The universality of renormalization-group limit-cycle behavior is illustrated with a simple discrete Hamiltonian model. A nonperturbative renormalization-group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound-state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero---also a consequence of the limit cycle. Wegner's eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegner's eigenvalues are independent of location on the limit cycle, in contrast with Wegner's operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonian's bound-state eigenvalues converge much more quickly than otherwise to an exact geometric series.
The universality of renormalization-group limit-cycle behavior is illustrated with a simple discrete Hamiltonian model. A nonperturbative renormalization-group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound-state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero---also a consequence of the limit cycle. Wegner's eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegner's eigenvalues are independent of location on the limit cycle, in contrast with Wegner's operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonian's bound-state eigenvalues converge much more quickly than otherwise to an exact geometric series.
Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.
Physics is often said to be universal when microscopically distinct systems show the same lowenergy behavior.In critical phenomena, the universality of the second-order phase transition is elucidated by the renormalization-group (RG) method, which provides a powerful computational tool for the critical exponents of the power-law behavior of observables.In particular, the attraction of different RG-flow trajectories to a small-dimensional sub-theory space accounts for the universal nature.The attraction structure motivates us to extract universal physical observables by investigating the sub-theory space called the renormalized trajectory, onto which RG flows are attracted.To put the motivation forward, we consider the universality in quantum few-body physics, in which the RG exhibits another characteristic flow than the fixed points.The Efimov physics, which is renowned for self-similar trimer states, shows an RG limit-cycle behavior that refers to a periodic RG-flow trajectory.In this thesis, we investigate the relation between the RG limit cycle and the universality in Efimov physics.Firstly, to verify that the RG limit cycle is the renormalized trajectory in Efimov physics, we demonstrate that the RG-flow trajectories of microscopically distinct systems universally exhibit the RG limit-cycle behavior at sufficiently low energy, thereby giving a numerical evidence of the attraction structure of the RG-flow trajectories to the RG limit cycle; in other words, we show that the RG limit cycle behaves as an attractor of various RG flows.To address this point unbiasedly, we perform an exact RG calculation for various effective models of identical bosons that features Efimov physics at low energy.We apply an exact functional renormalization-group (FRG) method to effective separable models for various systems of identical bosons with short-range resonant interactions.For a given microscopic three-body model, we numerically demonstrate for the first time that the RG-flow trajectory converges at the RG limit cycle in the low-energy regime.Physical observables such as Efimov's scaling factor and the three-body parameter can be extracted from the RG limit cycle.Secondly, we investigate the role of RG limit cycle in a four-body extension of Efimov physics, which has attracted considerable attention in recent years.Although the universality in four-body physics has been established both theoretically and experimentally, the relationship between the RG limit cycle and the four-body physics remains an open question.By performing an FRG calculation for a simple effective field theory, we obtain for the first time the RG limit cycle in the four-body sector that reproduces the universality of the four-body physics.As the first attempt, we develop a simple separable pole approximation of the four-body FRG equation and successfully reproduce qualitative features of the four-body physics.A systematic improvement of the approximation quantitatively reproduces the numbers and the binding energies of tetramer states that emerge universally at low energy.2 The restriction r 0 < ∼ R < ∼ a on the length scale limits the binding energy E t of the Efimov trimers as ℏ ma 2 < ∼ E t < ∼ ℏ mr 2 0 .3 A similar discrete scale-invariant phenomenon is known for the doubly excited H -ion, in which two valence electrons are excited from their ground states [16].In the doubly excited H -ion, an interaction between an electron-
Abstract Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is led to the important notion of relevant, irrelevant and marginal operators and then to the universality of the critical phenomena. Furthermore, the chapter also covers (as regards the RG) transformation laws, effective Hamiltonians, the Gaussian model, the Ising model, operators of quantum field theory, universal ratios, critical exponents and β-functions.
In this thesis we investigate Limit Cycles in Quantum Systems. Limit cycles are a renormalization group (RG) topology. When degrees of freedom are integrated out, the coupling constants flow periodically in a closed curve. The presence of limit cycles is restricted by the necessary condition of discrete scale invariance. A signature of discrete scale invariance and limit cycles is log-periodic behavior. The first part of this thesis is concerned with the study of limit cycles with the similarity renormalization group (SRG). Limit cycles are mainly investigated within conventional renormalization group frameworks, where degrees of freedom, which are larger than a given cutoff, are integrated out. In contrast, in the SRG potentials are unitarily transformed and thereby obtain a band-diagonal structure. The width of the band structure can be regarded as an effective cutoff. We investigate the appearance of limit cycles in the SRG evolution. Our aim is to extract signatures as well as the scaling factor of the limit cycle. We consider the 1/R2-potential in a two-body system and a three-body system with large scattering lengths. Both systems display a limit cycle. Besides the frequently used kinetic energy generator we apply the exponential and the inverse generator. In the second part of this thesis, Limit Cycles at Finite Density, we examine the pole structure of the scattering amplitude for distinguishable fermions at zero temperature in the medium. Unequal masses and a filled Fermi sphere for each fermion species are considered. We focus on negative scattering lengths and the unitary limit. The properties of the three-body spectrum in the medium and implications for the phase structure of ultracold Fermi gases are discussed.
Wegner's paradigm for constructing renormalization groups in statistical mechanics is applied to fluids. The coordinates of the fraction ($1\ensuremath{-}y$) of the molecules are integrated over and a system of interactions describing the remaining fraction y is constructed with the aid of the McMillan-Mayer solution theory. Renormalized Ursell functions are defined in terms of which the group operations are shown to be linear. The group operations commute with a scaling operation and it is found that a series of renormalization groups depending on a parameter x can be constructed by combining the operation of integrating over coordinates with the scaling operation. By comparing the renormalized cluster functions with the form of the critical cluster functions given by scaling it is shown that a fixed point of one of the series of groups corresponds to the critical point. The parameter x turns out to be the scaling exponent for the chemical potential. The question of the universality of critical phenomena is expressed as the question of the discreteness or continuity of the spectrum of values of the parameter x for which the linear fixed-point equations have a nontrivial solution. It is suggested that this question can be answered definitively only when a precise formulation is made of the function space in which the sequence of renormalized Ursell functions must lie.
We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with the local polynomial approximation (LPA), and discuss the advantages and shortcomings of these methods, and compare our results with the literature. In particular, convergence of the LPA is discussed in some detail. We finally integrate the flows numerically and find a c-function which determines these flows to be gradient in this approximation.