J. C. A. Barata

The Moore–Penrose Pseudoinverse: A Tutorial Review of the Theory

In the last decades, the Moore–Penrose pseudoinverse has found a wide range of applications in many areas of science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore–Penrose pseudoinverse and in related subjects, such as the singular value decomposition theorem for matrices. In this note, we present a tutorial review of the theory of the Moore–Penrose pseudoinverse. We present the first definitions and some motivations, and after obtaining some basic results, we center our discussion on the spectral theorem and present an algorithmically simple expression for the computation of the Moore–Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented, and for those who already have some working knowledge of the subject.

Strong-coupling theory of two-level atoms in periodic fields

We present a new convergent strong-coupling expansion for two-level atoms in external periodic fields, free of secular terms. As a first application, we show that the coherent destruction of tunneling is a third-order effect. We also present an exact treatment of the high-frequency region, and compare it with the theory of averaging. The qualitative frequency spectrum of the transition probability amplitude contains an effective Rabi frequency.

ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR A TWO-LEVEL SYSTEM WITH A HAMILTONIAN DEPENDING QUASI–PERIODICALLY ON TIME

We consider the Schrodinger equation for a class of two-level atoms in a quasi-periodic external field for large coupling, i.e. for which the energy difference 2∊ between the unperturbed levels is sufficiently small. We show that this equation has a solution in terms of a formal power series in ∊, with coefficients which are quasi-periodical functions of the time, in analogy to the Lindstedt–Poincare series in classical mechanics.

On the Distribution and Gap Structure of Lee–Yang Zeros for the Ising Model: Periodic and Aperiodic Couplings

In this work, we present some results on the distribution of Lee–Yang zeros for the ferromagnetic Ising model on the rooted Cayley Tree (Bethe Lattice), assuming free boundary conditions, and in the one-dimensional lattice with periodic boundary conditions. In the case of the Cayley Tree, we derive the conditions that the interactions between spins must obey in order to ensure existence or absence of phase transition at finite temperature (T≠0). The results are first obtained for periodic interactions along the generations of the lattice. Then, using periodic approximants, we are also able to obtain results for aperiodic sequences generated by substitution rules acting on a finite alphabet. The particular examples of the Fibonacci and the Thue-Morse sequences are discussed. Most of the results are obtained for a Cayley Tree with arbitrary order d. We will be concerned in showing whether or not the zeros become dense in the whole unit circle of the fugacity variable. Regarding the one-dimensional Ising model, we derive a general treatment for the structure of gaps (regions free of Lee–Yang zeros) around the unit circle.

Time evolution of two-level systems driven by periodic fields

Abstract We describe a method to compute perturbative expansions for two-level systems driven by periodic time-dependent fields. Our expansions do not contain secular terms being, therefore, convenient for the study of long-time properties. We apply the method to the case of ac–dc fields and analyse situations where an approximate effect of dynamical localisation is exhibited.

Griffiths' singularities in diluted ising models on the Cayley tree

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.

Aspects of two-level systems under external time-dependent fields

The dynamics of two-level systems in time-dependent backgrounds is under consideration. We present some new exact solutions in special backgrounds decaying in time. On the other hand, following ideas of Feynman, Vernon and Hellwarth, we discuss in detail the possibility to reduce the quantum dynamics to a classical Hamiltonian system. This, in particular, opens the possibility to directly apply powerful methods of classical mechanics (e.g. KAM methods) to study the quantum system. Following such an approach, we draw conclusions of relevance for “quantum chaos” when the external background is periodic or quasi-periodic in time.

Particle scattering in Euclidean lattice field theories

A Haag-Ruelle Scattering Theory for Euclidean Lattice Field Theories is developed.

Topological low-temperature limit of Z(2) spin-gauge theory

We study Z(2) lattice gauge theory on triangulations of compact d-manifolds. We reformulate the theory algebraically, describing it in terms of the structure constants of a bidimensional vector space H equipped with algebra and coalgebra structures, and prove that in the low-temperature limit H reduces to a Hopf algebra, in which case the theory becomes equivalent to a topological field theory. The degeneracy of the ground state is shown to be a topological invariant. This fact is used to compute the zeroth- and first-order terms in the low-temperature expansion of Z for arbitrary triangulations. In finite temperatures, the algebraic reformulation gives rise to new duality relations among classical spin models, related to changes of basis of H.

Electrically and magnetically charged states and particles in the 2+1-dimensional ℤ N -Higgs gauge model

Electrically as well as magnetically charged states are constructed in the 2+1-dimensional Euclidean ℤN-Higgs lattice gauge model, the former following ideas of Fredenhagen and Marcu [1] and the latter using duality transformations on the algebra of observables. The existence of electrically and of magnetically charged particles is also established. With this work we prepare the ground for the constructive study of anyonic statistics of multiparticle scattering states of electrically and magnetically charged particles in this model.