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1、Rong-Gen Cai (蔡榮根） Institute of Theoretical Physics Chinese Academy of Sciences Yukawa Institute for Theoretical Physics Kyoto University,Holographic Magnetism

2、from General Relativity,Kobe University，2016.11.09,Refs: arXiv: 1404.2856,1404.7737, 1410.5080, 1501.04481, 1504.00855,1505.03405, 1507.00546,1507.03105 1502.00437, 1601.029

3、36,2015: GR100,,GR is a theory about space and time!,引力（時空）的本質(zhì)？1500年前，奧古斯丁的名言：時間是什么？,It is more than that! It can also tell us something which is not relevant to gravity!,This is the goal of my talk:

4、 Magnetism from GR!The idea is that the gravity in AdS bulk is equivalent to aCFT at the AdS boundary, AdS/CFT correspondence, a realization of the holographic principle.,Black hole is a window to quantum grav

5、ity,Thermodynamics of black hole,S.Hawking, 1974, J. Bekenstein, 1973,1、Introduction: holographic principle,,Entropy in a system with surface area A :S<A/4G A system with gravity in d+1 dimensions can be described by

6、 a theory without gravity in d dimension!,(G. t’ Hooft),(L. Susskind),The world is a hologram？,Holography of Gravity,Why GR?,The planar black hole with AdS radius L=1:,where:,Temperature of the black hole:Energy of the

7、black hole:Entropy of the black hole:,The black hole behaves like a thermal gas in 2+1 dimensions in thermodynamics!,Topology theorem of black hole horizon:,AdS/CFT correspondence（1997, J. Maldacena）:,“Real conceptual

8、change in our thinking about Gravity.”(E. Witten, Science 285 (1999) 512,,,CFT,,AdS,AdS/CFT dictionary :,Herein the bulk: the boundary value of the field propagating in the bulk in the boundary theory:

9、 the source of the operator dual to the bulk field,Quantum field theory in d-dimensions operator Ο boundary,quantum gravitational theory in (d+1)-dimensions

10、 dynamical field φ bulk,,,(0909.3553, S. Hartnoll),AdS/CFT correspondence： gravity/gauge field 2) different spacetime dimension3) weak/strong duality4) classical/quantum,Applications in various field

11、s: low energy QCD (AdS/QCD), condensed matter theory (AdS/CMT) e.g., holographic superconductivity (non-) Fermion fluid,Paramagnetism-Ferromagnetism Phase Transition in a Dyonic Black Hole Phys. Rev. D 90,

12、081901 (2014) (Rapid Communication)2) Model for Paramagnetism/antiferromagnetism Phase Transition Phys. Rev. D 91, 086001 (2015)3) Coexistence and competition of ferromagnetism and p-wave superconductivity in

13、holographic model Phys. Rev. D 91, 026001 (2015)4) Holographic model for antiferromagnetic quantum phase transition induced by magnetic field Phys. Rev. D 92, 086001 (2015)5) Antisymmetric tensor field and

14、spontaneous magnetization in holographic duality Phys. Rev. D 92, 046001 (2015)6) Holographic antiferromganetic quantum criticality and AdS_2 scaling limit Phys. Rev. D 92, 046005 (2015)7) Massive 2-form field

15、and holographic ferromagnetic phase transition JHEP 1511 (2015) 0218) Insulator/metal phase transition and colossal magnetoresistance in holographic model Phys.Rev.D92 (2015)1060029) Introduction to Holographic

16、 Superconductor Models Sci. China Phys. Mech. Astron. 58 (2015) 060401,Holographic magnetism:,Outline:,1 Introduction2 Ferromagnetism/paramagnetism phase transition 3 Antiferromagnetism/paramagnetism pha

17、se transition4 Antiferromagnetic quantum phase transition5 Insulator/metal phase transition and colossal magnetoresistance effect6 Coexistence and competition between ferromagnetism and superconductivi

18、ty 7 Summary,how to build a holographic model of superconductors,CFT CFT/AdS Gravity Global symmetry Abelian gauge fieldScalar operator Scalar fieldTemperature

19、 Black holePhase transition High T/no hair Low T/ hairy BH,,G.T. Horowitz, 1002.1722,Building a holographic superconductor S. Hartn

20、oll, C.P. Herzog and G. Horowitz, arXiv: 0803.3295 PRL 101, 031601 (2008),High Temperature（black hole without hair):,Holographic superconductors,Consider the case of m^2L^2=-2，like a conformal scalar field.In the

21、 probe limit and A_t= Phi,At the large r boundary:,Scalar operator condensateO_i:,Boundary conduction:at the horizon: ingoing modeat the infinity:,,AdS/CFT,source:,Conductivity:,Conductivity,Maxwell equation with zer

22、o momentum :,current,A universal energy gap: ~ 10%,BCS theory: 3.5 K. Gomes et al, Nature 447, 569 (2007),,Summary:,The CFT has a global abelian symmetry corresponding a massless gauge field propagating in the

23、 bulk AdS space.Also require an operator in the CFT that corresponds to a scalar field that is charged with respect to this gauge field..3. Adding a black hole to the AdS describes the CFT at finite temper

24、ature.Looks for cases where there are high temperature black hole solutions with no charged scalar hair, but below some critical temperature black hole solutions with charged scalar hair and dominates th

25、e free energy.,arXiv: 1003.0010, PRD82 (2010) 045002,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

26、leads to the spatial rotational symmetry breaking spontaneously, the time reversal symmetry is not broken spontaneously in the magnetic ordered phase.,2、A model for ferromagnetism/paramagnetism transitionarXiv: 1404.

27、2856, PRD 90 (2014) 081901, Rapid Comm.,The model:,The reasons: The ferromagnetic transition breaks the time reversal symmetry, spatial rotating symmetry, but is not associated with any symmetry such as U(1), SU(2

28、).The magnetic moment is a spatial component of a tensor, 3)In weak external magnetic field, it is proportional to external magnetic field.,We are considering the probe limit, the background is,Temperature:,The ansatz:

29、,The boundary condition:,The off-shell free energy:,on shell:,Ising-like model:,arXiv: 1507.00546,Spontaneous magnetization: B=0,The response to external magnetic field:,Obey the Curie-Weiss Law,magnetic susceptibility:,

30、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 canno

31、t 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

32、states of magnetic moment can appear.,3、Faramagnetism/antiferromagnetism phase transition,arXiv:1404.7737,Antiferromagnetic material does not show any macroscopic magnetic moment when external magnetic field is absent, i

33、t 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, s

34、uch 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:,

35、Magnetic susceptibility:,Three minimal requirements to realize the holographic modelfor the phase transition of paramagnetism/antiferromagnetism.The antiparallel magnetic structure as T<T_NThe susceptibility behav

36、ior Breaking the time reversal symm & spatial rotating symm,Our model:,The probe limit,The ansatz:,Define:,The equations of motion:,The boundary conditions:,The parameter constraint:,The on-shell free energy:,alpha

37、_0 and beta_0 are initial values at the horizon!,The influence on strong external magnetic field,New model:,arXiv: 1504.00855,4、Antiferromagnetic quantum phase transition,critical magnetic field Ex:4.2,

38、Th:5.0 Dynamical exponent: 2,Er2-2xY2xTi2O7,5、Insulator/metal phase transition and colossal magnetoresistance in holographic massive gravity,Blake, Tong and Vegh, arXiv:1310.3832Blake and Tong, arXiv:1308.4970

39、Mefford and Horowitz, arXiv: 1406.4188,There is a position dependent mass,Our model:,Some magnetic materials such as manganitesexhibit the colossal magnetoresistance effect.,This measures the strength of inhomogeneity

40、,The black brane solution:,The ansatz:,The asymptotic solution at the boundary:,The perturbation:,DC conductivity:,The AdS boundary:,DC resistivity:,By the membrane paradigm:,Iqbal and Liu, arXiv: 0809.3808,The DC resist

41、ivity in the strong inhomogeneity limit:,Numerical results:,A. Urushibara et al, PRB51 (1995),Coexistence between superconductivity and Ferromagnetism!,6、Coexistence between ferromagnetism and p-wave order,6、Coexistence

42、between ferromagnetism and p-wave order,P-wave: Einstein-Maxwell-Complex vector model:,(arXiv:1410.5080),arXiv: 1309.4877, JHEP 1401 (2014) 032 “A holographic p-wave superconductor model”,Our model:,plays a c

43、rucial role,The probe limit:,The ansatz:,It is found that only the following case is consistent,Define:,1) Superconducting ferromagnet,2) Ferromagnetic superconductor,1) In the case that the ferromagnetic phase appears

44、 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 stat

45、e. 2) In the case that the superconducting phase appears first, the attractive interaction will lead to a magnetic p-wave superconducting phase in low temperatures. If the interaction is repulsive, the system will be i

46、n a pure p-wave superconducting phase or ferromagnetic phase when the temperature is lowered.,Conclusions:,7、Summary,1) present a holographic model for the paramagnetism -ferromagnetism phase transition 2) realiz

47、e the paramagnetism-antiferromagnetism transition antiferromagnetic quantum phase transition 4）insulator/metal phase transition and colossal magnetoresistance effect5) coexistence and competition between ferrom