Rong-Gen Cai ( 蔡荣根） Institute of Theoretical Physics

1. Kobe University，2016.11.09 HolographicMagnetism from GeneralRelativity Rong-Gen Cai(蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences Yukawa Institute for Theoretical Physics Kyoto University Refs: arXiv:1404.2856,1404.7737,1410.5080,1501.04481, 1504.00855,1505.03405,1507.00546,1507.03105 1502.00437, 1601.02936

2. 2015: GR100 GRisatheoryabout spaceandtime! 引力（时空）的本质？ 1500年前，奥古斯丁 的名言：时间是什么？

3. Itismorethanthat! Itcanalsotellussomethingwhichisnotrelevanttogravity! Thisisthegoalofmytalk: MagnetismfromGR! TheideaisthatthegravityinAdSbulkisequivalenttoa CFTattheAdSboundary,AdS/CFT correspondence, arealizationoftheholographic principle.

4. 1、Introduction: holographic principle Black hole is a window to quantum gravity Thermodynamics of black hole S.Hawking, 1974, J. Bekenstein, 1973

5. Holography of Gravity Entropy in a system with surface area A :S<A/4G A system with gravity in d+1 dimensions can be described by a theory without gravity in d dimension! (G.t’Hooft) (L.Susskind) The world is a hologram？

6. Why GR? TheplanarblackholewithAdSradiusL=1: where: • Temperatureoftheblackhole: • Energyoftheblackhole: • Entropyoftheblackhole: The black hole behaves likeathermalgas in2+1dimensionsinthermodynamics!

7. Topologytheoremofblackholehorizon:

8. AdS/CFT correspondence（1997, J.Maldacena）: CFT AdS “Real conceptual change in our thinking about Gravity.” (E.Witten, Science285(1999)512

9. AdS/CFTdictionary: Here in the bulk: the boundary value of the field propagating in thebulk in the boundary theory: the source of the operator dual to the bulk field

10. quantum gravitational theory in (d+1)-dimensions dynamical fieldφ bulk Quantum field theory in d-dimensions operatorΟ boundary (0909.3553, S. Hartnoll)

11. AdS/CFT correspondence： • gravity/gauge field • 2) different spacetime dimension • 3) weak/strong duality • 4) classical/quantum Applications in various fields: low energy QCD(AdS/QCD), condensedmattertheory(AdS/CMT) e.g.,holographicsuperconductivity (non-)Fermionfluid

12. Holographic magnetism: • Paramagnetism-Ferromagnetism Phase Transition in a Dyonic Black Hole • Phys. Rev. D 90, 081901 (2014)(RapidCommunication) • 2)Model for Paramagnetism/antiferromagnetism Phase Transition • Phys. Rev. D 91, 086001 (2015) • 3)Coexistence and competition of ferromagnetism and p-wave superconductivity in • 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 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 and holographic ferromagnetic phase transition • JHEP 1511 (2015) 021 • 8)Insulator/metal phase transition and colossal magnetoresistance in holographic model • Phys.Rev.D92(2015)106002 • 9) Introduction to Holographic Superconductor Models • Sci. China Phys. Mech. Astron. 58 (2015) 060401

13. Outline: 1Introduction 2Ferromagnetism/paramagnetismphase transition 3Antiferromagnetism/paramagnetismphasetransition 4Antiferromagnetic quantum phase transition 5Insulator/metalphasetransitionandcolossal magnetoresistanceeffect 6Coexistenceandcompetitionbetweenferromagnetism andsuperconductivity 7Summary

14. howtobuildaholographic 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, 1002.1722

15. Holographic superconductors Building a holographic superconductor S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv: 0803.3295 PRL 101, 031601 (2008) High Temperature（black hole without hair):

16. 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:

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

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

20. Summary: • The CFT has a global abelian symmetry corresponding a • massless gauge field propagating in the 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 • temperature. • 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 the free energy.

21. arXiv:1003.0010,PRD82 (2010) 045002 Breaking a globalSU(2) symmetryrepresenting spin into a U(1) subgroup. The symmetry breaking is triggered by condensation of a triplet scalarfield . This model leads to the spatial rotational symmetry breakingspontaneously, the time reversal symmetry is not brokenspontaneously in the magneticordered phase.

22. 2、A model for ferromagnetism/paramagnetismtransition arXiv:1404.2856,PRD90(2014)081901,Rapid Comm. The model: • Thereasons: • Theferromagnetictransitionbreaksthetimereversalsymmetry,spatialrotating • symmetry,butisnotassociatedwithanysymmetrysuchasU(1),SU(2). • Themagneticmomentisaspatialcomponentofatensor, • 3)Inweakexternalmagneticfield,itisproportionaltoexternalmagneticfield.

23. We are considering the probe limit, the background is Temperature: Theansatz:

24. Theboundarycondition:

25. Theoff-shellfreeenergy: Ising-likemodel: arXiv:1507.00546 onshell:

26. Spontaneousmagnetization:B=0

27. Theresponsetoexternalmagneticfield: magneticsusceptibility: ObeytheCurie-WeissLaw

28. Thehysteresisloopinasinglemagneticdomain: When T < Tc, the magnetic moment is not single valued. Theparts DE and BA are stable, which can be realized inthe external field. The part CF is unstable which cannotexist in the realistic system. The parts EF and CB aremetastable states, which may exist in some intermediateprocesses and can be observed in experiment. Whenthe external fieldcontinuouslychanges, the metastable states of magneticmoment can appear.

29. 3、Faramagnetism/antiferromagnetismphasetransition arXiv:1404.7737 Antiferromagnetic material does not show any macroscopic magnetic momentwhen external magnetic field is absent, it is still a kind of magnetic ordered materialwhentemperature 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:

30. Magneticsusceptibility:

31. Threeminimalrequirementstorealizetheholographicmodel • forthephasetransitionofparamagnetism/antiferromagnetism. • TheantiparallelmagneticstructureasT<T_N • Thesusceptibilitybehavior • Breakingthetimereversalsymm&spatialrotatingsymm Ourmodel:

32. The probe limit Theansatz: Define:

33. Theequationsofmotion: Theboundaryconditions:

34. Theparameterconstraint: Theon-shellfreeenergy:

35. alpha_0andbeta_0areinitialvaluesatthehorizon!

37. Theinfluenceonstrongexternalmagneticfield

38. arXiv:1504.00855 Newmodel:

41. 4、Antiferromagneticquantumphasetransition

42. Er2-2xY2xTi2O7 criticalmagneticfield Ex:4.2,Th:5.0 Dynamicalexponent:2

43. 5、Insulator/metalphasetransitionandcolossal magnetoresistanceinholographicmassivegravity Somemagneticmaterialssuchasmanganites exhibitthecolossalmagnetoresistanceeffect. Ourmodel: Blake, Tong and Vegh, arXiv:1310.3832 BlakeandTong,arXiv:1308.4970 MeffordandHorowitz,arXiv:1406.4188 Thereisapositiondependentmass Thismeasuresthestrengthofinhomogeneity

44. Theblackbranesolution: Theansatz: Theasymptoticsolutionattheboundary:

45. DCconductivity: Theperturbation: TheAdSboundary: DCresistivity:

46. Bythemembraneparadigm: IqbalandLiu,arXiv:0809.3808 TheDCresistivityinthestronginhomogeneitylimit:

47. Numericalresults: A.Urushibaraetal,PRB51(1995)

48. 6、Coexistencebetweenferromagnetismand p-waveorder Coexistence between superconductivity and Ferromagnetism!

49. 6、Coexistencebetweenferromagnetismand p-waveorder (arXiv:1410.5080) P-wave:Einstein-Maxwell-Complexvector model: arXiv:1309.4877,JHEP1401(2014)032 “A holographic p-wave superconductor model”

50. Ourmodel: playsacrucialrole