Topologia, Simetria e transições de fase em Modelos de Spin
AUTOR(ES)
Fernando Antonio Nóbrega Santos
DATA DE PUBLICAÇÃO
2009
RESUMO
We use a topological approach to describe the frustration- and field-induced phase transitions exhibited by the infinite-range XY model on the AB2 chain, including noncollinear spin structures. For this purpose, we have computed the Morse number and the Euler characteristic, as well as other topological invariants, which are found to behave similarly as function of the energy level in the context of Morse Theory. In particular, we use a method based on an analogy with statistical mechanics to compute the Euler characteristic, which proves to be quite feasible. We also introduce topological energies which help to clarify several properties of the transitions, both at zero and finite temperature. In addition, we establish a nontrivial direct connection between the thermodynamics of the systems, which have been solved exactly under the saddle point approach, and the topology of their configuration space. This connection allow us to identify the non-degeneracy condition under which the divergence of the density of Jacobians critical points (jl(E)) at the critical energy of a topology-induced phase transition, proposed by Kastner and Schnetz [Phys. Rev. Lett. 100, 160601 (2008)] as a necessary criterion, is suppressed. Our findings, and those available in the literature, suggest that the cusp-like singularity exhibited both by the Euler characteristic and the topological contribution for the entropy at the critical energy, put together with the divergence of jl(E), emerge as necessary and sufficient conditions for the occurrence of the finite-temperature topology-induced phase transitions examined in this work. The general character of this proposal should be subject to a more rigorous scrutiny. Finally, we discuss the concept of the integration with respect to the Euler characteristic and its relationship with thermodynamics and phase transitions. These ideas are used to study the the infinite-range XY model. In particular, combining statistical mechanics and Morse theory, we determine the phase transition critical temperature of the infinite-range XY model using the Euler characteristic. Moreover, we provide evidence that the information embedded in the Euler characteristic suffice to determine the magnetization, in the microcanonical ensemble, except for the metastable solutions.
ASSUNTO(S)
phase transitions, topology, noncollinear spin structures, euler characteristic fisica transições de fase topologia simetria modelos de spin característica de euler
