Quantum Criticality. S. Sachdev and B. Keimer, Physics Today, February Talk online: sachdev.physics.harvard.edu HARVARD. Thursday, May 5, 2011

Transcription

1 Quantum Criticality S. Sachdev and B. Keimer, Physics Today, February 2011 Talk online: sachdev.physics.harvard.edu HARVARD

2 What is a quantum phase transition? Non-analyticity in ground state properties as a function of some control parameter g

3 What is a quantum phase transition? Non-analyticity in ground state properties as a function of some control parameter g E g True level crossing: Usually a first-order transition

4 What is a quantum phase transition? Non-analyticity in ground state properties as a function of some control parameter g E E True level crossing: Usually a first-order transition g Avoided level crossing which becomes sharp in the infinite volume limit: second-order transition g

5 g Many levels are important near a second-order quantum phase transition

6 Why study quantum phase transitions? g c g The ground states at large and small g have wavefunctions which can usually be written as products of wavefunctions of local degrees of freedom i.e. they have negligible quantum entanglement. The quantum critical state at gc often has longrange quantum entanglement : the spooky non-local quantum correlations pointed out by Einstein, Podolsky, and Rosen survive in a macrosopic system at the longest distances.

7 Why study quantum phase transitions? g c g We are often able to describe the quantum state at gc by methods drawn from quantum field theory; expansion in g-gc then allows for a controlled theory in an intermediate coupling regime important for many experimental systems

8 Why study quantum phase transitions? T Quantum criticality g c g The quantum critical point controls properties over a wide regime of quantum criticality at non-zero temperatures. I will argue that this regime is the key to understanding the physical properties of a variety of modern electronic materials.

9 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

10 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

11 Degrees of freedom: j = 1 N qubits, N "large"!... # j, " j ( ) ( ) # + " $ = # % " j 1 1 or! =, j 2 2 j j j j

12 Degrees of freedom: j = 1 N qubits, N "large"!... # j, " j ( ) ( ) # + " $ = # % " j 1 1 or! =, j 2 2 j j j j Hamiltonian of decoupled qubits: H = " Jg #! 0 j x j 2Jg! j! j

13 Coupling between qubits: z z 1 j j 1 j H J!! + = " #

14 Coupling between qubits: z z 1 j j 1 j H J!! + = " # ( )( ) "! +! " "! +! " j j j+ 1 j+ 1

15 Coupling between qubits: z z 1 j j 1 j H J!! + = " # Prefers neighboring qubits are either " " or!! j j+ 1 j j+ 1 (not entangled)

16 Full Hamiltonian # ( g x z z ) j j j H = H + H =" J! +!! j g c g

17 Full Hamiltonian # ( g x z z ) j j j H = H + H =" J! +!! j g c g j... N 1 N Product state for large g

18 Full Hamiltonian # ( g x z z ) j j j H = H + H =" J! +!! j g c g j... N 1 N or j... N 1 N Product state for small g

19 Full Hamiltonian # ( g x z z ) j j j H = H + H =" J! +!! j g c g Entangled state at quantum critical point, involving complicated superposition of 2 N qubit configurations

20 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

21 c a b Quasi-1D Ising ferromagnet CoNb 2 O 6 Co 2+ spin chain along c-axis c Oxygen Ferromagnetic superexchange ~ 90 bond Co-O-Co ~ 20K ~ 2meV Co 2+ Co 2+ CoO 6 distorted octahedron Strong easy-axis (Ising) 30 mev Single crystal of CoNb 2 O 6 (Oxford image furnace) 4 cm R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010).

22 c a b Co 2+ spin chain along c-axis c Quasi-1D Ising ferromagnet CoNb 2 O 6 Magnetic long-range order Bragg peak Co 2+ Transverse field Jg (Tesla) R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010).

23 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

24 Superfluid-insulator transition of 87 Rb atoms in a magnetic trap and an optical lattice potential M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

25 Mott insulator of 87 Rb atoms in a magnetic trap and an optical lattice potential M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

26 Applying an electric field to the Mott insulator

27

28 Why is there a peak (and not a threshold) when E = U?

29

30

31 Resonant transition when E U

32 Virtual state

33

34 Virtual state

35

36 Resonant transition when E U

37 Resonant transition when E U

38

39

40

41

42

43 Phase diagram (E-U)/t S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, (2002)

44 Phase diagram Ising quantum phase transition (E-U)/t S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, (2002)

45 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane

46 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane

47 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane

48 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane

49 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane

50 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane Constraint: forbidden!

51 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model = up = down = x-y plane Constraint: forbidden! Include a term (J/4) i (σz i 1) σ z i+1 1 and send J. Infinite exchange interaction!

52 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model Δ=E-U <0 Paramagnetic state

53 Hamiltonian of resonant subspace Effective Hamiltonian can be written as spin model Δ=E-U > 0 Antiferromagnetic state, two fold degenerate

54 Hamiltonian of resonant subspace + Antiferromagnetic state, two fold degenerate

55 Phase diagram of spin model h z /J H = i J 4 σz i σ z i+1 h z 2 σz i h x 2 σx i J, h z = J +(U E), h x =2 2t h x /J hx =0: classical first order phase transition Finite hx: quantum phase transition, second order

56 J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

57 Quantum gas microscope J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

58

59

60 In-situ imaging of antiferromagnetic chains J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

61 In-situ imaging of antiferromagnetic chains = = J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

62 In-situ imaging of antiferromagnetic chains Antiferromagnetic order = = = 1/g J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

63 Hanbury-Brown-Twiss noise correlations measure Fourier transform of boson density J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

64 Hanbury-Brown-Twiss noise correlations measure Fourier transform of boson density Peaks from antiferromagnetic order J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature 472, 307 (2011)

65 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

66 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

67 The cuprate superconductors

68 Square lattice antiferromagnet H = ij J ij Si S j Ground state has long-range Néel order Order parameter is a single vector field ϕ = η isi η i = ±1 on two sublattices ϕ = 0 in Néel state.

69 Square lattice antiferromagnet H = ij J ij Si S j J J/λ Weaken some bonds to induce spin entanglement in a new quantum phase

70 Square lattice antiferromagnet H = ij J ij Si S j J J/λ Ground state is a quantum paramagnet with spins locked in valence bond singlets = 1 2

71 = 1 2 λ c λ

72 = 1 2 λ c λ Pressure in TlCuCl3 A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).

73 TlCuCl 3 An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.

74 TlCuCl 3 Quantum paramagnet at ambient pressure

75 TlCuCl 3 Neel order under pressure A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).

76 = 1 2 λ c λ

77 Excitation spectrum in the paramagnetic phase λ c Spin S =1 λ triplon

78 Excitation spectrum in the paramagnetic phase λ c Spin S =1 λ triplon

79 Excitation spectrum in the paramagnetic phase λ c Spin S =1 λ triplon

80 Excitation spectrum in the paramagnetic phase λ c Spin S =1 λ triplon

81 Excitation spectrum in the paramagnetic phase λ c Spin S =1 triplon λ

82 Excitation spectrum in the Néel phase λ c λ Spin waves

83 Excitation spectrum in the Néel phase λ c λ Spin waves

84 Excitation spectrum in the Néel phase λ c λ Spin waves

85 TlCuCl 3 with varying pressure Energy [mev] Pressure [kbar] Triplon in the quantum paramagnet. Spin waves and a new Higgs particle in the Néel phase: the latter represents longitudinal oscillations in the magnitude of the Néel order. Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, (2008)

86 TlCuCl 3 with varying pressure Energy [mev] Triplon energy gap (λ λ c ) zν Pressure [kbar] Triplon in the quantum paramagnet. Spin waves and a new Higgs particle in the Néel phase: the latter represents longitudinal oscillations in the magnitude of the Néel order. Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, (2008)

87 TlCuCl 3 with varying pressure Energy [mev] Spin waves Pressure [kbar] Triplon in the quantum paramagnet. Spin waves and a new Higgs particle in the Néel phase: the latter represents longitudinal oscillations in the magnitude of the Néel order. Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, (2008)

88 TlCuCl 3 with varying pressure Energy [mev] Higgs Pressure [kbar] Triplon in the quantum paramagnet. Spin waves and a new Higgs particle in the Néel phase: the latter represents longitudinal oscillations in the magnitude of the Néel order. Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, (2008)

89 TlCuCl 3 with varying pressure Energy [mev] Pressure [kbar] Triplon in the quantum paramagnet. Spin waves and a new Higgs particle in the Néel phase: the latter represents longitudinal oscillations in the magnitude of the Néel order. Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, (2008)

90 = 1 2 λ c λ Quantum critical point with non-local entanglement in spin wavefunction

91 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Classical spin waves Dilute triplon gas Neel order Pressure in TlCuCl3

92 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Triplon energy gap Classical spin waves (λ λ c ) zν ; crossover at k B T. Dilute triplon gas Neel order Pressure in TlCuCl3

93 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Classical Boltzmann theory of Classical spin waves triplon particles: Leads to relaxation and thermal equilibration times of order (/k B T )e /k BT Dilute triplon gas Neel order Pressure in TlCuCl3

94 Neutron scattering measurements of the collisions between triplons in Y 2 BaNiO 5 Guangyong Xu, C. Broholm, Yeong-Ah Soh, G. Aeppli, J. F. DiTusa, Ying Chen, M. Kenzelmann, C. D. Frost, T. Ito, K. Oka, H. Takagi, Science 317, 1049 (2007)

95 Neutron scattering measurements of the collisions between triplons in Y 2 BaNiO 5 Guangyong Xu, C. Broholm, Yeong-Ah Soh, G. Aeppli, J. F. DiTusa, Ying Chen, M. Kenzelmann, C. D. Frost, T. Ito, K. Oka, H. Takagi, Science 317, 1049 (2007) Theoretical prediction Γ =1.20k B Te /k BT K. Damle and S. Sachdev, Phys. Rev. B 57, 8307 (1998)

96 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Classical Boltzmann theory of Classical spin waves triplon particles: Leads to relaxation and thermal equilibration times of order (/k B T )e /k BT Dilute triplon gas Neel order Pressure in TlCuCl3

97 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Classical spin waves Quantum critical Classical non-linear spin waves, also with Dilute triplon gas relaxational and equilibration times /k B T Neel order Pressure in TlCuCl3

98 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Classical spin waves Dilute triplon gas Neel order Pressure in TlCuCl3

99 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Classical spin waves Dilute triplon gas Neel order Pressure in TlCuCl3

100 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994). Quantum critical Classical spin waves Strongly coupled dynamics and transport with no particle/wave interpretation, and relaxation thermal equilibration times are universally proportional to /k B T Neel order Dilute triplon gas Pressure in TlCuCl3

101 Quantum critical transport Quantum nearly perfect fluid with shortest possible relaxation time, τ R τ R = C k B T where C is a universal constant S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

102 Non-zero temperature crossovers for the quantum Ising chain 0 0 Color density plot of dξ 1 /dt,whereξ is the spin correlation length. This quantity measures the strength of the interactions between the thermal excitations.

103 Neutron scattering measurements on La 1.86 Sr 0.14 Cu O 4, showing scaling of the dynamic spin susceptibility at an incommensurate wavevector: χ (ω,t)= A ω T 2 η Φ k B T G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook and J. Kulda, Science, 278, 1432 (1997).

104 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

105 Outline 1. The quantum Ising chain A. The magnetic insulator CoNb2O6 B. Ultracold Rb atoms in an optical lattice 2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators 3. Higher temperature superconductors and strange metals Quantum criticality of fermions and Fermi surfaces

106 The cuprate superconductors

107 Square lattice antiferromagnet H = ij J ij Si S j Ground state has long-range Néel order Order parameter is a single vector field ϕ = η isi η i = ±1 on two sublattices ϕ = 0 in Néel state.

108 Holedoped Electrondoped

109 Electron-doped cuprate superconductors Holedoped Electrondoped

110 Electron-doped cuprate superconductors n 2 Holedoped Electrondoped Figure prepared by K. Jin and and R. L. Greene based on N. P. Fournier, P. Armitage, and R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010). Resistivity ρ 0 + AT n

111 Electron-doped cuprate superconductors n 2 Holedoped Electrondoped Figure prepared by K. Jin and and R. L. Greene based on N. P. Fournier, P. Armitage, and R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010). Resistivity ρ 0 + AT n

112 Electron-doped cuprate superconductors Strange Metal n Holedoped Electrondoped 1 Figure prepared by K. Jin and and R. L. Greene based on N. P. Fournier, P. Armitage, and R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010). Resistivity ρ 0 + AT n

113 Iron pnictides: a new class of high temperature superconductors

114 Iron pnictides: a new class of high temperature superconductors Ort Tet Ishida, Nakai, and Hosono arxiv: v1 50 AFM / Ort 0 SC S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, Physical Review Letters 104, (2010).

115 Temperature-doping phase diagram of the iron pnictides: BaFe 2 (As 1-x P x ) 2 T 0 T SDW T c SDW! " Resistivity ρ 0 + AT α Superconductivity 0 SDW= spin density wave = antiferromagnetism in a metal S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

116 Temperature-doping phase diagram of the iron pnictides: BaFe 2 (As 1-x P x ) 2 T 0 T SDW T c SDW! " Resistivity ρ 0 + AT α Superconductivity 0 SDW= spin density wave = antiferromagnetism in a metal S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

117 Temperature-doping phase diagram of the iron pnictides: Strange Metal BaFe 2 (As 1-x P x ) 2 T 0 T SDW T c! " SDW Resistivity ρ 0 + AT α Superconductivity 0 SDW= spin density wave = antiferromagnetism in a metal S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

118 Organic Bechgaard compounds Resistivity ρ 0 + AT α Doiron-Leyraud et al., PRB 80, (2009)

119 Organic Bechgaard compounds Resistivity ρ 0 + AT α Doiron-Leyraud et al., PRB 80, (2009)

120 Organic Bechgaard compounds Strange Metal Resistivity ρ 0 + AT α Doiron-Leyraud et al., PRB 80, (2009)

121 Lower Tc superconductivity in the heavy fermion compounds G. Knebel, D. Aoki, and J. Flouquet, arxiv:

122 Fermi surface Metal with large Fermi surface Momenta with electronic states empty Momenta with electronic states occupied

123 Fermi surface+antiferromagnetism Metal with large Fermi surface + The electron spin polarization obeys S(r, τ) = ϕ (r, τ)e ik r where K is the ordering wavevector.

124 Metal with large Fermi surface

125 Fermi surfaces translated by K =(π, π).

126 Electron and hole pockets in antiferromagnetic phase with ϕ =0

127 Fermi surface+antiferromagnetism ncreasing SDW ϕ =0 ϕ =0 Metal with electron and hole pockets Metal with large Fermi surface Increasing interaction S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

128 Quantum oscillations Nd 2 x Ce x CuO 4 T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. Lett. 103, (2009).

129 Increasing SDW order s Quantum oscillations Nd 2 x Ce x CuO 4 T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. Lett. 103, (2009).

130 Fermi surface+antiferromagnetism ncreasing SDW ϕ =0 ϕ =0 Metal with electron and hole pockets Metal with large Fermi surface Increasing interaction S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

131 Quantum criticality of antiferromagnetism and Fermi surfaces T * Fluctuating Fermi pockets Quantum Strange Critical Metal Large Fermi surface ncreasing SDW Spin density wave (SDW)

132 Quantum criticality of antiferromagnetism and Fermi surfaces T * Fluctuating Fermi pockets Quantum Strange Critical Metal Large Fermi surface ncreasing SDW Spin density wave (SDW)

133 Quantum criticality of antiferromagnetism and Fermi surfaces T * Fluctuating Fermi pockets Strange Quantum Metal Critical Large Fermi surface ncreasing SDW Spin density wave (SDW)

134 Conclusions Paradigm of quantum phase transitions: the quantum Ising chain, realized in the ferromagnetic insulator CoNb2O6, and ultracold atoms in tilted optical lattices

135 Conclusions Described excitations spectrum of the coupled-dimer antiferromagnet TlCuCl3. It has regimes of classical dynamics of triplon particles, and of non-linear spin waves, and a regime of quantum-criticality with characteristic equilibration time /k B T

136 Conclusions Quantum criticality of antiferromagnetism and Fermi surface reconstruction controls strange metal regime of quasi-two dimensional higher temperature superconductors