arXiv subjects

Volker Bach

Publications and source records attributed to Volker Bach.

31 records · Page 2Linked to original sources

arXiv2013

Continuous Renormalization Group Analysis of Spectral Problems in Quantum Field Theory

The isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory. It was introduced in 1995 [see \cite{BachFrohlichSigal1995}, \cite{BachFrohlichSigal1998}] and since then it has been used to prove several results for non-relativistic quantum electrodynamics. After the introduction of the method there have been many works in which extensions, simplifications or clarifications are presented (see \cite{BachChenFrohlichSigal2003}, \cite{GriesemerHasler2008}, \cite{FrohlichGriesemerSigal2009}). In this paper we present a new approach in which we construct a flow of operators parametrized by a continuous variable in the positive real axis. While this is in contrast to the discrete iteration used before, this is more in spirit of the original formulation of the renormalization group introduced in theoretical physics in 1974 \cite{KogutWilson1974}. The renormalization flow that we construct can be expressed in a simple way: it can be viewed as a single application of the Feshbach-Schur map with a clever selection of the spectral parameter. Another advantage of the method is that there exists a flow function for which the renormalization group that we present is the orbit under this flow of an initial Hamiltonian. This opens the possibility to study the problem using different techniques coming from the theory of evolution equations.

math-ph↗

arXiv2013

Representability Conditions by Grassmann Integration

Representability conditions on the one- and two-particle density matrix for fermion systems are formulated by means of Grassmann integrals. A positivity condition for a certain kind of Grassmann integral is established which, in turn, induces the well-known G-, P- and Q-Conditions of quantum chemistry by an appropriate choice of the integrand. Similarly, the T1- and T2-Conditions are derived. Furthermore, quasifree Grassmann states are introduced and it is shown that every so-called generalized one-particle density matrix which is bounded between 0 and 1 corresponds to a unique quasifree Grassmann state.

math-ph↗

arXiv2013

Generalized One-Particle Density Matrices and Quasifree States

In the spirit of the generalized one-particle density matrix for fermions, we introduce generalized one- and two-particle density matrices to state representability conditions up to second order for boson systems without assuming particle number-conservation. Furthermore, we show for both particle species that, for a semibounded Hamiltonian, the in?mum of the variation of the energy functional w.r.t quasifree states coincides with the one of a variation over pure quasifree states. Moreover, it is proven for fermions that only pure quasifree states have a generalized 1-pdm that is a projection, and a similar statement for bosons.

math-ph↗

arXiv2013

Existence and Construction of Resonances for Atoms Coupled to the Quantized Radiation Field

For a nonrelativistic atom, which is minimally coupled to the quantized radiation field, resonances emerging from excited atomic eigenstates are constructed by an iteration scheme inspired by \cite{Pizzo2003} and \cite{BachFrohlichPizzo2006}. This scheme successively removes an infrared cut off in momentum space and yields a convergent algorithm enabling us to calculate the resonance eigenvalues and eigenstates, to arbitrary order in the feinstructure constant $α\sim 1/137$, and is thus an alternative method of proof of a similar result obtained in \cite{Sigal2010}.

math-ph↗

arXiv2013

Minimization of the Energy of the Non-Relativistic One-Electron Pauli-Fierz Model over Quasifree States

In this article is proved the existence and uniqueness of a minimizer of the energy for the non-relativistic one electron Pauli-Fierz model, within the class of pure quasifree states. The minimum of the energy on pure quasifree states coincides with the minimum of the energy on quasifree states. Infrared and ultraviolet cutoffs are assumed, along with sufficiently small coupling constant and momentum of the dressed electron. A perturbative expression of the minimum of the energy on quasifree states for a small momentum of the dressed electron and small coupling constant is then given. We also express the Lagrange equation for the minimizer, in terms of the generalized one particle density matrix of the pure quasifree state.

math-ph↗

arXiv2012

Effective dynamics of an electron coupled to an external potential in non-relativistic QED

In the framework of non-relativistic QED, we show that the renormalized mass of the electron (after having taken into account radiative corrections) appears as the kinematic mass in its response to an external potential force. Specifically, we study the dynamics of an electron in a slowly varying external potential and with slowly varying initial conditions and prove that, for a long time, it is accurately described by an associated effective dynamics of a Schrödinger electron in the same external potential and for the same initial data, with a kinetic energy operator determined by the renormalized dispersion law of the translation-invariant QED model.

math-ph↗

arXiv2012

Fermion Correlation Inequalities derived from G- and P-Conditions

It is shown in this paper that the G-Condition and the P-Condition from representability imply the fermion correlation estimate from [1] which, in turn, is known to yield a nontrivial bound on the accuracy of the Hartree-Fock approximation for large Coulomb systems.

math-ph↗

arXiv2010

The Integrated Density of States for the Wilson Dirac Operator

It is shown that gauge field-dependent fermion Dirac operators from lattice QCD form an ergodic operator family in the probabilistic sense, provided the gauge field is an ergodic random field. As a consequence, the integrated density of states of such Dirac operators in the thermodynamic limit exists and is almost surely independent of the chosen gauge field configuration.

math-ph↗

arXiv2007

Dynamical Symmetries in Supersymmetric Matrix Models

We reveal a dynamical SU(2) symmetry in the asymptotic description of supersymmetric matrix models. We also consider a recursive approach for determining the ground state, and point out some additional properties of the model(s).

hep-th↗

arXiv2006

Ferromagnetism of the Hubbard Model at Strong Coupling in the Hartree-Fock Approximation

As a contribution to the study of Hartree-Fock theory we prove rigorously that the Hartree-Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential mu) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U, and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, {S}_z = sum_x n_{x,\uparrow} - n_{x,\downarrow}, and we find that the choice 2{S}_z = N = particle number gives the lowest energy at fixed 0 < mu < 4d.

cond-mat.str-el↗

arXiv2005

The Renormalized Electron Mass in Non-Relativistic Quantum Electrodynamics

This work addresses the problem of infrared mass renormalization for a scalar electron in a translation-invariant model of non-relativistic QED. We assume that the interaction of the electron with the quantized electromagnetic field comprises a fixed 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 $σ\to0$, 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 using the isospectral smooth Feshbach map. It provides an explicit, finite algorithm that determines the renormalized electron mass at $p=0$ to any given precision.

math-ph↗

arXiv2002

Mathematical analysis of the photoelectric effect

We study the photoelectric effect on the example of a simplified model of an atom with a single bound state, coupled to the quantized electromagnetic field. For this model, we show that Einstein's prediction for the photoelectric effect is qualitatively and quantitatively correct to leading order in the coupling parameter. More specifically, considering the ionization of the atom by an incident photon cloud consisting of $N$ photons, we prove that the total ionized charge is additive in the $N$ involved photons. Furthermore, if the photon cloud is approaching the atom from a large distance, the kinetic energy of the ejected electron is shown to be given by the difference of the photon energy of each single photon in the photon cloud and the ionization energy.

math-ph↗

arXiv1993

Generalized Hartree-Fock Theory and the Hubbard Model

The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.

cond-mat↗