Finite-size scaling exponents in the interacting boson model

Abstract

We investigate the finite-size scaling exponents for the critical point at the shape-phase transition from U(5) (spherical) to O(6) (deformed γ -unstable) dynamical symmetries of the interacting boson model, making use of the Holstein-Primakoff boson expansion and the continuous unitary transformation technique. We compute exactly the leading-order correction to the ground-state energy, the gap, the expectation value of the d-boson number in the ground state and the E2 transition probability from the ground state to the first excited state and determine the corresponding finite-size scaling exponents. The interest in the study of quantum phase transitions (QPT) has kept growing in the last years in different branches of quantum many-body physics, ranging from macroscopic systems such as quantum magnets, high-T c superconductors [1] or dilute Bose and Fermi gases [2] to mesoscopic systems such as atomic nuclei or molecules [3]. Although, strictly speaking, QPT occurs only in macroscopic systems, there is a renewed interest in studying structural changes in finitesize systems where precursors of the transition are already observed [4]. The understanding of the modifications on the characteristics of the QPT induced by finite-size effects is of crucial importance to extend the concept of phase transitions to finite systems.In the present study, we analyze these finite-size corrections in the interacting boson model (IBM) of nuclei [5], but the same technique can be applied to other boson systems, for instance, to the molecular vibron model [6] or to a multilevel boson model of Bose-Einstein condensates where similar QPT take place [7].The IBM is a two-level boson model that includes an angular momentum L = 0 boson (scalar s boson) and five angular momentum L = 2 bosons (quadrupole d-bosons) separated by an energy gap. The s and d bosons represent s-and d-wave idealized Cooper nucleon pairs. The algebraic structure of this model is governed by the U(6) group and the model has three dynamical symmetries in which the Hamiltonian, written in terms of the invariant (Casimir) operators of a nested chain of subgroups of U(6), is analytically solvable. The dynamical symmetries are named by the first subgroup in the chain: U(5), SU(3), and O(6). The classical or thermodynamic limit of the model was investigated by using an intrinsic state formalism that introduces the shape variables β and γ [8][9][10]. Within this geometric picture the U(5), SU(3), and O(6) dynamical symmetries correspond to spherical, axially deformed, and deformed γ -unstable shapes, respectively. Transition between two of these dynamical symmetry limits are described in terms of a Hamiltonian with a control parameter that mixes the Casimir operators of the two dynamical symmetries. As a function of the control parameter, the system crosses smoothly a region of structural changes in the ground-state wave function for finite number N of bosons. In the large N limit, the smooth crossover turns into a sharp QPT between two well-defined shape phases [9,[11][...

Emergent dynamical symmetry at the triple point of nuclear deformations

“…It can be clearly observed from Fig. 2 that the low-lying spectrum of 108 Pd can be well described by the Eu(5) pattern determined by (40). Particularly, the relative B(E2) strengths in the Eu(5) DS are independent of the parameters, while the data of 108 Pd seem to be well reproduced by those of the Eu(5) DS.…”

First-order quantum phase transitions: Test ground for emergent chaoticity, regularity and persisting symmetries

“…In the present work, we choose β 0 = √ 2, for which the intrinsic Hamiltonian interpolates between the U(5) and SU(3) dynamical symmetries and various expressions simplify, since η = 0 in Eq. (40). For convenience, Table 2 lists the values of the relevant control and order parameters when β 0 = √ 2.…”

Parity-symmetry-adapted coherent states and entanglement in quantum phase transitions of vibron models

“…[34]. However, this symmetry goes unnoticed when proposing ansatzs to approximate the ground state energy in terms of (nonsymmetric) 'projective' [16] (or 'Hartree axial' [31]) CSs…”