Kavli Affiliate: Nathan Lewis
| First 5 Authors: Kassahun H. Betre, Kassahun H. Betre, , ,
| Summary:
Motivated by applications in background-independent quantum gravity, we
discuss the quantization of labeled and unlabeled finite multigraphs with a
maximum edge count. We provide a unified way to represent quantum multigraphs
with labeled or unlabeled vertices, which enables the study of quantum
multigraphs as dynamical microscopic degrees of freedom and not just as
representations of relations among quantum states of particles. The quantum
multigraphs represent a quantum mechanical treatment of the relations
themselves and give rise to Hilbert space realizations of relations. After
defining the Hilbert space, we focus on quantum simple graphs and explore the
thermodynamics resulting from two simple models, a free Hamiltonian and an
Ising-type Hamiltonian (with interactions among nearest-neighbor edges). We
show that removing the distinction among vertices by considering unlabeled
vertices gives rise to a qualitatively different thermodynamics. We find that
the free theory of labeled quantum simple graphs is the
ErdHos–R’enyi–Gilbert $G(N,p)$ model of random graphs. This model has
analytic free energy and hence no thermodynamic phase transition. On the other
hand, the unlabeled quantum graphs give rise to proper thermodynamic phase
transitions in both the free and the ferromagnetic Ising models, characterized
by divergence in the specific heat and critical slowing near the critical
temperature. The thermodynamic phase transition has an order parameter given as
the fraction of vertices in the largest connected component. Although this is
similar to the phase transition in the $G(N,p)$ model, in this case it
represents the actual thermodynamic phase transition.
| Search Query: ArXiv Query: search_query=au:”Nathan Lewis”&id_list=&start=0&max_results=3