Application of Gauge/Gravity Duality to Application of Gauge/Gravity Duality to Condensed matter systemsCondensed matter systems

曾化碧（渤海大学）（渤海大学）

From Black Hole to Superconductor

An introduction

Content

• What is AdS/CFT?

• How it works?

• Two typical examples of AdS/CMT

• Conclusion and future

Applied AdS/CFT

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) Electromagnetismb) Weak nuclearc) Strong nuclear

d)

Standard ModelQuantum Gravity: String Theory, Loop Quantum Gravity.

Problem: Strong Coupled system in QCD. CMT.

Gauge Field Theory and Quantum Gravity

a) Electromagnetismb) Weak nuclearc) Strong nuclear

d)

Standard ModelQuantum Gravity: String Theory, Loop Quantum Gravity.

Problem: Strong Coupled system in QCD. CMT.

?challenge VS chance

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].

String theory from SU(N) gauge field theoryG. 't Hooft, \A planar diagram theory for strong

interactions," Nucl. Phys. B 72, 461(1974).

G. ‘t Hooft

Large N (size of gauge matrices), tohave enough states. ‘t Hooft (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.

gauge theory

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].

Surprise and Profound

Another way to unify them?

The Conclusion Of Gauge/Gravity (AdS/CFT) Duality

QFT in p D Quantum Gravity in p+1 D

Large N QFT in p D Classic Gravity in p+1 D

p=3 AdS4/CFT3

Maldacena 1997Witten 1998….

Strongly Coupled Weakly coupled

Gauge/Gravity Duality

Special Relativity

Quantum MechanicsQuantum Field Theory

Black Hole

String Theory

Condensed MatterQuantum Information

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

GR

People in every fields pay attention to it!

Thermodynamics

GR from holographic principle , Wei Gu, Miao Li, Rong-Xin Miao

arXiv:1011.3419v2

分子动力学

Two important concepts

• Holographic principle Black hole Gauge Gravity (spacetime)

• Emergent (浮现) Condensed matterholographic

emergent

Xiao-gang Wen, string-net condensation

Xiao-liang Qi, exactly holographic pictureAnother 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 LABBlack 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)

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.

Supersymmetry emerged from a 2+1 dimension lattice model arXiv:1009.5127.

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.101:031601,2008 (Cited by 750+times)

A Neutral AdS Black Hole coupled with a charged scalar field and a Maxwell field will superconduct .

Some Details of CalculationThe Lagrange:

The black hole background:

The Hawking temperature:

The Ansartz:

The equations to solve:

Asymptotic bahavior on the boundary:

The condensation

Chemical potential

Charge density

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.

• 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

Connecting holographic superconductor and G-L theory (BCS theory?)

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,

D.Tong,JHEP 04. 120. 2010. A class of non-Fermi liquid discovered by 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

• The conductivity

Fit well with the ARPES experiment when v_k=0.5

• The geometry of the horizon is very important for the appearance of the Non-Fermi Liquid behavior.

We may say that the strange metal phase in cuprates is a strong coupled fermion system with the same scale behavior of the horizon.

Thomas Faulkner, Nabil Iqbal, Hong Liu, John McGreevy, arXiv:1003.1728.

HRG and effective field theory of holographic non-fermion Liquid

T.~Faulkner and J.~Polchinski,``Semi-Holographic Fermi Liquids,'' JHEP {\bf 1106}, 012 (2011) [arXiv:1001.5049 [hep-th]].

Semi-Holographic Non-fermion Liquid

An general AdS/CMT theory

Solve the equations of motion for the fields then we can compute the one point and two point green function, then we know the information of the boundary theory.

Problems: 1.How to find a proper AdS/CMT theory? (trial-and- error?)

2.Some times the physics is not very clear. We can not know the exact action of the field theory.

Other Voices

• P.W. Anderson..... (Strange theory )• Quantum Gravity...• Future?(numerical relativity... )

Some other important AdS/CMT works

• Non-equilibruim process M. J. Bhaseen, Jerome P. Gauntlett, B. D. Simons, Julian Sonner, Toby Wiseman, Holographic Superfluids and the Dynamics of Symmetry Breaking,Phys.Rev.Lett. 110 (2013) 015301; Allan Adams, Paul M. Chesler, Hong Liu,Holographic Vortex Liquids and Superfluid Turbulence ,Science 341 (2013) 368-372 .

• Electron transport properties: Yi Ling, Chao Niu, Jianpin Wu, Zhuoyu Xian, Hongbao Zhang, Metal-insulator Transition by Holographic Charge Density Waves, Phys. Rev. Lett. 113, 091602 (2014)

Aristomenis Donos, Sean A. Hartnoll,Interaction-driven localization in holography ,Nature Phys. 9 (2013) 649-655 .

Our Recent works

• Nonequilibrium superconductor : JHEP 07(2014)096 , JHEP06(2014)019

• Anderson Localization: Phys.Rev. D88 (2013) 12, 126004

• Conductivity in Cuprates: Phys. Rev. D 90, 046001 (2014)

• Impruity effect in superconductor: arXiv:1411.3955

• One dimensional superconductor: Phys.Rev. D87 (2013) 4, 046009,JHEP 1211 (2012) 112

Future directions: QCD; Electron transport theory; non-equilibruim statistics;HRG and make a bridge between holographic models and field theory.

Summary and Comments• The duality has not been proved strictly, there are

also voices of doubt. However, it works well.• What we learn now and how to make the method

more powerful. (rules to find a gravity theory for a given system?) (Symmetry)

• Problems: Try, like a black box...• Good work need more communication between

different fields.(Non-equilibruim statistics...)• Ultimate goal: to solve the strong coupled

problems? Need more breakthrough.(RG theory in AdS/CFT)

Thanks very much!