Searcharxiv⌕ Search

arXiv subjects

Wolfgang Spitzer

Publications and source records attributed to Wolfgang Spitzer.

29 records · Page 2Linked to original sources
arXiv2012

Counterexamples to Ferromagnetic Ordering of Energy Levels

The Heisenberg ferromagnet has symmetry group ${\rm SU}(2)$. The property known as ferromagnetic ordering of energy levels (FOEL) states that the minimum energy eigenvalue among eigenvectors with total spin $s$ is monotone decreasing as a function of $s$. While this property holds for certain graphs such as open chains, in this note we demonstrate some counterexamples. We consider the spin 1/2 model on rings of length $2n$ for $n=2,3,...,8$, and show that the minimum energy among all spin singlets is less than or equal to the minimum energy among all spin triplets, which violates FOEL. This also shows some counterexamples to the "Aldous ordering" for the symmetric exclusion process. We also review some of the literature related to these examples.

math-ph↗
arXiv2010

A geometric approach to absolutely continuous spectrum for discrete Schrödinger operators

We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schrödinger operators developed in \cite{FHS1,FHS2,FHS3,FHS4}. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees. The latter implies ac spectrum for a percolation model on trees. Finally, we introduce certain loop tree models which lead to some interesting open problems.

math-ph↗
arXiv2010

Free Fermions Violate the Area Law For Entanglement Entropy

We show that the entanglement entropy associated to a region grows faster than the area of its boundary surface. This is done by proving a special case of a conjecture due to Widom that yields a surprisingly simple expression for the leading behaviour of the entanglement entropy.

hep-th↗
arXiv2009

On the AC spectrum of one-dimensional random Schroedinger operators with matrix-valued potentials

We consider discrete one-dimensional random Schroedinger operators with decaying matrix-valued, independent potentials. We show that if the l^2-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schroedinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schroedinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon, Simon, and Souillard.

math-ph↗
arXiv2008

Transport of interface states in the Heisenberg chain

We demonstrate the transport of interface states in the one-dimensional ferromagnetic Heisenberg model by a time dependent magnetic field. Our analysis is based on the standard Adiabatic Theorem. This is supplemented by a numerical analysis via the recently developed time dependent DMRG method, where we calculate the adiabatic constant as a function of the strength of the magnetic field and the anisotropy of the interaction.

cond-mat.stat-mech↗
arXiv2008

Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops

We consider random Schrödinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models. The first model is the usual binary tree with certain strongly correlated random potentials. These potentials are of interest since for complete correlation they exhibit localization at all disorders. In the second model we change the tree graph by adding all possible edges to the graph inside each sphere, with weights proportional to the number of points in the sphere.

math-ph↗
arXiv2005

Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions

We give a precise definition for excitations consisting of a droplet of size n in the XXZ chain with various choices of boundary conditions, including kink boundary conditions and prove that, for each n, the droplet energies converge to a boundary condition independent value in the thermodynamic limit. We rigorously compute an explicit formula for this limiting value using the Bethe ansatz.

math-ph↗
arXiv2003

Ferromagnetic Ordering of Energy Levels

We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb-Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.

math-ph↗
arXiv1998

Hydrodynamics for quasi-free quantum systems

We consider quasi-free quantum systems and we derive the Euler equation using the so-called hydrodynamic limit. We use Wigner's well-known distribution function and discuss an extension to band distribution functions for particles in a periodic potential. We investigate the Bosonic system of hard rods and calculate fluctuations of the density. Keywords: Euler equation, quantum distribution function, hydrodynamic limit

cond-mat.stat-mech↗