ApplicationApplication ofof Gauge/GravityGauge/Gravity DualityDuality toto CondensedCondensed mattermatter systemssystems An introduction From Black Hole to Superconductor 曾化碧 （渤海大学） Content • What is AdS/CFT? Applied AdS/CFT • How it works? • Two typical examples of AdS/CMT • Conclusion and future What is Gauge/Gravity duality and the position of it • An important progress in theoritical physics in 20 years. (change the usual understanding of nature) The duality between the largest scale theory and smallest scale theory. Gravity and Quantum field theory Gauge Field Theory and (Quantum) Gravity a) Electromagnetism b) Weak nuclear d) c) Strong nuclear Quantum Gravity: Standard Model String Theory, Loop Quantum Problem: Strong Coupled system in QCD. CMT. Gravity. Gauge Field Theory and Quantum Gravity challenge VS chance a) Electromagnetism b) Weak nuclear d) c) Strong nuclear ? Quantum Gravity: Standard Model String Theory, Loop Quantum Problem: Strong Coupled system in QCD. CMT. Gravity. Motivations • The holographic principle (‘t Hooft, Susskind): quantum gravity in any space can be formulated in terms of degrees of freedom living on the boundary of the space. This is motivated by the Bekenstein-Hawking black hole entropy, S~A(the area of the horizon). G. 't Hooft, \Dimensional reduction in quantum gravity," arXiv:gr-qc/9310026. L. Susskind, \The World As A Hologram," J. Math. Phys. 36, 6377 (1995) [arXiv:hep-th/9409089]. G. ‘t Hooft String theory from SU(N) gauge field theory Large N (size of gauge matrices), to G. 't Hooft, \A planar diagram theory for strong have enough states. ‘t Hooft (1974) interactions," Nucl. Phys. B 72, 461(1974). showed that one gets an interesting limit if one replaced the 3 colors of QCD with a large number N. The remarkable discovery is that these two kinds of theory are dual. Surprise and Profound gauge theory Another way to unify them? Any (quantum) gravity theory in a (d + 1)-dimensional asymptotically anti-de Sitter spacetime is dual to a d-dimensional quantum field theory ‘‘living at its boundary.’’ J. M. Maldacena, \The large N limit of superconformal field theories and supergravity,"Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200]; (引用：10000+) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, \Gauge theory cor-relators from non-critical string theory," Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]; E. Witten, \Anti-de Sitter space and holography," Adv. Theor. Math.Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. The Conclusion Of Gauge/Gravity (AdS/CFT) Duality Maldacena 1997 Witten 1998…. QFT in p D Quantum Gravity in p+1 D Strongly Coupled Weakly coupled Large N QFT in p D Classic Gravity in p+1 D p=3 AdS4/CFT3 People in every fields pay attention to it! Special Relativity Thermodynamics Quantum Field Theory GR from holographic principle , Wei Quantum Mechanics Gu, Miao Li, Rong-Xin Miao arXiv:1011.3419v2 String Theory Gauge/Gravity Duality Black Hole GR Condensed Matter Quantum Information 分子动力学 Jacob D. Bekenstein, Quantum Information and Quantum Black Holes, arXiv:gr-qc/0107049. Xian-Hui Ge, You-Gen Shen, Relating Quantum Information to Charged Black Holes, arXiv:gr-qc/0503094. Baocheng Zhang, Qing-yu Cai, Ming-sheng Zhan, Li You , An interpretation for the entropy of a black hole, arXiv:1102.5144 Two important concepts • Holographic principle Black hole emergent Gauge Gravity (spacetime) holographic • Emergent (浮现) Condensed matter Xiao-gang Wen, string-net condensation Xiao-liang Qi, exactly holographic picture Another example: Area laws for the entanglement entropy - a review，J. Eisert, M. Cramer, M.B. Plenio， arXiv:0808.3773 Ancient thoughts • 天地万物莫大于秋毫。《庄子》 • 一花一世界。 • 全息（holographic） 和 浮现(emergent)，同一个事物的两方面？？ How It Works? • Anti- De sitter Space and the Extra Dimension Holographic Renormalization Group Holographic RG and effective theory The Duality Dictionary Finite Temperature Gauge/Gravity Duality as a LAB Black hole full of answers (Nature 448, 1000-1001 (30 August 2007)) Some Applications • AdS/QCD arXiv:1101.0618 • AdS/Hydrodynamic P. K. Kovtun, D. T. Son, and A. O. Starinets, Phys.Rev.Lett. 94 (2005) 111601… • Holographic Universe Paul McFadden, Kostas Skenderis, The Holographic Universe, arXiv:1001.2007. • AdS/CM (Condensed Matter). CM/Gravity? (The Concept of Emergent in CM) Supersymmetry emerged from a 2+1 dimension lattice model arXiv:1009.5127. A review from condensed matter theorist: Subir Sachdev, Condensed matter and AdS/CFT, arXiv:1002.2947. A review from General Relativity theorist: Gary T. Horowitz, Surprising Connections Between General Relativity and Condensed Matter, arXiv:1010.2784. Why AdS/CMT ？ • Understand strongly coupled quantum many body systems is difficult. (Non-fermion liquid, High T SC, interplay between disorder and interaction, metal/insulator phase transition) • Far-from equilibruim statistics. (Tuburlance, superfluid/superconductor dynamics,non-equilibruim phase transition...) • AdS/CMT have already covered many fields in CMT. The partitial function of a quantum many-body systemcan be obtained by solving classic field equations of a gravity theory. Works very well!!!! (1000+) Some Applications via holography • FQHE JHEP 0809:130,2008 ; JHEP 0906:066,2009 ; JHEP 0811:020,2008 .(Chern-Simons on the boundary) • Quantum Critical Point Phys.Rev.B76:144502,2007 . PhysRevD.75.085020. • Topological Insulator Phys.Rev.D82:126003,2010; Phys.Rev.D82:086014,2010 . • (Topological) entanglement entropy Phys.Rev.Lett.106:141301,2011; Phys.Rev.Lett. 96 (2006) 181602 . • Disorder system JHEP 0812:065,2008; arXiv:1103.6068. • 1+1 dimension system JHEP 1011:055,2010 ; JHEP 1001:114,2010 • Lattice model Hubbard model New J.Phys.13:035004,2011; Phys.Rev.D81:026007,2010 Two Examples with details • Superconductor and theory of phase transitions • Non-fermion Liquid from AdS/CFT Holographic Superconductors • Motivation: superconducting black holes. • Method: add matter fields to a black hole background to make it superconduct. Steven S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys.Rev.D78:065034,2008 S-wave Superconductor From Gauge/Gravity Duality Sean A. Hartnoll, Christopher P. Herzog, Gary T. Horowitz, Building an AdS/CFT superconductor,Phys.Rev.Lett.101:0316 01,2008 (Cited by 750+times) A Neutral AdS Black Hole coupled with a charged scalar field and a Maxwell field will superconduct . Some Details of Calculation The Lagrange: The black hole background: The Hawking temperature: The Ansartz: The equations to solve: The condensation Asymptotic bahavior on the boundary: Chemical Charge density potential What the model get ? Order parameter：(Strongly coupled) ～ The same as the G-L also BCS theory • Conductivity Goes to infinite at zero frequency, a sign of superconducting • Coherence Length and penetration Phys. Rev. D 78, 106006 (2008) . • Vortex solution under magnetic field Phys.Rev.D81:126012,2010 ; Phys.Rev.D78:046004,2008 ; JHEP0809:121,2008 ; Phys.Rev.D80:126009,2009 ; Phys.Rev.D81:026002,2010 ; PhysRevLett.103.091601. • Supercurrent JHEP07(2010)060 . • Josephson Junction arXiv:1101.3326 . • ……….. All the same like G-L theory P-wave and D-wave models: add different matter fields in the black hole back ground. • P-wave: Einstein-Yang-Mills Theory Phys. Rev. Lett.101, 191601 (2008); JHEP;0811, 033 (2008); JHEP 0808, 035 (2008) D-wave: Charged Tensor Field in the Black Hole Phys. Rev.D81:106008, 2010; JHEP 1011:137,2010 ; arXiv:1103.1714 Jiunn-Wei Chen NTU What we have done in the first two years of AdS/CMT ? • Properties of p-wave and d-wave holographic superconductor under magnetic field. Phys. Rev. D 80, 066001 (2009); Phys. Rev. D 82, 126008 (2010). All these results are the same as the G-L（BCS） theory • Some important properties of d-wave and –p-wave model at the phase transition point Phys. Rev. D 81, 106001 (2010); Phys. Rev. D 82, 126014 (2010). The conclusion: The coherence length for both models are divergent like the G-L theory , the behavior of penetration depth also the same as G-L theory. Coherence length: Landon equation: Superfluidity density: • Supercurrent in p-wave holographic superconductor Phys.Rev.D83:046010,2011 The conclusion: We expected the same results for d-wave model. Connecting holographic superconductor and G-L theory (BCS theory?) • Try to understand the universal critical exponents for holographic supercondutor models H.B. Zeng, Xin Gao, Yu Jiang, and Hong-Shi Zong, Analytical Computation of Critical Exponents in Several Holographic superconductors, arXiv:1012.5564, JHEP 1105:002,2011. The conclusion: It is the fundamental symmetry decides the universal behaviors of holographic superconductors just like G-L theory Lei Yin, Defu Hou, Hai-cang Ren, PhysRevD.91.026003 . We need HRG to understand these models better (fixed point, Mermin-Wagner theorem, B-K-T phase transition and so on). Things to do. Ginzburg-Landau-Wilson ？ Comments： these models reproduce the well understand properties for second order phase transition and support the Gauge/Gravity duality . Are more than these? One example: Non-Fermion Liquid Strange Metal Transport Realized by Gauge/Gravity Duality Bottom-up: Fermionic field in a black hole background. Hong Liu, John McGreevy, David Vegh , PhysRevD.83.065029. Thomas Faulkner, et al. Science 329, 1043 (2010). Thomas Faulkner, Nabil Iqbal, Hong Liu, John McGreevy, arXiv:1003.1728. Top-down: Probe D-brane. S.A.Hartnoll, J.Polchinski, E.Silverstein, A class of non-Fermi liquid discovered by D.Tong,JHEP 04. 120. 2010. holographic duality. The strange metal behavior • The linear temperature dependence of the resistivity. Still no satisfactory theoritical explanation. • ARPES: The low energy excitation near Fermi surface are not Fermi-liquid like quasiparticles. Another example: the heavy fermion system near a quamtum phase transition. The model The charged black hole background: The perturbative spin field : Results • The spectrum function Fit well with the ARPES experiment when v_k=0.5 • The conductivity • The geometry of the horizon is very important for the appearance of the Non-Fermi Liquid behavior.