We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.
We discuss the quantum mechanics of a particle restricted to the half-line >x 0 with potential energy a = V x 2 for a -< < 1 4 0. It is known that two scale-invariant theories may be defined.By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)space of Hamiltonians.A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points.Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points.Using RG arguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory.By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value.It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator.Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions.In the former example, V appears as part of an expectation value in the Feynman-Kac formula.In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.
In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.
In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.
The renormalization-group (RG) method is applied to the problem of formation of a localized state of a particle moving in a given potential. It is shown that RG transformation on the particle's Green's function can be performed exactly. The fixed-point equations yield information on the critical binding strength, while the transformation equations near the fixed points give scaling laws and critical exponents, e.g., for the dependence of the localization radius on the energy and the potential strength. The general theory is illustrated by considering the simplest case of the Slater-Koster problem in detail. We find that it is possible to take all the irrelevant variables into account and thus obtain the exact result for the critical binding point. Critical exponents and scaling laws are, however, universal in character. The method is then extended to the case of motion in random potentials, and some applications are pointed out. In particular it is shown that the RG properties of the averaged absolute value squared (〈∣G∣2〉) of the Green's function do not correspond to a simple second-order phase transition. The analysis illustrates certain features of RG methods which have not heretofore been encountered in other problems and which are likely to be found in treatments of disordered materials. In particular, the runaway behavior observed in the RG transformation of 〈G〉 and 〈∣G∣2〉 as well as in many random-spin systems, might be associated with the formation of bound states in the band tail for any amount of randomness.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].