Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics

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1 Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics Dmitry Bagrets Nucl. Phys. B 9, 9 (06) arxiv: Alexander Altland Univ. zu Köln Alex Kamenev Univ. of Minnesota PCS IBS Workshop, Daejeon, South Korea, September 5-9, 06

2 J ijkl χ χ Ν χ This talk Sachdev-Ye-Kitaev model χ Ν - infra-red emergent conformal symmetry - soft-mode action & mapping to Liouville quantum mechanics - IR universal assymptotics of correlation functions Holographic principle & NAdS /NCFT correspondence χ i

3 Sachdev-Ye-Kitaev model S. Sachdev, PRX 5 (05) 0405 A. Kitaev, talks at KITP, Spring 05 Hˆ N = 4! ijkl J χ χ χ χ ijkl i j k l χ Ν J ijkl χ χ - Couplings J s are quenched random Gaussian variables: χ Ν χ i ( ) 3 J J J ijkl = 0; ijkl = 3! N Class D Majorana wire {χ i, χ j }=δ ij

4 Effective action S. Sachdev 05; J. Maldacena & D. Stanford 05 (R-times) replicated Matsubara action NJ exp exp ' R R 4 ˆ a ab H dτ = Gττ ' dτ dτ a= 8 a, b = ab a b -point Green s function: G ( τ ) ( τ ') ττ ' χi χi N i

5 Effective action S. Sachdev 05; J. Maldacena & D. Stanford 05 (R-times) replicated Matsubara action NJ exp exp ' R R 4 ˆ a ab H dτ = Gττ ' dτ dτ a= 8 a, b -point Green s function: Gττ = χ χ ( τ ) ( τ ') ab a b ' i i N i Resolution of identity ab a b T = δ NGττ ' + χi ( τ ) χi ( τ ') DG = exp tr N G χi χi D( G, ) i Σ Σ Σ i

6 Effective action S. Sachdev 05; J. Maldacena & D. Stanford 05 - integrating out Majoranas Self-energy N J ab 4 ba ab S[ G, Σ ] = tr ln ( τ + Σ) + Gττ ' dτ dτ ' + Στ ' τgττ ' dτ dτ ' 4

7 Effective action S. Sachdev 05; J. Maldacena & D. Stanford 05 - integrating out Majoranas Self-energy N J ab 4 ba ab S[ G, Σ ] = tr ln ( τ + Σ) + Gττ ' dτ dτ ' + Στ ' τgττ ' dτ dτ ' 4 Emergent conformal symmetry in the IR limit - reparametrization of time: τ = f ( t) - Green s function: G% f G f - Self-energy: [ ] / 4 [ ] / 4 t t = '( t) τ τ '( t) Σ% Σ τ τ [ ] 3/ 4 [ ] 3/ 4 t t = f '( t) f '( t) τ f(t) t

8 Saddle point Self-consistent Dyson equation (S. Sachdev, J. Ye 993) ( ) ab ab 3 =, Σττ ' J Gττ ' + Σ G = τ Mean-field solution G δ τ τ ' ab ab τ τ ' / J (!) breaks IR conformal symmetry down to SL(,R)

9 Conformal symmetry (d=) A. Kitaev 05 Infinitesimal reparametrization transformation τ = t + a + a t + a t + a < < g τ = 0 g( t) + ( ) k ( ) k = 0..., + a t g t + k t k... Virasoro algebra (d=) [ Lˆ, Lˆ ] = q p Lˆ, Lˆ : = t p, p 0 ( ) + p q p q p t Subgroup H=SL(,R) at + b τ = h ( t) - spanned by generators with p=0,, ct + d

N N tt f f τ } dτ 4 J 0 ' ' symmetry breaking term + ε+ω/ - the Schwarzian derivative is

10 IR soft-mode action J. Maldacena & D. Stanford 05 Goldstone s action on G/H S ( ) log [ f ] = N tr ˆ [ ] ˆ ' [ ] {, tgtt f t G N N tt f f τ } dτ 4 J 0 ' ' symmetry breaking term + ε+ω/ - the Schwarzian derivative is defined by { f, τ } f ''' 3 f '' f ' f ' - SL(,R) invariance of the soft action at + b h f, τ =, τ if h( t) = SL(,R) ct + d { o } { f } ω ε ω/

11 Green s function DB, Altland, Kamenev 06 Q: What is the IF limit of Green s function? / 4 / 4 [ f '( τ ] [ ] ) f '( τ ) S0 G( τ τ ) m Df ( τ ) e / G / H f ( τ) f ( τ ) - average the mean field result over Goldstone s soft modes - orthogonality catastrophe is expected Phase representation f(τ ) [ f ] S 0 + N log N [ ϕ] ϕ ', ' J ( τ ) dτ f ( τ ) non-compact phase = ( ) e ϕ τ τ

12 Green s function DB, Altland, Kamenev 06 Q: What is the IF limit of Green s function? dα G( τ τ ) m Dϕ [ τ ] e e e α + G / H 0 τ ϕ ( τ ) ϕ ( τ ) S0 [ ϕ ] α exp[ ϕ ( τ )] dτ 4 4 τ Liouville potential Phase representation f(τ ) S 0 + N log N [ ϕ] ϕ ', ' J ( τ ) dτ f ( τ ) non-compact phase = ( ) e ϕ τ τ

13 Liouvillian QM Effective Hamiltonian ϕ k ˆ ϕ ϕ H = + αe, M ~ N ln N M J αe ϕ effective mass Green s function k + R ϕ + dα ϕ ϕ 4 τ k M 4 G( τ ) m 0 e k e k e 0 α 0 k

14 Green s function G ( ε ) / ε ε / - mean field Julia code N=4 ε J ~ J N ln N ~ M G( ε ) ( π ) ( i ) / sin 4 J M 0 + iε + k k Γ + k ( k M ) ε 4 dk

15 Green s function G ( ε ) / ε ε / - mean field Julia code N=4 Time domain: ε J ~ J N ln N ~ M G( τ ) / τ, τ < / ± 3/ J τ, τ > /

16 Zero-bias anomaly ε Yu. Nazarov, JETP 89; L.Levitov & A.Shytov, JETP 97 A.Kamenev & A.Andreev, PRB 99 I ε G ( ε ) + G ( ε ) T > < tunneling conductance S( t) + iε t G> ( ε ) = t e dt ~ ν 0 exp ln 8gπ ε S t = 8gπ ln ( ) t - electron s action of tunnelling under Coulomb barrier

17 Four-point Green s function G N τ, τ, τ, τ = χ ( τ ) χ ( τ ) χ ( τ ) χ ( τ ) ( ) i i j 3 j 4 N i, j Time ordering: τ, τ 0 4 τ, τ τ ± ± 3 G ( τ ) τ, τ < / τ, τ > / 4 / 3/ ~ J N ln N single-partilce level spasing universal long-time decay

18 (p)-point Green s function p,0 = ( ) ( 0) G ( τ ) χ τ χ p α β α = β = p 3 τ 3 p p ϕ ( τ ) ϕ ( 0) 4 4 ( p 3) / 3/ p( τ ) S[ ϕ ] ~ τ, τ / G < e e > > universal long-time decay

Quantum Majorana wire at criticality Statistics of zero-energy wave

19 Random mass Dirac model L. Balents & M.Fisher 97, D. Shelton & A. Tsvelik 98 Hˆ iu x m( x) = m( x) iu x m( x) m( x ') δ ( x x ') Quantum Majorana wire at criticality Statistics of zero-energy wave functions dis ψ ( x) ψ ( ) ~ p dis L x 3/ universal (p-independent) decay

20 Holography principle: Outlook - SYK model is dual to nearly AdS gravity S. Sachdev 05; Kitaev 06; J. Maldacena, D. Stanford & Yang 06; K. Jensen 06; - The Schwarzian action naturaly (?) emerges in nearly AdS gravity Quantum chaos: out-of-time-order 4-point correlator χi ( 0) χ j ( t) χi ( 0 ) χ j ( t) ~ exp ( π β ) t β N maximal Lyapunov exponent Q: what happens at low T<?

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