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 $SO(2J+1)$-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 $S$-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