Toward a Holographic Model of d-wave Superconductors Chen-Pin Yeh National Taiwan University With Jiunn-Wei Chen, Ying-Jer Kao, Debaprasad Maity and Wen-Yu Wen.

Outline Introduction to high temperature superconductor How does AdS/CFT work for s-wave superconductor A model for the holographic d-wave superconductor

Introduction (normal SuperConductor) Many metals are superconducting when lowered to certain temperature, Tc (~few Kelvin) Properties: Zero D.C resistance, Perfect diamagnetic, Critical frequency

Introduction (normal SC) Explained by the BCS theory (almost fifty years after the discovery) Fermi surface Why Tc small? k -k BCS works because of this hierarchy Normal state: Fermi Liquid Paring SC state

Introduction (High Tc SC) BCS theory is so successful that it is embarrassed to discover the high Tc material, Cuprate in 1986, for Tc~35K Why Tc high?

High Tc physics Cuprate Two dimension layer made of copper-oxygen plane Un-doped parent compound: Mott Insulator and anti-ferromagnetic.

High Tc physics t-j model xt>>j Fermi-liquid metal ? xt<<j t: hopping energy U: on-site energy J=4(t^2)/U:exchange energy t-j model xt>>j Fermi-liquid metal ? xt<<j Pseudo-gap (spin liquid?) SC

d-wave symmetry High Tc ground state has d-wave symmetry Quasi particle spectrum in BCS state

Experimental evidence of d-wave symmetry s-wave d-wave Tunneling from Normal metal to SC

Why AdS/CFT can help In general sense, AdS/CFT is a duality between gravity theory with isometry identified as conformal symmetry in gauge theory Quantum critical point is a good starting place to apply AdS/CFT Study of gauge theory in strong coupled region (like high Tc) is usually difficult. AdS/CFT provides a alternative

holographic model for non-Fermi liquid Non-Fermi liquid behavior is found in charged AdS black hole Liu, McGreevy and Vegh 09’ One thing people can get holographically is the correlation functions of the dual gauge theory in large N, large coupling.

holographic model for s-wave SC Charged scalar in (3+1) AdS black hole Hartnoll, Herzog and Horowitz 08’ Linear response to electric perturbations Order parameter is dual to ψ(r)

Simple model for d-wave SC Landau-Ginsberg type model N. Mermin 73’ Order parameter is the mix of all d-wave states Particular choice of potential will give symmetry with

Holographic model for d-wave SC Put the following In the (3+1) AdS Black hole background (probe limit) Order parameter in field theory is dual to Bij (i,j=x,y)

Holographic model for d-wave SC Background solution ansatz The equations of motion The asymptotic behavior

Holographic model for d-wave SC Condensate f1 Near the critical point T/Tc Signal the second order phase transition

Electrical perturbation Need to turn on perturbations of B accordingly Linearized equations of motion

Conductivity Ax equation of motion can be solved independently. Equations of motion are the same for x y Conductivity is isotropic gap Re[σ] In-falling at horizon. Near the boundary: ω/Tc

Perturbation of B For m^2<2 we have the following asymptotic behavior After imposing in-falling condition for b at horizon, it is hard to make b normalizable. However configuration of finite energy can be found.

Issues of negative energy When B is uncharged, the negative kinetic energy can be Removed by imposing When B is charged, the constrain is unknown to us yet

Two possibility The d-wave background solution in this set up doesn’t exist after including the constrains. The background ansatz automatically satisfies the constrains (this is true if B is uncharged). And the perturbation of A is still decoupled from B and consistent solutions for B exit. So the calculation of conductivity still holds.

Conclusion Currently done in AdS/CFT Properties of the strange metal Strange metal-SC transition Insulator-SC transition at zero temperature… Our attempt Strange metal-SC transition with the d-wave symmetry order parameter.

Conclusion Future works Magnetic interaction is important in the high Tc materials; gravity theory should account for it. How to understand the doping of Mott insulator and pseudo-gap region holographically. It is still hard to identify the boundary degrees of freedom. Thus less can be said about quasi-particle spectrum, how and why the paring formed.