Algebraic approach to two-dimensional systems: Shape phase transitions, monodromy, and thermodynamic quantities

Abstract

We analyze shape phase transitions in two-dimensional algebraic models. We apply our analysis to linearto-bent transitions in molecules and point out what observables are particularly sensitive to the transition ͑order parameters͒. We study numerically the scaling behavior of observables and confirm the dependence of the energy gap for phase transitions of U͑n͒-SO͑n +1͒ type. We calculate energies of excited states and show their unusual behavior for some values of the Hamiltonian control parameter. This behavior is due to the doublehumped nature of the potential and can be associated with the concept of monodromy. Finally, we compute numerically thermodynamic quantities, in particular heat capacities, and show their large variation at and around the critical value of the control parameter.

Excited state quantum phase transitions in many-body systems

“…This requires the identification of physical systems described by algebraic Hamiltonians with ξ > ξ c for which states with quantum numbers k >> 1 can be observed. The most promising situations thus far are in molecules described by s-b boson models [30]. Further examples are needed to fully understand the experimental implications of the ESQPT.…”

“…For states in the vicinity of the ground state QPT, the gap vanishes more quickly than N −1 , as the power law ∆ ∼ N −4/3 . This has been established both numerically and analytically for the various models under consideration [30,44,[49][50][51][52]. 7 The gap at the excited state QPT also approaches zero more rapidly than ∆ ∼ N −1 .…”

“…11(d)]. Here the wave functions for the eigenstates with respect to the U(2) basis are known analytically [30,55]. The probability distribution for each state is reflection symmetric about N b /N = 0.5, peaked at two symmetric extreme values.…”

“…At the SO(3) limit, the entire spectrum falls below E = 0 and constant downward concavity follows from the exact formula [28] for the eigenvalues, quadratic in k. Although here we are considering the dip in ∂E/∂k as a property of the ESQPT in a manybody interacting boson model, it should be noted that the dip arising for the associated two-dimensional Schrödinger equation is well known as the "Dixon dip" [48], with applications to molecular spectroscopy (see also Ref. [30]). …”

“…It will be convenient to make extensive use of the U(3) two-dimensional vibron model [28][29][30] for illustration in this article. The U(3) vibron model is the simplest two-level model which still retains a nontrivial angular momentum or seniority quantum number (unlike the Lipkin model).…”