1. Holographic Magnetism from General Relativity
KITPC program on holographic duality for Condensed matter physics , 2015
Holographic Magnetism from General Relativity
Rong-Gen Cai
Institute of Theoretical Physics
Chinese Academy of Sciences
Refs: arXiv: , PRD90 (2014) (Rapid Comm)
arXiv: , PRD 91 (2015)
arXiv: , PRD91 (2015)
arXiv: ; arXiv: (PRD in press)
arXiv: (PRD in press);
arXiv: ; arXiv:

2. Outline: 1、Introduction
2、 Ferromagnetism/paramagnetism phase transition
3、Antiferromagnetism/paramagnetism phase transition
4、Antiferromagnetic quantum phase transition
5、Insulator/metal phase transition and colossal
magnetoresistance
6、Coexistence and competition between ferromagnetism
and superconductivity
7、Summary

3. Why GR?
The planar black hole with AdS radius L=1:
where:
Temperature of the black hole:
Energy of the black hole:
Entropy of the black hole:
The black hole behaves like a thermal gas
in 2+1 dimensions in thermodynamics!

4. AdS/CFT correspondence（1997, J. Maldacena）:
“Real conceptual change
in our thinking about Gravity.”
(E. Witten, Science 285 (1999) 512)

5. Topology theorem of black hole horizon:

6. Holographic model of superconductors
CFT CFT/AdS Gravity
Global symmetry Abelian gauge field
Scalar operator Scalar field
Temperature Black hole
Phase transition High T/no hair
Low T/ hairy BH
G.T. Horowitz,

7. Holographic superconductors
Building a holographic superconductor
S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv:
PRL 101, (2008)
High Temperature（black hole without hair):

8. Consider the case of m^2L^2=-2，like a conformal scalar field.
In the probe limit and A_t= Phi
At the large r boundary:
Scalar operator condensate
O_i:

10. Maxwell equation with zero momentum :
Conductivity
Maxwell equation with zero momentum :
Boundary conduction:
at the horizon: ingoing mode
at the infinity:
AdS/CFT
current
source:
Conductivity:

11. A universal energy gap: ~ 10%
BCS theory: 3.5
K. Gomes et al, Nature 447, 569 (2007)

12. arXiv: , PRD82 (2010)
Breaking a global SU(2) symmetry representing spin into a U(1) subgroup.
The symmetry breaking is triggered by
condensation of a triplet scalar field .
This model leads to the spatial rotational
symmetry breaking spontaneously, the time reversal symmetry is not broken spontaneously in the magnetic ordered phase.

13. 2、A Model for ferromagnetism/paramagnetism transition
arXiv: , PRD 90 (2014) , Rapid Comm.
The Model:
Motivation is as follows: 1) time reversal symmetry and rotating symmetry are broken
2) In MFT, the order parameter is a pseudovector and is coupled to external magnetic field;
3) In weak external magnetic field, it is proportional to external magnetic field.

14. We are considering the probe limit, the background is
Temperature:
The ansatz:

15. The boundary condition:

16. The off-shell free energy:
arXiv:
on shell:

17. Spontaneous magnetization: B=0

18. The response to external magnetic field
Obey the Curie-Weiss Law

19. The hysteresis loop in a single magnetic domain:
When T < Tc, the magnetic moment is not single valued. The parts DE and BA are stable, which can be realized in the external field. The part CF is unstable which cannot exist in the realistic system. The parts EF and CB are metastable states, which may exist in some intermediate processes and can be observed in experiment. When the external field continuously changes, the metastable states of magnetic moment can appear.

20. 3、Faramagnetism/antiferromagnetism phase transition
arXiv:
Antiferromagnetic material does not show any macroscopic magnetic moment when external magnetic field is absent, it is still a kind of magnetic ordered material when temperature is below the Neel temperature T_N. The conventional picture, due to L.
Neel, represents a macroscopic antiferromagnetism as consisting of two sublattices, such that spins on one sublattice point opposite to that of the other sublattice. The order parameter is the staggered magnetization, as the diference between the two magnetic moments associated with the two sublattices:

21. Magnetic susceptibility:

22. Three minimal requirements to realize the holographic model
for the phase transition of paramagnetism/antiferromagnetism.
The antiparallel magnetic structure as T<T_N
The susceptibility behavior
Breaking the time reversal symm & spatial rotating symm
Our model:

23. The probe limit
The ansatz:
Define:

24. The equations of motion:
The boundary conditions:

25. The parameter constraint:
The on-shell free energy:

26. alpha_0 and beta_0 are initial values at the horizon!

28. The influence on strong external magnetic field

29. arXiv:
New model:

32. 4、Antiferromagnetic quantum phase transition

34. Linear response theory:
if the spectrum function has a pole at

37. Neither power-law nor
square root, but it is expected
in strong coupling case.
Indeed in the case d=z=2,
the quantum critical theory
is not in general a weak
coupling theory.
arXiv:

38. 5、Insulator/metal phase transition and colossal
magnetoresistance in holographic massive gravity
Some magnetic materials such as manganites
exhibit the colossal magnetoresistance effect.
Our model:
Blake, Tong and Vegh, arXiv:
Blake and Tong, arXiv:
Mefford and Horowitz, arXiv:
There is a position dependent mass
This measures the strength of inhomogeneity

39. The black brane solution:
The ansatz:
The asymptotic solution at the boundary:

40. DC conductivity:
The perturbation:
The AdS boundary:
DC resistivity:

41. By the membrane paradigm:
Iqbal and Liu, arXiv:
The DC resistivity in the strong inhomogeneity limit:

42. Numerical results:
A. Urushibara et al, PRB51 (1995)

43. 6、Coexistence between ferromagnetism and p-wave order
(arXiv: )
P-wave: Einstein-Maxwell-Complex vector model:
arXiv: , JHEP 1401 (2014) 032

45. Our model:

46. The probe limit:
The ansatz:
It is found that only the following case is consistent

47. Define:

48. 1) Superconducting ferromagnet

52. 2) Ferromagnetic superconductor

55. Conclusions:
1) In the case that the ferromagnetic phase appears first,
if the interaction is attractive, the system shows the ferromagnetism and superconductivity can coexist in low temperatures. If the interaction is repulsive, the system will only be in a pure ferromagnetic state.
2) In the case that the superconducting phase appears first, the attractive interaction will leads to a magnetic p-wave superconducting phase in low temperatures. If the interaction is repulsive, the system will be in a pure p-wave superconducting phase or ferromagnetic phase when the temperature is lowered.

56. 7、Summary
1) Present a holographic model for the paramagnetism
-ferromagnetism phase transition
2) The paramagnetism-antiferromagnetism transition
Antiferromagnetic quantum phase transition
4）Insulator/metal phase transition and
colossal magnetoresistance effect
5) Coexistence and competition between ferromagnetism
and p-wave superconductivity

57. Thanks !

58. 全息原理在弦论中的实现(1997)： 典型例子： （J. Maldacena) 在反德西特时空（带一个紧致空间）上的
这个反德西特时空边界上的一个共形场论。
典型例子：
IIB superstring theory on AdS5 x S5
N=4 SYM Theory
（J. Maldacena)