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Lifshitz scaling effects on the holographic paramagnetic-ferromagnetic phase transition


We disclose the effects of Lifshitz dynamical exponent z on the properties of holographic paramagnetic-ferromagnetic phase transition in the background of Lifshitz spacetime. To preserve the conformal invariance in higher dimensions, we consider the Power-Maxwell (PM) electrodynamics as our gauge field. We introduce a massive 2-form coupled to the PM field and perform the numerical shooting method in the probe limit by assuming the PM and the 2-form fields do not back-react on the background geometry. The obtained results indicate that the critical temperature decreases with increasing the strength of the power parameter q and dynamical exponent z. Besides, the formation of the magnetic moment in the black hole background is harder in the absence of an external magnetic field. At low temperatures, and in the absence of an external magnetic field, our result show the spontaneous magnetization and the ferromagnetic phase transition. We find that the critical exponent takes the universal value \(\beta = 1/2\) regardless of the parameters qzd, which is in agreement with the mean field theory. In the presence of an external magnetic field, the magnetic susceptibility satisfies the Curie-Weiss law.

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    Since our chosen masses should satisfy the Breitenlohner-Freedman(BF) bound, \(4m^{2}-[z-(6-d)]^{2}>0\), therefore by considering \(m^{2}=1/8\), the dynamical exponent should be confined with \((6-d)-2m<z<2m+(6-d)\). So we have chosen the values for dynamical exponent as \(z=3/2, 7/4\) for \(d=4\). For \(d=5\), we have considered \(z=3/2, 17/10\) for dynamical exponent. Thus, for these choice of the parameter m and d, the values of Lifshitz dynamical exponent is restricted \(z<2\).


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We are grateful to the referee for constructive comments which helped us improve our paper. We thank the Research Council of Shiraz University. The work of A.S. has been supported financially by the Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran.

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Ghotbabadi, B.B., Sheykhi, A. & Bordbar, G.H. Lifshitz scaling effects on the holographic paramagnetic-ferromagnetic phase transition. Gen Relativ Gravit 53, 88 (2021).

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  • Paramagnetic-ferromagnetic
  • Phase Transition
  • Lifshitz