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arXiv:quant-ph/0604011 · Europhysics Letters (EPL)

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

Francesco Plastina, G Liberti, Angelo C. M. Carollo

Published Sep 19, 2006• 45 citations
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Abstract

We discuss the thermodynamic and finite-size scaling properties of the geometric phase in the adiabatic Dicke model, describing a quantum 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 orders in the 1/ N expansion of the Berry phase and its critical exponent.

DOI
10.1209/epl/i2006-10270-x

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We discuss the thermodynamic and finite-size scaling properties of the geometric phase in the adiabatic Dicke model, describing a quantum 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 orders in the 1/ N expansion of the Berry phase and its critical exponent.
We obtain the ground-state energy level and associated geometric phase in the Dicke model analytically by means of the Holstein-Primakoff transformation and the boson expansion approach in the thermodynamic limit. The nonadiabatic geometric phase induced by the photon field is derived with the time-dependent unitary transformation. It is shown that the quantum phase transition characterized by the nonanalyticity of the geometric phase is remarkably of the first order. We also investigate the scaling behavior of the geometric phase at the critical point, which can be measured in a practical experiment to detect the quantum phase transition.
We discuss the finite size behavior of the adiabatic Dicke model, describing the collective coupling of a set of NNN two-level atoms (qubits) to a faster (electromagnetic) oscillator mode. The energy eigenstates of this system are shown to be directly related to those of another widely studied collective spin model, the uniaxial one. By employing an approximate continuum approach, we obtain a complete characterization of the properties of the latter, which we then use to evaluate the scaling properties of various observables for the original Dicke model near its quantum phase transition.
We study the Berry phase of the ground state and excited states in the Lipkin model. Using the Schwinger representation of spin operators, we get the expression for the Berry phase of the ground state in the thermodynamic limit and numerically calculate that of the excited states in finite sizes. We find that the Berry phase of the ground state is nonanalytic, but its derivative (with respect to the control parameter $\ensuremath{\alpha}$, see text) is finite even in the thermodynamic limit. Whereas, for the excited states, the Berry phase tends to be nonanalytic, and its derivative approaches $\ensuremath{\infty}$ in the thermodynamic limit and in the vicinity of the critical point of the excited-state quantum phase transition.
A laser-driven Bose-Einstein condensate of ultracold atoms loaded into a lossy high-finesse optical resonator exhibits critical behavior and, in the thermodynamic limit, a phase transition between stationary states of different symmetries. The system realizes an open-system variant of the celebrated Dicke model. We study the transition for a finite number of atoms by means of a Hartree-Fock-Bogoliubov method adapted to a damped-driven open system. The finite-size scaling exponents are determined and a clear distinction between the nonequilibrium and equilibrium quantum criticality is found.
We study the phase diagram of a class of models in which a generalized cluster interaction can be quenched by Ising exchange interaction and external magnetic field. We characterize the various phases through winding numbers. They may be ordinary phases with local order parameter or exotic ones, known as symmetry protected topologically ordered phases. Quantum phase transitions with dynamical critical exponents z = 1 or z = 2 are found. Quantum phase transitions are analyzed through finite-size scaling of the geometric phase accumulated when the spins of the lattice perform an adiabatic precession. In particular, we quantify the scaling behavior of the geometric phase in relation with the topology and low energy properties of the band structure of the system.