173 research outputs found
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
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
Dispersive Estimates for Harmonic Oscillator Systems
We consider a large class of harmonic systems, each defined as a quasi-free
dynamics on the Weyl algebra over . In contrast to
recently obtained, short-time locality estimates, known as Lieb-Robinson
bounds, we prove a number of long-time dispersive estimates for these models
Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems
We prove Lieb-Robinson bounds for the dynamics of systems with an infinite
dimensional Hilbert space and generated by unbounded Hamiltonians. In
particular, we consider quantum harmonic and certain anharmonic lattice
systems
Approximating open quantum system dynamics in a controlled and efficient way: A microscopic approach to decoherence
We demonstrate that the dynamics of an open quantum system can be calculated
efficiently and with predefined error, provided a basis exists in which the
system-environment interactions are local and hence obey the Lieb-Robinson
bound. We show that this assumption can generally be made. Defining a dynamical
renormalization group transformation, we obtain an effective Hamiltonian for
the full system plus environment that comprises only those environmental
degrees of freedom that are within the effective light cone of the system. The
reduced system dynamics can therefore be simulated with a computational effort
that scales at most polynomially in the interaction time and the size of the
effective light cone. Our results hold for generic environments consisting of
either discrete or continuous degrees of freedom
Ferromagnetic Ordering of Energy Levels for Symmetric Spin Chains
We consider the class of quantum spin chains with arbitrary
-invariant nearest neighbor interactions, sometimes
called for the quantum deformation of , for
. We derive sufficient conditions for the Hamiltonian to satisfy the
property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the
property that the ground state energy restricted to a fixed total spin subspace
is a decreasing function of the total spin. Using the Perron-Frobenius theorem,
we show sufficient conditions are positivity of all interactions in the dual
canonical basis of Lusztig. We characterize the cone of positive interactions,
showing that it is a simplicial cone consisting of all non-positive linear
combinations of "cascade operators," a special new basis of
intertwiners we define. We also state applications to
interacting particle processes.Comment: 23 page
Finitely Correlated States on Quantum Spin Chains
We study a construction, which yields a class of translation invariant states on quantum spin chains, characterised by the property that the correlations across any bond can be modelled on a finite dimensional vector space. These states, which are dense in the set of all translation invariant states, can be considered as generalised valence bond states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic "Néel ordered" states. Ergodic finitely correlated states have exponential decay of correlations. All states considered can be considered as "functions" of states of a special kind, so-called "purely generated states", which are shown to be ground states for suitably chosen interactions. We show that all these states have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group, but the isotropic ground states of some one-dimensional antiferromagnets, recently studied by Affleck, Kennedy, Lieb, and Tasaki fall in this class
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.Comment: 10 page
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