The existence of the Efimov effect is drastically affected by the dimensionality of the space in which the system is embedded. The effective spatial dimension containing an atomic cloud can be continuously modified by compressing it in one or two directions. In the present article we determine for a general AAB system formed by two identical bosons A and a third particle B in the two-body unitary limit, the dimensionsality D for which the Efimov effect can exist for different values of the mass ratio A = mB/mA. In addition, we provide a prediction for the Efimov discrete scaling factor, exp (π/s), as a function of a wide range of values of A and D, which can be tested in experiments that can be realized with currently available technology.Introduction. The Efimov effect [1] appears over an incredibly large variety of systems and spans a wide range of scales-atoms, molecules, exotic nuclei, condensed matter systems and even the DNA-turning into a quite rich research area in physics [2]. Such a surprising phenomenon that can occur in a system of three particles with finite-range forces was discovered by V. Efimov in 1970 within the nuclear physics context, by showing that the system can have an infinite sequence of geometrically spaced energy levels when the pairs of the three particles have infinite two-body scattering length. Such a spectrum is closely related to the collapse of the three-body binding energy, discovered by L.H. Thomas in 1935 [3], by decreasing the range of the interaction with respect to its scattering length.The Thomas collapse and the Efimov effect are intimately related to the dimension where the system is inserted. Both Thomas and Efimov considered only a three dimensional environment (D = 3). However, the flexibility of manipulating ultracold atomic traps brought the study of few-body systems to a new era: the Efimov effect was experimentally confirmed [4] after almost 40 years since the original prediction; the use of the Feshbach resonance technique allows a fine tuning of the two-body interactions [5]; and the possibility to compress and expand the atomic cloud by changing the lasers and magnetic fields can create effectively two-and onedimensional situations [6].Despite of the many advances in theory and experiment, there is an almost unexplored issue: the effect of a continuous changing of the spatial dimension on the different observables. For integer dimensions, we know that the Efimov effect does not exist in D = 2 or D = 1. More precisely, it has been demonstrated for identical bosons that the Efimov effect survives for 2.3 < D < 3.8 [7]. However, the possibilities in engineering atomic traps with heteronuclear species raises the question on how the spatial dimensionality affects mass-imbalanced threebody systems. Such systems provide a more favorable situation for the experimental investigation of the Efimov effect [8][9][10][11][12][13][14][15]. Notwithstanding the fact that an atomic

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We address the reliability of the Optimized Perturbation Theory (OPT) in the context of the 0-dimensional O(N ) scalar field model. The effective potential, the self-energy and the 1PI four-point Green's function for the model are computed using different optimization schemes and the results contrasted to the exact results for the model. Our results are also compared to those obtained with the 1/N -expansion and with those from ordinary perturbation theory. The OPT results are shown to be stable even at large couplings and to have better convergence properties than the ones produced in the 1/N -expansion. It is also shown that the principle of minimal sensitive optimization procedure used in conjunction with the OPT method tends to always produce better results, in particular when applied directly to the selfenergy.

We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born-Oppenheimer approximation for an arbitrary dimension D. We restrict D to the interval 2 < D < 4, and derive the heavy-heavy D-dimensional effective potential proportional to 1/R 2 (R is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy-heavy effective potential 1/R 2 approaches the limit −(D − 2) 2 /4. We obtained the scaling function for the 133 Cs-133 Cs-6 Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of D and the heavy-light binding energy E D 2 . In addition, we found that the energy of the (N + 1) th excited state reaches the twobody continuum independently of the dimension D when E D 2 /E (N ) 3 = 0.89, where E (N ) 3is the N th excited three-body binding energy.

The mean field approximation results are obtained in a five-site cluster on the diamond lattice from the Bogoliubov inequality. Spin correlation identities for the Blume-Capel model on diamond lattice are derived from a five-site cluster and used to obtain an effective field approximation. The free-energy, magnetization, critical frontiers and tricritical points are obtained from the mean field approximation and the effective field approximation and are compared to those obtained by other methods. From the mean-field approximation, we also studied the unstable and metastable states besides the stable states present in the model.