Quantum Phase Transitions
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1 Quantum Phase Transitions Subir Sachdev Talks online at
2 What is a phase transition? A change in the collective properties of a macroscopic number of atoms
3 What is a quantum phase transition? Change in the nature of entanglement in a macroscopic quantum system.
4 Entanglement Hydrogen atom: Hydrogen molecule: = _ = 1 2 ( ) Superposition of two electron states leads to non-local correlations between spins
5 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
6 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
7 Chinese Terracotta warriors ( BC) Han Purple BaCuSi 2 O 6 M. Jaime et al., Phys. Rev. Lett. 93, (2004)
8 Han Purple BaCuSi 2 O 6 Each Cu 2+ has a single free electron spin M. Jaime et al., Phys. Rev. Lett. 93, (2004)
9 Weak magnetic field Han Purple BaCuSi 2 O 6 = 1 2 ( ) M. Jaime et al., Phys. Rev. Lett. 93, (2004)
10 Strong magnetic field Han Purple BaCuSi 2 O 6 Magnetic field, H M. Jaime et al., Phys. Rev. Lett. 93, (2004)
11 Han Purple BaCuSi 2 O 6 SPM QPM XY-AFM M. Jaime et al., Phys. Rev. Lett. 93, (2004)
12 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
13 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
14 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
15 Han Purple BaCuSi 2 O 6 Each Cu 2+ has a single free electron spin. Vary the ratio J/J M. Jaime et al., Phys. Rev. Lett. 93, (2004)
16 Neel Spin gap J/J Vary the ratio J/J
17 Temperature, T Neel Spin gap 0 J/J Vary the ratio J/J
18 Temperature, T 0 Spin wave J/J Vary the ratio J/J
19 Temperature, T Neel Spin gap 0 J/J Vary the ratio J/J
20 Temperature, T Neel Spin gap 0 Triplet magnon J/J Vary the ratio J/J
21 Temperature, T Neel Spin gap 0 J/J Vary the ratio J/J
22 Temperature, T Neel Spin gap 0 J/J Vary the ratio J/J S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
23 Temperature, T Quantum Criticality 0 J/J Vary the ratio J/J S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
24 Temperature, T Quantum Criticality Thermal excitations interact via a universal S matrix. Neel Spin gap 0 J/J Vary the ratio J/J S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
25 Temperature, T Decoherence time = kt B Neel Spin gap 0 J/J Vary the ratio J/J S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
26 Temperature, T Quantum critical transport Spin diffusion constant Neel D s 2 c = Θ kt where Θ is a universal number B Spin gap 0 J/J Vary the ratio J/J A.V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B 49, (1994).
27 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
28 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
29 Trap for ultracold 87 Rb atoms
30 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
31 The Bose-Einstein condensate in a periodic potential G = + + BEC Lowest energy state for many atoms = G G G = terms Large fluctuations in number of atoms in each potential well superfluidity (atoms can flow without dissipation)
32 Breaking up the Bose-Einstein condensate G = + + Lowest energy state for many atoms MI = 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
33 Velocity distribution of 87 Rb atoms Superfliud M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
34 Velocity distribution of 87 Rb atoms Insulator M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
35 Non-zero temperature phase diagram Superfluid Insulator Depth of periodic potential
36 Non-zero temperature phase diagram Dynamics of the classical Gross-Pitaevski equation Superfluid Insulator Depth of periodic potential
37 Non-zero temperature phase diagram Dilute Boltzmann gas of particle and holes Superfluid Insulator Depth of periodic potential
38 Non-zero temperature phase diagram No wave or quasiparticle description Superfluid Insulator Depth of periodic potential
39 Resistivity of Bi films Conductivity σ σ T 0 = Superconductor ( ) σ Insulator T 0 = 0 ( ) σ Quantum critical point T 0 ( ) 4e h 2 D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)
40 Non-zero temperature phase diagram Superfluid Insulator Depth of periodic potential
41 Non-zero temperature phase diagram Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) Superfluid Insulator Depth of periodic potential K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
42 Non-zero temperature phase diagram Needed: Cold atom experiments in this regime Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) Superfluid Insulator Depth of periodic potential K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
43 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.
44 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. 3+1 dimensional AdS space 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
45 Hydrodynamics of a conformal field theory (CFT) Hydrodynamics of a CFT Waves of gauge fields in a curved background
46 Hydrodynamics of a conformal field theory (CFT) The scattering cross-section of the thermal excitations is universal and so transport coefficients are universally determined by k B T Charge diffusion constant Conductivity D c 2 c = Θ kt B 4e σ =Θ h 2 K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
47 Hydrodynamics of a conformal field theory (CFT) 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: Spin diffusion constant Spin conductivity D s = σ = 2 3 c 4π kt N 3/2 32π B P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, hep-th/
48 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
49 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
50 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
51 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
52 Valence bonds in benzene Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling)
53 Valence bonds in benzene Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling)
54 Valence bonds in benzene Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling)
55 Temperature-doping phase diagram of the cuprate superconductors
56 Antiferromagnetic (Neel) order in the insulator H = J Sii Sj ; Si spin operator with S=1/2 ij
57 Induce formation of valence bonds by e.g. ring-exchange interactions H J S i S + K 4-spin exchange = i j ij A. W. Sandvik, cond-mat/
58 As in H 2 and benzene, each electron wants to pair up with another electron and form a valence bond 1 2 = ( - )
59 1 2 = ( - )
60 1 2 = ( - )
61 1 2 = ( - )
62 1 2 = ( - )
63 Entangled liquid of valence bonds (Resonating valence bonds RVB) 1 2 = ( - ) P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).
64 Valence bond solid (VBS) 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
65 Valence bond solid (VBS) 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
66 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
67 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
68 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
69 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
70 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
71 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
72 Valence bond solid (VBS) More possibilities for entanglement with nearby states 1 2 = ( - ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, (2001).
73 Excitations of the RVB liquid 1 2 = ( - )
74 Excitations of the RVB liquid 1 2 = ( - )
75 Excitations of the RVB liquid 1 2 = ( - )
76 Excitations of the RVB liquid 1 2 = ( - )
77 Excitations of the RVB liquid 1 2 = ( - ) Electron fractionalization: Excitations carry spin S=1/2 but no charge
78 Excitations of the VBS 1 2 = ( - )
79 Excitations of the VBS 1 2 = ( - )
80 Excitations of the VBS 1 2 = ( - )
81 Excitations of the VBS 1 2 = ( - )
82 Excitations of the VBS 1 2 = ( - ) Free spins are unable to move apart: no fractionalization, but confinement
83 Phase diagram of square lattice antiferromagnet H J S i S + K 4-spin exchange = i j ij A. W. Sandvik, cond-mat/
84 Phase diagram of square lattice antiferromagnet Neel order VBS order K/J H J S i S + K = i j ij 4-spin exchange T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
85 Phase diagram of square lattice antiferromagnet Neel order VBS order RVB physics appears at the quantum critical point which has fractionalized excitations: deconfined criticality H J S i S + K = i j ij 4-spin exchange K/J T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
86 Phase diagram of square lattice antiferromagnet Neel order VBS order K/J T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
87 Temperature, T Quantum criticality of fractionalized excitations 0 K/J
88 Phases of nuclear matter
89 Observation of a valence bond solid (VBS) X[Pd(dmit) 2 ] 2 Pd S C One free electron spin on each vertex of a triangular lattice M. Tamura et al., J. Phys. Soc. Jpn. 75, (2006)
90 Observation of a valence bond solid (VBS) RVB (?) Pressure-temperature phase diagram of ETMe 3 P[Pd(dmit) 2 ] 2 Y. Shimizu et al. cond-mat/
91 Temperature-doping phase diagram of the cuprate superconductors
92 Temperature-doping phase diagram of the cuprate superconductors Neel order VBS order Deconfined quantum critical point (DQCP) K/J
93 Temperature-doping phase diagram of the cuprate superconductors Neel order DQCP Neel order + d-wave superconductivity Superconducting algebraic holon liquid d-wave superconductivity Hole concentration R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil, to appear
94 Temperature-doping phase diagram of the cuprate superconductors Quantum critical phases with enhanced VBS correlations Neel order DQCP Neel order + d-wave superconductivity Superconducting algebraic holon liquid d-wave superconductivity Hole concentration R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil, to appear
95 Temperature-doping phase diagram of the cuprate superconductors STM in zero field
96 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)
97 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)
98 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)
99 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)
100 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)
101 Glassy Valence Bond Solid (VBS) 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)
102 Temperature-doping phase diagram of the cuprate superconductors Glassy Valence Bond Solid (VBS)
103 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
104 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
105 Conclusions 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.
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