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arXiv:cond-mat/0510281 · Europhysics Letters (EPL)

Finite-size scaling exponents in the Dicke model

Julien Vidal, S Dusuel

Published May 2, 2006• 164 citations
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Abstract

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.

DOI
10.1209/epl/i2006-10041-9

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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.
We analyze the finite-size scaling exponents in the Lipkin-Meshkov-Glick model by means of the Holstein-Primakoff representation of the spin operators and the continuous unitary transformations method. This combination allows us to exactly compute the leading corrections to the ground-state energy, the gap, the magnetization, and the two-spin correlation functions. We also present numerical calculations for large system size which confirm the validity of this approach. Finally, we use these results to discuss the entanglement properties of the ground state focusing on the (rescaled) concurrence that we compute in the thermodynamical limit.
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.
The Hamiltonian of the two-photon Dicke model is diagonalized for the normal phase and super-radiant phase beyond the mean-field method respectively, giving the critical coupling strength. Besides a spectral collapse, the super-radiant phase transition is presented by the vanishing of the excitation energy, which is lower than the mean-field results. Finite-size scaling exponents for the ground-state energy and the atomic angular momentum are analytically derived from scaling hypothesis, belonging to the same scaling universality of the one-photon Dicke model.
We investigate the finite-size scaling exponents for the critical point at the shape-phase transition from U(5) (spherical) to O(6) (deformed \ensuremath{\gamma}-unstable) dynamical symmetries of the interacting boson model, making use of the Holstein-Primakoff boson expansion and the continuous unitary transformation technique. We compute exactly the leading-order correction to the ground-state energy, the gap, the expectation value of the d-boson number in the ground state and the E2E2E2 transition probability from the ground state to the first excited state and determine the corresponding finite-size scaling exponents.
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.