Searcharxiv⌕ Search

arXiv subjects

Bruno Nachtergaele

Publications and source records attributed to Bruno Nachtergaele.

At least 73 records · Page 4Linked to original sources
arXiv2007

Isolated Eigenvalues of the Ferromagnetic Spin-J XXZ Chain with Kink Boundary Conditions

We investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Delta and kink boundary conditions. Since the third component of the total magnetization, M, is conserved, it is meaningful to study the spectrum for each fixed value of M. We prove that for J>= 3/2 the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain. In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum.

math-ph↗
arXiv2006

Entanglement in Finitely Correlated Spin States

We derive bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure finitely correlated state (FCS) on a chain of spins of any magnitude. Finitely correlated states are otherwise known as matrix product states or generalized valence-bond states. The bounds become exact in the limit of the entanglement of a single spin and the half-infinite chain to the right (or the left) of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for non-vanishing entanglement in terms of a single spin and the ``memory'' of the FCS, is also sufficient . Our result also generalizes the study of entanglement in the ground state of the AKLT model by Fan, Korepin, and Roychowdhury. Our result permits one to calculate more efficiently, numerically and in some cases even analytically, the entanglement of arbitrary finitely correlated quantum spin chains.

math-ph↗
arXiv2006

Propagation of Correlations in Quantum Lattice Systems

We provide a simple proof of the Lieb-Robinson bound and use it to prove the existence of the dynamics for interactions with polynomial decay. We then use our results to demonstrate that there is an upper bound on the rate at which correlations between observables with separated support can accumulate as a consequence of the dynamics.

math-ph↗
arXiv2006

Quantum Spin Systems after DLS1978

In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Ne'el order at positive temperature for the spin-S Heisenberg antiferromagnet on the d-dimensional hypercubic lattice when either S >= 1 and d >= 3 or S=1/2 and d is sufficiently large. This was the first proof of spontaneous breaking of a continuous symmetry in a quantum model at finite temperature. Since then the ideas of DLS have been extended and adapted to a variety of other problems. In this paper I will present an overview of the most important developments in the study of the Heisenberg model and related quantum lattice systems since 1978, including but not restricted to those directly related to the paper by DLS.

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↗
arXiv2005

Ordering of Energy Levels in Heisenberg Models and Applications

In a recent paper we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.

math-ph↗
arXiv2005

A Ferromagnetic Lieb-Mattis Theorem

We prove ferromagnetic ordering of energy levels for XXX Heisenberg chains of any spin and XXZ spin chains with all spins equal to 1/2. Ferromagnetic ordering means that the minimum energies in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result provides a ferromagnetic analogue of the well-known theorem by Lieb and Mattis about ordering of energy levels in antiferromagnetic and ferrimagnetic systems on bipartite graphs.

math-ph↗
arXiv2005

A model with simultaneous first and second order phase transitions

We introduce a two dimensional nonlinear XY model with a second order phase transition driven by spin waves, together with a first order phase transition in the bond variables between two bond ordered phases, one with local ferromagnetic order and another with local antiferromagnetic order. We also prove that at the transition temperature the bond-ordered phases coexist with a disordered phase as predicted by Domany, Schick and Swendsen. This last result generalizes the result of Shlosman and van Enter (cond-mat/0205455). We argue that these phenomena are quite general and should occur for a large class of potentials.

cond-mat.stat-mech↗
arXiv2004

Quantum Spin Systems

This article is a short introduction to the general topic of quantum spin systems. After a brief sketch of the history of the subject, the standard mathematical framework for formulating problems and results in quantum spin systems is described. Then, three short sections are devoted to Spontaneaous Symmetry Breaking, Phase transitions, and Dynamcis.

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↗
arXiv2003

The large-spin asymptotics of the ferromagnetic XXZ chain

We present new results and give a concise review of recent previous results on the asymptotics for large spin of the low-lying spectrum of the ferromagnetic XXZ Heisenberg chain with kink boundary conditions. Our main interest is to gain detailed information on the interface ground states of this model and the low-lying excitations above them. The new and most detailed results are obtained using a rigorous version of bosonization, which can be interpreted as a quantum central limit theorem.

math-ph↗
arXiv2003

Derivation of the Euler Equations from Quantum Dynamics

We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the Fermion Hamiltonian.

math-ph↗
arXiv2003

Derivation of the Euler equations from many-body quantum mechanics

The Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions is shown to converge to the compressible Euler equations in the hydrodynamic limit. The pressure function is given by the standard formula from quantum statistical mechanics with the two-body potential under consideration. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. No intermediate description, such as a Boltzmann equation or semi-classical approximation, is used in our proof. We require some technical conditions on the dynamics, which can be considered as interesting open problems in their own right.

math-ph↗
arXiv2002

On the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations

We study the time evolution of interfaces of the ferromagnetic XXZ chain in a magnetic field. A scaling limit is introduced where the strength of the magnetic field tends to zero and the microscopic time to infinity while keeping their product constant. The leading term and its first correction are determined and further analyzed in more detail for the case of a uniform magnetic field.

math-ph↗