Scaling of Berry's phase close to the Dicke quantum phase transition

Abstract

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...

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