Objective Interactions in a many body quantum system are encoded in a Hamiltonian, where the physical intuition that particles can only interact with those which are closeby is formally imposed as a local structure in the Hamiltonian, and homogeneity in space is imposed by a translational invariant structure on a given regular lattice in one, two or three dimensions. The first main aim of this proposal is to characterize the existence of a uniform (with the system size) lower bound on the gap between the two lowest eigenvalues of a given local translational invariant Hamiltonian.There are many reasons which motivate this study, coming from different fields, and hence many potential applications. We will concentrate here on those coming from quantum information theory and from condensed matter physics and mainly, as the second main aim of this proposal, on classifying the different possible quantum phases arising in this type of models. Fields of science natural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencescomputer and information sciencescomputer securitycryptographynatural sciencesphysical sciencesquantum physicsquantum field theorynatural sciencescomputer and information sciencesartificial intelligencemachine learningnatural sciencesphysical scienceselectromagnetism and electronicssuperconductivity Keywords Spectrum of quantum Hamiltonians classification of quantum phases locally interacting quantum spin systems dissipation in quantum systems computability tensor networks Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-CoG-2014 - ERC Consolidator Grant Call for proposal ERC-2014-CoG See other projects for this call Funding Scheme ERC-COG - Consolidator Grant Host institution UNIVERSIDAD COMPLUTENSE DE MADRID Net EU contribution € 1 462 750,00 Address AVENIDA DE SENECA 2 28040 Madrid Spain See on map Region Comunidad de Madrid Comunidad de Madrid Madrid Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 462 750,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSIDAD COMPLUTENSE DE MADRID Spain Net EU contribution € 1 462 750,00 Address AVENIDA DE SENECA 2 28040 Madrid See on map Region Comunidad de Madrid Comunidad de Madrid Madrid Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 462 750,00