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A two-level boson model to study the Casten triangle

Julien Vidal, J. M. Arias, J. Dukelsky, J. E. García-Ramos

Published Feb 22, 2006• 0 citations
OpenAlex

Abstract

We introduce a simple two-level boson model with the same energy surface as the Q-consistent Interacting Boson Model Hamiltonian. The model can be diagonalized for large number of bosons and the results used to check analytical finite-size corrections to the energy gap and the order parameter in the critical region.

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The phase diagram of two-level boson Hamiltonians, including the interacting boson model (IBM), is studied beyond the standard mean field approximation using the Holstein-Primakoff mapping. The limitations of the usual intrinsic state (mean field) formalism concerning finite-size effects are pointed out. The analytic results are compared to numerics obtained from exact diagonalizations. Excitation energies and occupation numbers are studied in different model space regions (Casten triangle for IBM) and especially at the critical points.
Two-level boson systems displaying a quantum phase transition from a spherical (symmetric) to a deformed (broken) phase are studied. A formalism to diagonalize Hamiltonians with O(2L+1)O(2L+1)O(2L+1) symmetry for large number of bosons is worked out. Analytical results beyond the simple mean-field treatment are deduced by using the continuous unitary transformations technique. In this scheme, a 1/N1/N1/N expansion for different observables is proposed and allows one to compute the finite-size scaling exponents at the critical point. Analytical and numerical results are compared and reveal the power of the present approach to compute the finite-size corrections in such a context.
The evolution of system observables at critical points of quantum phase transitions in atomic nuclei is explored within the context of the interacting boson model-1. The evolution of energies and transition strengths in the ground state band at the infinite-size critical point are examined with respect to system size. Properties of first-order phase transitions away from the U(5)-SU(3) leg of the Casten triangle are examined, and the effects of angular momentum on the evolution of system observables at the critical point are investigated for the first time.
Exact numerical results of the interacting boson model Hamiltonian along the integrable line from U(5) to O(6) are obtained by diagonalization within boson seniority subspaces. The matrix Hamiltonian reduces to a block tridiagonal form that can be diagonalized for large number of bosons. We present results for the low-energy spectrum and the transition probabilities for systems up to 10,000 bosons, which confirm that at the critical point the system is equally well described by the Bohr Hamiltonian with a ${\ensuremath{\beta}}^{4}$ potential.