Explaining Fermi Liquid stability with AdS Black holes Koenraad Schalm Institute Lorentz for Theoretical Physics Leiden University Mihailo Cubrovic, Jan Zaanen, Koenraad Schalm arxiv/0904.1993 arxiv/1011.XXXX . . . GGI Firenze 2010 Crete Sep 2010 Friday, November 5, 2010
AdS/CMT Black Hole physics Rich = Friday, November 5, 2010
Charged Black-hole Hair • The origin of rich black hole physics – Instability of charged Black holes [e.g. Gubser @ Strings 2009] � . . . + ¯ φ ( A 0 ) 2 φ + . . . m 2 φ ∼ − A 2 S = ⇔ 0 – The holographic superconductor [Hartnoll, Herzog, Horowitz] r = ∞ + + + + + + r = r H ++++++++++++ + + + + + + + Friday, November 5, 2010
BH Stability • AdS/CFT: Ground state stability is BH stability Z CFT ( φ ) = exp S on − shell ( φ ( φ ∂ AdS = J )) AdS Friday, November 5, 2010
Bosons vs. Fermions • What about Fermions? � √− g � R + 6 − 1 � A Γ A ( D M + igA M ) Ψ − m ¯ 4 F 2 − ¯ Ψ e M S = + S bnd ΨΨ – No perturbative instability (no superradiance) Friday, November 5, 2010
Searching for the Fermi-liquid groundstate Conjecture AdS-RN BH with Fermions is metastable. • – AdS-RN cannot be the true groundstate − Large groundstate entropy S CF T = A H 4 G N − ω 2 dt 2 + d ω 2 � � ω 2 + dx i dx i ds 2 L 2 = Near horizon 2 – Should exist 1st order transition to AdS Fermi-hair BH − Similar indications from [Faulkner et al (unpublished)] , [Hartnoll, Polchinski, Silverstein, Tong] , [ Kraus et al.] Friday, November 5, 2010
Searching for the Fermi-liquid groundstate • What is the true dual groundstate ? – “Free Fermi-gas” in the bulk. PROBLEM: – Fermi-Dirac statistics demands non-local behavior in AdS [always true for multiple Fermions] Non-local RG flow?! Friday, November 5, 2010
Searching for the Fermi-liquid groundstate • What is the true dual groundstate ? – “Free Fermi-gas” in the bulk. PROBLEM: – Fermi-Dirac statistics demands non-local behavior in AdS [always true for multiple Fermions] SOLUTION: – Need some local approximation: − Integrate out the fermions [Kraus..] − Fluid/Thomas Fermi approximation [de Boer et al, Hartnoll et al.] − Single fermion In all cases we wish to know how the fields behave at ∂ AdS to read off what happens in the CFT Friday, November 5, 2010
Fermi gas in a confining potential • In all cases we wish to know how the fields behave at ∂ AdS to read off what happens in the CFT. – AdS acts like a confining potential well. What is it the behavior of the Fermi gas at infinity? − Consider a spinless fermion in a d -dim harmonic oscillator potential well − ∂ 2 2 m + m ω 2 r 2 ¯ x � Ψ = E Ψ , ρ ( r ) = Ψ E ( r ) Ψ E ( r ) 2 E<E F • Thomas-Fermi: • Exact answer fluid is confined and has a long range tail: [Brack, an edge L 2 = 2 /m ω van Zyl] ω − r 2 E F / ω ρ T F ∝ ( E F a i L i (4 r 2 L 2 ) d/ 2 L 2 ) e − 2 r 2 /L 2 � ρ exact ∝ i =0 Friday, November 5, 2010
Fermi gas in a confining potential • In all cases we wish to know how the fields behave at ∂ AdS to read off what happens in the CFT. – AdS acts like a confining potential well. What is it the behavior of the Fermi gas at infinity? − Consider a spinless fermion in a d -dim harmonic oscillator potential well − ∂ 2 2 m + m ω 2 r 2 ¯ � x Ψ = E Ψ , ρ ( r ) = Ψ E ( r ) Ψ E ( r ) 2 E<E F – Far away, confined composite “gas” should approximate a “point particle” 1.003 1.002 Normalized Ratio of fluid density to single particle den- 1.001 Out[18]= − sity in same V ( r ) with 1.000 n = E F / 2 ω for n = 1 , 20 , 100 0.999 0 200 400 600 800 1000 Friday, November 5, 2010
Take it and run... • We wish to know how the behavior at ∂ AdS to read off what happens in the CFT. Conjecture: Study a single Dirac particle in the presence of a BH – This “hydrogen atom” captures the dynamics at ∂ AdS Single particle but large Backreaction — if charge is macroscopic – Expect a Lifschitz S = 0 solution for any charge q F Friday, November 5, 2010
Building an holographic Fermi-liquid • Instead of Ψ , work directly with probability density ± ≡ ¯ J µ Ψ ± i γ µ Ψ ± – Obtain dynamics for composite fields... ± ≡ ¯ Ψ ± i γ µ Ψ ± , I = ¯ J µ Ψ + Ψ − , A 0 = Φ – Infer “equations of motion” from EOM of Ψ ± ∓ Φ ( ∂ z + 2 A ± ) J 0 = f I. ± 2 Φ f ( J 0 + − J 0 ( ∂ z + A + + A − ) I = − ) 1 ∂ 2 2 z 3 √ f ( J 0 + + J 0 = − ) z Φ − − 1 � 3 − zf ′ � ± mL Recall = A ± z √ f 2 z 2 f Friday, November 5, 2010
Building an holographic Fermi-liquid • “Entropy Collapse” to a Lifshitz BH [also Hartnoll, Polchinski, Silverstein, Tong] – Boundary conditions at the horizon z = 1 ... J ± (1 − z ) − 1 / 2 + . . . J 0 = ± I hor (1 − z ) − 1 / 2 + . . . = I hor (1 − z ) ln( z − 1) + Φ (2) (1) = Φ hor (1 − z ) + . . . . Φ – Φ (1) hor corresponds to a “source” on the horizon, (infinite backreaction) – Dynamically Φ (1) hor = 0 J ± = 0 = I hor → “Dirac Hair” requires (mild) backreaction at the horizon ⇒ Friday, November 5, 2010
Building a holographic Fermi-liquid • A holographic Migdal’s relation – Boundary behavior of the Dirac field A + z 3 / 2 − m + B + z 5 / 2+ m + . . . Ψ + = A − z 5 / 2+ m + B − z 5 / 2 − m + . . . Ψ − = – Recall that for bosons Φ = Jz ∆ + � O � J z d − ∆ + . . . – Spontaneous symmetry breaking [Gubser, Hartnoll, Herzog, Horowitz] : solution with J = 0 , � O � � = 0 . J = 0 is the quasinormal mode. Friday, November 5, 2010
Building a holographic Fermi-liquid • A holographic Migdal’s relation – Boundary behavior of the Dirac field A + z 3 / 2 − m + B + z 5 / 2+ m + . . . = Ψ + A − z 5 / 2+ m + B − z 5 / 2 − m + . . . = Ψ − – For fermions the Green’s function ω − v F ( k − k F ) + reg = B − Z G ( ω , k ) = A + – For A ± ( k F ) = 0 , B − ( k F ) cannot be a fermionic vev. Instead A + ( k F ) = | B − ( k F ) | 2 Z ≃ B − ( k F ) n F ( k ) Migdal: n F : Z ∂ ω W k → Friday, November 5, 2010
The meaning of J − • The Green’s function (bulk extension) G ( z ) = Ψ − ( z ) S Ψ − 1 + ( z ) , D + A ± Ψ ± = − / / T Ψ ∓ – Convenient way to solve G directly T Ψ + ( z ) S Ψ − 1 ∂ z G = ( A + − A − ) G + G/ T G − / + ( z ) – Consider however the combinations ( Γ I = { 1 1 , γ i , γ ij , . . . } ) ± ( z ) = ¯ + ( z 0 )¯ J I Ψ − 1 Ψ ± ( z ) Γ I Ψ ± ( z ) Ψ − 1 + ( z 0 ) G I ( z ) = ¯ + ( z 0 )¯ Ψ − 1 Ψ + ( z ) Γ I Ψ − ( z ) S Ψ − 1 + ( z ) If T i has only a single nonvanishing component, these are the same equations as before. Friday, November 5, 2010
The meaning of J − • The Green’s function (bulk extension) G ( z ) = Ψ − ( z ) S Ψ − 1 + ( z ) , D + A ± Ψ ± = − / / T Ψ ∓ – Consider the combinations ( Γ I = { 1 1 , γ i , γ ij , . . . } ) ± ( z ) = ¯ + ( z 0 )¯ J I Ψ − 1 Ψ ± ( z ) Γ I Ψ ± ( z ) Ψ − 1 + ( z 0 ) G I ( z ) = ¯ + ( z 0 )¯ Ψ − 1 Ψ + ( z ) Γ I Ψ − ( z ) S Ψ − 1 + ( z ) – For generic ω , k : + ) − 1 ¯ J I − ( z 0 ) = ( J 1 1 G Γ I G Friday, November 5, 2010
The meaning of J − • The Green’s function (bulk extension) G ( z ) = Ψ − ( z ) S Ψ − 1 + ( z ) , D + A ± Ψ ± = − / / T Ψ ∓ – Consider the combinations ( Γ I = { 1 1 , γ i , γ ij , . . . } ) ± ( z ) = ¯ + ( z 0 )¯ J I Ψ − 1 Ψ ± ( z ) Γ I Ψ ± ( z ) Ψ − 1 + ( z 0 ) G I ( z ) = ¯ + ( z 0 )¯ Ψ − 1 Ψ + ( z ) Γ I Ψ − ( z ) S Ψ − 1 + ( z ) – For pole ω ( k ) : J I Γ I G | on − shell = − Tr γ 0 G F ( ω ( k )) | on − shell = f F D ( T, ω ( k )) ρ states ( ω ( k )) . J − ( ω ( k ) , k ) = n F ( k ) Friday, November 5, 2010
Building an holographic Fermi-liquid “Proof” AdS-RN BH with Fermions is metastable. • � 4 1.2 x 10 n F density 1 0.8 Re J 0.6 0.4 T c ≈ 0.075 ( µ /T) c ≈ 3 0.2 0 0.02 0.04 0.06 0.08 0.1 (vdW liquid to FL transition T as in He 3 .) Friday, November 5, 2010
Building an holographic Fermi-liquid “Proof” AdS-RN BH with Fermions is metastable. • � 4 1.2 x 10 ! + 023/4-536 ! ,75/897,:;./,3<3,:;. n F density -==/7>/>23/?36@:/3;36.A/897,3 1 ! * ! ) 0.8 ! '( ,-./ ! 1 ( " Re J 0.6 ! '' ! 1 " / # /0 / ! '#&& 0.4 T c ≈ 0.075 ! '& ( µ /T) c ≈ 3 0.2 ! '% ! '$ 0 ! ! ! "#" ! " ! $#" ! $ ! %#" ! % ! &#" ! & 0.02 0.04 0.06 0.08 0.1 ,-./0 T Friday, November 5, 2010
Building an holographic Fermi-liquid “Proof” AdS-RN BH with Fermions is metastable. • � 4 1.2 x 10 !"$ - n F density ! 1 ! !"$ 0.8 ! !"& Re J 0.6 ! !"( . / ! 010 2 ! 3 ! / 0.4 ! !"* T c ≈ 0.075 . " .2. / 3. " ( µ /T) c ≈ 3 ! # 0.2 ! #"$ - 0 ! !"!# !"!$ !"!% !"!& !"!' !"!( !"!) 0.02 0.04 0.06 0.08 0.1 T +, ! Friday, November 5, 2010
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