Renormalization | EPFL Graph Search

Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian. For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathemati

Hidden and explicit quantum scale invariance

Sander Johannes Nicolaas Mooij, Thibault Voumard

There exist renormalization schemes that explicitly preserve the scale invariance of a theory at the quantum level. Imposing a scale-invariant renormalization breaks renormalizability and induces new nontrivial operators in the theory. In this work, we study the effects of such scale-invariant renormalization procedures. On the one hand, an explicitly quantum scale-invariant theory can emerge from the scale-invariant renormalization of a scale-invariant Lagrangian. On the other hand, we show how a quantum scale-invariant theory can equally emerge from a Lagrangian visibly breaking scale invariance renormalized with scale-dependent renormalization (such as the traditional (MS) over bar scheme). In this last case, scale invariance is hidden in the theory, in the sense that it only appears explicitly after renormalization.

2019

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive delta-impurities symmetrically situated around the origin

Fabio Rinaldi

In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive delta-interaction centered at the origin or by a pair of identical attractive delta-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Bruning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction in one dimension, fully investigated in [2, 3], given the existence in both models of the remarkable spectral phenomenon called "level crossing". The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of "positional disorder", introduced in [1] in the case of a quantum dot with a single delta-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single delta-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive delta-interactions symmetrically situated at the same distance from the origin.

St Petersburg Natl Research Univ Information Technologies, Mech & Optics2016

Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with

N\in\mathbb{N}

. There are also vertical edges connecting two nearest neighbor vertices on distinct layers

\langle(x,i),(x,i+1)\rangle

for

x,i\in\mathbb{Z}

. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability

\lambda/(2N)

with

\lambda\geq 1

, while vertical edges are open with probability

\epsilon

to be suitably tuned as

N

grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature. We first deal with the critical case when

\lambda=1

. If

\epsilon=\kappa N^{-2/5}

, we see a phase transition in

\kappa

: positive and finite constants

C_1,C_2

exist so that there is no percolation if

\kappaC_2

. The derivation of the scaling limit is inspired by works on the long range contact process. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process, which is interesting by itself. A renormalization argument is used for the percolative regime. We then deal with the supercritical case when

\lambda>1

. If

\epsilon=e^{-\kappa N}

, we can also see a phase transition in

\kappa

. The horizontal and vertical edges can be discovered through subordinate process in each regime. The proof is based on the analysis of supercritical branching random walk but several levels of attritions are introduced to make sure the independent structure. The comparison between our original percolation and the percolation on the inhomogeneous square lattice is used in the renormalization scheme.

EPFL2021

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive delta-impurities symmetrically situated around the origin

Fabio Rinaldi

St Petersburg Natl Research Univ Information Technologies, Mech & Optics2016

Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with

N\in\mathbb{N}

. There are also vertical edges connecting two nearest neighbor vertices on distinct layers

\langle(x,i),(x,i+1)\rangle

for

x,i\in\mathbb{Z}

. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability

\lambda/(2N)

with

\lambda\geq 1

, while vertical edges are open with probability

\epsilon

to be suitably tuned as

N

grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature. We first deal with the critical case when

\lambda=1

. If

\epsilon=\kappa N^{-2/5}

, we see a phase transition in

\kappa

: positive and finite constants

C_1,C_2

exist so that there is no percolation if

\kappaC_2

\lambda>1

. If

\epsilon=e^{-\kappa N}

, we can also see a phase transition in

\kappa

EPFL2021