Holographic superconductors and superfluids: effect of backreaction - PowerPoint PPT Presentation

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Holographic superconductors and superfluids: effect of backreaction Betti Hartmann Jacobs University Bremen, Germany Gauge/Gravity Duality MPI Munich, 1st August 2013 Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Collaborations and References Work done in collaboration with: Yves Brihaye - Université de Mons, Belgium References: Y. Brihaye and B. Hartmann, Phys.Rev. D81 (2010) 126008 Y. Brihaye and B. Hartmann, JHEP 1009 (2010) 002 Y. Brihaye and B. Hartmann, Phys. Rev. D83 (2011) 126008 Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Outline Motivation 1 The model 2 Gauss-Bonnet Holographic superconductors 3 Holographic superfluids away from probe limit 4 Summary 5 Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Outline Motivation 1 The model 2 Gauss-Bonnet Holographic superconductors 3 Holographic superfluids away from probe limit 4 Summary 5 Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Gauge/gravity duality d-dim String Theory ⇔ (d-1)-dim SU(N) gauge theory (Maldacena; Witten; Gubser, Klebanov & Polyakov (1998)) Couplings λ = ( ℓ/ l s ) 4 = g 2 N g s ∼ g 2 , λ : ’t Hooft coupling N : rank of gauge group ℓ : bulk scale/AdS radius l s : string scale g : gauge coupling g s : string coupling classical gravity limit: g s → 0 l s /ℓ → 0 dual to strongly coupled QFT with N → ∞ Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Application: high temperature superconductivity Bismuth-strontium-calcium-copper-oxide superconductivity associated Bi 2 Sr 2 Ca Cu 2 O 8+x to CuO 2 -planes T c ≈ 95 K energy gap ω g / T c = 7 . 9 ± 0 . 5 (Gomes et al., 2007) ⇒ compare to BCS value: ω g / T c = 3 . 5 ⇒ Strongly coupled electrons in high-T c superconductors? Taken from wikipedia.org Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Holographic phase transitions T : temperature, µ : chemical potential AdS black hole AdS black hole at at T  T c , ≠ 0 T  T c , ≠ 0 scalar hair formation Holographic Holographic superconductor/ phase conductor/ superfluid transition fluid  small phase − 1 T   transition AdS soliton AdS soliton scalar hair at  c ,T  0 at 0  c ,T  0 formation Holographic phase Holographic superconductor/ transition insulator superfluid Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Higher order curvature corrections Coleman-Mermin-Wagner theorem : no spontaneous symmetry breaking of continuous symmetry in (2+1) dim at finite temperature BUT: holographic superconductors have been constructed using (3+1)-dim black hole solutions of classical gravity (Hartnoll, Herzog & Horowitz (2008) and many more...) ⇒ higher order curvature corrections on gravity side ??? Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Probe limit vs. Backreaction G = 0 ( e = ∞ ) “probe limit” : fixed space-time background ⇒ coupled scalar & gauge field equations G � = 0 ( e < ∞ ): backreaction of matter fields on space-time ⇒ coupled gravity, scalar & gauge field equations results in systems of coupled, nonlinear ordinary or partial differential equations that have to be solved numerically Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary The model Gauss-Bonnet gravity in d-dimensional Anti-de Sitter (AdS d ) � S = 1 d d x √− g � R − 2 Λ + α � 2 L GB + 2 γ L m + S ct 2 γ Gauss-Bonnet Lagrangian � R MNKL R MNKL − 4 R MN R MN + R 2 � L GB = S ct : boundary counterterm γ = 8 π G : gravitational coupling Λ = − ( d − 1 )( d − 2 ) / ( 2 ℓ 2 ) : cosmological constant ℓ : AdS radius α : Gauss–Bonnet coupling Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary The model Lagrangian of charged complex scalar field : L m = − 1 4 F MN F MN − ( D M ψ ) ∗ D M ψ − m 2 ψ ∗ ψ , M , N = 0 , 1 , 2 , 3 , d − 1 U(1) field strength tensor F MN = ∂ M A N − ∂ N A M covariant derivative D M ψ = ∂ M ψ − ieA M ψ e : electric charge m : mass of scalar field Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Outline Motivation 1 The model 2 Gauss-Bonnet Holographic superconductors 3 Holographic superfluids away from probe limit 4 Summary 5 Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary The Ansatz For d ≤ ( 3 + 1 ) Gauss-Bonnet terms do not contribute Metric in d = ( 4 + 1 ) f ( r ) dr 2 + r 2 � dx 2 + dy 2 + dz 2 � 1 ds 2 = − f ( r ) a 2 ( r ) dt 2 + Electric field only A M dx M = φ ( r ) dt Gauge freedom: scalar field chosen to be real ψ = ψ ( r ) Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Equations 2 r 2 r 2 /ℓ 2 − f f ′ = r 2 − 2 α f � 2 e 2 φ 2 ψ 2 + f ( 2 m 2 a 2 ψ 2 + φ ′ 2 ) + 2 f 2 a 2 ψ ′ 2 � γ r 3 − ( r 2 − 2 α f ) 2 fa 2 γ r 3 ( e 2 φ 2 ψ 2 + a 2 f 2 ψ ′ 2 ) a ′ = af 2 ( r 2 − 2 α f ) � 3 � φ ′ + 2 e 2 ψ 2 r − a ′ φ ′′ = − φ a f � 3 � � � m 2 − e 2 φ 2 r + f ′ f + a ′ ψ ψ ′ + ψ ′′ = − fa 2 a f � �� � m 2 eff Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Conditions at the horizon Horizon r = r h f ( r h ) = 0 , a ( r h ) finite Regularity of matter fields on horizon � m 2 ψ r 2 � φ ( r h ) = 0 , ψ ′ ( r h ) = � 4 r − γ r 3 � � m 2 ψ 2 + φ ′ 2 / ( 2 a 2 ) � � r = r h Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Conditions on the AdS boundary Electric potential φ ( r ≫ 1 ) = µ − q / r 2 µ : chemical potential q : charge density Scalar field ψ ( r ≫ 1 ) = ψ − r λ − + ψ + r λ + with � 2 α 4 − 3 ( ℓ eff /ℓ ) 2 , ℓ 2 λ ± = 2 ± eff = � 1 − 4 α/ℓ 2 1 − Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Parameters Equations invariant under rescalings r → λ r , t → λ t , ℓ → λℓ , e → e /λ , α → λ 2 α ( 1 ) : φ → λφ , ψ → λψ , γ → γ/λ 2 , e → e /λ ( 2 ) : ⇒ e = 1 , ℓ = 1 without loss of generality choose m 2 = − 3 ℓ 2 > m 2 BF = − 4 ℓ 2 (close to, but above Breitenlohner-Freedman (BF) bound) Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary AdS black hole without scalar hair ψ ( r ) ≡ 0 , φ ( r ) = q − q r 2 , a ( r ) ≡ 1 r 2 h �  � � � � � f ( r ) = r 2 1 − r 4 − 4 αγ q 2 1 − r 2 � 1 − 4 α � h  1 −  ℓ 2 r 4 r 6 r 2 2 α h (Boulware, Deser, 1985; Cai, 2002) Betti Hartmann Holographic superconductors and superfluids: backreaction

Motivation The model Gauss-Bonnet Holographic superconductors Holographic superfluids away from probe limit Summary Formation of scalar hair on AdS black hole Pioneering example: electrically charged AdS black hole close to T = 0 unstable to form scalar hair (Gubser, 2008) Example (Y. Brihaye & B.Hartmann, Phys.Rev. D81 (2010) 126008) d=4+1 BF bound m 2 eff AdS unstable AdS stable Betti Hartmann Holographic superconductors and superfluids: backreaction