E. M. Santangelo

Spectral asymmetry for bag boundary conditions

We give an expression, in terms of boundary spectral functions, for the spectral asymmetry of the Euclidean Dirac operator in two dimensions, when its domain is determined by local boundary conditions and the manifold is of product type. As an application, we explicitly evaluate the asymmetry in the case of a finite-length cylinder and check that the outcome is consistent with our general result. Finally, we study the asymmetry in a disc, which is a non-product case, and propose an interpretation.

Planar QED at finite temperature and density: Hall conductivity, Berry's phases and minimal conductivity of graphene

We study one-loop effects for massless Dirac fields in two spatial dimensions, coupled to homogeneous electromagnetic backgrounds, both at zero and at finite temperature and density. In the case of a purely magnetic field, we analyze the relationship among the invariance of the theory under large gauge transformations, the appearance of Chern–Simons terms and of different Berrys phases. In the case of a purely electric background field, we show that the effective Lagrangian is independent of the chemical potential and of the temperature. More interesting, we show that the minimal conductivity, as predicted by the quantum field theory, is the right multiple of the conductivity quantum and is, thus, consistent with the value measured for graphene, with no extra factor of π in the denominator.

Strong ellipticity and spectral properties of chiral bag boundary conditions

We prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds. From a knowledge of the heat kernel in an infinite cylinder, some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension.

The quantum Hall effect in graphene samples and the relativistic Dirac effective action

We study the Euclidean effective action per unit area and the charge density for a Dirac field in a two-dimensional (2D) spatial region, in the presence of a uniform magnetic field perpendicular to the 2D plane, at finite temperature and density. In the limit of zero temperature we reproduce, after performing an adequate Lorentz boost, the Hall conductivity measured for different kinds of graphene samples, depending upon the phase choice in the fermionic determinant.

Finite temperature properties of the Dirac operator under local boundary conditions

We study the finite-temperature free energy and fermion number for Dirac fields in a one-dimensional spatial segment, under two different members of the family of local boundary conditions defining a self-adjoint Euclidean Dirac operator in two dimensions. For one of such boundary conditions, compatible with the presence of a spectral asymmetry, we discuss in detail the contribution of this part of the spectrum to the zeta-regularized determinant of the Dirac operator and, thus, to the finite-temperature properties of the theory.

The finite-temperature relativistic Landau problem and the relativistic quantum Hall effect

This paper presents a study of the free energy and particle density of the relativistic Landau problem, and their relevance to the quantum Hall effect. First we study the zero-temperature Casimir energy and fermion number for Dirac fields in a (2+1)-dimensional Minkowski spacetime, in the presence of a uniform magnetic field perpendicular to the spatial manifold. Then, we go to the finite-temperature problem, with a chemical potential, introduced as a uniform zero component of the gauge potential. By performing a Lorentz boost, we obtain Halls conductivity in the case of crossed electric and magnetic fields.

Quantum Hall effect in graphene: A functional determinant approach

We start the paper with a brief presentation of the main characteristics of graphene, and of the Dirac theory of massless fermions in 2+1 dimensions obtained as the associated low-momentum effective theory, in the absence of external fields. We then summarize the main steps needed to obtain the Hall conductivity in the effective theory at finite temperature and density, with emphasis on its dependence on the phase of the Dirac determinant selected during the evaluation of the effective action. Finally, we discuss the behavior, under gauge transformations, of the contribution due to the lowest Landau level, and interpret gauge transformations as rotations of the corresponding spinors around the magnetic field.

Heat trace asymptotics and the Gauss?Bonnet theorem for general connections

We examine the local super trace asymptotics for the de Rham complex defined by an arbitrary super connection on the exterior algebra. We show, in contrast to the situation in which the connection in question is the Levi-Civita connection, that these invariants are generically non-zero in positive degree and that the critical term is not the Pfaffian.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowkers 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

Spectral asymmetry on the ball and asymptotics of the asymmetry kernel

Let be the Dirac operator on a D = 2d dimensional ball with radius R. We calculate the spectral asymmetry for D = 2 and D = 4, when local chiral bag boundary conditions are imposed. With these boundary conditions, we also analyse the small-t asymptotics of the heat trace where P is an operator of Dirac type and F is an auxiliary smooth smearing function.

Searching for a

We present a detailed analytic study on the three-dimensional sphere of the most popular candidates for