arXiv subjects

Tobias Verhulst

Publications and source records attributed to Tobias Verhulst.

7 recordsLinked to original sources

arXiv2011

A multiplet analysis of spectra in the presence of broken symmetries

We introduce the notion of a generalised symmetry M of a hamiltonian H. It is a symmetry which has been broken in a very specific manner, involving ladder operators R and R*. In Theorem 1 these generalised symmetries are characterised in terms of repeated commutators of H with M. Breaking supersymmetry by adding a term linear in the supercharges is discussed as a motivating example. The complex parameter gamma which appears in the definition of a generalised symmetry is necessarily real when the spectrum of M is discrete. Theorem 2 shows that gamma must also be real when the spectrum of H is fully discrete and R and R* are bounded operators. Any generalised symmetry induces a partitioning of the spectrum of H in what we call M-multiplets. The hydrogen atom in the presence of a symmetry breaking external field is discussed as an example. The notion of stability of eigenvectors of H relative to the generalised symmetry M is discussed. A characterisation of stable eigenvectors is given in Theorem 3.

math-ph↗

arXiv2010

A mechanism for pair formation in strongly correlated systems

We start from a Hamiltonian describing non-interacting fermions and add bosons to the model, with a Jaynes-Cummings-like interaction between the bosons and fermions. Because of the specific form of the interaction the model can be solved exactly. In the ground state, part of the electrons form bound pairs with opposite momentum and spin. The model also shows a gap in the kinetic energy of the fermions, but not in the spectrum of the full Hamiltonian. This gap is not of a mean-field nature, but is due to the Pauli exclusion principle.

cond-mat.str-el↗

arXiv2010

Electron pair forming in an integrable model

We consider a lattice model in which phonons scatter with pairs of electrons. All eigenvalues and eigenvectors can be obtained analytically. For a suitable choice of parameters the ground state consists of a Fermi sea of non-interacting electrons, with on top a layer of paired electrons. The binding energy of the electron pairs is partly cancelled by an increased kinetic energy. This results in a gap in the kinetic energy spectrum of the electrons.

cond-mat.str-el↗

arXiv2009

Analysis of the N=4 Hubbard ring using a counting operator

We prove three theorems about the use of a counting operator to study the spectrum of model Hamiltonians. We analytically calculate the eigenvalues of the Hubbard ring with four lattice positions and apply our theorems to describe the observed level crossings.

cond-mat.str-el↗

arXiv2009

Counting operator analysis of the discrete spectrum of some model Hamiltonians

The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R^\dagger which share some of the properties of creation and annihilation operators, and such that $M$ becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by the symmetries of H, but by those of a reference Hamiltonian H_ref, which is defined by H_ref=H-R-R^\dagger, and which commutes with M. Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.

cond-mat.str-el↗

arXiv2007

Ensemble averaged entanglement of two-particle states in Fock space

Recent results, extending the Schmidt decomposition theorem to wavefunctions of identical particles, are reviewed. They are used to give a definition of reduced density operators in the case of two identical particles. Next, a method is discussed to calculate time averaged entanglement. It is applied to a pair of identical electrons in an otherwise empty band of the Hubbard model, and to a pair of bosons in the the Bose-Hubbard model with infinite range hopping. The effect of degeneracy of the spectrum of the Hamiltonian on the average entanglement is emphasised.

quant-ph↗

arXiv2006

Entanglement of a microcanonical ensemble

We replace time-averaged entanglement by ensemble-averaged entanglement and derive a simple expression for the latter. We show how to calculate the ensemble average for a two-spin system and for the Jaynes-Cummings model. In both cases the time-dependent entanglement is known as well so that one can verify that the time average coincides with the ensemble average.

quant-ph↗