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arXiv:2109.06227 · Physical Review A

Revisiting the phase transitions of the Dicke model

Pragna Das, Auditya Sharma

Published Mar 30, 2022• 16 citations
arXivHTMLPDF

Abstract

The Dicke model exhibits a variety of phase transitions. The quantum phase transition from the normal phase to the super-radiant phase is marked by a dramatic change in the scaling of the participation ratio. We find that the ground state in the super-radiant phase exhibits multifractality manifest in the participation ratio scaling as the square root of the full Hilbert-space dimension. The thermal phase transition temperature, for which an exact analytical expression may be obtained using the partition function, is strikingly captured by the mutual information between two spins. In the excited-state quantum phase transition within the super-radiant phase, there are two cutoff energies; the central energy bands between the lower and upper cutoff energies show distinctly different behaviors. While the level statistics of the central band is Wigner-Dyson, the lower and upper bands show mixed behavior that is closer to Poisson than Wigner-Dyson. This finding is corroborated with the aid of several eigenvector properties: von Neumann entanglement entropy between spins and bosons, the mean photon number, concurrence between two spins, and participation ratio.

DOI
10.1103/physreva.105.033716

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Comment on “Revisiting the phase transitions of the Dicke model”

Ángel L. Corps, Armando Relaño

In the work of Das and Sharma [Phys. Rev. A 105, 033716 (2022)] the phase transitions of the Dicke model are studied. Its main result is that, besides the well-known quantum phase transition, excited-state quantum phase transition, and thermal phase transition exhibited by the model, there exists an upper bound energy ${E}_{*}$ beyond which the model ceases to exhibit quantum chaotic behavior and the structure of the eigenfunctions changes. Based on this finding, a number of well-established results about the Dicke model are called into question. We argue that this result and all its consequences are spurious numerical effects resulting from an improper truncation of the infinite-dimensional Hilbert space necessary for numerical diagonalization.

arXiv:2209.14660•Physical Review A•Oct 13, 2022•3 citations

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Lina Bao, Feng Pan, Jing Lu +1 more

Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation.

Entropy•Jul 16, 2015•7 citations
The Dicke model exhibits a variety of phase transitions. The quantum phase transition from the normal phase to the super-radiant phase is marked by a dramatic change in the scaling of the participation ratio. We find that the ground state in the super-radiant phase exhibits multifractality manifest in the participation ratio scaling as the square root of the full Hilbert-space dimension. The thermal phase transition temperature, for which an exact analytical expression may be obtained using the partition function, is strikingly captured by the mutual information between two spins. In the excited-state quantum phase transition within the super-radiant phase, there are two cutoff energies; the central energy bands between the lower and upper cutoff energies show distinctly different behaviors. While the level statistics of the central band is Wigner-Dyson, the lower and upper bands show mixed behavior that is closer to Poisson than Wigner-Dyson. This finding is corroborated with the aid of several eigenvector properties: von Neumann entanglement entropy between spins and bosons, the mean photon number, concurrence between two spins, and participation ratio.
We systematically analyze the various phase transitions of the anisotropic Dicke model that is endowed with both rotating and counterrotating light-matter couplings. In addition to the ground-state quantum phase transition (QPT) from the normal to the superradiant phase, the anisotropic Dicke model also exhibits other transitions, namely, the excited state quantum phase transition (ESQPT), ergodic to nonergodic transition (ENET), and the temperature-dependent phase transition. We show that these phase transitions are profitably studied not only with the standard consecutive level spacing ratio, but also with the aid of various eigenvector quantities such as von Neumann entanglement entropy, the participation ratio, multifractal dimension, and mutual information. For ENET, both the statics and dynamics of the participation ratio offer a consistent and useful picture. An exciting finding from our work is that the ESQPT and the ENET are closely related to each other. We show this with the aid of two characteristic energies in the spectrum corresponding to jumps in von Neumann entropy.
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 introduce and study the disordered Dicke model in which the spin-boson couplings are drawn from a random distribution with some finite width. Regarding the quantum phase transition we show that when the standard deviation σσσ of the coupling strength gradually increases, the critical value of the mean coupling strength μμμ gradually decreases and after a certain σσσ there is no quantum phase transition at all; the system always lies in the super-radiant phase. We derive an approximate expression for the quantum phase transition in the presence of disorder in terms of μμμ and σσσ, which we numerically verify. Studying the thermal phase transition in the disordered Dicke model, we obtain an analytical expression for the critical temperature in terms of the mean and standard deviation of the coupling strength. We observe that even when the mean of the coupling strength is zero, there is a finite temperature transition if the standard deviation of the coupling is sufficiently high. Disordered couplings in the Dicke model will exist in quantum dot superlattices, and we also sketch how they can be engineered and controlled with ultracold atoms or molecules in a cavity.
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.
In the work of Das and Sharma [Phys. Rev. A 105, 033716 (2022)] the phase transitions of the Dicke model are studied. Its main result is that, besides the well-known quantum phase transition, excited-state quantum phase transition, and thermal phase transition exhibited by the model, there exists an upper bound energy E∗{E}_{*}E∗​ beyond which the model ceases to exhibit quantum chaotic behavior and the structure of the eigenfunctions changes. Based on this finding, a number of well-established results about the Dicke model are called into question. We argue that this result and all its consequences are spurious numerical effects resulting from an improper truncation of the infinite-dimensional Hilbert space necessary for numerical diagonalization.
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation.