Holographic superconductors

Transcription

1 Holographic superconductors Sean Hartnoll Harvard University Work in collaboration with Chris Herzog and Gary Horowitz : , Frederik Denef : Frederik Denef and Subir Sachdev : 0907.????. Strings 09 June 09 Rome Sean Hartnoll (Harvard U) Superconductors June 09 Rome 1 / 17

2 Plan of talk Motivation 1 What is a superconductor? 2 What does a typical theory of superconductivity look like? 3 What defines a nonconventional superconductor? Holographic superconductors 1 Ingredients for a holographic superconductor 2 Black hole instabilities 3 Electrical conductivity 4 Landscape of superconducting membranes Sean Hartnoll (Harvard U) Superconductors June 09 Rome 2 / 17

3 Motivation 1 What is a superconductor? 2 What does a typical theory of superconductivity look like? 3 What defines a nonconventional superconductor? Sean Hartnoll (Harvard U) Superconductors June 09 Rome 3 / 17

4 What is a superconductor? Spontaneously broken global U(1) symmetry Goldstone boson with transformation: θ θ + Λ. Gauge invariance free energy in background potential A: F = d d x f [A dθ]. Current generated by a small field J = δf δa = f [0]δA (London equation). A=dθ+δA It follows that the electrical conductivity diverges as ω 0 J = if [0] δe σ(ω)δe. ω Sean Hartnoll (Harvard U) Superconductors June 09 Rome 4 / 17

5 What does a typical theory of superconductivity look like? In a textbook on superconductivity one finds the BCS Hamiltonian H = k,σ ɛ k c kσ c kσ g eff 2 k,k c k c k c k c k + k A k J k. Interaction term is generated by the exchange of a soft phonon between two effective electrons. Need ɛ k ɛ F, ɛ k ɛ F ω D. Theory predicts the symmetry breaking condensate g eff 2 c k c k = 2ω D e 1/ g eff 2 g(ɛ F ). ω D is Debye frequency (energy scale of phonons) and g(ɛ F ) density of states at Fermi energy. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 5 / 17

6 More on BCS theory Theory predicts the critical temperature in terms of the condensate 2 T C = The electrical conductivity in the superconducting phase is computed from the current two point function σ(ω) = i J xj x R (ω) ω. The U(1) symmetry is global in BCS theory photons not important. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 6 / 17

7 Phase diagram of cuprate High - T c superconductors Sean Hartnoll (Harvard U) Superconductors June 09 Rome 7 / 17

8 Two senses of non-bcs Ingredients of BCS theory Weakly coupled excitations: glue (phonons) + dressed electrons. Interactions: pairing mechanism. Simple sense of non-bcs The glue is not phonons. Effective theory of electrons similar to BCS. Example: heavy fermion compounds paramagnons. A stronger sense of non-bcs There are no quasiparticles! Strongly interacting soup with spontaneous symmetry breaking. Possibly relevant for High-T c. Use AdS/CFT as a solvable example. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 8 / 17

9 Holographic superconductors 1 Ingredients for a holographic superconductor 2 Black hole instabilities 3 Electrical conductivity 4 Landscape of superconducting membranes Sean Hartnoll (Harvard U) Superconductors June 09 Rome 9 / 17

10 Minimal ingredients for a holographic superconductor Minimal ingredients Continuum theory have T µν need bulk g ab. Conserved charge have J µ need bulk A a. Cooper pair operator have O need bulk φ. Write a minimal phenomenological bulk Lagrangian L 1+3 = 1 2κ 2 R + 3 L 2 κ 2 1 4g 2 F abf ab φ iqaφ 2 m 2 φ 2. There are four dimensionless quantities in this action. Newton s constant central charge of the CFT: c = 192L 2 /κ 2. Maxwell coupling DC conductivity σ xx = 1 g 2. Mass scaling dimension ( 3) = (ml) 2. Charge q is the charge of the dual operator O. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 10 / 17

11 Two instabilities of a charged AdS black hole By dimensional analysis T c µ. The dual geometry is therefore Reissner-Nordstrom-AdS. RN-AdS can be unstable against a (charged) scalar for two reasons. Reason 1 [Gubser 08]: Background charge shifts mass: m 2 eff. m2 q 2 A 2 t. Reason 2 [SAH-Herzog-Horowitz 08]: Near extremality AdS 2 throat with mbf-2 2 = 1 4L 2 = 3 2 2L 2 > 9 4L 2 = m2 BF 4. Precise criterion for instability at T = 0 [Denef-SAH 09, Gubser 08] q 2 γ ( 3), γ 2 = 2g 2 L 2 κ 2. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 11 / 17

12 Γ q [Denef-SAH 09] Sean Hartnoll (Harvard U) Superconductors June 09 Rome 12 / 17

13 Endpoint hairy black holes [SAH-Herzog-Horowitz 08] Endpoint of instability is a hairy black hole: ds 2 = g(r)e χ(r) dt 2 + dr 2 g(r) + L2 r 2 ( dx 2 + dy 2), A = A t (r)dt, φ = φ(r). Solve numerically (take = 2). Can obtain O : 8 q O 2 T c Compare 8 to 3.5 for BCS and 5 8 for High-T C. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 13 / 17

14 Electrical conductivity [SAH-Herzog-Horowitz 08] Computed the conductivity. At T 0, typical curves Re Σ Re Σ Ω O 1 If the gap is 2 then we found that Re σ(ω 0) e α /T Ω O2 Generally α 1, unlike BCS theory, no weakly coupled picture in terms of Cooper pairs. Exact gap means the Goldstone boson is not contributing. Sean Hartnoll (Harvard U) Superconductors June 09 Rome 14 / 17

15 Landscape of superconducting membranes [Denef-SAH 09] Many examples in Sasaki-Einstein compactifications of M theory. Distribution of critical temperatures 1000 Solutions Sean Hartnoll (Harvard U) Superconductors June 09 Rome 15 / 17

16 Probing the normal state with magnetic fields [Denef-SAH-Sachdev in preparation] The secret of superconductivity lies in the normal state. The charged-ads black hole is dual to a new exotic state of matter. It is not a weakly coupled Fermi liquid. What is it? Magnetic fields are a powerful tool to characterise Fermi surfaces: de Haas-van Alphen oscillations. Looking for these in AdS/CFT requires one loop in the bulk (1/N). Sean Hartnoll (Harvard U) Superconductors June 09 Rome 16 / 17

17 Summary Theories of superconductivity: weak coupling pairing of electrons. May not apply to nonconventional superconductors, e.g. cuprates. AdS/CFT: strongly coupled system exhibiting superconductivity. Showed results for condensate and electrical conductivity. landscape of superconductors in M theory. Key question: how to think about the normal state? Sean Hartnoll (Harvard U) Superconductors June 09 Rome 17 / 17