Random-Matrix Approach to RPA equations. I

We study the RPA equations in their most general form by taking the matrix elements appearing in the RPA equations as random. This yields either a unitarily or an orthogonally invariant random-matrix model which is not of the Cartan type. The average spectrum of the model is studied with the help of a generalized Pastur equation. Two independent parameters govern the behaviour of the system: The strength $\alpha^2$ of the coupling between positive- and negative-energy states and the distance between the origin and the centers of the two semicircles that describe the average spectrum for $\alpha^2 = 0$, the latter measured in units of the equal radii of the two semicircles. With increasing $\alpha^2$, positive- and negative-energy states become mixed and ever more of the spectral strength of the positive-energy states is transferred to those at negative energy, and vice versa. The two semicircles are deformed and pulled toward each other. As they begin to overlap, the RPA equations yield non--real eigenvalues: The system becomes unstable. We determine analytically the critical value of the strength for the instability to occur. Several features of the model are illustrated numerically.Comment: 29 pages, 6 figure

Structure of trajectories of complex matrix eigenvalues in the Hermitian-non-Hermitian transition

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.Comment: 12 pages, 3 figure

¿El Polo Norte geográfico terrestre es un polo norte magnético o un polo sur?

Presentamos una actividad alternativa a una práctica de laboratorio habitual en la enseñanza universitaria. Su intención es acercar a los estudiantes a la metodología científica mediante el planteamiento de preguntas relevantes.Palabras claves: Recursos didácticos; Física; investigación escolar; actividad de laboratorio.Is the earth’s North Pole a magnetic north pole or a south pole?We present an activity that differs from what is generally taught in practicals in university teaching laboratories. The aim is to approach scientific methodology by posing relevant questions.Keywords: teaching resources, physics, school research, laboratory activities

Connection between low energy effective Hamiltonians and energy level statistics

We study the level statistics of a non-integrable one dimensional interacting fermionic system characterized by the GOE distribution. We calculate numerically on a finite size system the level spacing distribution $P(s)$ and the Dyson-Mehta $\Delta_3$ correlation. We observe that its low energy spectrum follows rather the Poisson distribution, characteristic of an integrable system, consistent with the fact that the low energy excitations of this system are described by the Luttinger model. We propose this Random Matrix Theory analysis as a probe for the existence and integrability of low energy effective Hamiltonians for strongly correlated systems.Comment: REVTEX, 5 postscript figures at the end of the fil

Time evolution of a quantum many-body system: transition from integrability to ergodicity in thermodynamic limit

Numerical evidence is given for non-ergodic (non-mixing) behavior, exhibiting ideal transport, of a simple non-integrable many-body quantum system in the thermodynamic limit, namely kicked $t-V$ model of spinless fermions on a ring. However, for sufficiently large kick parameters $t$ and $V$ we recover quantum ergodicity, and normal transport, which can be described by random matrix theory.Comment: 4 pages in RevTex (6 figures in PostScript included

Level density for deformations of the Gaussian orthogonal ensemble

Formulas are derived for the average level density of deformed, or transition, Gaussian orthogonal random matrix ensembles. After some general considerations about Gaussian ensembles we derive formulas for the average level density for (i) the transition from the Gaussian orthogonal ensemble (GOE) to the Poisson ensemble and (ii) the transition from the GOE to $m$ GOEs.Comment: 7 pages revtex4, 5 eps figures, submitted to Phys. Rev.

Duality Between the Weak and Strong Interaction Limits for Randomly Interacting Fermions

We establish the existence of a duality transformation for generic models of interacting fermions with two-body interactions. The eigenstates at weak and strong interaction U possess similar statistical properties when expressed in the U=0 and U=infinity eigenstates bases respectively. This implies the existence of a duality point U_d where the eigenstates have the same spreading in both bases. U_d is surrounded by an interval of finite width which is characterized by a non Lorentzian spreading of the strength function in both bases. Scaling arguments predict the survival of this intermediate regime as the number of particles is increased.Comment: RevTex4, 4 pages, 4 figures. Accepted for publication at Phys. Rev. Let

Electronic properties of disordered corner-sharing tetrahedral lattices

We have examined the behaviour of noninteracting electrons moving on a corner-sharing tetrahedral lattice into which we introduce a uniform (box) distribution, of width W, of random on-site energies. We have used both the relative localization length and the spectral rigidity to analyze the nature of the eigenstates, and have determined both the mobility edge trajectories as a function of W, and the critical disorder, Wc, beyond which all states are localized. We find (i) that the mobility edge trajectories (energies Ec vs. disorder W) are qualitatively different from those found for a simple cubic lattice, and (ii) that the spectral rigidity is scale invariant at Wc and thus provides a reliable method of estimating this quantity -- we find Wc/t=14.5. We discuss our results in the context of the metal-to-insulator transition undergone by LiAlyTi{2-y}O4 in a quantum site percolation model that also includes the above-mentioned Anderson disorder, and show that the effects produced by Anderson disorder are far less important than those produced by quantum site percolation, at least in the determination of the doping concentration at which the metal-to-insulator transition is predicted to occur

Disordered ensembles of random matrices

It is shown that the families of generalized matrix ensembles recently considered which give rise to an orthogonal invariant stable L\'{e}vy ensemble can be generated by the simple procedure of dividing Gaussian matrices by a random variable. The nonergodicity of this kind of disordered ensembles is investigated. It is shown that the same procedure applied to random graphs gives rise to a family that interpolates between the Erd\"{o}s-Renyi and the scale free models.Comment: 8 pages, 4 figure

Scaling in Relativistic Thomas-Fermi Approach for Nuclei

By using the scaling method we derive the virial theorem for the relativistic mean field model of nuclei treated in the Thomas-Fermi approach. The Thomas-Fermi solutions statisfy the stability condition against scaling. We apply the formalism to study the excitation energy of the breathing mode in finite nuclei with several relativistic parameter sets of common use.Comment: 13 page