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arXiv:0809.4385 · Physical Review A

Numerically exact solution to the finite-size Dicke model

Qing‐Hu Chen, Yuyu Zhang, Tao Liu, Ke-Lin Wang

Published Nov 3, 2008• 156 citations
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Abstract

By using extended bosonic coherent states, a technique to solve the Dicke model exactly is proposed in the numerical sense. The accessible system size is two orders of magnitude higher than that reported in the literature. Finite-size scaling for several observables, such as the ground-state energy, Berry phase, and concurrence, is analyzed. A scaling exponent for ground-state energy is found. An existing discrepancy in the scaling exponent of the concurrence is reconciled.

DOI
10.1103/physreva.78.051801

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By using extended bosonic coherent states, a technique to solve the Dicke model exactly is proposed in the numerical sense. The accessible system size is two orders of magnitude higher than that reported in the literature. Finite-size scaling for several observables, such as the ground-state energy, Berry phase, and concurrence, is analyzed. A scaling exponent for ground-state energy is found. An existing discrepancy in the scaling exponent of the concurrence is reconciled.
We consider the finite-size corrections in the Dicke model and determine the scaling exponents at the critical point for several quantities such as the ground-state energy or the gap. Therefore, we use the Holstein-Primakoff representation of the angular momentum and introduce a canonical transformation to diagonalize the Hamiltonian in the normal phase. As already observed in several systems, these corrections turn out to be singular at the transition point and thus lead to nontrivial exponents. We show that for the atomic observables, these exponents are the same as in the Lipkin-Meshkov-Glick model, in agreement with numerical results. We also investigate the behavior of the order parameter related to the radiation mode and show that it is driven by the same scaling variable as the atomic one.
Within the numerically exact solution to the Dicke model proposed previously, we study the quantum criticality in terms of the ground-state (GS) energy, fidelity, and the order parameter. The finite size scaling analysis for the average fidelity susceptibility (FS) and second derivative of GS energy are performed. The correlation length exponent is obtained to be $\ensuremath{\nu}=2/3$, which is the same as that in Lipkin-Meshkov-Glick model obtained previously, suggesting the same universality. It is observed that average FS and second derivative of GS energy show similar critical behavior, demonstrating the intrinsic relation in the Dicke model. The scaling behavior for the order parameter and the singular part of the GS energy at the critical point are also analyzed and the obtained exponents are consistent with the previous scaling hypothesis in 1/N1/N1/N expansion scheme.
We consider an exactly solvable inhomogeneous Dicke model which describes an interaction between a disordered ensemble of two-level systems with single mode boson field. The existing method for evaluation of Richardson–Gaudin equations in the thermodynamical limit is extended to the case of Bethe equations in Dicke model. Using this extension, we present expressions both for the ground state and lowest excited states energies as well as leading-order finite-size corrections to these quantities for an arbitrary distribution of individual spin energies. We then evaluate these quantities for an equally-spaced distribution (constant density of states). In particular, we study evolution of the spectral gap and other related quantities. We also reveal regions on the phase diagram, where finite-size corrections are of particular importance.
We perform a Schrieffer-Wolff transformation to the two-photon Dicke model by keeping the leading-order correction with a quartic term of the field, which is crucial for finite-size scaling analysis. Besides a spectral collapse as a consequence of two-photon interaction, the super-radiant phase transition is indicated by the vanishing of the excitation energy and the uniform atomic polarization. The scaling functions for the ground-state energy and the atomic pseudospin are derived analytically. The scaling exponents of the observables are the same as those in the standard Dicke model, indicating they are in the same universality class.