In this paper, we consider three-dimensional massive gravity’s rainbow and obtain black hole solutions in three different cases of Born–Infeld, logarithmic, and exponential theories of nonlinear electrodynamics. We discuss the horizon structure and geometrical properties. Then, we study thermodynamics of these models by considering the first-order quantum correction effects, which appear as a logarithmic term in the black hole entropy. We discuss such effects on the black hole stability and phase transitions. We find that due to the quantum corrections, the second-order phase transition happens in Born–Infeld and logarithmic models. We obtain the modified first law of black hole thermodynamics in the presence of logarithmic corrections.

中文翻译：

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大质量引力彩虹中非线性带电 BTZ 黑洞的量子校正热力学

在本文中，我们考虑了三维大质量引力的彩虹，并在非线性电动力学的 Born-Infeld、对数和指数理论的三种不同情况下获得了黑洞解。我们讨论了视界结构和几何特性。然后，我们通过考虑一阶量子校正效应来研究这些模型的热力学，它在黑洞熵中表现为对数项。我们讨论了这种对黑洞稳定性和相变的影响。我们发现由于量子校正，二阶相变发生在 Born-Infeld 和对数模型中。我们在存在对数校正的情况下获得了修正后的黑洞热力学第一定律。