Crossover phenomena and resonances in quantum systems

Abstract

tonian that exhibit resonances. We show the existence of a crossover phenomenon for the energy spectrum; the transition from a bound state to a continuum is a ‘‘second-order phase transition’’ in one region and a ‘‘firstorder phase transition’’ in another region. As the parameters varied, the numerical value of the critical exponent of the energy levels changed from two to one. In the zone where the critical exponent equals one, the system has narrow resonances and they disappear when the exponent is two. The method has potential applicability in predicting stable and metastable atomic and molecular states. We also show that finite-size scaling methods are capable of detecting multicritical points. DOI: 10.1103/PhysRevA.64.062502 PACS number~s!: 31.15.2p The finite-size scaling method ~FSS! was formulated in statistical mechanics to extrapolate information obtained from a finite system to the thermodynamic limit @1,2# .I n quantum mechanics, when using variation methods, one encounters the same finite-size problem in studying the critical behavior of a quantum Hamiltonian H(l 1 , . .., l k )a s a function of its set of parameters $l i%. In this context, critical means the values of $l i% for which a bound-state energy is nonanalytic. In this paper, a critical point l c is defined as the point where a bound-state energy becomes absorbed or degenerate with a continuum. In the variational calculations, the finite-size corresponds to the number of elements in a complete basis set used to expand the exact wave function of a given Hamiltonian @3#. The problem of what happens when the continuum ‘‘swallows’’ a bound state has a long history with many important results and was reviewed in Ref. @3#. Recently, Neirotti, Serra, and Kais have developed the FSS method to calculate atomic and molecular critical parameters @4,5#. In particular, FSS calculations for two and three electron atoms gave very accurate results for the critical nuclear charges @6,7#. These results have shown that the analytical behavior of the energy as a function of the nuclear charge for lithiumlike atoms is completely different from that of heliumlike atoms. Analogy with standard phase transitions show that for helium, the transition from a bound state to a continuum is ‘‘first-order,’’ while lithium exhibits a ‘‘secondorder phase transition.’’ For the helium sequence, the question of whether or not the radius of convergence, of the 1/Z expansion for the energy, is the same as the variational critical energy was discussed by several authors @ 8‐1 0#. The coincidence between the critical point and the radius of convergence was conjectured by Reinhardt using arguments based on dilatation analyticity @11# and later rigorously confirmed, analytically and numerically, by Baker et al. @12#. Our results for the helium sequence using the finite-size scaling method were in complete agreement with the Reinhardt