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arXiv:2310.18978 · Entropy

Quantum Phase Transitions in a Generalized Dicke Model

Wen Liu, Liwei Duan

Published Oct 29, 2023• 4 citations
arXivHTMLPDF

Abstract

We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.

DOI
10.3390/e25111492

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We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
We study the coupled-top model with three large spins located on a triangle. Depending on the coupling strength, there exist three phases: disordered paramagnetic phase, ferromagnetic phase, and frustrated antiferromagnetic phase, which can be distinguished by the mean-field approach. The paramagnetic-ferromagnetic phase transition is accompanied by the breaking of the global Z2{Z}_{2}Z2​ symmetry, whereas the paramagnetic-antiferromagnetic phase transition is accompanied by the breaking of both the global Z2{Z}_{2}Z2​ symmetry and the translational symmetry. Exact analytical results of higher-order quantum effects beyond the mean-field contribution, such as the excitation energy, quantum fluctuation, and von Neumann entropy, can be achieved by the Holstein-Primakoff transformation and symplectic transformation in the thermodynamic limit. Near the quantum critical point, the energy gap closes, along with the divergence of the quantum fluctuation in certain quadrature and von Neumann entropy. Particular attention should be paid to the antiferromagnetic phase, where geometric frustration takes effect. The critical behaviors in the antiferromagnetic phase are quite different from those in the paramagnetic and ferromagnetic phases, which highlight the importance of geometric frustration. The triangular coupled-top model provides a simple and feasible platform to study the quantum phase transition and the novel critical behaviors induced by geometric frustration.
We gain quantitative insights on effects of light-matter interactions on correlated quantum matter by quantum Monte Carlo simulations. We introduce a wormhole algorithm for the paradigmatic Dicke-Ising model which combines the light-matter interaction of the Dicke model with Ising interactions. The quantum phase diagrams for ferro- and antiferromagnetic interactions on the chain and the square lattice are determined. The occurring superradiant phase transitions are in the same universality class as the Dicke model, leading to a well-known peculiar finite-size scaling that we elucidate in terms of scaling above the upper critical dimension. For the ferromagnetic case, the transition between the normal and the superradiant phase is of second order with Dicke criticality (first order) for large (small) longitudinal fields separated by a multicritical point. For antiferromagnetic interactions, we establish the light-matter analog of a lattice supersolid with off-diagonal superradiant and diagonal magnetic order and determine the nature of all transition lines. Our achievements allow to quantitatively enter the domain of mesoscopic light-matter systems, which is an increasingly prominent topic at the interface of quantum optics and condensed matter physics.
In the vicinity of ground-state phase transitions quantum correlations can display nonanalytic behavior and critical scaling. This signature of emergent collective effects has been widely investigated within a broad range of equilibrium settings. However, under nonequilibrium conditions, as found in open quantum many-body systems, characterizing quantum correlations near phase transitions is challenging. Moreover, the impact of local and collective dissipative processes on quantum correlations is not broadly understood. Here we consider as a paradigmatic setting the superradiant phase transition of the open quantum Dicke model and characterize quantum and classical correlations across the phase diagram. We develop an approach to quantum fluctuations which allows us to show that local dissipation, which cannot be treated within the commonly employed Holstein-Primakoff approximation, rather unexpectedly leads to an enhancement of collective quantum correlations, and to the emergence of a nonequilibrium superradiant phase in which the bosonic and spin degrees of freedom of the Dicke model are entangled.
The Dicke model describes the collective behavior of a subwavelength-size ensemble of two-level atoms (i.e., spin-1/2) interacting identically with a single quantized radiation field of a cavity. Across a critical coupling strength it exhibits a zero-temperature phase transition from the normal state to the superradiant phase where the field is populated and the collective spin acquires a nonzero x component, which can be imagined as ferromagnetic ordering of the atomic spins along x. Here we introduce a variant of this model where two subwavelength-size ensembles of spins interact with a single quantized radiation field with different strengths. Subsequently, we restrict ourselves to a special case where the coupling strengths are opposite (which is unitarily equivalent to equal-coupling strengths). Because of the conservation of the total spin in each ensemble individually, the system supports two distinct superradiant states with x-ferromagnetic and x-ferrimagnetic spin ordering, coexisting with each other in a large parameter regime. The stability and dynamics of the system in the thermodynamic limit are examined using a semiclassical approach, which predicts nonstationary behaviors due to the multistabilities. At the end, we also perform small-scale full quantum-mechanical calculations, with results consistent with the semiclassical ones.
The occurrence of a second-order quantum phase transition in the Dicke model is a well-established feature. On the contrary, a comprehensive understanding of the corresponding open system, particularly in the proximity of the critical point, remains elusive. When approaching the critical point, the system inevitably enters first the system-bath ultrastrong coupling regime and finally the deepstrong coupling regime, causing the failure of usual approximations adopted to describe open quantum systems. We study the interaction of the Dicke model with bosonic bath fields in the absence of additional approximations, which usually relies on the weakness of the system-bath coupling. We find that the critical point is not affected by the interaction with the environment. Moreover, the interaction with the environment is not able to affect the system ground-state condensates in the superradiant phase, whereas the bath fields are i n f e c t e d by the system and acquire macroscopic occupations. The obtained reflection spectra display lineshapes which become increasingly asymmetric, both in the normal and superradiant phases, when approaching the critical point.
In the thermodynamic limit, the Dicke-Ising chain maps exactly onto an effective self-consistent matter Hamiltonian with the photon field acting solely as a self-consistent effective field. As a consequence, no quantum correlations between photons and spins are needed to understand the quantum phase diagram. This enables us to determine the quantum phase diagram in the thermodynamic limit using numerical linked-cluster expansions combined with density matrix renormalization group calculations (NLCE+DMRG) to solve the resulting self-consistent matter Hamiltonian. This includes magnetically ordered phases with significantly improved accuracy compared to previous estimates. For ferromagnetic Ising couplings, we refine the location of the multicritical point governing the change in the order of the superradiant phase transition, reaching a relative accuracy of 10−410^{-4}10−4. For antiferromagnetic Ising couplings, we confirm the existence of the narrow antiferromagnetic superradiant phase in the thermodynamic limit. The effective matter Hamiltonian framework identifies the antiferromagnetic superradiant phase as the many-body ground state of an antiferromagnetic transverse-field Ising model with longitudinal field. This phase emerges through continuous Dicke-type polariton condensation from the antiferromagnetic normal phase, followed by a first-order transition to the paramagnetic superradiant phase. Thus, NLCE+DMRG provides a precise determination of the Dicke-Ising phase diagram in one dimension by solving the self-consistent effective matter Hamiltonian.