Phase diagram of the two-fluid Lipkin model: A “butterfly” catastrophe

Abstract

Background:In the past few decades quantum phase transitions have been of great interest in nuclear physics. In this context, two-fluid algebraic models are ideal systems to study how the concept of quantum phase transition evolves when moving into more complex systems, but the number of publications along this line has been scarce up to now. Purpose: We intend to determine the phase diagram of a two-fluid Lipkin model that resembles the nuclear protonneutron interacting boson model Hamiltonian using both numerical results and analytic tools, i.e., catastrophe theory, and compare the mean-field results with exact diagonalizations for large systems. Method: The mean-field energy surface of a consistent-Q-like two-fluid Lipkin Hamiltonian is studied and compared with exact results coming from a direct diagonalization. The mean-field results are analyzed using the framework of catastrophe theory. Results:The phase diagram of the model is obtained and the order of the different phase-transition lines and surfaces is determined using a catastrophe theory analysis. Conclusions: There are two first-order surfaces in the phase diagram, one separating the spherical and the deformed shapes, while the other separates two different deformed phases. A second-order line, where the later surfaces merge, is found. This line finishes in a transition point with a divergence in the second-order derivative of the energy that corresponds to a tricritical point in the language of the Ginzburg-Landau theory for phase transitions.

Excited-state quantum phase transitions in a two-fluid Lipkin model

“…The three last cases correspond to Hamiltonians with y = 0, i.e., with y 1 = −y 2 , and, therefore, located on the gray vertical plane of the phase diagram. As we proved in [21], for values of y < 1 a second order QPT appears at x c = 4/5, for y = 1 the QPT shows a divergence in d 2 E/dx 2 also at x = 4/5, but for y > 1 the QPT becomes of first order. Thus, next we try to disentangle whether or not there is a different qualitative behaviour between these three situations in terms of analyses of density of states, Peres lattice, Poincaré section, and participation ratio.…”

“…This line is, in fact, a triple point where three degenerated minima coexist (one spherical, and two deformed with different deformation). The point y = 0, y = 1 shows a unique behaviour because presents a divergence [21] in the second derivative of the energy with respect to the control parameter. At this point, the spinodal and the antispinodal lines merge with the first order phase transition surface giving rise to a tricritical point.…”

“…In this work we will use the u 1 (1) ⊗ u 2 (1) basis in which the quantum numbers n t 1 and n t 2 are specified, as explained in Ref. [21].…”

“…The aim of this work is to extend the study of the two-fluid Lipkin model, whose QPTs and phase diagram were studied in detail in [21], to the excited states realm, in other words, to study the excited-state quantum phase transitions (ESQPTs) of the model. The term ESQPT was first coined in [22] and studied in detail in [23].…”

Phase diagram of an extended Agassi model

“…In addition, there is a line, χ = Σ with Λ = 1+χ 2 2χ , in which four phases, HF, BCS, HF-BCS and the closed valley solutions, are degenerated, plus a single point, χ = Σ = Λ = 1, in which the five solutions (same as before plus the spherical) are degenerated. Such a rich phase diagram do not appear even in the case of more complex systems, such as the proton-neutron interacting boson model [16], the two-fluid Lipkin model [17] or for Hamiltonians with up two three-body interactions [18].…”