Flávio Cordeiro

Extended supersymmetric σ-model based on the SO(2N+1) Lie algebra of the fermion operators

Extended supersymmetric σ-model is given, standing on the SO(2N+1) Lie algebra of fermion operators composed of annihilation–creation operators and pair operators. Canonical transformation, the extension of the SO(2N) Bogoliubov transformation to the SO(2N+1) group, is introduced. Embedding the SO(2N+1) group into an SO(2N+2) group and using SO(2N+2)U(N+1) coset variables, we investigate a new aspect of the supersymmetric σ-model on the Kahler manifold of the symmetric space SO(2N+2)U(N+1). We construct a Killing potential which is just the extension of the Killing potential in the SO(2N)U(N) coset space given by van Holten et al. to that in the SO(2N+2)U(N+1) coset space. To our great surprise, the Killing potential is equivalent with the generalized density matrix. Its diagonal-block matrix is related to a reduced scalar potential with a Fayet–Ilipoulos term. The reduced scalar potential is optimized in order to see the behaviour of the vacuum expectation value of the σ-model fields and a proper solution for one of the SO(2N+1) group parameters is obtained. We give bosonization of the SO(2N+2) Lie operators and vacuum functions in terms of the SO(2N+2)U(N+1) coset variables, a U(1) phase and the corresponding Kahler potential.

The Buck–Sukumar model described in terms of su(2) ⊗ su(1, 1) coherent states

The Buck–Sukumar model, which describes an assembly of A identical two-level atoms in interaction with a monochromatic radiation field, is investigated using su(2) ⊗ su(1, 1) coherent states, in the framework of conventional mean-field many-body approaches. In particular, the super-radiant phase transition is studied. We find that results based on the mean-field method compare favorably with exact results. We also find that the results are much improved if the constant of motion of the model is implemented exactly, with the help of appropriate projection techniques, instead of being implemented only in the average. Since the Hamiltonian of the Buck–Sukumar model is unbounded from below, i.e., it lacks a ground state, a stabilized version of the model is also studied.

Schwinger representation approach to the Lipkin model

The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The groundstate, first excited state and RPA energies are obtained within several variant types of coherent states. It has been found that generalized coherent states defined in consonance with the parity symmetry of the model describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone.

The two-level pairing model in the Schwinger representation

The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su(2) algebra (BCS theory).

Anomaly-free supersymmetric \frac{{{\text{SO}}\left( {2N + 2} \right)}}{{{\text{U}}\left( {N + 1} \right)}} σ-model based on the SO(2N + 1) Lie algebra of the fermion operators

The extended supersymmetric (SUSY) σ-model has been proposed on the bases of SO(2N + 1) Lie algebra spanned by fermion annihilation-creation operators and pair operators. The canonical transformation, extension of an SO(2N) Bogoliubov transformation to an SO(2N + 1) group, is introduced. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using SO(2N + 2)/U(N + 1) coset variables, we have investigated the SUSY σ-model on the Kähler manifold, the coset space SO(2N + 2)/U(N + 1). We have constructed the Killing potential, extension of the potential in the SO(2N)/U(N) coset space to that in the SO(2N + 2)/U(N + 1) coset space. That Killing potential is equivalent to the generalized density matrix whose diagonal-block part is related to a reduced scalar potential with a Fayet-Ilipoulos term. We rescale the Goldstone fields by a parameter f (inverse of mass mσ). The f-deformed reduced scalar potential is optimized with respect to vacuum expectation value of the σ-model fields and a solution for one of the SO(2N + 1) group parameters has been obtained. The solution, however, is only a small part of all solutions obtained from anomaly-free SUSY coset models. To construct the coset models consistently, we must embed a coset coordinate in an anomaly-free spinor representation (rep) of SO(2N + 2) group and give corresponding Kähler and Killing potentials for an anomaly-free SO(2N + 2)/U(N + 1) model based on each positive chiral spinor rep. Using such mathematical manipulation we construct successfully the anomaly-free SO(2N + 2)/U(N + 1) SUSY σ-model and investigate new aspects of such a SUSY σ-model which have never been seen in the usual SUSY σ-model on the Kähler coset space SO(2N)/U(N). Then we reach a f-deformed reduced scalar potential in the anomaly-free SO(2N + 2)/U(N + 1) SUSY σ-model. It is minimized with respect to the vacuum expectation value of anomaly-free SUSY σ-model fields. Thus we find an interesting f-deformed solution very different from the previous solution for an anomaly-free SO(2 ⋅ 5 + 2)/(SU(5 + 1) × U(1)) SUSY σ-model.

Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective subma- nifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the con- ventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective vari- able differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the -HF theory has been recon- structed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent ex- pansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a -dependent potential with a -periodicity. A bilinear equation for the -HF theory has been transcribed onto the corresponding -function using the regu- lar representation for the group and the Schur-polynomials. The -HF SCF theory on an infinite-dimensional Fock space F1 leads to a dynamics on an infinite-dimensional Grass- mannian Gr1 and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr1 which is affiliated with the group manifold obtained by reducting gl(1) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.

The Bonn nuclear quark model revisited

We present the exact solutions to the equations of the lowest energy states of the colored and color-symmetric sectors of the Bonn quark model, which is SU(3) symmetric and is defined in terms of an effective pairing force with su(4) algebraic structure. We show that the groundstate of the model is not color symmetrical except for a narrow interval in the range of possible quark numbers. We also study the performance of the Glauber coherent state, as well as of superconducting states of the BCS type, with respect to the description, not only of the absolute (colored) groundstate, but also of the minimum energy state of the color-symmetrical sector, finding that it is remarkably good. We use the model to discuss, in a schematic context, some controversial aspects of the conventional treatment of color superconductivity.