We analyze the quantum phase transition for a set of N -two level systems interacting with a bosonic mode in the adiabatic regime. Through the Born-Oppenheimer approximation, we obtain the finite-size scaling expansion for many physical observables and, in particular, for the entanglement content of the system. PACS numbers: 64.60. Fr, 03.65.Ud, 05.70.Jk, 73.43.Nq Introduction. Many-body systems have become of central interest in the realm of quantum information theory as training grounds to study static, dynamical and sharing properties of quantum correlations. It has been natural, then, to employ the entanglement as a tool to analyze quantum phase transitions, one of the most striking consequences of quantum correlations in many body systems [1,2,3]. In this letter, we study the Dicke superradiant phase transition and obtain the finite size scaling behavior of the quantum correlations at criticality.The Dicke model (DM) describes the interaction of N two-level systems (qubits) with a single bosonic mode [4], and has become a paradigmatic example of collective quantum behavior. It exhibits a second-order phase transition [5], which has been studied extensively [6,7,8]. The continued interest in the DM arises from its broad application range [9] and from its rich dynamics, displaying many non-classical features [10,11,12,13]. The ground state entanglement of the DM has been recently analyzed [14,15,16], and some aspect of its finite size behavior has been obtained numerically. Finally, Vidal and Dusuel,[17], obtained the critical exponents by a modified Holstein-Primakoff approach.The exact treatment of the finite-size corrections to the Dicke transition is quite complicated and the study of some limiting cases can be useful. In this Letter we analyze the case of N qubits coupled to a slow oscillator. We discuss on equal footings both the finite size and the thermodynamic limit of the model, thus allowing to obtain the phase transition as well as its precursors at finite N . Indeed, we obtain the dominant scaling behavior for some entanglement measures and the entire 1/N expansion for all of the relevant physical observables, such as the order parameter. Concerning the quantum correlations, we argue that what is really relevant for the critical behavior is the bi-partite entanglement between the oscillator mode and the set of two level systems.Adiabatic limit. The interaction of N identical qubits with a bosonic mode is described by the Hamiltonian

We discuss the thermodynamic and finite size scaling properties of the geometric phase in the adiabatic Dicke model, describing the super-radiant phase transition for an N qubit register coupled to a slow oscillator mode. We show that, in the thermodynamic limit, a non zero Berry phase is obtained only if a path in parameter space is followed that encircles the critical point. Furthermore, we investigate the precursors of this critical behavior for a system with finite size and obtain the leading order in the 1/N expansion of the Berry phase and its critical exponent.PACS numbers: 03.65. Vf, 42.50.Fx A considerable understanding of the formal description of quantum mechanics has been achieved after Berry's discovery [1] of a geometric feature related to the motion of a quantum system. He showed that the wave function of a quantum system retains a memory of its evolution in its complex phase, which only depends on the "geometry" of the path traversed by the system. Known as the geometric phase, this contribution originates from the very heart of the structure of quantum mechanics. Historically, the first implicit derivation of geometric phase is to be found in the work of H. C. Longuet-Higgins et al.[2] and G. Herzberg and H. C. Longuet-Higgins [3] on the study of the E ⊗ e Jahn-Teller problem in molecular systems. They showed that the Born-Oppenheimer wave function, if required to be real and smoothly varying, undergoes a sign change (a π geometric phase shift) when the nuclear configuration is brought around a crossing point of the energy levels, i.e. a point where the energy levels become degenerate. The widespread interest in geometric phases is unquestionably due to the work of Berry [1] who put the concept of geometric phase in an unexpectedly general scenario, showing that it is a property of the evolution of any quantum mechanical system. It is then not too surprising how this idea has been so broadly studied and applied to a variety of contexts [4,5].An apparently unrelated area in which a connection with Berry Phase (BP) has been recently drawn is the study of Quantum Phase Transitions (QPT) in manybody systems [6,7,8]. It is a well known fact that QPT [9] are accompanied by a qualitative change in the nature of correlations in the ground state of a quantum system, and describing these changes is clearly one of the major interests in condensed matter physics. In particular, critical points are associated with a non-analytical behavior of the system energy density [9] . It is expected that such drastic changes are reflected in the geometry of the Hilbert space. The geometric phase is able to capture singular behaviors of the wave function, and is therefore expected to signal the presence of QPT. The appearance of non-trivial BP in presence of criticality is also heuris-tically motivated by the fact that QPT arise in correspondence of energy level crossings (in the thermodynamical limit) between ground and excited states [9]. As implicitly suggested by the early work of H. C. , level crossings generate...

SummaryOf the well known risk factors for thrombosis protein S deficiency is one of the most difficult to diagnose with certainty. Reliable estimates for the prevalence of protein S deficiency in the general population are not available and the risk of thrombosis is a controversial issue. It has been shown that levels of protein S fluctuate over time. However the determinants of low levels of protein S in the healthy population are not clear. Therefore, we evaluated the influence of sex, age and hormonal state on the antigen levels of protein S in 474 healthy control subjects of the Leiden Thrombophilia Study (LETS). In univariate analysis, sex, age, oral contraceptive (OC) use and post-menopausal state all influenced protein S antigen levels. In a multivariate model for the whole sample only menopausal state and OC use had still an effect on the levels of total protein S and only menopausal state had an independent effect on the values of free protein S. On the basis of this analysis we established different cut-off levels for these subgroups and we re-evaluated in the Leiden Thrombophilia Study the risk of thrombosis for individuals with low protein S using these different reference ranges. With these specific cut-off points, we did not observe an increase in the risk of thrombosis in patients deficient of total protein S (OR 1.2, 95% CI 0.5-2.9) or free protein S (OR 1.3, 95% CI 0.5-3.5). When men and women were analyzed separately, the risk in women was 1.5 (95% CI 0.4-5.4) and 2.4 (95% CI 0.6-9.2) for total and free protein S deficiencies, respectively; and there was no increase in thrombotic risk for men. We conclude that it may be helpful to apply separate cut-off levels in the assessment of protein S levels. This does not, however explain the differences between our results and those of others in the estimate of thrombotic risk of protein S deficiency.

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