Ground states and associated path measures in the renormalized Nelson model

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson’s operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

Accurate analytic solution for ideal boson gases in a highly anisotropic two-dimensional harmonic trap

Motivated by quantum statistical mechanics, we propose an accurate analytical solution to the problem of Bose–Einstein condensation (BEC) of ideal bosons in a two-dimensional anisotropic harmonic trap. The study reveals that the number of noncondensed bosons is characterized by an analytical function, which relates to a series expansion of q-digamma functions in mathematics. The q-digamma function is a function of temperature, boson number, and anisotropic parameter. The analytical solution describes fully the experimental results of the BEC of ideal bosons in a two-dimensional anisotropic harmonic trap. We derive the analytical expressions of the critical temperature and the condensate fraction in the thermodynamic limit. The first main conclusion is that for a fixed temperature and boson number, there is a critical anisotropic parameter, which is the precise onset of BEC in this harmonically trapped two-dimensional system. The second main conclusion is that the critical temperature in a two-dimensional anisotropic harmonic trap is larger than that in a two-dimensional isotropic harmonic trap.

Line of approximate SU(3) symmetry inside the symmetry triangle of the Interacting Boson Model

The U(5), SU(3), and O(6) symmetries of the Interacting Boson Model (IBM) have been traditionally placed at the vertices of the symmetry triangle, while an O(5) symmetry is known to hold along the U(5)–O(6) side of the triangle. We construct [1] for the first time a symmetry line in the interior of the triangle, along which the SU(3) symmetry is preserved. This is achieved by using the contraction of the SU(3) algebra to the algebra of the rigid rotator in the large boson number limit of the IBM. The line extends from the SU(3) vertex to near the critical line of the first order shape/phase transition separating the spherical and prolate deformed phases. It lies within the Alhassid–Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices amidst chaotic regions, thus providing an explanation for its existence.

What can we learn from the Interacting Boson Model in the limit of large boson numbers?

Over the years, studies of collective properties of medium and heavy mass nuclei in the framework of the Interacting Boson Approximation (IBA) model have focused on finite boson numbers, corresponding to valence nucleon pairs in specific nuclei. Attention to large boson numbers has been motivated by the study of shape/phase transitions from one limiting symmetry of IBA to another, which become sharper in the large boson number limit, revealing in parallel regularities previously unnoticed, although they survive to a large extent for finite boson numbers as well. Several of these regularities will be discussed. It will be shown that in all of the three limiting symmetries of the IBA [U(5), SU(3), and O(6)], energies of 0+ states grow linearly with their ordinal number. Furthermore, it will be proved that the narrow transition region separating the symmetry triangle of the IBA into a spherical and a deformed region is described quite well by the degeneracies E(0^+_2 ) = E(6^+_1 ), E(0^+_3 ) = E(10^+_1 ), E(0^+_4 ) = E(14^+_1 ), the energy ratio E(6^+_1 )/E(0^+_2 ) turning out to be a simple, empirical, easy-to-measure effective order parameter, distinguishing between first- and second-order transitions. The energies of 0+ states near the point of the first order shape/phase transition between U(5) and SU(3) will be shown to grow as n(n+3), where n is their ordinal number, in agreement with the rule dictated by the relevant critical point symmetries studied in the framework of special solutions of the Bohr Hamiltonian. The underlying dynamical and quasi-dynamical symmetries are also discussed.

Approximate symmetries in the Interacting Boson Model

Dynamical symmetries have played a central role for many years in the study of nuclear structure. Recently, the concepts of Partial Dynamical Symmetry (PDS) and Quasi-Dynamical Symmetry (QDS) have been introduced. We shall discuss examples of PDS and QDS appearing in the large boson number limit of the Interacting Boson Mod

Study of shape coexistence in the 180−190Hg isotopes by SO(6) representation of eigenstates

In this paper, we have studied the shape coexistence in the [Formula: see text]Hg isotopes. The SO(6) representation of eigenstates and a transitional Hamiltonian in the Interacting Boson Model (IBM) are used to consider the evolution from prolate to oblate shapes for systems with total boson number [Formula: see text]. Parameter free (up to overall scale factors) predictions for energy spectra and quadrupole transition rates are found to be in good agreement with experimental counterparts. The results for the control parameter of transitional Hamiltonian offer a combination of spherical and deformed shapes in these Hg isotopes and also more deviation from SO(6) limit is observed when the quadrupole deformation is decreased. Also, there are some suggestions about the expectation values of the [Formula: see text] operator which are determined in the first state of ground, beta and gamma bounds and the control parameter of model.

Study of Yb isotopes with the coherent state formalism

Coherent state approach (CSA) is applied to the interacting boson model (IBM) in the rotational region. States of the [Formula: see text] particle system are built up in a most general form as a function of the shape parameters [Formula: see text] and [Formula: see text]. The parameter [Formula: see text] occurring in the quadrupole operator which determines the degree of the [Formula: see text]-softness is taken to be varying with the boson number. The excitation energy equations are formed by using the moment of inertia of a rotating system which is obtained from the solution of the cranking problem. For each isotope, the deformation parameter is found by minimizing the ground state energy equation. The results are used to fit the experimental excitation energy spectrum and the electric quadrupole transition ratios of Yb isotopes. It is found that CSA works well to predict the energy of the states up to [Formula: see text]. But the predictions about the electric quadrupole transitions are nearly [Formula: see text] times greater than the experimental data.

Finding the boson-number distributions in superconducting thin-film rings

A theory of the infrared (IR)-field-induced single-photon generation by the narrow thin-film superconducting rings made of the isotropic s-wave pairing type-II superconductors is presented. It is shown that statistical measurements of the energies of photons emitted by the same current-carrying ring prepared initially in the same quantum state, allow to find the number distribution of Cooper pairs in the superconductor.