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Quantum Phase Transitions Subir Sachdev Talks online at http://sachdev.physics.harvard.edu

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What is a “phase transition” ? A change in the collective properties of a macroscopic number of atoms

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What is a “quantum phase transition” ? Change in the nature of entanglement in a macroscopic quantum system.

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Entanglement Hydrogen atom: Hydrogen molecule: = _ Superposition of two electron states leads to non-local correlations between spins

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Chinese Terracotta warriors (479-221 BC) M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Han Purple – BaCuSi 2 O 6

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M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Each Cu 2+ has a single free electron spin Han Purple – BaCuSi 2 O 6

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M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Weak magnetic field Han Purple – BaCuSi 2 O 6

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M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Strong magnetic field Magnetic field, H Han Purple – BaCuSi 2 O 6

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QPM XY-AFM SPM M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Han Purple – BaCuSi 2 O 6

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Each Cu 2+ has a single free electron spin. Han Purple – BaCuSi 2 O 6 Vary the ratio J/J’

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J/J’ Vary the ratio J/J’ Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 Spin wave

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J/J’ Vary the ratio J/J’ Temperature, T 0 Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 “Triplet magnon” Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989) Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 Quantum Criticality S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

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J/J’ Vary the ratio J/J’ Temperature, T 0 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). Thermal excitations interact via a universal S matrix. Quantum Criticality Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 Decoherence time S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). Neel Spin gap

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J/J’ Vary the ratio J/J’ Temperature, T 0 A.V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B 49, 11919 (1994). Quantum critical transport Spin diffusion constant where is a universal number Neel Spin gap

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Trap for ultracold 87 Rb atoms

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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 Bose-Einstein condensate in a periodic potential Lowest energy state for many atoms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

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By tuning repulsive interactions between the atoms, states with multiple atoms in a potential well can be suppressed. The lowest energy state is then a Mott insulator – it has negligible number fluctuations, and atoms cannot “flow” Breaking up the Bose-Einstein condensate Lowest energy state for many atoms

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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Velocity distribution of 87 Rb atoms Superfliud

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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Velocity distribution of 87 Rb atoms Insulator

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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential

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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Dynamics of the classical Gross-Pitaevski equation

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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Dilute Boltzmann gas of particle and holes

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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential No wave or quasiparticle description

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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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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential

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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). Needed: Cold atom experiments in this regime

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Hydrodynamics of a conformal field theory (CFT) Maldacena’s AdS/CFT correspondence relates the hydrodynamics of CFTs to the quantum gravity theory of the horizon of a black hole in Anti-de Sitter space.

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Holographic representation of black hole physics in a 2+1 dimensional CFT at a temperature equal to the Hawking temperature of the black hole. Black hole 3+1 dimensional AdS space Hydrodynamics of a conformal field theory (CFT)

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Hydrodynamics of a CFT Waves of gauge fields in a curved background

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Hydrodynamics of a conformal field theory (CFT) The scattering cross-section of the thermal excitations is universal and so transport co- efficients are universally determined by k B T Charge diffusion constant Conductivity K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

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Hydrodynamics of a conformal field theory (CFT) P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, hep-th/0701036 Spin diffusion constant Spin conductivity For the (unique) CFT with a SU(N) gauge field and 16 supercharges, we know the exact diffusion constant associated with a global SO(8) symmetry:

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling) Valence bonds in benzene

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Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling) Valence bonds in benzene

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Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling) Valence bonds in benzene

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Temperature-doping phase diagram of the cuprate superconductors

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Antiferromagnetic (Neel) order in the insulator

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Induce formation of valence bonds by e.g. ring-exchange interactions A. W. Sandvik, cond-mat/0611343

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= As in H 2 and benzene, each electron wants to pair up with another electron and form a valence bond

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=

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=

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=

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=

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= Entangled liquid of valence bonds (Resonating valence bonds – RVB) P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS)

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS)

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

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= Excitations of the RVB liquid

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=

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=

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=

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= Electron fractionalization: Excitations carry spin S=1/2 but no charge

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= Excitations of the VBS

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=

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=

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=

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= Free spins are unable to move apart: no fractionalization, but confinement

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Phase diagram of square lattice antiferromagnet A. W. Sandvik, cond-mat/0611343

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Phase diagram of square lattice antiferromagnet VBS order K/J Neel order T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Phase diagram of square lattice antiferromagnet VBS order K/J Neel order T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004). RVB physics appears at the quantum critical point which has fractionalized excitations: “deconfined criticality”

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Phase diagram of square lattice antiferromagnet VBS order K/J Neel order T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

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Temperature, T 0 Quantum criticality of fractionalized excitations K/J

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Phases of nuclear matter

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X[Pd(dmit) 2 ] 2 Pd C S One free electron spin on each vertex of a triangular lattice M. Tamura et al., J. Phys. Soc. Jpn. 75, 093701 (2006) Observation of a valence bond solid (VBS)

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Pressure-temperature phase diagram of ETMe 3 P[Pd(dmit) 2 ] 2 Y. Shimizu et al. cond-mat/0612545 RVB (?) Observation of a valence bond solid (VBS)

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Temperature-doping phase diagram of the cuprate superconductors

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VBS order K/J Neel order Deconfined quantum critical point (DQCP)

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Temperature-doping phase diagram of the cuprate superconductors Hole concentration Neel order + d-wave supercon- ductivity “Supercon- ducting algebraic holon liquid” d-wave supercon- ductivity R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil, to appear DQCP

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Temperature-doping phase diagram of the cuprate superconductors Hole concentration Neel order + d-wave supercon- ductivity “Supercon- ducting algebraic holon liquid” d-wave supercon- ductivity R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil, to appear DQCP Quantum critical phases with enhanced VBS correlations

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Temperature-doping phase diagram of the cuprate superconductors STM in zero field

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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

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Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007) “Glassy” Valence Bond Solid (VBS)

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Temperature-doping phase diagram of the cuprate superconductors “Glassy” Valence Bond Solid (VBS)

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

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Studies of new materials and trapped ultracold atoms are yielding new quantum phases, with novel forms of quantum entanglement. Some materials are of technological importance: e.g. high temperature superconductors. Real-world studies on the entanglement of large numbers of qubits: insights may be important for quantum cryptography and quantum computing. Tabletop “laboratories for the entire universe”: quantum mechanics of black holes, quark-gluon plasma, neutrons stars, and big-bang physics. Conclusions

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