199 research outputs found
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.Comment: 20 pages, 2 figures, uses conm-p-l.cls. 1 reference adde
Lieb-Robinson Bounds for the Toda Lattice
We establish locality estimates, known as Lieb-Robinson bounds, for the Toda
lattice. In contrast to harmonic models, the Lieb-Robinson velocity for these
systems do depend on the initial condition. Our results also apply to the
entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable
assumptions, our methods also yield a finite velocity for certain perturbations
of these systems
A Multi-Dimensional Lieb-Schultz-Mattis Theorem
For a large class of finite-range quantum spin models with half-integer
spins, we prove that uniqueness of the ground state implies the existence of a
low-lying excited state. For systems of linear size L, of arbitrary finite
dimension, we obtain an upper bound on the excitation energy (i.e., the gap
above the ground state) of the form (C\log L)/L. This result can be regarded as
a multi-dimensional Lieb-Schultz-Mattis theorem and provides a rigorous proof
of a recent result by Hastings.Comment: final versio
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.Comment: 20 pages, contribution for the proceedings of QMATH9, Giens,
September 200
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
Gapped ground states of quantum spin systems have been referred to in the
physics literature as being `in the same phase' if there exists a family of
Hamiltonians H(s), with finite range interactions depending continuously on , such that for each , H(s) has a non-vanishing gap above its
ground state and with the two initial states being the ground states of H(0)
and H(1), respectively. In this work, we give precise conditions under which
any two gapped ground states of a given quantum spin system that 'belong to the
same phase' are automorphically equivalent and show that this equivalence can
be implemented as a flow generated by an -dependent interaction which decays
faster than any power law (in fact, almost exponentially). The flow is
constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we
give a proof extended to infinite-dimensional Hilbert spaces. In addition, we
derive a general result about the locality properties of the effect of
perturbations of the dynamics for quantum systems with a quasi-local structure
and prove that the flow, which we call the {\em spectral flow}, connecting the
gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a
result, we obtain that, in the thermodynamic limit, the spectral flow converges
to a co-cycle of automorphisms of the algebra of quasi-local observables of the
infinite spin system. This proves that the ground state phase structure is
preserved along the curve of models .Comment: Updated acknowledgments and new email address of S
A quantum central limit theorem for non-equilibrium systems: Exact local relaxation of correlated states
We prove that quantum many-body systems on a one-dimensional lattice locally
relax to Gaussian states under non-equilibrium dynamics generated by a bosonic
quadratic Hamiltonian. This is true for a large class of initial states - pure
or mixed - which have to satisfy merely weak conditions concerning the decay of
correlations. The considered setting is a proven instance of a situation where
dynamically evolving closed quantum systems locally appear as if they had truly
relaxed, to maximum entropy states for fixed second moments. This furthers the
understanding of relaxation in suddenly quenched quantum many-body systems. The
proof features a non-commutative central limit theorem for non-i.i.d. random
variables, showing convergence to Gaussian characteristic functions, giving
rise to trace-norm closeness. We briefly relate our findings to ideas of
typicality and concentration of measure.Comment: 27 pages, final versio
Non-equilibrium states of a photon cavity pumped by an atomic beam
We consider a beam of two-level randomly excited atoms that pass one-by-one
through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no
leaking of photons from the cavity, the pumping by the beam leads to an
unlimited increase in the photon number in the cavity. We derive an expression
for the mean photon number for all times. Taking into account leaking of the
cavity, we prove that the mean photon number in the cavity stabilizes in time.
The limiting state of the cavity in this case exists and it is independent of
the initial state. We calculate the characteristic functional of this
non-quasi-free non-equilibrium state. We also calculate the energy flux in both
the ideal and open cavity and the entropy production for the ideal cavity.Comment: Corrected energy production calculations and made some changes to
ease the readin
Quantum harmonic oscillator systems with disorder
We study many-body properties of quantum harmonic oscillator lattices with
disorder. A sufficient condition for dynamical localization, expressed as a
zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the
eigenfunction correlators for an effective one-particle Hamiltonian. We show
how state-of-the-art techniques for proving Anderson localization can be used
to prove that these properties hold in a number of standard models. We also
derive bounds on the static and dynamic correlation functions at both zero and
positive temperature in terms of one-particle eigenfunction correlators. In
particular, we show that static correlations decay exponentially fast if the
corresponding effective one-particle Hamiltonian exhibits localization at low
energies, regardless of whether there is a gap in the spectrum above the ground
state or not. Our results apply to finite as well as to infinite oscillator
systems. The eigenfunction correlators that appear are more general than those
previously studied in the literature. In particular, we must allow for
functions of the Hamiltonian that have a singularity at the bottom of the
spectrum. We prove exponential bounds for such correlators for some of the
standard models
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