NEW HORIZONS IN QUANTUM MATTER

Transcription

1 The 34 th Jerusalem School in Theoretical Physics NEW HORIZONS IN QUANTUM MATTER 27.12, , 2017 Photo credit Frans Lanting / Modern quantum materials realize a remarkably rich set of electronic phases. This school will explore the many new concepts and methods which have been developed in recent years, moving beyond the traditional paradigms of Fermi liquid theory and spontaneous symmetry breaking. In particular, longrange quantum entanglement appears in topological and quantum-critical states, and the school will discuss new techniques required to describe their observable properties. For more details: General Director: David Gross UCSB Director: Subir Sachdev Harvard University Codirectors: Erez Berg Weizmann Institute of Science Dror Orgad The Hebrew University Speakers: Erez Berg Weizmann Institute of Science Andrey Chubukov University of Minnesota Antoine Georges Collège de France Sean Hartnoll Stanford University Steve Kivelson Stanford University Subir Sachdev Harvard University Nati Seiberg IAS, Princeton Senthil Todadri MIT

2 The SYK model of non-fermi liquids and black holes Applications of Gauge-Gravity Duality 2016 Chalmers University of Technology Gothenburg, Sweden, October 4, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD

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

4 Quantum matter without quasiparticles Quasiparticles are long-lived excitations which can be combined to yield the complete low-energy many-body spectrum Quasiparticles need not be electrons: they can be emergent excitations which involve non-local changes in the wave function of the underlying electrons e.g. Laughlin quasiparticles, visons... How do we rule out quasiparticle excitations? Examine the time it takes to reach local thermal equilibrium. Equilibration takes a long time while quasiparticles collide (in Fermi liquids, 1/T 2 ;in gapped systems, e /T ). Systems without quasiparticles saturate a (conjectured) lower bound on the local-equilibration/de-phasing/ transition-to-quantum-chaos time ' C ~ k B T where C is a T -independent constant.

5 Quantum matter without quasiparticles Quasiparticles are long-lived excitations which can be combined to yield the complete low-energy many-body spectrum Quasiparticles need not be electrons: they can be emergent excitations which involve non-local changes in the wave function of the underlying electrons e.g. Laughlin quasiparticles, visons... How do we rule out quasiparticle excitations? Examine the time it takes to reach local thermal equilibrium. Equilibration takes a long time while quasiparticles collide (in Fermi liquids, 1/T 2 ;in gapped systems, e /T ). Systems without quasiparticles saturate a (conjectured) lower bound on the local-equilibration/de-phasing/ transition-to-quantum-chaos time ' C ~ k B T where C is a T -independent constant.

6 Quantum matter without quasiparticles Quasiparticles are long-lived excitations which can be combined to yield the complete low-energy many-body spectrum Quasiparticles need not be electrons: they can be emergent excitations which involve non-local changes in the wave function of the underlying electrons e.g. Laughlin quasiparticles, visons... How do we rule out quasiparticle excitations? Examine the time it takes to reach local thermal equilibrium. Equilibration takes a long time while quasiparticles collide (in Fermi liquids, 1/T 2 ;in gapped systems, e /T ). Systems without quasiparticles saturate a (conjectured) lower bound on the local-equilibration/de-phasing/ transition-to-quantum-chaos time ' C ~ k B T where C is a T -independent constant. S. Sachdev, Quantum Phase Transitions (1999) K. Damle and Sachdev, PRB 56, 8714 (1997)

7 Quantum matter without quasiparticles Shortest possible local-equilibration/de-phasing/ transition-to-quantum-chaos with ' C ~ s D v 2 b L 1 k B T ~ 4 k B C ~ k B T 2 ~ k B T S. Sachdev, Quantum Phase Transitions (1999) K. Damle and Sachdev, PRB 56, 8714 (1997) P. Kovtun, D.T. Son, and A.O. Starinets, PRL 94, (2005) Saturation requires fixed point with disorder and interactions S. A. Hartnoll, Nature Physics 11, 54 (2015) M. Blake, PRL 117, (2016) J. Maldacena, S. H. Shenker and D. Stanford, JHEP 08 (2016)106 In Fermi liquids, 1/T 2 ; in gapped systems, e /T.

8 1 = k BT ~ J. A. N. Bruin, H. Sakai, R. S. Perry, A. P. Mackenzie, Science. 339, 804 (2013)

9 Theories of non-fermi liquids Sachdev-Ye-Kitaev (SYK) model Coupled SYK models: diffusive metals without quasiparticles Holographic Einstein-Maxwell-axion theory with momentum dissipation

10 Theories of non-fermi liquids Sachdev-Ye-Kitaev (SYK) model Coupled SYK models: diffusive metals without quasiparticles Holographic Einstein-Maxwell-axion theory with momentum dissipation

11 SYK model 3 4 H SYK = (2N) 3/2 5 6 J 3,5,7,13 J 4,5,6,11 10 NX J ij;k` c i c j c k c` µ X i,j,k,`=1 i Q = 1 X c i N c i i c i c i A fermion can move only by entangling with another fermion: the Hamiltonian has nothing but entanglement J 8,9,12, Cold atom realization: I. Danshita, M. Hanada, and M. Tezuka, arxiv: S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015)

12 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)

13 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)

14 SYK model T = 0 Green s function G 1/ p S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993) T>0 Green s function implies conformal invariance G 1/(sin( T )) 1/2 Non-zero entropy as T! 0, S(T! 0) = NS These features indicate that the SYK model is dual to the low energy limit of a quantum gravity theory of black holes with AdS 2 near-horizon geometry. The Bekenstein- Hawking entropy is NS 0. There is a scalar zero mode associated with the breaking of reparameterization invariance down to SL(2,R). The same pattern of symmetries is present in gravity theories on AdS 2. The dependence of S 0 on the density Q matches the be-

15 SYK model T = 0 Green s function G 1/ p T>0Green sfunction implies conformal invariance G 1/(sin( T )) 1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999) Non-zero entropy as T! 0, S(T! 0) = NS These features indicate that the SYK model is dual to the low energy limit of a quantum gravity theory of black holes with AdS 2 near-horizon geometry. The Bekenstein- Hawking entropy is NS 0. There is a scalar zero mode associated with the breaking of reparameterization invariance down to SL(2,R). The same pattern of symmetries is present in gravity theories on AdS 2. The dependence of S 0 on the density Q matches the be-

16 SYK model T = 0 Green s function G 1/ p T>0 Green s function implies conformal invariance G 1/(sin( T )) 1/2 Non-zero entropy as T! 0, S(T! 0) = NS A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B 63, (2001) These features indicate that the SYK model is dual to the low energy limit of a quantum gravity theory of black holes with AdS 2 near-horizon geometry. The Bekenstein- Hawking entropy is NS 0. There is a scalar zero mode associated with the breaking of reparameterization invariance down to SL(2,R). The same pattern of symmetries is present in gravity theories on AdS 2. The dependence of S 0 on the density Q matches the be-

17 SYK model large N exact ED N=8 ED N=12 ED N= Boson N=12 Boson N=16 Fermion N=12 Fermion N=16 p!jim(g) Im(G)J !/J W. Fu and S. Sachdev, PRB 94, (2016) Large N solution of equations for G and agree well with exact diagonalization of the finite N Hamiltonian ) no spin-glass order FIG. 10. Imaginary part of Green s function for hardcore boson and fermion model. The center gets much higher in the boson model when system size gets larger. The inset figure near! = 0. However, exact diagonalization of the same model with hard-core bosons indicates the presence of spin-glass order in the ground state. Unlike the fermionic case, P 2 =1forallN in the bosonic model. We can apply simi argument as in Ref. [14]: for the half-filled sector (only in even N cases), the level st the Wigner-Dyson distribution of Gaussian orthogonal random matrix ensembles, w filling sectors, it obeys distribution of Gaussian unitary random matrix ensembles. Our thermal entropy results for bosons are similar to the fermionic results: althoug eventually approaches 0 at zero temperature, there is still a trend of a larger low entropy residue as the system size gets larger.!/j

18 After integrating the fermions, the partition function can be written as a path integral with an action S analogous to a Luttinger- Ward functional Z Z = DG( 1, 2 )D ( 1, 2 )exp( SYK model NS) S =lndet[ ( 1 2 )(@ 1 + µ) ( 1, 2 )] Z + d 1 d 2 ( 1, 2 ) G( 2, 1 )+(J 2 /2)G 2 ( 2, 1 )G 2 ( 1, 2 ) A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B 63, (2001) At frequencies J, the time derivative in the determinant is less important, and without it the path integral is 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 )] where f( ) and g( ) are arbitrary functions. 1/4 g( 1 ) g( 2 ) G( 1, 2) 3/4 g( 1 ) g( 2 ) ( 1, 2) A. Georges and O. Parcollet PRB 59, 5341 (1999) A. Kitaev, unpublished S. Sachdev, PRX 5, (2015)

19 After integrating the fermions, the partition function can be written as a path integral with an action S analogous to a Luttinger- Ward functional Z Z = DG( 1, 2 )D ( 1, 2 )exp( SYK model NS) S =lndet[ ( 1 2 )(@ X 1 + Xµ) ( 1, 2 )] Z + d 1 d 2 ( 1, 2 ) G( 2, 1 )+(J 2 /2)G 2 ( 2, 1 )G 2 ( 1, 2 ) A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. B 63, (2001) At frequencies J, the time derivative in the determinant is less important, and without it the path integral is 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 )] where f( ) and g( ) are arbitrary functions. 1/4 g( 1 ) g( 2 ) G( 1, 2) 3/4 g( 1 ) g( 2 ) ( 1, 2) A. Georges and O. Parcollet PRB 59, 5341 (1999) A. Kitaev, unpublished S. Sachdev, PRX 5, (2015)

20 (and similarly for ) and obtain an e ective action for f( ). This action does not vanish because of the time derivative in the determinant which is not reparameterization invariant. J. Maldacena and D. Stanford, arxiv: See also A. Kitaev, unpublished, and J. Polchinski and V. Rosenhaus, arxiv: SYK model 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 ))

21 (and similarly for ) and obtain an e ective action for f( ). This action does not vanish because of the time derivative in the determinant which is not reparameterization invariant. J. Maldacena and D. Stanford, arxiv: See also A. Kitaev, unpublished, and J. Polchinski and V. Rosenhaus, arxiv: 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 model 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.

22 SYK model 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 does not vanish because of the time derivative in the determinant which is not reparameterization invariant. J. Maldacena and D. Stanford, arxiv: See also A. Kitaev, unpublished, and J. Polchinski and V. Rosenhaus, arxiv:

23 SYK model 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 f 0 The couplings are given by thermodynamics ( is the grand potential) K 2 T, +4 2 E 2 K = 2 E 0 2 µ Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished

24 SYK model 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 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 = T h( E( ) µ Q( )) Q(0)i h( E( ) µ Q( ))( E(0) µ Q(0))i /T s f 0 where s is the static susceptibility matrix given by (@ s 2 /@µ 2 ) 2 2 T (@ 2 /@T 2. ) µ Wenbo Fu, Yingfei Gu, S. Sachdev, unpublished

25 Theories of non-fermi liquids Sachdev-Ye-Kitaev (SYK) model Coupled SYK models: diffusive metals without quasiparticles Holographic Einstein-Maxwell-axion theory with momentum dissipation

26 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:

27 SYK model 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.

28 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 )

29 Theories of non-fermi liquids Sachdev-Ye-Kitaev (SYK) model Coupled SYK models: diffusive metals without quasiparticles Holographic Einstein-Maxwell-axion theory with momentum dissipation

30 SYK model T = 0 Green s function G 1/ p T>0 Green s function implies conformal invariance G 1/(sin( T )) 1/2 Non-zero entropy as T! 0, S(T! 0) = NS These features indicate that the SYK model is dual to the low energy limit of a quantum gravity theory of black holes with AdS 2 near-horizon geometry. The Bekenstein- Hawking entropy is NS 0. There is a scalar zero mode associated with the breaking of reparameterization invariance down to SL(2,R). The same pattern of symmetries is present in gravity theories on AdS 2. The dependence of S 0 on the density Q matches the be-

31 SYK model T = 0 Green s function G 1/ p T>0 Green s function implies conformal invariance G 1/(sin( T )) 1/2 Non-zero entropy as T! 0, S(T! 0) = NS These features indicate that the SYK model is dual to the low energy limit of a quantum gravity theory of black holes with AdS 2 near-horizon geometry. The Bekenstein- Hawking entropy is NS 0. S. Sachdev, PRL 105, (2010) There is a scalar zero mode associated with the breaking of reparameterization invariance down to SL(2,R). The same pattern of symmetries is present in gravity theories on AdS 2. The dependence of S 0 on the density Q matches the be-

32 SYK and AdS2 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 (2010) P H Y S I C A L R E V I E W L E T T E R S w 8 OC 105, (2010) Holographic Metals and the Fractionalized Fermi Liquid Subir Sachdev Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 23 June 2010; published 4 October 2010) We show that there is a close correspondence between the physical properties of holographic metals near charged black holes in anti de Sitter (AdS) space, and the fractionalized Fermi liquid phase of the lattice Anderson model. The latter phase has a small Fermi surface of conduction electrons, along with a spin liquid of local moments. This correspondence implies that certain mean-field gapless spin liquids are states of matter at nonzero density realizing the near-horizon, AdS 2 R 2 physics of Reissner- Nordström black holes.

33 SYK model T = 0 Green s function G 1/ p T>0 Green s function implies conformal invariance G 1/(sin( T )) 1/2 Non-zero entropy as T! 0, S(T! 0) = NS These features indicate that the SYK model is dual to the low energy limit of a quantum gravity theory of black holes with AdS 2 near-horizon geometry. The Bekenstein- Hawking entropy is NS 0. There is a scalar zero mode associated with the breaking of reparameterization invariance down to SL(2,R). The same pattern of symmetries is present in gravity theories on AdS 2. A. Kitaev, KITP talk, 2015 The dependence of S 0 on the density Q matches the be-

34 on AdS 2. SYK model The dependence of S 0 on the density Q matches the behavior of the Wald-Bekenstein-Hawking entropy of AdS 2 horizons in a large class of gravity theories. The scalar zero mode leads to a linear-in-t specific heat S(T! 0) = NS 0 + N T +... S. Sachdev PRX 5, (2015) An identical scalar zero model is also present in the low energy limit of theories of quantum gravity on AdS 2. The Lyapunov time to quantum chaos saturates the lower bound both in the SYK model and in quantum gravity. L = 1 2 ~ k B T

35 on AdS 2. SYK model The dependence of S 0 on the density Q matches the behavior of the Wald-Bekenstein-Hawking entropy of AdS 2 horizons in a large class of gravity theories. The scalar zero mode leads to a linear-in-t specific heat S(T! 0) = NS 0 + N T +... An identical scalar zero model is also present in the low energy limit of theories of quantum gravity on AdS 2. J. Maldacena and D. Stanford, arxiv: The Lyapunov time to quantum chaos saturates the lower bound both in the SYK model and in quantum gravity. L = 1 2 ~ k B T

36 on AdS 2. SYK model The dependence of S 0 on the density Q matches the behavior of the Wald-Bekenstein-Hawking entropy of AdS 2 horizons in a large class of gravity theories. The scalar zero mode leads to a linear-in-t specific heat S(T! 0) = NS 0 + N T +... An identical scalar zero model is also present in the low energy limit of theories of quantum gravity on AdS 2. The Lyapunov time to quantum chaos saturates the lower bound both in the SYK model and in quantum gravity. L = 1 2 ~ k B T A. Kitaev, KITP talk, 2015 J. Maldacena and D. Stanford, arxiv:

37 It would be nice to have a solvable model of holography. theory bulk dual anom. dim. chaos solvable in 1/N SYM Einstein grav. large maximal no O(N) Vasiliev 1/N 1/N yes SYK `s `AdS O(1) maximal yes black hole yes no yes Slide by D. Stanford at Strings 2016, Beijing column added by SS

38 Einstein-Maxwell-axion theory 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. S = Z d 4 x p ĝ ˆR +6/L X i=1 AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 (@ ˆ' i ) ˆF µ ˆF µ charge T 2 density Q ~x!, For ˆ' i = 0, we obtain the Reissner-Nördstrom-AdS charged black hole, with a near-horizon AdS 2 T 2 near-horizon geometry. For ˆ' i = kx i, we obtain a similar solution but with momentum dissipation (a bulk massive graviton).

39 Einstein-Maxwell-axion theory 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. S = Z d 4 x p ĝ ˆR +6/L X i=1 AdS 2 T 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 (@ ˆ' i ) ˆF µ ˆF µ charge T 2 density Q ~x!, In the small torus limit, T 1/R, wherer is the size of the torus, the theory dimensionally reduces to an Einstein- Maxwell-dilaton theory in two dimensions S = Z d 2 x p g e R + e /2 (6/L 2 ) m 2 e /2 1 4 e3 /2 F ab F ab, A. Almheiri and J. Polchinski, JHEP 1511 (2015) 014; A. Almheiri and B. Kang, arxiv: ; M. Cvetic and I. Papadimitriou, arxiv:

40 Einstein-Maxwell-axion theory The Einstein-Maxwell-dilaton theory of the small torus limit, T 1/R, is equivalent on its boundary to the Schwarzian theory discussed earlier for the SYK model S,f = K 2 Z 1/T 0 d (@ ) Z 1/T 0 d {f, } + i L 4 2 Z 1/T 0 d (@ ){f, }. So 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 = T h( E( ) µ Q( )) Q(0)i h( E( ) µ Q( ))( E(0) µ Q(0))i /T s where s is the static susceptibility matrix given by (@ s 2 /@µ 2 ) 2 2 T (@ 2 /@T 2. ) µ A. Kitaev, unpublished; J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv: ; K. Jensen, arxiv: ; J. Engelsoy, T.G. Mertens, and H. Verlinde, arxiv:

41 Einstein-Maxwell-axion theory Finally, in the large torus limit, T 1/R, we have the behavior of the di usive metal without quasiparticles found in the coupled SYK models hq; Qi k,! he µq; Qi k,! /T = i!( i! + Dk 2 ) 1 +1 s he µq; Qi k,! he µq; E µqi k,! /T where the di usivities are related to the thermoelectric conductivities by the Einstein relations 1 D = T apple s. 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.

42 Non-Fermi liquids Shortest possible phase coherence time, fastest possible local equilibration time, or fastest possible Lyapunov time towards quantum chaos, all of order ~ k B T Realization in solvable SYK model, which saturates the lower bound on the Lyapunov time. Coupled SYK models realize di usive metal without quasiparticles. Remarkable holographic match to Einstein-Maxwell-axion theories with momentum dissipation via the Schwarzian e ective action.