Holographic d-wave superconductors Francesco Benini Princeton - PowerPoint PPT Presentation

GGI workshop, 13 October 2010 Holographic d-wave superconductors Francesco Benini Princeton University with Chris Herzog, Rakibur Rahman, Amos Yarom based on 1006.0731 1007.1981

Outline ● Holographic superconductors ● d-wave superconductors charged massive spin-2 fields ● Fermionic operators and spectral function ● d-wave gap, Dirac nodes, Fermi arcs ● Future directions

Real superconductors ● Superconductor: a system characterized by a transition to a state with zero resistivity (below T c ) It can be modeled by spontaneous breaking of U(1) e.m. Gizburg Landau ● BCS: weakly coupled description. E.g. cuprates look strongly coupled ● Phenomenology depends on the nature of the order parameter ● Cuprates: d-wave superconductors (spin-2 order parameter) ● Interesting phenomenology, ARPES & STM: d-wave gap, Dirac nodes, Fermi arcs, pseudo-gap, ... ● Motivation: how far can we go without using the details of the atomic structure, but only “symmetries” and basic features?

Experimental results ● Normal phase: Fermi surface ● Superconducting phase: Fermi surface is gapped ● d-wave: anisotropic gap ~ |cos 2θ| ● 4 nodes ● Dirac cones at the nodes ● In pseudo-gap phase: nodes open into Fermi arcs

Holographic Superconductors ● Holographic superfluid: a field theory (CFT) at temperature T ≥ 0 and non-zero chemical potential ρ , with U(1) global symmetry and charged order parameter ψ ● Holographic superconductor: weakly gauge the U(1) (photon) ● Study the CFT at strong coupling via AdS/CFT ● Study the behavior of extra operators (not directly involved in condensation), e.g. fermionic operators ● Bottom-up approach: focus on subset of fields

AdS/CFT Map CFT d , T mn Gravity (Einstein-Hilbert) in g μν asymptotically AdS d+1 U(1) global, J m U(1) gauge symmetry, A μ Charged order parameter charged (massive) field O of dimension Δ ψ of mass m 2 z 2 ~ 2  − dt 2  CFT at T > 0 BH in AdS 2  d  2  dz ds x d − 1 z  0 L Turn on chemical potential  s  z d −− 1 〈 J m 〉 z − 1 A m ~ J m source J 0 (s) for J m  s  z # 〈 O 〉 z # No source O (s) , read off VEV O ~ O Causal CFT regular & infalling b.c. at horizon

d-wave ● The order parameter is d-wave massive charged spin-2 field in the bulk → (graviton: massless neutral) ● Various problems could arise: wrong number of d.o.f. ghosts faster than light signals on non-trivial background ● What action?

Spin-2 fields    − m 2     d , 1 ● E.g. : L =−∂    ∂ ℝ in ● Number of d.o.f.: symmetric φ μν massive spin-2 particle constraints  d  1  d  2  d  d  1  − 1 d  2 2 2 ● The extra modes contain ghosts .

Fierz-Pauli action ● Fierz-Pauli action (unique quadratic and 2 derivatives): 2  ∣   ∣ 2  2  2 ∣   ∣ 2 − 2  2 − m 2 −  ∂   ∣ ∂   ∣ L FP =− ∣ ∂    ∣ where    and ≡     ≡∂ ● Get the equations: 2   0 =□− m } d  2 constraints 0 =  0 = ● Correct number of d.o.f., no ghosts, causal propagation

Fierz-Pauli action 2  ∣   ∣ 2  2  2 ∣   ∣ 2 − 2   ∂   ∣ ∂   ∣ 2 − m 2 − L FP =− ∣ ∂    ∣ ● Coefficients uniquely fixed by either: require d+2 constraint equations use Stückelberg formalism and require not higher derivative terms    h   1 m ∂  B  ∂  B  − 1 2 ∂  ∂  X m  h  =∂    ∂     B  =∂  − m    X = 2m  require no ghosts nor tachyons in the propagator

Charged Spin-2 field on background ● Covariant derivative: ∂   D  =∂    − iq A  → get the problems back! a L a − i q F  [ D  , D  ]= R  Solved by coupling to curvatures R μνρλ & F μν ● Write down most general quadratic 2-derivative action (up to dim d+1 operators) Require d+2 constraint equations ● Metric: background must be Einstein (vacuum) → probe limit Buchbinder Gitman Pershin ● F μν : background can be generic Federbush

Charged spin-2 field on background ● Action: ∗ D   c.c. − m 2  2 ∣   ∣ 2  ∣ D   ∣ 2 − 2 − ∣  ∣ 2  2  ∣   ∣ L spin 2 =− ∣ D    ∣ 1 2 − i q F   ∗   − ∗    d  1 R ∣  ∣  2 R   ● Einstein background → probe limit L tot = R −− 1   L spin 2 4 F  F   =    / q A  =  ● Large q and small ρ : A  / q = / q matter & gauge fields do not L mat =  L mat / q backreact on the metric

Charged Spin-2 field on background ∗ D   c.c. − m 2  2 ∣   ∣ 2  ∣ D   ∣ 2 − 2 − ∣  ∣ 2  2  ∣   ∣ L spin 2 =− ∣ D    ∣ 1 ∗   − 2 − i q F   ∗    d  1 R ∣  ∣  2 R   ● F μν new problem: faster than light signals → Velo 2  v max ≃  1  q ∣ F  ∣ m at large momenta (hyperbolic for small F μν ) ● Argyres-Nappi: causal action on 26-dimensional flat spacetime & constant F μν . Each term is non-linear function of F μν .

Charged spin-2 field on background ● Argyres-Nappi action: ● We think of our action as first terms in expansion in q ∣ F  ∣ ≪ 1  ≫ 1 2 m

AdS/CFT ● Field/operator correspondence: φ μν ↔ O mn  s  z 2 L − 2  O mn d −− 2 2 =− d   mn z  0 〈 O mn 〉 z ~  d m 2  − f  z  dt f  z   ● Ansatz: f  z = 1 −  z h  d 2 2 2 = L 2  dz z d 2  d  T = ds x d 4  z h z A = z  dt 2  xy = L ⇒ D    == 0 2  z  2 z ● Equations: same as s-wave! ● Boundary conditions: chemical potential ρ critical temperature → T c

AdS/CFT ● There exist a critical temperature T c ● T > T c : normal state (charged BH) A = [ 1 −  ] dt z h  d − 2 z   = 0 ● T < T c : superconducting phase (condensate) φ xy ≠ 0 d = 2+1: ● Compute conductivity σ mn : in d = 2+1 isotropic at leading order

Fermions ● In AdS/CFT only gauge-invariant operators: study fermionic operators bulk spinor Ψ composite fermionic operator ↔ O Ψ (from p.o.v. of weakly gauged U(1) “composite electron”) ● Compute retarded Green's function & spectral function:  ,  k = Tr Im G R  ,    0 }〉 G R  t ,  x = i  t 〈{ O   t ,  x  ,O  k  ● Direct connection with ARPES ● Green's function used to detect Fermi surface in normal phase Liu, McGreevy, Vegh Cubrovic, Zaanen, Schalm in the following d = 2+1

Fermionic action ● What action? Write down all terms up to dimension 5 (on background): ∗   D ∗   c    h.c.  D  − m   2  c 3  i c 4  5   i ∣   ∣ L  = i  ∗  ∗  c  c 1  c 2  5  h.c.   ● We use: D  =∂    − i q ∗  c   D  − m    D ∗     h.c. L  = i  2 A  Faulkner, Horowitz, McGreevy, Roberts, Vegh ● Majorana-like term: considered for s-wave it gives rise to a gapped Fermi surface

Retarded Green's function ● 2-point function:   0 }〉 G R  t ,  x = i  t 〈{ O   t ,  x  ,O  0 =  D  − m   2i     D   c ● EOM − i  t  i   ,  i  t − i  − , − probe k ⋅ k   z  e k ⋅ k   z  x   x  = e asymptotic =   2  − m  L z   O  z   e z  0   1 S    e  1 O  z  R  m  L z   ~ infalling b.c.  ,   ,  − , − G R  ,  k    k  = M   S   S  k  c t k =− i M  R  M  ● Spectral function (density of states):  ,  k = Tr Im G R  ,  k  sharp peaks dispersion relation ω( k ) of quasi-normal modes →

The gap – E.g. s-wave ● Peaks in spectral func. → ω( k ) quasi-normal modes ● η = 0 : Fermi surface 0 = D  1   1 = E  ⇒ k  0 = D  2   2

The gap – E.g. s-wave ● Peaks in spectral func. → ω( k ) quasi-normal modes ● η ≠ 0 : gap ∗ = E  0 = D  1   1  2 ⇒  : k  c : =− E − ∗  k  0 = D  2   2  1 Ψ: Ψ c :

d-wave spectral function  D c       Coupling: ● E.g. : spectral func. at θ = fixed ω ● Exp procedure: for every θ 1) identify Fermi momentum k 2) draw EDC ω k x k y

EDC's ● Compare Energy Distribution Curves with exp's: Kanigel et al, PRL 99 (2009) 157001 Underdoped Bi 2 Sr 2 CaCuO 8