23 research outputs found
The ground state construction of bilayer graphene
We consider a model of half-filled bilayer graphene, in which the three
dominant Slonczewski-Weiss-McClure hopping parameters are retained, in the
presence of short range interactions. Under a smallness assumption on the
interaction strength as well as on the inter-layer hopping , we
construct the ground state in the thermodynamic limit, and prove its
analyticity in , uniformly in . The interacting Fermi surface is
degenerate, and consists of eight Fermi points, two of which are protected by
symmetries, while the locations of the other six are renormalized by the
interaction, and the effective dispersion relation at the Fermi points is
conical. The construction reveals the presence of different energy regimes,
where the effective behavior of correlation functions changes qualitatively.
The analysis of the crossover between regimes plays an important role in the
proof of analyticity and in the uniform control of the radius of convergence.
The proof is based on a rigorous implementation of fermionic renormalization
group methods, including determinant estimates for the renormalized expansion
Plate-nematic phase in three dimensions
We consider a system of anisotropic plates in the three-dimensional
continuum, interacting via purely hard core interactions. We assume that the
particles have a finite number of allowed orientations. In a suitable range of
densities, we prove the existence of a uni-axial nematic phase, characterized
by long range orientational order (the minor axes are aligned parallel to each
other, while the major axes are not) and no translational order. The proof is
based on a coarse graining procedure, which allows us to map the plate model
into a contour model, and in a rigorous control of the resulting contour
theory, via Pirogov-Sinai methods.Comment: 29 pages, 4 figure
A Pfaffian formula for monomer-dimer partition functions
We consider the monomer-dimer partition function on arbitrary finite planar
graphs and arbitrary monomer and dimer weights, with the restriction that the
only non-zero monomer weights are those on the boundary. We prove a Pfaffian
formula for the corresponding partition function. As a consequence of this
result, multipoint boundary monomer correlation functions at close packing are
shown to satisfy fermionic statistics. Our proof is based on the celebrated
Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the
authors, and a careful labeling and directing procedure of the vertices and
edges of the graph.Comment: Added referenc