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The critical behavior of entanglement near a quantum phase transition has been studied intensely over the past few years. Few-body quantum systems show critical behavior near the ionization point. We investigate the scaling properties of the von Neumann entropy for an atomiclike system near the ionization threshold. Using finite size scaling methods we calculate the critical charge and the critical exponent associated to the von Neumann entropy. The parallelism between the behavior of entanglement near a quantum phase transition and the behavior of the von Neumann entropy in a critical few-body quantum system is analyzed.

We solve a model of polymers represented by self-avoiding walks on a lattice which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical endpoint and tricritical lines, as well as a multicritical point.From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical endpoints and a line of tricritical points, separated by the multicritical point, is always continuous.This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.

Bound and resonance states of a two-electron quantum dot are studied using a variational expansion with real basis-set functions. The two-electron entanglement ͑von Neumann entropy͒ is calculated as a function of the quantum-dot size at both sides of the critical size, where the ground ͑bound͒ state becomes a resonance ͑unbound͒ state. The use of von Neumann entropy is proposed as a method for the determination of the energy of a resonance.

The delocalization phenomenon was discovered by Hatano et al. and Miller et al. for a class of non-Hermitian quantum-mechanical problems. We show that the delocalization is only one example of many possible critical phenomena which are associated with the selforthogonality of an eigenstate of the non-Hermitian Hamiltonian. It is shown that in this class of problems the self-orthogonality occurs at the series of branch points in the complex energy plane that serve as gates for the "particle" to hop from one Bloch energy band to another one.

We present finite-size scaling and phenomenological renormalization equations for calculations of the critical points of the electronic structure of atoms and molecules. Results show that the method is efficient and very accurate for estimating the critical screening length for one-electron screened Coulomb potentials and the critical nuclear charge for two-electron atoms. The method has potential applicability for many-body quantum systems. [S0031-9007 (97)04408-6] PACS numbers: 31.15. -p, 05.70.Jk

We solve a model of self-avoiding walks with up to two monomers per site on the Bethe lattice. This model, inspired in the Domb-Joyce model, was recently proposed to describe the collapse transition observed in interacting polymers [J. Krawczyk, Phys. Rev. Lett. 96, 240603 (2006)]. When immediate self-reversals are allowed (reversion-allowed model), the solution displays a phase diagram with a polymerized phase and a nonpolymerized phase, separated by a phase transition which is of first order for a nonvanishing statistical weight of doubly occupied sites. If the configurations are restricted forbidding immediate self-reversals (reversion-forbidden model), a richer phase diagram with two distinct polymerized phases is found, displaying a tricritical point and a critical end point.

The finite-size scaling method is used to calculate the critical parameters for the lithium isoelectronic series. The critical nuclear charge, which is the minimum charge necessary to bind three electrons, for the ground state was found to be Z c Ӎ 2. Results show that the analytical behavior of the energy as a function of the nuclear charge for lithium is completely different from that of helium. 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 "second order phase transition." [S0031-9007(98)06405-9] PACS numbers: 05.70.Jk The study of the critical parameters of a quantum Hamiltonian and quantum phase transitions have attracted much interest in recent years [1]. These transitions take place at the absolute zero of temperature, where phase transition means that the quantum ground state of the system changes in some fundamental way as some microscopic parameters change in the Hamiltonian. In atomic and molecular physics, it has been suggested that there are possible analogies between critical phenomena and singularities of the energy [2][3][4]. Analogies between symmetry breaking of electronic structure configurations and standard phase transitions have been established for many electron atoms and simple molecular systems by studying the corresponding large dimension limit Hamiltonian [5,6].Recently, we presented the finite-size scaling (FSS) method to calculate the critical parameters for electronic structure [7,8]. In statistical mechanics, the FSS method provides a way to extrapolate information obtained from a finite system to the thermodynamic limit [9,10]. In our applications, the finite size corresponds to the number of elements in a complete basis set used to expand the exact eigenfunction of a given Hamiltonian. In the FSS method we assumed that the two lowest eigenvalues of the quantum Hamiltonian could be taken as the leading eigenvalues of a transfer matrix of a classical pseudosystem. The phenomenological renormalization equation was used to obtain the critical properties of the system. In this approach one has to rely on the analogy to classical statistical mechanics, a more direct finite-size scaling approach without the need to make such an analogy was established by a systematic expansion in a finite (truncated) basis set [11].The analytical behavior of the energy as a function of parameters for a given system has been the subject of study for many years. In particular, the study of the analytical behavior of the energy as a function of the nuclear charge, Z. Morgan and co-workers [12] have performed a 401-order perturbation calculation to resolve the controversy over the radius of convergence of the l 1͞Z expansion for the ground state energy of the heliumlike ions. Such high order calculations were necessary to study the asymptotic behavior of the perturbation series and to determine that the radius of convergence, l ‫ء‬ , is equal to l c , the critical value of l for which the Hamiltonia...