Dynamical aspects of lattice fermions with long-range interactions

• Grant number: 14/26137-5
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2015
• End date: February 28, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
• Principal Investigator: Walter Alberto de Siqueira Pedra
• Grantee: Rafael Sussumu Yamaguti Miada
• Host Institution: Instituto de Física (IF). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

[1] studies fermions on the lattice with long-range interactions and shows that, exactly as in the case of short-range interactions [2, 3], equilibrium states can be properly defined by minimization of a free-energy density functional on a convenient locally convex space. Here, the term long-range interaction refers to the presence of mean-field components in the interaction. Equilibrium states of short-range interactions can alternatively be defined by the KMS property w.r.t. the group of automorphisms generated by the interaction [2, 3]. It is well-known that KMS states which are invariant (w.r.t. space translations) are exactly those invariant states minimizing the free-energy density [2]. Hence, in the case of short-range interactions, both approaches to equilibrium states are equivalent, in which concerns invariant states. If the interaction is long-ranged, the use of the KMS property, which is a dynamical property, to define equilibrium states is not immediate: in general, it is not clear from the beginning whether the finite volume dynamics converges to a strongly continuous dynamics for the whole observable algebra in the limit of large volume; see, for instance, [7]. However, as observed by Haag [8], the dynamics does possess a well-defined large volume limit for appropriate representations of the observable algebra, in the sense of the strong operator topology, even in the case of long-range interactions. Van Hemmen [9] used this fact in order to define equilibrium states via the KMS property for infinite volume systems with long-range interactions.The main aim of Rafael Miada will be to show, for a large class of long-range interactions, that the minimizers of the free-energy density are equilibrium states in the sense of van Hemmen. Other questions related to this fact will also be considered. For instance, it would be desirable to understand the relation between the large volume limit of time-correlation functions for (finite volume) Gibbs states and such functions for (infinite volume) equilibrium states in the sense of van Hemmen. Such relations were already investigated for an exactly solvable model [7]. But, as far as we know, still present, there is no general result. We believe to be able to give some contribution to the unterstanding of the questions described above, by using a general result of [1] on the structure of minimizers of the free-energy for a very general class of long-range interactions. Indeed, [1] shows that such minimizers can be always represented as convex integrals on minimizers of the free-energy density of (effective) short-range interactions. By this reason, one can expect that such representations of minimizers of the free-energy density for long-range interactions will make possible to use what is known for KMS states of short-range interactions also for a class of long-range models. (AU)