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448 C.-Y. Zhang et al. / Nuclear Physics B 914 (2017) 446–460
and exponential (ENE) [51–53] Lagrangian. The authors of Ref. [54] observed that the expo-
nential form of nonlinear electrodynamics has stronger effect on the condensation formation and
conductivity for the holographic conductors in the backgrounds of AdS black hole by considering
three types of typical nonlinear electrodynamics. However, in the AdS soliton background the
critical chemical potentials are independent of the e
xplicit form of the nonlinear electrodynamics,
i.e., the ENE, BINE and LNE correction do not effect on the critical potentials. So based on the
research about holographic superconductor with the nonlinear electrodynamics, we have studied
the effect of BI coupling parameter on the paramagnetism–ferromagnetism phase transition [55].
And no
w it is interesting to investigate how the other two types of nonlinear electrodynamics
influence the paramagnetism–ferromagnetism phase transition.
The structure of this wo
rk is as follows. In section 2, we introduce the basic field equations
of holographic ferromagnetism model with ENE and LEN in the Schwarzschild-AdS black hole
which have not been studied as far as we know, and compare them with the BINE holographic
paramagnetism model. In section 3 by the numerical method we obtain the critical temperature
and study the magnetic moment in the presence of the three kinds of typical nonlinear electrody-
namics. Magnetic susceptibility density and h
ysteresis loop will be shown in section 4. Finally
in the last section we will include our summary and discussion.
2. Holographic model
In this paper, the model we are considering is just general relativity with a negative cosmo-
logical constant =−3/L
2
, a U(1) field A
μ
and a massive 2-form field M
μν
in 4-dimension
space-time. The ghost free action
S =
1
2κ
2
d
4
x
√
−g(L
1
+λ
2
L
2
) (1)
L
1
= R +6/L
2
+L(F ), (2)
L
2
=−
1
12
(dM)
2
−
m
2
4
M
μν
M
μν
−
1
2
M
μν
F
μν
−
J
8
V(M) (3)
where dM is the exterior differential of 2-form field M
μν
, m
2
is the squared mass of 2-form field
M
μν
being greater than zero (see Ref. [21] for detail), λ and J are two real model parameters
with J<0for producing the spontaneous magnetization, λ
2
characterizes the back reaction of
the 2-form field M
μν
to the background geometry and to the Maxwell field strength, and V(M)is
a nonlinear potential of the 2-form field describing the self-interaction of the polarization tensor.
For simplicity, we take the form of V(M)as follows,
V(M)= (
∗
M
μν
M
μν
)
2
=[
∗
(M ∧ M)]
2
, (4)
where
∗
is the Hodge-star operator. As shown in Ref. [21], this potential shows a global minimum
at some nonzero value of ρ. Meanwhile, L(F ) is the Lagrangian of three classes of Born–Infeld-
like nonlinear electrodynamics
L(F ) =
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1
b
ln(1 +bF ), LNE
1
b
(1 −
√
1 +2bF ), BINE
−
1
4b
(e
−4bF
−1). ENE
(5)