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Physical Review A

Finite-size corrections in the bosonic algebraic approach to two-dimensional systems

P. Pérez-Fernández, J. M. Arias, J. E. García-Ramos, Francisco Pérez-Bernal

Published Jun 30, 2011• 18 citations
Publisher

Abstract

Analytical results beyond the mean-field-limit approximation for several observables of the two-dimensional limit of the vibron model are presented. Finite-size scaling exponent values are also analytically derived. The computed corrections and scaling exponents are compared to numerical calculations for this algebraic model, based on a bosonic u(3) spectrum-generating algebra, obtaining good agreement.

DOI
10.1103/physreva.83.062125

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Analytical results beyond the mean-field-limit approximation for several observables of the two-dimensional limit of the vibron model are presented. Finite-size scaling exponent values are also analytically derived. The computed corrections and scaling exponents are compared to numerical calculations for this algebraic model, based on a bosonic u(3) spectrum-generating algebra, obtaining good agreement.
We study in detail the relation between the two-dimensional Euclidean dynamical E(2) symmetry and the quantum phase transition in the two-dimensional limit of the vibron model, called the U(3) vibron model. Both geometric and algebraic descriptions of the U(3) vibron model show that structures of low-lying states at the critical point of the model with a quartic potential as its classical limit can be approximately described by the E(2) symmetry. We also fit the finite-size scaling exponent of the energy levels and E1 transition rates in the F(2) model, which is exactly the E(2) model but with truncation in its Hilbert subspace, as well as those at the critical point in the U(3) vibron model. The N-scaling power law around the critical point shows that the E(2) symmetry is well preserved even for cases with finite number of bosons. In addition, two kinds of experimentally accessible effective order parameters, such as the energy ratios E21/E11, E31/E 11 and E1 transition ratios B(E1;21→1 1)/B(E1;11→11), B(E1;02→ 11)/B(E1;11→01), are proposed to identify the second-order phase transition in such systems. Possible empirical examples exhibiting approximate E(2) symmetry are also presented. © 2010 IOP Publishing Ltd.
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 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.