39 research outputs found
Landau levels on a torus
Landau levels have represented a very rich field of research, which has
gained widespread attention after their application to quantum Hall effect. In
a particular gauge, the holomorphic gauge, they give a physical implementation
of Bargmann's Hilbert space of entire functions. They have also been recognized
as a natural bridge between Feynman's path integral and Geometric Quantization.
We discuss here some mathematical subtleties involved in the formulation of the
problem when one tries to study quantum mechanics on a finite strip of sides
L_1, L_2 with a uniform magnetic field and periodic boundary conditions. There
is an apparent paradox here: infinitesimal translations should be associated to
canonical operators [p_x,p_y] \propto i\hslash B, and, at the same time, live
in a Landau level of finite dimension B L_1L_2/(hc/e), which is impossible from
Wintner's theorem. The paper shows the way out of this conundrum.Comment: 10 pages, two color eps-file
The Numerical Sausage
The renormalization group equation describing the evolution of the metric of
the non linear sigma models poses some nice mathematical problems involving
functional analysis, differential geometry and numerical analysis. We describe
the techniques which allow a numerical study of the solutions in the case of a
two-dimensional target space (deformation of the --model. Our
analysis shows that the so-called sausages define an attracting manifold in the
U(1) symmetric case, at one-loop level. The paper describes i) the known
analytical solutions, ii) the spectral method which realizes the numerical
integrator and allows to estimate the spectrum of zero--modes, iii) the
solution of variational equations around the solutions, and finally iv) the
algorithms which reconstruct the surface as embedded in .Comment: 15 pages, uuencoded postscript fil
High energy gravitational scattering: a numerical study
The S-matrix in gravitational high energy scattering is computed from the
region of large impact parameters b down to the regime where classical
gravitational collapse is expected to occur. By solving the equation of an
effective action introduced by Amati, Ciafaloni and Veneziano we find that the
perturbative expansion around the leading eikonal result diverges at a critical
value signalling the onset of a new regime. We then discuss the main features
of our explicitly unitary S-matrix down to the Schwarzschild's radius R=2G
s^(1/2), where it diverges at a critical value b ~ 2.22 R of the impact
parameter. The nature of the singularity is studied with particular attention
to the scaling behaviour of various observables at the transition. The
numerical approach is validated by reproducing the known exact solution in the
axially symmetric case to high accuracy.Comment: 11 pages, 6 figure
perturbation of hydrogen degenerate levels and so 4
We present a short note about the perturbative correction to Rydberg energies under a perturbation cosΞ/rΌ and discuss the role of SO(4) symmetry
Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
Liouville field theory on a sphere is considered. We explicitly derive a
differential equation for four-point correlation functions with one degenerate
field . We introduce and study also a class of four-point
conformal blocks which can be calculated exactly and represented by finite
dimensional integrals of elliptic theta-functions for arbitrary intermediate
dimension. We study also the bootstrap equations for these conformal blocks and
derive integral representations for corresponding four-point correlation
functions. A relation between the one-point correlation function of a primary
field on a torus and a special four-point correlation function on a sphere is
proposed
Phylogeography and genomic epidemiology of SARS-CoV-2 in Italy and Europe with newly characterized Italian genomes between February-June 2020
SPLITTING LANDAU LEVELS ON THE 2D TORUS BY PERIODIC PERTURBATIONS
We study the spectrum of the Schroedinger operator for a particle constrained on a two-dimensional flat torus under the combined action of a transverse magnetic field and a conservative force. A numerical method is presented which allows to compute the spectrum with high accuracy. The method employs a fast Fourier transform to accurately represent the momentum variables and takes into account the twisted boundary conditions required by the presence of the magnetic field. An accuracy of 12 digits is attained even with coarse grids. Landau levels are reproduced in the case of a uniform magnetic field satisfying Dirac's condition. A new fine structure of levels within the single Landau level is formed when the field has a sinusoidal component with period commensurable to the integer magnetic charge. This fact is interpreted in terms of the peculiar symmetry Z(N) x Z(N) which holds in the unperturbed case