Scaling Properties and Asymptotic Spectra of Finite Models of Phase Transitions as They Approach Macroscopic Limits

The asymptotic spectra and scaling properties of a mixed-symmetry Hamiltonian, which exhibits a second-order phase transition in its macroscopic limit, are examined for a system of $N$ interacting bosons. A second interacting boson-model Hamiltonian, which exhibits a first-order phase transition, is also considered. The latter shows many parallel characteristics and some notable differences, leaving it open to question as to the nature of its asymptotic critical-point properties.

Figure 1

Excitation energies of the Hamiltonian ${\widehat{H}}_{12}(\alpha )$ of Eq. (1) for a system of 60 bosons as a function of $\alpha$. Each energy level is an SO(5) multiplet of states and is labeled by an SO(5) angular momentum $v$ known as seniority.

Figure 2

Excitation energies for levels of $v\le 4$ in units of the lowest $v=1$ excitation energy for $\alpha =0.5$: (a) for the Hamiltonian ${\widehat{H}}_{12}(\alpha )$ of Eq. (1) and a range of $N$ values; (b) for the collective model Hamiltonian ${\widehat{H}}_{\mathrm{C}\mathrm{M}}(\alpha )$ of Eq. (10); and (c) for an infinite square-well potential (from Ref. 6).

Figure 3

Excitation energies of the Hamiltonian ${\widehat{H}}_{13}(\alpha )$ of Eq. (13) for a system of 32 bosons as a function of $\alpha$.

Figure 4

The ratio $B(E2:4\to 2)/B(E2:2\to 0)$ for $N=10$, 32, and 100.

Figure 5

Excitation energies for some low-lying states of the Hamiltonian ${\widehat{H}}_{13}$ at the critical point for $N=10$ and 60, in units of the excitation energy of the first excited state.

Figure 6

A log-log plot of the excitation energies of the first excited state of ${\widehat{H}}_{13}$ at the critical point as a function of $N$.