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Hydrodynamic transport near quantum critical points and the AdS/CFT correspondence

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Sean Hartnoll, KITP Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Sean Hartnoll, KITP Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Markus Mueller, Harvard Subir Sachdev, Harvard Markus Mueller, Harvard Subir Sachdev, Harvard Particle theorists Condensed matter theorists

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

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The insulator:

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Excitations of the insulator:

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Graphene

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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Wave oscillations of the condensate (classical Gross- Pitaevski equation)

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Dilute Boltzmann gas of particle and holes

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CFT at T>0

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D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) Resistivity of Bi films M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

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Density correlations in CFTs at T >0

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K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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Density correlations in CFTs at T >0 K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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Collisionless-hydrodynamic crossover in graphene L. Fritz, M. Mueller, J. Schmalian and S. Sachdev, to appear.

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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For experimental applications, we must move away from the ideal CFT e.g. A chemical potential A magnetic field B CFT

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Cuprate superconductors

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Nernst measurements Cuprate superconductors

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Nernst experiment Vortices move in a temperature gradient Phase slip generates Josephson voltage 2eV J = 2 h n V E J = B x v H eyey HmHm Nernst signal e y = E y /| T |

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Use coupling g to induce a transition to a VBS insulator Cuprate superconductors

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Nernst measurements Cuprate superconductors

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S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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Conservation laws/equations of motion S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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Constitutive relations which follow from Lorentz transformation to moving frame S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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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 76 144502 (2007)

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For experimental applications, we must move away from the ideal CFT e.g. A chemical potential A magnetic field B An impurity scattering rate 1/ imp (its T dependence follows from scaling arguments) CFT

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S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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Solve initial value problem and relate results to response functions (Kadanoff+Martin)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping: S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B 76 144502 (2007)

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LSCO Experiments Measurement of (T small) Y. Wang et al., Phys. Rev. B 73, 024510 (2006).

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LSCO Experiments Measurement of (T small) Y. Wang et al., Phys. Rev. B 73, 024510 (2006).

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56 LSCO Experiments Measurement of (T small) Y. Wang et al., Phys. Rev. B 73, 024510 (2006). T-dependent cyclotron frequency! 0.035 times smaller than the cyclotron frequency of free electrons (at T=35 K) Only observable in ultra-pure samples where. → Prediction for c :

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Theory for LSCO Experiments Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, 024510 (2006). -dependence

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Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, 024510 (2006). Theory for LSCO Experiments

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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1. Model systems (i) Superfluid-insulator transition of lattice bosons, (ii) graphene 2. Quantum-critical transport at integer filling, zero magnetic field, and with no impurities Collisionless-t0-hydrodynamic crossover of CFT3s 3. Quantum-critical transport at integer generic filling, nonzero magnetic field, and with impurities Nernst effect and a hydrodynamic cyclotron resonance 4. The AdS/CFT correspondence Quantum criticality and dyonic black holes Outline

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

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Objects so massive that light is gravitationally bound to them. Black Holes The region inside the black hole horizon is causally disconnected from the rest of the universe.

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Black Hole Thermodynamics Bekenstein and Hawking discovered astonishing connections between the Einstein theory of black holes and the laws of thermodynamics

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Black Hole Thermodynamics Bekenstein and Hawking discovered astonishing connections between the Einstein theory of black holes and the laws of thermodynamics

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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 A 2+1 dimensional system at its quantum critical point Black hole 3+1 dimensional AdS space Maldacena

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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 Quantum criticality in 2+1 D Black hole 3+1 dimensional AdS space Strominger, Vafa Black hole temperature = temperature of quantum criticality

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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 Black hole 3+1 dimensional AdS space Maldacena Dynamics of quantum criticality = waves in curved gravitational background Quantum criticality in 2+1 D

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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 Black hole 3+1 dimensional AdS space Son “Friction” of quantum critical dynamics = black hole absorption rates Quantum criticality in 2+1 D

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Application of the AdS/CFT correspondence

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Hydrodynamic theory for thermoelectric response functions of quantum critical systems Applications to the cuprates and graphene. Exact solutions via black hole mapping have yielded first exact results for transport co-efficients in interacting many- body systems, and were valuable in determining general structure of hydrodynamics. Hydrodynamic theory for thermoelectric response functions of quantum critical systems Applications to the cuprates and graphene. Exact solutions via black hole mapping have yielded first exact results for transport co-efficients in interacting many- body systems, and were valuable in determining general structure of hydrodynamics. Conclusions

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P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) Collisionless to hydrodynamic crossover of SYM3

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P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) Collisionless to hydrodynamic crossover of SYM3

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Universal constants of SYM3 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) C. Herzog, JHEP 0212, 026 (2002)

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Electromagnetic self-duality

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