In this work we consider a superradiant phase transition problem for the Dicke-Ising model, which generalizes the Dicke and Ising models for annealed complex networks presuming spin-spin interaction. The model accounts for the interaction between a spin-1/2 (two-level) system and external classical (magnetic) and quantized (transverse) fields. We examine regular, random, and scale-free network structures characterized by the $\delta$ function, random (Poisson), and power-law exponent $[p\left(k\right)\propto k^{-\gamma }]$ degree distributions, respectively. To describe paramagnetic (PM)-ferromagrenic (FM) and superradiant (SR) phase transitions we introduce two order parameters: the total weighted spin $z$ component and the normalized transverse field amplitude, which correspond to the spontaneous magnetization in $z$ and $x$ directions, respectively. For the regular networks and vanishing external field we demonstrate that these phase transitions generally represent prerequisites for the crossover from a disordered spin state to the ordered one inherent to the FM and/or SR phase. Due to the interplay between the spin interaction and the finite-size effects in networks we elucidate novel features of the SR state in the presence of the PM-FM phase transition. In particular, we show that the critical temperature may be high enough and essentially depends on parameters which characterize statistical properties of the network structure. For the scale-free networks we demonstrate that the network architecture, characterized by the particular value of $\gamma$, plays a key role in the SR phase transition problem. Within the anomalous regime scale-free networks possess a strong effective spin-spin interaction supporting fully ordered FM state, which is practically nonsensitive to variations of the quantum transverse field or moderate classical magnetic field. In a scale-free regime the networks exhibit vanishing of the collective spin component in $z$ direction with increasing $\gamma$ accompanied by establishing spontaneous magnetization in the transverse field. The SR phase transition occurs in the presence of some FM state. We establish the conditions for the network parameters, classical and quantum field features to obtain a quantum phase transition in the spin system when the critical temperature approaches zero.

中文翻译：

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复杂网络中超辐射相变的平均场理论

在这项工作中，我们考虑了 Dicke-Ising 模型的超辐射相变问题，该模型将 Dicke 和 Ising 模型推广到假定自旋相互作用的退火复杂网络。该模型解释了自旋 1/2（两能级）系统与外部经典（磁）和量化（横向）场之间的相互作用。我们检查了规则的、随机的和无标度的网络结构，其特征是$\delta$ 函数、随机（泊松）和幂律指数 $[磷\left(克\right)\propto {克}^{-\gamma }]$度分布，分别。为了描述顺磁 (PM)-铁磁 (FM) 和超辐射 (SR) 相变，我们引入了两个阶参数：总加权自旋$z$ 分量和归一化横向场振幅，它们对应于 $z$ 和 $X$方向，分别。对于规则网络和消失的外场，我们证明这些相变通常代表从无序自旋状态到 FM 和/或 SR 相固有的有序状态的交叉的先决条件。由于自旋相互作用和网络中有限尺寸效应之间的相互作用，我们阐明了在 PM-FM 相变存在下 SR 状态的新特征。特别是，我们表明临界温度可能足够高，并且基本上取决于表征网络结构统计特性的参数。对于无标度网络，我们证明了网络架构，其特征在于$\gamma$, 在 SR 相变问题中起着关键作用。在异常状态下，无标度网络具有强大的有效自旋-自旋相互作用，支持完全有序的 FM 状态，这实际上对量子横向场或中等经典磁场的变化不敏感。在无标度机制中，网络表现出集体自旋分量的消失$z$ 方向随着增加 $\gamma$伴随着在横向场中建立自发磁化。SR 相变发生在某些 FM 状态下。我们建立了网络参数、经典和量子场特征的条件，以在临界温度接近零时获得自旋系统中的量子相变。