptual ideas that underlie the general theory of renormalization and illustrate how each of these ideas is realized in the real space and the field theoretic formulations.