Analytical approximation of weakly nonlinear continuous systems using renormalization group method

A direct method based on renormalization group method (RGM) is proposed for determining the analytical approximation of weakly nonlinear continuous systems. To demonstrate the application of the method, we use it to analyze some examples. First, we analyze the vibration of a beam resting on a nonlinear elastic foundation with distributed quadratic and cubic nonlinearities in the cases of primary and subharmonic resonances of the nth mode. We apply the RGM to the discretized governing equation and also directly to the governing partial differential equations (PDE). The results are in full agreement with those previously obtained with multiple scales method. Second, we obtain higher order approximation for free vibrations of a beam resting on a nonlinear elastic foundation with distributed cubic nonlinearities. The method is applied to the discretized governing equation as well as directly to the governing PDE. The proposed method is capable of producing directly higher order approximation of weakly nonlinear continuous systems. It is shown that the higher order approximation of discretization and direct methods are not in general equal. Finally, we analyze the previous problem in the case that the governing differential equation expressed in complex-variable form. The results of second order form and complex-variable form are not in agreement. We observe that in use of RGM in higher order approximation of continuous systems, the equations must not be treated in second order form.     

Short distance freedom of quantum gravity

Fourth order derivative gravity in 3+1$3+1$ dimensions is perturbatively renormalizable and is shown to describe a unitary theory of gravitons in a limited coupling parameter space. The running gravitational constant which includes graviton contribution is computed. Generically, gravitational Newton?s constant vanishes at short distances in this perturbatively renormalizable and unitary theory.     

Renormalizability conditions for almost-commutative geometries

We formulate conditions for almost-commutative (spacetime) manifolds under which the asymptotically expanded spectral action is renormalizable. These conditions are of a graph-theoretical nature, involving the Krajewski diagrams that classify such geometries. This applies in particular to the Standard Model of particle physics, giving a graph-theoretical argument for its renormalizability. A promising potential application is in the selection of physical (renormalizable) field theories described by almost-commutative geometries, thereby going beyond the Standard Model.     

Quantum censorship in two dimensions

It is pointed out that increasingly attractive interactions, represented by partially concave local potential in the Lagrangian, may lead to the degeneracy of the blocked, renormalized action at the gliding cutoff scale by tree-level renormalization. A quantum counterpart of this mechanism is presented in the two-dimensional sine-Gordon model. The presence of Quantum Censorship is conjectured which makes the loop contributions pile up during the renormalization and thereby realize an approximate semiclassical effect.     

The tensor renormalization group for pure and disordered two-dimensional lattice systems

The tensor renormalization group (TRG) is a powerful new approach for coarse-graining classical two-dimensional (2D) lattice Hamiltonians. It uses the intuitive framework of traditional position space renormalization group methods-analyzing flows in the space of Hamiltonian parameters-but can be systematically improved to yield thermodynamic properties at much higher precision. We present initial results demonstrating that the TRG can be generalized to quenched random systems, applying it to obtain the phase diagram of a bond-diluted triangular lattice Ising ferromagnet. This opens a variety of potential future applications, most prominently spin glasses.     

From low-momentum interactions to nuclear structure

We present an overview of low-momentum two-nucleon and many-body interactions and their use in calculations of nuclei and infinite matter. The softening of phenomenological and effective field theory (EFT) potentials by renormalization group (RG) transformations that decouple low and high momenta leads to greatly enhanced convergence in few- and many-body systems, while maintaining a decreasing hierarchy of many-body forces. This review surveys the RG-based technology and results, discusses the connections to chiral EFT, and clarifies various misconceptions.     

Coherent transport in linear arrays of quantum dots: The effects of period doubling and of quasi-periodicity

We evaluate the phase-coherent transport of electrons along linear structures of varying length, which are made from two types of potential wells set in either a periodic or a Fibonacci quasi-periodic sequence. The array is described by a tight-binding Hamiltonian and is reduced to an effective dimer by means of a decimation-renormalization method, extended to allow for connection to external metallic leads, and the transmission coefficient is evaluated in a T-matrix-scattering approach. Parallel behaviors are found for the energy dependence of the density of electron states and of the transmittivity of the array. In particular, we explicitly show that on increasing its length the periodic array undergoes a metal–insulator transition near single occupancy per dot, whereas prominent pseudo-gaps emerge away from the band center in the Fibonacci-ordered array.     

The renormalized electron mass in non-relativistic quantum electrodynamics

This work addresses the problem of infrared mass renormalization for a non-relativistic electron minimally coupled to the quantized electromagnetic field (the standard, translationally invariant system of an electron in non-relativistic QED). We assume that the interaction of the electron with the quantized electromagnetic field is subject to an ultraviolet regularization and an infrared regularization parametrized by σ>0. For the value p=0 of the conserved total momentum of electron and photon field, bounds on the renormalized mass are established which are uniform in σ→0, and the existence of a ground state is proved. For |p|>0 sufficiently small, bounds on the renormalized mass are derived for any fixed σ>0. A key ingredient of our proofs is the operator-theoretic renormalization group based on the isospectral smooth Feshbach map. It provides an explicit, finite algorithm for determining the renormalized electron mass at p=0 to any given precision.