[Paper Review] Massive $2$-form field and holographic ferromagnetic phase transition

Rong-Gen Cai, Run-Qiu Yang+2 more|arXiv (Cornell University)|Jul 2, 2015

Black Holes and Theoretical Physics38 references4 citations

TL;DR

This paper investigates holographic ferromagnetic phase transitions in an AdS4 black brane background using a massive 2-form field coupled to the Maxwell field. In two probe limits—neglecting backreaction of the 2-form field or all matter fields—it finds spontaneous magnetization and second-order phase transitions at low temperatures, with DC resistivity showing colossal magnetic resistance behavior. The model exhibits Ising-like critical behavior and analytically derives a Reissner-Nordström-like black brane solution with negative cosmological constant.

ABSTRACT

In this paper, we investigate in some detail the holographic ferromagnetic phase transition in an AdS${_4}$ black brane background by introducing a massive 2-form field coupled to the Maxwell field strength in the bulk. In the two probe limits, one is to neglect the back reaction of the 2-form field to the background geometry and to the Maxwell field, and the other to neglect the back reaction of both the Maxwell field and the 2-form field, we find that the spontaneous magnetization and the ferromagnetic phase transition always happen when the temperature gets low enough with similar critical behavior. We calculate the DC resistivity in a semi-analytical method in the second probe limit and find it behaves as the colossal magnetic resistance effect in some materials. In the case with the first probe limit, we obtain the off-shell free energy of the holographic model near the critical temperature and compare with the Ising-like model. We also study the back reaction effect and find that the phase transition is always second order. In addition, we find an analytical Reissner-Norström-like black brane solution in the Einstein-Maxwell-2-form field theory with a negative cosmological constant.

Motivation & Objective

• To explore ferromagnetic phase transitions in holographic models using a massive 2-form field coupled to the Maxwell field in an AdS4 black brane background.
• To investigate the critical behavior of the system in two probe limits: one neglecting 2-form field backreaction, the other neglecting all matter field backreactions.
• To examine the DC resistivity in the ferromagnetic phase and compare it with the colossal magnetic resistance effect in real materials.
• To analyze the phase transition with full backreaction and determine its order, and to derive an analytical Reissner-Nordström-like black brane solution.
• To compare the holographic model's behavior near the critical temperature with the Ginzburg-Landau theory for the Ising universality class.

Proposed method

• Formulate a bulk action in Einstein-Maxwell-2-form field theory with a negative cosmological constant, introducing a massive 2-form field coupled to the Maxwell field strength.
• Apply two probe limits: (1) neglect backreaction of the 2-form field on geometry and Maxwell field; (2) neglect backreaction of both Maxwell and 2-form fields.
• Use semi-analytical methods to compute DC resistivity in the second probe limit, solving the linearized equations of motion near the critical temperature.
• Derive the on-shell free energy and analyze the phase transition using the Ginzburg-Landau formalism, comparing with the Ising model near $T_c$.
• Construct tensor hairy black brane solutions by including backreaction effects and solve the coupled equations numerically and analytically.
• Perform semi-analytic calculations of spontaneous magnetization and susceptibility near $T_c$, using eigenfunction expansions and eigenvalue decomposition.

Experimental results

Research questions

• RQ1 Does the introduction of a massive 2-form field in an AdS4 black brane background lead to a spontaneous magnetization and a second-order phase transition?
• RQ2 How does the DC resistivity behave in the ferromagnetic phase, and does it exhibit the colossal magnetic resistance effect observed in certain materials?
• RQ3 What is the critical behavior of the system near the phase transition, and how does it compare with the Ising universality class in Ginzburg-Landau theory?
• RQ4 How does the inclusion of backreaction from the 2-form and Maxwell fields affect the nature of the phase transition?
• RQ5 Can an analytical Reissner-Nordström-like black brane solution be derived in the Einstein-Maxwell-2-form field theory with negative cosmological constant?

Key findings

• Spontaneous magnetization and a second-order ferromagnetic phase transition occur in both probe limits when temperature drops below a critical value $T_c$.
• The DC resistivity in the second probe limit exhibits a colossal magnetic resistance effect, with resistivity increasing significantly in the ferromagnetic phase.
• Near the critical temperature, the spontaneous magnetization scales as $M ightarrow a_0 (T_c - T) / T_c$ with $a_0 \approx 1.1286$ for $k=4$, matching numerical results.
• The magnetic susceptibility diverges as $T \rightarrow T_c^+$, with $\lambda^2 \chi^{-1}/\mu_c \simeq 4.0520(T/T_c - 1)$, closely matching numerical data.
• The phase transition remains second order even when backreaction of the 2-form and Maxwell fields is included, as confirmed by the on-shell free energy analysis.
• An analytical Reissner-Nordström-like black brane solution is derived in the Einstein-Maxwell-2-form field theory with negative cosmological constant, supporting the existence of the phase transition.