SYK models and black holes

Transcription

1 SYK models and black holes Black Hole Initiative Colloquium Harvard, October 25, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD

2 Wenbo Fu, Harvard Yingfei Gu, Stanford Richard Davison, Harvard

3 Conventional quantum matter: 1. Ground states connected adiabatically to independent electron states 2. Boltzmann-Landau theory of quasiparticles Metals E In Metal k

4 Topological quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Boltzmann-Landau theory of quasiparticles The fractional quantum Hall effect: the ground state is described by Laughlin s wavefunction, and the excitations are quasiparticles which carry fractional charge.

5 Quantum matter without quasiparticles: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle 2. No quasiparticles structure of excited states Strange metals: Such metals are found, most prominently, near optimal doping in the the cuprate high temperature superconductors. But how can we be sure that no quasiparticles exist in a given system? Perhaps there are some entangled quasiparticles inaccessible to current experiments..

6 Quantum matter without quasiparticles: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle 2. No quasiparticles structure of excited states Strange metals: Such metals are found, most prominently, near optimal doping in the the cuprate high temperature superconductors. But how can we be sure that no quasiparticles exist in a given system? Perhaps there are some exotic quasiparticles inaccessible to current experiments..

7 Local thermal equilibration or phase coherence time, ' : There is an lower bound on ' in all many-body quantum systems of order ~/(k B T ), ' >C ~ k B T, and the lower bound is realized by systems without quasiparticles. In systems with quasiparticles, ' is parametrically larger at low T ; e.g. in Fermi liquids ' 1/T 2, and in gapped insulators ' e /(k BT ) where energy gap. is the K. Damle and S. Sachdev, PRB 56, 8714 (1997) S. Sachdev, Quantum Phase Transitions, Cambridge (1999)

8 A bound on quantum chaos: The time over which a many-body quantum system becomes chaotic is given by L = 1/ L,where L is the Lyapunov exponent determining memory of initial conditions. This Lyapunov time obeys the rigorous lower bound L 1 2 ~ k B T A. I. Larkin and Y. N. Ovchinnikov, JETP 28, 6 (1969) J. Maldacena, S. H. Shenker and D. Stanford, arxiv:

9 A bound on quantum chaos: The time over which a many-body quantum system becomes chaotic is given by L = 1/ L,where L is the Lyapunov exponent determining memory of initial conditions. This Lyapunov time obeys the rigorous lower bound L 1 2 ~ k B T Quantum matter without quasiparticles fastest possible many-body quantum chaos

10 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010) Resistivity 0 + AT Strange Metal BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic 1.0 Superconductivity 0

11 Strange metals 1 = k BT ~ J. A. N. Bruin, H. Sakai, R. S. Perry, A. P. Mackenzie, Science. 339, 804 (2013)

12 LIGO September 14, 2015

13 LIGO September 14, 2015 Black holes have a ring-down time, r, in which they radiate energy, and stabilize to a featureless spherical object. This time can be computed in Einstein s general relativity theory. For this black hole r =7.7 milliseconds. (Radius of black hole = 183 km; Mass of black hole = 62 solar masses.)

14 LIGO September 14, 2015 Featureless black holes have a Bekenstein-Hawking entropy, and a Hawking temperature, T H.

15 LIGO September 14, 2015 Expressed in terms of the Hawking temperature, the ring-down time is r ~/(k B T H )! For this black hole T H 1nK.

16 The Sachdev-Ye-Kitaev Figure credit: L. Balents (SYK) model: A theory of a strange metal Dual theory of gravity on AdS 2 Fastest possible quantum chaos with L = ~ 2 k B T

17 The Sachdev-Ye-Kitaev Figure credit: L. Balents (SYK) model: A theory of a strange metal Dual theory of gravity on AdS 2 Fastest possible quantum chaos with L = ~ 2 k B T A. Kitaev, unpublished J. Maldacena and D. Stanford, arxiv:

18 SYK model H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` µ X i c i c i c i c j + c j c i =0, c i c j + c j c i = ij Q = 1 X c N i c i 3 4 i J 3,5,7,13 J 4,5,6, J 8,9,12,14 J ij;k` are independent random variables with J ij;k` = 0 and J ij;k` 2 = J 2 N!1yields critical strange metal. 14 A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015) 12 S. Sachdev and J. Ye, PRL 70, 3339 (1993)

19 SYK model Feynman graph expansion in J ij.., and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) = J 2 G 2 ( )G( ) Low frequency analysis shows that the solutions must be gapless and obey = (z) =µ 1 Ap z +..., G(z) = A pz for some complex A. The ground state is a non-fermi liquid, with a continuously variable density Q. S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

20 SYK model Feynman graph expansion in J ij.., and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) = J 2 G 2 ( )G( ) Low frequency analysis shows that the solutions must be gapless and obey (z) =µ 1 Ap z +..., G(z) = A pz for some complex A. The ground state is a non-fermi liquid, with a continuously variable density Q. S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

21 T = 0 Green s function G 1/ p for > 0 e 2 E / p for < 0 T>0 Green s function has conformal invariance G e 2 ET /(sin( T )) 1/2 Non-zero GPS entropy as T! 0, S(T! 0) = NS SYK models are are states of matter at non-zero density realizing the near-horizon, AdS 2 R 2 physics of Reissner- Nördstrom black holes. The Bekenstein-Hawking entropy is NS 0 (GPS = BH). SYK model E is identified with the electric field on AdS 2. The relationship (@S/@Q) =2 E is obeyed both by the GPS T entropy, and by the BH entropy of AdS 2 horizons in a large class of gravity theories. S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

22 T = 0 Green s function G 1/ p for > 0 e 2 E / p for < 0 T>0 Green s function has conformal invariance G e 2 ET /(sin( T )) 1/2 Non-zero GPS entropy as T! 0, S(T! 0) = NS SYK models are are states of matter at non-zero density realizing the near-horizon, AdS 2 R 2 physics of Reissner- Nördstrom black holes. The Bekenstein-Hawking entropy is NS 0 (GPS = BH). SYK model E is identified with the electric field on AdS 2. The relationship (@S/@Q) =2 E is obeyed both by the GPS T entropy, and by the BH entropy of AdS 2 horizons in a large class of gravity theories. A. Georges and O. Parcollet PRB 59, 5341 (1999)

23 T = 0 Green s function G 1/ p for > 0 e 2 E / p for < 0 T>0 Green s function has conformal invariance G e 2 ET /(sin( T )) 1/2 Non-zero GPS entropy as T! 0, S(T! 0) = NS SYK models are are states of matter at non-zero density realizing the near-horizon, AdS 2 R 2 physics of Reissner- Nördstrom black holes. The Bekenstein-Hawking entropy is NS 0 (GPS = BH). SYK model A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B 63, (2001) E is identified with the electric field on AdS 2. The relationship (@S/@Q) =2 E is obeyed both by the GPS T entropy, and by the BH entropy of AdS 2 horizons in a large class of gravity theories.

24 T = 0 Green s function G 1/ p for > 0 e 2 E / p for < 0 T>0 Green s function has conformal invariance G e 2 ET /(sin( T )) 1/2 Non-zero GPS entropy as T! 0, S(T! 0) = NS SYK models are are states of matter at non-zero density realizing the near-horizon, AdS 2 R 2 physics of Reissner- Nördstrom black holes. The Bekenstein-Hawking entropy is NS 0 (GPS = BH). SYK and AdS2 E is identified with the electric field on AdS 2. S. Sachdev, PRL 105, (2010) The relationship (@S/@Q) =2 E is obeyed both by the GPS T entropy, and by the BH entropy of AdS 2 horizons in a large class of gravity theories. S. Sachdev, PRX 5, (2015)

25 SYK and AdS2 GPS BH entropy entropy AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt charge T 2 density Q = 1 ~x Einstein-Maxwell theory + cosmological constant Mapping to SYK applies when temperature 1/(size of T 2 )

26 SYK and AdS2 1 G(i!) =, ( ) = J 2 G 2 ( )G( ) i! + µ (i!) 1 A (z) =µ z +..., G(z) = pz Ap S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

27 SYK and AdS2 1 G(i!) =, ( ) = J 2 G 2 ( )G( ) i! X + Xµ (i!) 1 A (z) =µ z +..., G(z) = pz Ap X At frequencies J, thei! + µ can be dropped, and without it equations are invariant under the reparametrization and gauge transformations = f( ) G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] ( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] 1/4 g( 1 ) g( 2 ) G( 1, 2) 3/4 g( 1 ) g( 2 ) ( 1, 2) where f( ) and g( ) are arbitrary functions. A. Kitaev, unpublished S. Sachdev, PRX 5, (2015)

28 SYK and AdS2 Let us write the large N saddle point solutions of S as G s ( 1 2 ) ( 1 2 ) 1/2, s ( 1 2 ) ( 1 2 ) 3/2. These are not invariant under the reparametrization symmetry but are invariant only under a SL(2,R) subgroup under which f( ) = a + b c + d, ad bc =1. So the (approximate) reparametrization symmetry is spontaneously broken. Reparametrization zero mode Expand about the saddle point by writing G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] 1/4 G s (f( 1 ) f( 2 )) (and similarly for ) and obtain an e ective action for f( ). This action must vanish for f( ) 2 SL(2,R). A. Kitaev, unpublished

29 Let us write the large N saddle point solutions of S as G s ( 1 2 ) ( 1 2 ) 1/2, s ( 1 2 ) ( 1 2 ) 3/2. Connections of SYK to gravity and AdS 2 These are not invariant under the reparametrization symmetry but are invariant only under a SL(2,R) subgroup under which horizons SYK and AdS2 f( ) = a + b c + d, ad bc =1. Reparameterization and gauge invariance are the symmetries of So the (approximate) reparametrization symmetry is spontaneously broken. the Einstein-Maxwell theory of Reparametrization zero mode Expand about the saddle point by writing gravity and electromagnetism G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] 1/4 G s (f( 1 ) f( 2 )) SL(2,R) is the isometry group of AdS 2. (and similarly for ) and obtain an e ective action for f( ). This action must vanish for f( ) 2 SL(2,R). A. Kitaev, unpublished

30 SYK and AdS2 Let us write the large N saddle point solutions of S as G s ( 1 2 ) ( 1 2 ) 1/2, s ( 1 2 ) ( 1 2 ) 3/2. These are not invariant under the reparametrization symmetry but are invariant only under a SL(2,R) subgroup under which f( ) = a + b c + d, ad bc =1. So the (approximate) reparametrization symmetry is spontaneously broken. Reparametrization zero mode Expand about the saddle point by writing G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] 1/4 G s (f( 1 ) f( 2 )) (and similarly for ) and obtain an e ective action for f( ). This action must vanish for f( ) 2 SL(2,R). J. Maldacena and D. Stanford, arxiv: See also A. Kitaev, unpublished, and J. Polchinski and V. Rosenhaus, arxiv:

31 SYK and AdS2 With g( ) =e i ( ), the action for ( ) and f( ) = 1 T fluctuations is tan( T ( + ( )) S,f = K 2 Z 1/T 0 d (@ + i(2 ET )@ ) Z 1/T 0 d {f, }, where {f, } is the Schwarzian: {f, } f f 00 2 f 0 2 f 0. The couplings are given by thermodynamics: is the grand potential, S 0 is the GPS entropy, and Q is the K 2, +4 2 E 2 2 K 2 T 2 E In holography: the term in the action has been obtained from theories on AdS 2 ; E is the electric field, and has the same relationship to S 0. J. Maldacena and D. Stanford, arxiv: ; R. Davison, Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished; S. Sachdev, PRX 5, (2015); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv: ; K. Jensen, arxiv: ; J. Engelsoy, T.G. Mertens, and H. Verlinde, arxiv: µ

32 SYK and AdS2 With g( ) =e i ( ), the action for ( ) and f( ) = 1 T fluctuations is tan( T ( + ( )) S,f = K 2 Z 1/T 0 d (@ + i(2 ET )@ ) Z 1/T 0 d {f, }, where {f, } is the Schwarzian: {f, } f f 0 2 f 00 f 0 2. The correlators of the density fluctuations, Q( ), and the energy fluctuations E µ Q( ) are time independent and given by h Q( ) Q(0)i h( E( ) µ Q( )) Q(0)i /T h( E( ) µ Q( )) Q(0)i h( E( ) µ Q( ))( E(0) µ Q(0))i /T = T s where s is the static susceptibility matrix given by (@ s 2 /@µ 2 ) 2 2 T (@ 2 /@T 2 ) µ. R. Davison, Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished

33 Coupled SYK models J jklm J 0 jklm j k l m j k l m Figure 1: A chain of coupled SYK sites: each site contains N 1 fermion with SYK interaction. The coupling between nearest neighbor sites are four fermion interaction with two from each site. Yingfei Gu, Xiao-Liang Qi, and D. Stanford, arxiv:

34 SYK models The correlators of the density fluctuations, Q( ), and the energy fluctuations E µ Q( ) are time independent and given by h Q( ) Q(0)i h( E( ) µ Q( )) Q(0)i /T h( E( ) µ Q( )) Q(0)i h( E( ) µ Q( ))( E(0) µ Q(0))i /T = T s where s is the static susceptibility matrix given by (@ s 2 /@µ 2 ) 2 2 T (@ 2 /@T 2 ) µ. Coupled SYK models hq; Qi k,! he µq; Qi k,! /T he µq; Qi k,! he µq; E µqi k,! /T = i!( i! + Dk 2 ) 1 +1 s where the di usivities are related to the thermoelectric conductivities by the Einstein relations 1 D = T apple s. R. Davison, Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished

35 Coupled SYK models hq; Qi k,! he µq; Qi k,! /T he µq; Qi k,! he µq; E µqi k,! /T = i!( i! + Dk 2 ) 1 +1 s where the di usivities are related to the thermoelectric conductivities by the Einstein relations 1 D = T apple s. The coupled SYK models realize a diffusive metal with no quasiparticle excitations. (a strange metal ) R. Davison, Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished

36 Holography:Einstein-Maxwell-axion theory S = Z d 4 x p ĝ ˆR +6/L AdS 2 R 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 2X i=1 (@ ˆ' i ) ˆF µ ˆF µ charge density Q R2 ~x For ˆ' i = 0, we obtain the Reissner-Nördstrom-AdS charged black hole, with a near-horizon AdS 2 R 2 near-horizon geometry. For ˆ' i = kx i, we obtain a similar solution but with momentum dissipation (a bulk massive graviton). This yields the same di usive metal correlators as the coupled SYK models, and the same relationship between and. Y. Bardoux, M. M. Caldarelli, and C. Charmousis, JHEP 05 (2012) 054; D. Vegh, arxiv: ; R. A. Davison, PRD 88 (2013) ; M. Blake and D. Tong, PRD 88 (2013), ; T. Andrade and B. Withers, JHEP 05 (2014) 101; M. Blake, PRL 117, (2016); R. Davison, Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished!,

37 Entangled quantum matter without quasiparticles Is there a connection between strange metals and black holes? Yes, e.g. the SYK model. Why do they have the same equilibration time ~/(k B T )? Strange metals don t have quasiparticles and thermalize rapidly; Black holes are fast scramblers. Theoretical predictions for strange metal transport in graphene agree well with experiments