232 research outputs found
Books Received
We investigate the persistence of spectral gaps of one-dimensional frustration free quantum lattice systems under weak perturbations and with open boundary conditions. Assuming that the interactions of the system satisfy a form of local topological quantum order, we prove explicit lower bounds on the ground state spectral gap and higher gaps for spin and fermion chains. By adapting previous methods using the spectral flow, we analyze the bulk and edge dependence of lower bounds on spectral gaps
Long-range order for the spin-1 Heisenberg model with a small antiferromagnetic interaction
We look at the general SU(2) invariant spin-1 Heisenberg model. This family
includes the well known Heisenberg ferromagnet and antiferromagnet as well as
the interesting nematic (biquadratic) and the largely mysterious
staggered-nematic interaction. Long range order is proved using the method of
reflection positivity and infrared bounds on a purely nematic interaction. This
is achieved through the use of a type of matrix representation of the
interaction making clear several identities that would not otherwise be
noticed. Using the reflection positivity of the antiferromagnetic interaction
one can then show that the result is maintained if we also include an
antiferromagnetic interaction that is sufficiently small.Comment: 15 pages, 1 figur
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.Comment: PACS 03.67.Mn, 05.50.+q. Minor typos in v1 corrected. In v2: expanded
Introduction and Discussion. Simplified proof of the main resul
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
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
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
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
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