Talk online: sachdev.physics.harvard.edu

Transcription

1 Talk online: sachdev.physics.harvard.edu

2 Particle theorists Condensed matter theorists

3 Quantum Entanglement Hydrogen atom: Hydrogen molecule: = _ = 1 2 ( ) Superposition of two electron states leads to non-local correlations between spins

4 Quantum Phase Transition Change in the nature of entanglement in a macroscopic quantum system. Familiar phase transitions, such as water boiling to steam, also involve macroscopic changes, but in thermal motion

5 Quantum Criticality The complex and non-local entanglement at the critical point between two quantum phases

6 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

7 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

8 The cuprate superconductors

9 Antiferromagnetic (Neel) order in the insulator No entanglement of spins

10 Antiferromagnetic (Neel) order in the insulator Excitations: 2 spin waves (Goldstone modes)

11 Weaken some bonds to induce spin entanglement in a new quantum phase

12 Ground state is a product of pairs of entangled spins.

13 Excitations: 3 S=1 triplons

14 Excitations: 3 S=1 triplons

15 Excitations: 3 S=1 triplons

16 Excitations: 3 S=1 triplons

17 Excitations: 3 S=1 triplons

18 Phase diagram as a function of the ratio of exchange interactions, λ λ c λ Quantum critical point with non-local entanglement in spin wavefunction

19 TlCuCl3

20 Phase diagram as a function of the ratio of exchange interactions, λ λ c Pressure in TlCuCl3 λ

21 TlCuCl 3 at ambient pressure triplon N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B (2001).

22 TlCuCl 3 with varying pressure Observation of 3 2 low energy modes, emergence of new longitudinal mode in Néel phase, and vanishing of Néel temperature at the quantum critical point Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, arxiv:

23 Quantum phase transition with full square lattice symmetry H = J ij S i S j ; Si spin operator with S =1/2

24 Quantum phase transition with full square lattice symmetry H = J ij S i S j + K four spin exchange A. W. Sandvik, Phys. Rev. Lett. 98, (2007)

25 Quantum phase transition with full square lattice symmetry H = J ij S i S j + K four spin exchange K/J A. W. Sandvik, Phys. Rev. Lett. 98, (2007) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

26 Why should we care about the entanglement at an isolated critical point in the parameter space?

27 Temperature, T Conformal field theory (CFT) at T>0 0 Neel VBS K/J

28 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

29 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

30 Black Holes Objects so massive that light is gravitationally bound to them.

31 Black Holes Objects so massive that light is gravitationally bound to them. The region inside the black hole horizon is causally disconnected from the rest of the universe. Horizon radius R = 2GM c 2

32 Black Hole Thermodynamics Bekenstein and Hawking discovered astonishing connections between the Einstein theory of black holes and the laws of thermodynamics Entropy of a black hole S = k BA 4l 2 P where A is the area of the horizon, and G l P = is the Planck length. c 3 The Second Law: da 0

33 Black Hole Thermodynamics Bekenstein and Hawking discovered astonishing connections between the Einstein theory of black holes and the laws of thermodynamics Horizon temperature: 4πk B T = 2 2Ml 2 P

34 AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space A 2+1 dimensional system at its quantum critical point Black hole Maldacena, Gubser, Klebanov, Polyakov

35 AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Black hole Quantum criticality in 2+1 D Black hole temperature = temperature of quantum criticality Strominger, Vafa

36 AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Black hole entropy = entropy of quantum criticality in 2+1 dimensions Black hole Quantum criticality in 2+1 D Strominger, Vafa

37 AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Black hole Quantum criticality in 2+1 D Dynamics of quantum criticality = waves in curved gravitational background Maldacena, Gubser, Klebanov, Polyakov

38 AdS/CFT correspondence The quantum theory of a black hole in a 3+1- dimensional negatively curved AdS universe is holographically represented by a CFT (the theory of a quantum critical point) in 2+1 dimensions 3+1 dimensional AdS space Black hole Quantum criticality in 2+1 D Friction of quantum critical dynamics = black hole absorption rates Son

39 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

40 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

41 Dope the antiferomagnets with charge carriers of density x by applying a chemical potential μ Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y

42 Superconductor T μ

43 T Superconductor μ Scanning tunnelling microscopy

44 STM studies of the underdoped superconductor Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y

45 Topograph Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

46 di/dv Spectra Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y Intense Tunneling-Asymmetry (TA) variation are highly similar Y. Kohsaka et al. Science 315, 1380 (2007)

47 Topograph Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

48 Tunneling Asymmetry (TA)-map at E=150meV Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

49 Tunneling Asymmetry (TA)-map at E=150meV Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

50 Tunneling Asymmetry (TA)-map at E=150meV Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y 12 nm Indistinguishable bond-centered TA contrast with disperse 4a 0 -wide nanodomains Y. Kohsaka et al. Science 315, 1380 (2007)

51 TA Contrast is at oxygen site (Cu-O-Cu bond-centered) R map (150 mv) Ca 1.88 Na 0.12 CuO 2 Cl 2, 4 K 4a 0 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

52 TA Contrast is at oxygen site (Cu-O-Cu bond-centered) R map (150 mv) Ca 1.88 Na 0.12 CuO 2 Cl 2, 4 K 12 nm S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999). 4a 0 Evidence for a predicted valence bond supersolid

53 T Superconductor μ Scanning tunnelling microscopy

54 T g Superconductor μ Insulator x =1/8

55 T g Superconductor μ Insulator x =1/8

56 T t or c u d n o Superc g μ Insulator x =1/8

57 Nernst measurements T t or c u d n o Superc g μ Insulator x =1/8

58 Nernst experiment e y H m H

59 T Nernst measurements Superconductor μ g Insulator x =1/8

60 Non-zero temperature phase diagram VBS Supersolid Superfluid VBS Insulator Coulomb interactions

61 Non-zero temperature phase diagram VBS Supersolid Quantum-critical dynamics in a magnetic field, at generic density, and with impurities Superfluid VBS Insulator Coulomb interactions

62 To the CFT of the quantum critical point, we add A chemical potential μ A magnetic field B After the AdS/CFT mapping, we obtain the Einstein-Maxwell theory of a black hole with An electric charge A magnetic charge A precise correspondence is found between general hydrodynamics of vortices near quantum critical points and solvable models of black holes with electric and magnetic charges S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

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64 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

65 Conservation laws/equations of motion S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

66 Constitutive relations which follow from Lorentz transformation to moving frame S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

67 Single dissipative term allowed by requirement of positive entropy production. There is only one independent transport co-efficient S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

68 Momentum relaxation from impurities S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

69 From these relations, we obtained results for the transport co-efficients, expressed in terms of a cyclotron frequency and damping: Transverse thermoelectric co-efficient ( ) ( ) 2 h α xy =Φ s B (k B T ) 2 2πτimp ρ 2 +Φ σ Φ ε+p (k B T ) 3 /2πτ imp 2ek B Φ 2 ε+p (k BT ) 6 + B 2 ρ 2 (2πτ imp / ), 2 where B = Bφ 0 /( v) 2 ; ρ = ρ/( v) 2. S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

70 LSCO Experiments Measurement of (T small) Y. Wang et al., Phys. Rev. B 73, (2006).

71 LSCO Experiments Measurement of (T small) Y. Wang et al., Phys. Rev. B 73, (2006).

72 LSCO Experiments Measurement of (T small) Y. Wang et al., Phys. Rev. B 73, (2006). Prediction for c : T-dependent cyclotron frequency! times smaller than the cyclotron frequency of free electrons (at T=35 K) Only observable in ultra-pure samples where.

73 Theory for LSCO Experiments -dependence Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, (2006).

74 LSCO Experiments Theory for Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, (2006).

75 To the CFT of the quantum critical point, we add A chemical potential μ A magnetic field B After the AdS/CFT mapping, we obtain the Einstein-Maxwell theory of a black hole with An electric charge A magnetic charge A precise correspondence is found between general hydrodynamics of vortices near quantum critical points and solvable models of black holes with electric and magnetic charges S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

76 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

77 Outline 1. Entanglement of spins Experiments on antiferromagnetic insulators 2. Black Hole Thermodynamics Connections to quantum criticality 3. Nernst effect in the cuprate superconductors Quantum criticality and dyonic black holes 4. Quantum criticality in graphene Hydrodynamic cyclotron resonance and Nernst effect

78 t Graphene

79 Graphene

80 Cyclotron resonance in graphene M. Mueller, and S. Sachdev, arxiv: Conditions to observe resonance } Negligible Landau quantization Hydrodynamic, collison-dominated regime Negligible broadening Relativistic, quantum critical regime

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