Max-Planck-Institut für Physik komplexer Systeme Dresden, May 22, Subir Sachdev

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1 HARVARD Quantum matter without quasiparticles Max-Planck-Institut für Physik komplexer Systeme Dresden, May 22, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu

2 Foundations of quantum many body theory: 1. Ground states connected adiabatically to independent electron states 2. Quasiparticle structure of excited states Metals E In Metal k

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

4 Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Boltzmann-Landau theory of quasiparticles Famous example: The fractional quantum Hall effect of electrons in two dimensions (e.g. in graphene) in the presence of a strong magnetic field. The ground state is described by Laughlin s wavefunction, and the excitations are quasiparticles which carry fractional charge.

5 Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle 2. No quasiparticles structure of excited states

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 Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

7 Quantum matter without quasiparticles: Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

8 Quantum matter without quasiparticles: Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

9 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Xibo Zhang, Chen-Lung Hung, Shih-Kuang Tung, and Cheng Chin, Science 335, 1070 (2012) Superfluid-insulator transition Ultracold 87 Rb atoms - bosons

10 On-site repulsion between bosons = U Tunneling amplitude between sites = t T Quantum critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g U/t

11 T Quantum critical T KT Superfluid Insulator 0 g c c g U/t

12 T T KT Superfluid Quantum critical Insulator Boltzmann theory of quasiparticles/holes 0 g c c g U/t

13 T Boltzmann theory of Nambu- Goldstone T KT phonons and vortices 0 Superfluid Quantum critical Insulator g c c g U/t

14 Conformal field theory T (CFT3) at T>0 Quantum critical T KT Superfluid Insulator 0 g c c g U/t

15 CFT3 at T>0 T T KT Quantum Quantum matter critical without quasiparticles Superfluid Insulator 0 g c c g U/t

16 CFT3 at T>0 T Quantum Shortest possible phase coherence or local thermal critical equilibration time T ~ KT k B T Superfluid Insulator 0 g c c g U/t A.V. Chubukov, S. Sachdev, and J. Ye, PRB 49, (1994); K. Damle and S. Sachdev, PRB 56, 8714 (1997); S. Sachdev, Quantum Phase Transitions, Cambridge (1999)

17 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 S. Sachdev, Quantum Phase Transitions, Cambridge (1999)

18 A bound on quantum chaos: The time over which a many-body quantum system becomes chaotic is given by S = 1/ L,where L is the Lyapunov exponent determining memory of initial conditions. This scrambling time obeys the rigorous lower bound S 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:

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

20 Quantum matter without quasiparticles: Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

21 Philip Kim Jesse Crossno Kin Chung Fong Andrew Lucas

22 Graphene k y Same Hubbard model as for ultracold atoms, but for electrons on the honeycomb lattice k x

23 Graphene Electron Fermi surface

24 Graphene Hole Fermi surface Electron Fermi surface

25 Graphene T (K) Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 D. E. Sheehy and J. Schmalian, PRL 99, (2007) M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

26 T (K) Graphene Quantum critical Dirac liquid Predicted strange metal without quasiparticles Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

27 Fermi liquids: quasiparticles moving ballistically between impurity (red circles) scattering events

28 Fermi liquids: quasiparticles moving ballistically between impurity (red circles) scattering events Strange metals: electrons scatter frequently off each other, so there is no regime of ballistic quasiparticle motion. The electron liquid then flows around impurities

29 Wiedemann-Franz ;< metals and alloys. Up to a certain temperature, inelastic scattering determines the Lorenz number value, and below this the scattering is elastic which is due to impurities. Supression of the electronic contribution to thermal conductivity and hence the separation of the lattice and electronic parts of conductivity can be done by application of a transverse magnetic field and hence the Lorenz number can be evaluated. The deviation of the Lorenz number in some degenerate semi8 conductors is attributed to phonon drag. In some c Fluid in Graphene Thermal and electrical with quasiparticles 10 ~ I I 101 I 0o Wiedemann-Franz Law.I conductivity 10 2 Relative electrical conductivity Figure I l Relative thermal conductivities, A, measured by Fermi liquid: Wiedemann-Franz law in a Wiedemann and Franz (AAg assumed to be = 100) and relative electrical conductivities, ~, measured by ( 9 Riess, (A) Becquerel, and (V) Lorenz. C~Agassumed to be After Wiedemann and Franz [1]. L0 = T I ~ ' I I 1 ~ 30 c Nc- 2.5 SiBe3 Se i " 2.0 I I r 10 3 \ SiGe1 I t I ; H~ Y,,L T, Yb ~o1 I t 10 4 f I [ 10 6 AI /. / Au W1 Cs2 Mg Nb~b.,~o/Cu Fi 9 i e As \." SiG% O I Cs Sb SiGez ~oe \ t SiGe2 Bi ~ W. K2 Carrier c o n c e n t r a t i o n (cm 3) ~ ~'v3"0t 2.5 O kb 3e2 Ag Fe Ir Coz Li2 AUz Au3 Ga AI2 All \\\!! o y t w, p, t F e q z, ~ / Cr '' Er / / 2 As t31~ Bi2 BiI \ X~., I t I [ I 10 e Electrical c o n d u c t i v i t y i I 10 7 Sb ~ K Pt,\ Rh Cq I Ni2 I 10 8 I I ua ~" I 10 9 I! g I 101o (~,~-1 cm-1) Figure 2 Experimental Lorenz number of elemental metals in the low-temperature residual resistance regime, see Table I. Also shown are our G.(Table S. Kumar, G. Prasad, R.O. Pohl,conductivity J. Mat. Sci.and 28,also 4261 (1993) own data points on a doped, degenerate semiconductor III). Data are plottedand versus electrical versus carrier

30 For a strange metal Transport in Strange Metals with a relativistic Hamiltonian, hydrodynamic, holographic, and memory function methods yield Lorentz ratio L = apple/(t ) = v2 F H imp T 2 Q 1 (1 + e 2 v 2 F Q2 imp /(H Q )) 2 Q! electron density; H! enthalpy density Q! quantum critical conductivity imp! momentum relaxation time from impurities. Note that for a clean system ( imp!1first), the Lorentz ratio diverges L 1/Q 4, as we approach zero electron density at the Dirac point. S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) M. Müller and S. Sachdev, PRB 78, (2008)

31 Graphene Predicted T (K) strange metal Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

32 Graphene Predicted T (K) strange metal Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

33 Graphene Predicted T (K) strange metal Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 Impurity scattering dominates M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

34 J. Crossno et al., Science 351, 1058 (2016) Thermal Conductivity (nw/k) mm 0 1 C 75 K 40 K 20 K V g (V) D E T dis κ e T el-ph Red dots: data σtl 0 Blue line: value for L = L 0 0 V -0.5 V

35 J. Crossno et al., Science 351, 1058 (2016) Thermal Conductivity (nw/k) mm 0 1 C 75 K 40 K 20 K V g (V) D E T dis κ e T el-ph Red dots: data σtl 0 Blue line: value for L = L 0 0 V -0.5 V

36 J. Crossno et al., Science 351, 1058 (2016) Thermal Conductivity (nw/k) mm 0 1 C 75 K 40 K 20 K V g (V) D E T dis κ e T el-ph Red dots: data σtl 0 Blue line: value for L = L 0 0 V -0.5 V

37 Graphene Predicted T (K) strange metal Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 Impurity scattering dominates M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

38 J. Crossno et al., Science 351, 1058 (2016) Dirac Fluid in Graphene 28 Strange metal in graphene Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 Wiedemann-Franz [Crossno et al, submitted] obeyed

39 J. Crossno et al., Science 351, 1058 (2016) Dirac Fluid in Graphene 28 Strange metal in graphene Wiedemann-Franz Law Violations in Experiment Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 Wiedemann-Franz [Crossno et al, submitted] violated!

40 V) n (109 cm-2) L/L L / L0 Tbath (K) 70 Temperature (K) C 10 8 H e H (ev/µm2) 100 Temperature (K) C h V e h -V T (K) 4 6 n (1010 cm 2) Tdis Tel-ph 8 D 6 κe 4 2 σtl 0 0V 0 E 6 10 mm V Tbath (K) FIG. 3.LDisorder Lorentz ratio = /(Tin )the Dirac fluid. (A) Minimum carrier 2 density as a function of temperature for all three samvples. imp At low temperature each 1sample is limited by disorder. F H = At high temperature2all2samples become limited 2by thermal 2 2 T Q (1 + e lines vf Q /(H )) (B) The Q eye. excitations. Dashed are aimp guide to the Lorentz ratio of all three samples as a function of bath temq! electron density; H! enthalpy density perature. The largest WF violation is seen in the cleanest sample. (C) The gateconductivity dependence of the Lorentz ratio is well critical Q! quantum fit to hydrodynamic theory of Ref. [5, 6]. Fits of all three imp! momentum relaxation from impurities samples are shown at 60 K. Alltime samples return to the Fermi liquid value (black dashed line) at high density. Inset shows S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) the fitted enthalpy density as a function of temperature and the theoretical value in clean graphene (black dashed line). Schematic inset illustrates the di erence between heat and J. current Crossno al., Science 351, 1058 (2016) charge in theetneutral Dirac plasma.

41 Strange metal in graphene Negative local resistance due to viscous electron backflow in graphene D. A. Bandurin1, I. Torre2,3, R. Krishna Kumar1,4, M. Ben Shalom1,5, A. Tomadin6, A. Principi7, G. H. Auton5, E. Khestanova1,5, K. S. NovoseIov5, I. V. Grigorieva1, L. A. Ponomarenko1,4, A. K. Geim1, M. Polini3,6 1 School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 2 National Enterprise for nanoscience and nanotechnology, Scuola Normale Superiore, I Pisa, Italy 3 Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I Genova (Italy) 4 Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom 5 National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom 6 National Enterprise for nanoscience and nanotechnology, Istituto Nanoscienze Consiglio Nazionale delle Ricerche and Scuola Normale Superiore, I Pisa, Italy 7 Radboud University, Institute for Molecules and Materials, NL 6525 AJ Nijmegen, The Netherlands Graphene hosts a unique electron system that due to weak electron phonon scattering allows micrometer scale ballistic transport even at room temperature whereas the local equilibrium is provided by frequent electron electron collisions. Under these conditions, electrons can behave as a viscous liquid and exhibit hydrodynamic phenomena similar to classical liquids. Here we report unambiguous evidence for this long sought transport regime. In particular, doped graphene exhibits L. Levitov and G. Falkovich, Nature Physics online FIG. 1: Current streamlines and injection potential mapwhich for visfig.to2:theno an anomalous (negative) voltage drop arxiv: , near current contacts, is attributed cous and ohmic whirlpools flows. White lines show current streamviscosity formation of submicrometer size in the electron flow. The viscosity of graphene s electron r

42 Strange metal in graphene Science 351, 1055 (2016) Negative local resistance due to viscous electron backflow in graphene D. A. Bandurin1, I. Torre2,3, R. Krishna Kumar1,4, M. Ben Shalom1,5, A. Tomadin6, A. Principi7, G. H. Auton5, E. Khestanova1,5, K. S. NovoseIov5, I. V. Grigorieva1, L. A. Ponomarenko1,4, A. K. Geim1, M. Polini3,6 1 School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 2 National Enterprise for nanoscience and nanotechnology, Scuola Normale Superiore, I Pisa, Italy 3 Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I Genova (Italy) 4 Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom 5 National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom 6 National Enterprise for nanoscience and nanotechnology, Istituto Nanoscienze Consiglio Nazionale delle Ricerche and Scuola Normale Superiore, I Pisa, Italy 7 Radboud University, Institute for Molecules and Materials, NL 6525 AJ Nijmegen, The Netherlands Graphene hosts a unique electron system that due to weak electron phonon scattering allows micrometer scale ballistic transport even at room temperature whereas the local equilibrium is provided by frequent electron electron collisions. Under these conditions, electrons can behave as a Figure 1. Viscous backflow in doped graphene. (a,b) Steady state distribution of current injected through liquids. we report viscous exhibit hydrodynamic phenomena to classical a narrowliquid slit for and a classical conducting medium with zero (a)similar and a viscous Fermi liquid (b). Here (c) Optical unambiguous evidence long sought transport regime. In particular,geometry doped graphene micrograph of one of ourfor SLGthis devices. The schematic explains the measurement for vicinityexhibits resistance. (d,e) (negative) Longitudinalvoltage conductivity forinjection this device as a function induced by to the an anomalous drop nearandcurrent contacts, whichof is attributed applying gate 0.3 A; whirlpools 1 m. Forinmore detail, seeflow. Supplementary Information. formation of voltage. submicrometer size the electron The viscosity of graphene s electron

43 Search for signatures of Navier-Stokes hydrodynamic flow Signature of Navier-Stokes hydrodynamic flow in PdCoO2 Experiment: Successively narrow the channel in factors of 2, measuring the resistance after every step. P.J.W.$Moll,$P.$Kushwaha,$N.$Nandi,$B.$Schmidt$and$A.P.$Mackenzie,$Science$ 351,$1061$(2016)$

44 Graphene: a metal that behaves like water

45 Quantum matter without quasiparticles: Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

46 Infinite-range model with quasiparticles H = 1 (N) 1/2 NX i,j=1 t ij c i c j +... c i c j + c j c i =0, c i c j + c j c i = ij 1 X c i N c i = Q i t ij are independent random variables with t ij = 0 and t ij 2 = t 2

47 Infinite-range model with quasiparticles H = 1 (N) 1/2 NX i,j=1 t ij c i c j +... c i c j + c j c i =0, c i c j + c j c i = ij 1 X c i N c i = Q i t ij are independent random variables with t ij = 0 and t ij 2 = t 2 Im G(!) µ! Fermions occupying eigenstates with a semi-circular density of states

48 Infinite-range (SYK) model of a strange metal 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)

49 Infinite-range (SYK) model of a strange metal 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 A fermion can move only by entangling with another fermion: the Hamiltonian has nothing but entanglement. 14 A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015) 12 S. Sachdev and J. Ye, PRL 70, 3339 (1993)

50 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6, J 8,9,12, Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E A. Sen, arxiv:hep-th/ ; S. Sachdev, arxiv:

51 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Georges, O. Parcollet, and A. S. Sen, Sachdev arxiv:hep-th/ ; PRB 63, S. (2001) Sachdev, arxiv:

52 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Georges, O. Parcollet, and A. S. Sen, Sachdev arxiv:hep-th/ ; PRB 63, S. (2001) Sachdev, arxiv:

53 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 Boundary area A b ; charge density Q The scrambling time of the SYK model saturates the bound on quantum chaos ~x L = D + m Local fermion density of states! 2 k 1/2, B! T> 0 (!) e 2 E! 1/2,! < 0. A. Kitaev, unpublished J. Polchinski Equation and V. Rosenhaus, of state relating arxiv: E J. Maldacena and Q depends and D. Stanford, upon the arxiv: geometry K. Jensen, of spacetime arxiv: far from the AdS 2 L = 1 Black hole thermodynamics (classical general relativity) =2 E S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Georges, O. Parcollet, and A. S. Sen, Sachdev arxiv:hep-th/ ; PRB 63, S. (2001) Sachdev, arxiv: ~

54 Quantum matter without quasiparticles: Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

55 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt Boundary area A b ; charge density Q The SYK strange metal is holographically dual to the gravity theory of the AdS 2 near-horizon geometry of ~x charged black holes = 1 L = D + m Local fermion density of states! 1/2 S. Sachdev,,! > 0 (!) e 2 E! 1/2,! < 0. Phys. Rev. Lett. 105, (2010) Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E A. Sen, arxiv:hep-th/ ; S. Sachdev, arxiv:

56 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2

57 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q Microscopic zero temperature entropy density, =2 E 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E A. Sen hep-th/ ; S. Sachdev PRX 5, (2015)

58 H = 1 (2N) 3/2 NX i,j,k,`=1 J ij;k` c i c j c k c` Einstein-Maxwell theory + cosmological constant J 3,5,7,13 J 4,5,6,11 Q = 1 X D E c i N c i. i Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < J 8,9,12,14 14 Known equation of state determines E as a function of Q 12 Horizon area A h ; AdS 2 R d ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt = 1 Microscopic zero temperature Evidence for entropy density, S, AdS 2 =2 E dual of H L = D + m Boundary area A b ; charge density Q ~x Local fermion density of states! 1/2,! > 0 (!) e 2 E! 1/2,! < 0. Equation of state relating E and Q depends upon the geometry of spacetime far from the AdS 2 Black hole thermodynamics (classical general relativity) =2 E S. Sachdev, PRL 105, (2010); PRX 5, (2015)

59 The scrambling times of the SYK model and of black holes in Einstein gravity saturate the bound on quantum chaos L = 1 2 ~ k B T S. Shenker and D. Stanford, arxiv: ; J. Maldacena, S. H. Shenker and D. Stanford, arxiv: ; A. Kitaev, unpublished; J. Polchinski and V. Rosenhaus, arxiv: ; Antal Jevicki, Kenta Suzuki, and Junggi Yoon, arxiv: J. Maldacena and D. Stanford, arxiv: ; K. Jensen, arxiv:

60 Quantum matter without quasiparticles: Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice Graphene Solvable random fermion Sachdev-Ye-Kitaev (SYK) model Charged black hole horizons in anti-de Sitter space Strange metals in high temperature superconductors

61 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

62 M. Platé, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy, S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, (2005) SM FL A conventional metal: the Fermi liquid with Fermi surface of size 1+p

63 SM FL Pseudogap metal at low p Many experimental indications that this metal behaves like a Fermi liquid, but with Fermi surface size p and not 1+p.

64 S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D.A. Bonn, W.N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust, Nature 531, 210 (2016). Pseudogap metal SM FL at low p Many experimental indications that this metal behaves like a Fermi liquid, but with Fermi surface size p and not 1+p. Recent experiments show the PG metal is also present at low T in high magnetic field

65 Y. Kohsaka et al., Science 315, 1380 (2007) M. H. Hamidian et al., Nature Physics 12, 150 (2016) SM Density wave (DW) order at low T and p FL

66 M. A. Metlitski and S. Sachdev, PRB 82, (2010).S.SachdevR.LaPlaca,PRL111, (2013). K. Fujita, M. H Hamidian, S. D. Edkins, Chung Koo Kim, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, H. Eisaki, S. Uchida, A. Allais, M. J. Lawler, E.-A. Kim, S. Sachdev, and J. C. Davis, PNAS 111, E3026(2014) SM C Identified as a d-form factor density wave, an instability of the PG described as FL*: a metal with small Fermi surfaces and emergent gauge fields. Z(r,150mV) FL a 0 y x 10Å

67 SM FL* FL

68 SM Transition from Z2-FL* to FL as FL a theory of the strange metal (SM)

69 Quantum critical point at optimal doping Transition is primarily topological. Main change is in the size of the Fermi surface. Symmetry-breaking and Landau order parameters appear to play a secondary role. The main symmetry breaking which appears co-incident with the transition is Ising-nematic ordering. But this symmetry cannot change the size of the Fermi surface; similar comments apply to time-reversal symmetry. Need a gauge theory for transition from topological to confined state.

70 Quantum critical point at optimal doping Transition is primarily topological. Main change is in the size of the Fermi surface. Symmetry-breaking and Landau order parameters appear to play a secondary role. The main symmetry breaking which appears co-incident with the transition is Ising-nematic ordering. But this symmetry cannot change the size of the Fermi surface; similar comments apply to time-reversal symmetry. Need a gauge theory for transition from topological to confined state.

71 Quantum critical point at optimal doping Transition is primarily topological. Main change is in the size of the Fermi surface. Symmetry-breaking and Landau order parameters appear to play a secondary role. The main symmetry breaking which appears co-incident with the transition is Ising-nematic ordering. But this symmetry cannot change the size of the Fermi surface; similar comments apply to time-reversal symmetry. Need a gauge theory for transition from topological to confined state.

72 Quantum critical point at optimal doping Transition is primarily topological. Main change is in the size of the Fermi surface. Symmetry-breaking and Landau order parameters appear to play a secondary role. The main symmetry breaking which appears co-incident with the transition is Ising-nematic ordering. But this symmetry cannot change the size of the Fermi surface; similar comments apply to time-reversal symmetry. Need a gauge theory for transition from topological to confined state.

73 SM FL Proposed a SU(2) gauge theory for transtion from Z2- FL* to FL. This phase transition is beyond the Landau-Ginzburg- Wilson paradigm, and is instead a Higgs-confinement transition in a SU(2) gauge theory

74 Entangled quantum matter without quasiparticles No quasiparticle excitations Shortest possible phase coherence time, fastest possible local equilibration time, or fastest possible scrambling ~ towards quantum chaos, all of order k B T Theory built from hydrodynamics/holography /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene. Related experiments in ultraclean Fermi liquid PdCoO 2. Future work: detection of hydrodynamic flow in other strange metals......

75 Entangled quantum matter without quasiparticles No quasiparticle excitations Shortest possible phase coherence time, fastest possible local equilibration time, or fastest possible scrambling ~ towards quantum chaos, all of order k B T Theory built from hydrodynamics/holography /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene. Related experiments in ultraclean Fermi liquid PdCoO 2. Future work: detection of hydrodynamic flow in other strange metals......

76 Entangled quantum matter without quasiparticles No quasiparticle excitations Shortest possible phase coherence time, fastest possible local equilibration time, or fastest possible scrambling ~ towards quantum chaos, all of order k B T Theory built from hydrodynamics/holography /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene. Related experiments in ultraclean Fermi liquid PdCoO 2. Future work: detection of hydrodynamic flow in other strange metals......

77 Entangled quantum matter without quasiparticles No quasiparticle excitations Shortest possible phase coherence time, fastest possible local equilibration time, or fastest possible scrambling ~ towards quantum chaos, all of order k B T Theory built from hydrodynamics/holography /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene. Related experiments in ultraclean Fermi liquid PdCoO 2. Future work: detection of hydrodynamic flow in other strange metals......

78 Entangled quantum matter without quasiparticles No quasiparticle excitations Shortest possible phase coherence time, fastest possible local equilibration time, or fastest possible scrambling ~ towards quantum chaos, all of order k B T Theory built from hydrodynamics/holography /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene. Related experiments in ultraclean Fermi liquid PdCoO 2. Future work: detection of hydrodynamic flow in other strange metals......