Singularities in the Hamiltonian at electronic degeneracies

Journal of Physics A: Mathematical and General, 2001

In this work we study the Landau levels in the presence of topological defects.

The realization of a genuine phase transition in quantum mechanics requires that at least one of the Kato's exceptional-point parameters becomes real. A new family of finite-dimensional and time-parametrized quantum-lattice models with such a property is proposed and studied. All of them exhibit, at a real exceptional-point time t=0, the Jordan-block spectral degeneracy structure of some of their observables sampled by Hamiltonian H(t) and site-position Q(t). The passes through the critical instant t=0 are interpreted as schematic simulations of non-equivalent versions of the Big-Bang-like quantum catastrophes.

Modern Physics Letters B, 2005

We review investigations on defects in systems described by real scalar fields in (D, 1) space-time dimensions. We first work in one spatial dimension, with models described by one and two real scalar fields, and in higher dimensions. We show that when the potential assumes specific form, there are models which support stable global defects for D arbitrary. We also show how to find first-order differential equations that solve the equations of motion, and how to solve models in D dimensions via soluble problems in D = 1. We illustrate the procedure examining specific models and showing how they may be used in applications in different contexts in condensed matter physics, and in other areas.

Zeitschrift für Physik B Condensed Matter, 1996

A large number of energy levels in different invariant subspaces of the anisotropic XY -model are quasidegenerate in a wide parameter range, i.e., their spacing is much smaller than the mean level spacing of the system. These quasi-degeneracies can be interpreted in two ways: (i) as tunnel splitting, (ii) as weak level splitting related to a parametric point of exact degeneracy. Starting from the second interpretation we calculate the tunnel splitting by use of perturbative methods.

Basics and Highlights in Fundamental Physics - Proceedings of the International School of Subnuclear Physics, 2001

Phys Rev B, 2001

We consider the two-dimensional t-t' Hubbard model with the Fermi level being close to Van Hove singularities. The phase diagram of the model is discussed. Far from the quantum phase transition (QPT) into ferro- (antiferro-) magnetic states, the self-energy at the singularity points has a nearly linear energy dependence, the density of states being proportional to ln3\|ε\|. In the quantum-disordered and quantum-critical regimes near the QPT a scaling approach is used. At small energies the self-energy demonstrates a powerlike behavior with the exponent changing from unity to zero in a narrow energy interval. Provided that the system is close to ferromagnetic instability which occurs at t'/t-->-1/2, the quasiparticle spectral weight has, instead of a narrow peak, a broad maximum and is nonmonotonous at small energies. The application of the results to cuprates is discussed.

Journal of Physics A: Mathematical and General, 2001

The Hamiltonian H = p 2 +x 4 +iAx, where A is a real parameter, is investigated. The spectrum of H is discrete and entirely real and positive for |A| < 3.169. As |A| increases past this point, adjacent pairs of energy levels coalesce and then become complex, starting with the lowest-lying energy levels. For large energies, the values of A at which this merging occurs scale as the three-quarters power of the energy. That is, as |A| → ∞ and E → ∞, at the points of coalescence the ratio a = |A|E −3/4 approaches a constant whose numerical value is a crit = 1.1838363072914 · · ·. Conventional WKB theory determines the high-lying energy levels but cannot be used to calculate a crit . This critical value is predicted exactly by complex WKB theory.

Doklady Physics, 2005

2021

Singularity of the potential function makes quantum tunneling, the case 0 ( ) /  V x V x  of which is the subject of this article, mathematically underdetermined. To circumvent the difficulties it introduced in physics, a potential singularity cutoff is often used, followed by a reverse limit transition, or is a suitable self-adjoint extension of the Hamiltonian along the entire coordinate axis made. However, both of them somehow affect the singular nature of the problem, and so I discuss here how quantum tunneling will behave if the original singular nature of the Schrodinger equation left untouched. To do this, I use the property of the probability density current that the singularities are mutually destroyed in it. It is found that the mildly singular potential with 0 1    has a finite, but unusual tunnelling transparency, in particular, a non-zero value at zero energy of the incident particle. Тhe tunneling of 1D Coulomb potential ( 1   ) exhibits infinitely fast and comp...

Physical Review D, 1994

c) Address after October 1st 1993: School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, U.K.