110 research outputs found
On ground state phases of quantum spin systems
In this short note, I review some recent results about gapped ground state
phases of quantum spin systems and discuss the notion of topological order.Comment: Note written for the News Bulletin of the International Association
of Mathematical Physics (IAMP); IAMP News Bulletin, July 201
Product vacua with boundary states and the classification of gapped phases
We address the question of the classification of gapped ground states in one
dimension that cannot be distinguished by a local order parameter. We introduce
a family of quantum spin systems on the one-dimensional chain that have a
unique gapped ground state in the thermodynamic limit that is a simple product
state but which on the left and right half-infinite chains, have additional
zero energy edge states. The models, which we call Product Vacua with Boundary
States (PVBS), form phases that depend only on two integers corresponding to
the number of edge states at each boundary. They can serve as representatives
of equivalence classes of such gapped ground states phases and we show how the
AKLT model and its -invariant generalizations fit into this
classification
On gapped phases with a continuous symmetry and boundary operators
We discuss the role of compact symmetry groups, G, in the classification of
gapped ground state phases of quantum spin systems. We consider two
representations of G on infinite subsystems. First, in arbitrary dimensions, we
show that the ground state spaces of models within the same G-symmetric phase
carry equivalent representations of the group for each finite or infinite
sublattice on which they can be defined and on which they remain gapped. This
includes infinite systems with boundaries or with non-trivial topologies.
Second, for two classes of one-dimensional models, by two different methods,
for G=SU(2) in one, and G\subset SU(d), in the other we construct explicitly an
`excess spin' operator that implements rotations of half of the infinite chain
on the GNS Hilbert space of the ground state of the full chain. Since this
operator is constructed as the limit of a sequence of observables, the
representation itself is, in principle, experimentally observable. We claim
that the corresponding unitary representation of G is closely related to the
representation found at the boundary of half-infinite chains. We conclude with
determining the precise relation between the two representations for the class
of frustration-free models with matrix product ground states.Comment: Minor changes and correction
Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems
We consider translation invariant gapped quantum spin systems satisfying the
Lieb-Robinson bound and containing single-particle states in a ground state
representation. Following the Haag-Ruelle approach from relativistic quantum
field theory, we construct states describing collisions of several particles,
and define the corresponding -matrix. We also obtain some general
restrictions on the shape of the energy-momentum spectrum. For the purpose of
our analysis we adapt the concepts of almost local observables and
energy-momentum transfer (or Arveson spectrum) from relativistic QFT to the
lattice setting. The Lieb-Robinson bound, which is the crucial substitute of
strict locality from relativistic QFT, underlies all our constructions. Our
results hold, in particular, in the Ising model in strong transverse magnetic
fields
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