Quantum criticality and high temperature superconductivity

Transcription

1 Quantum criticality and high temperature superconductivity University of Waterloo February 14, 2014 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD

2 William Witczak-Krempa Perimeter Erik Sorensen McMaster Sean Hartnoll Stanford Raghu Mahajan Stanford Matthias Punk Innsbruck

3 Lauren Hayward Roger Melko David Hawthorn Erez Berg Max Metlitski Rolando La Placa Jay Deep Sau HARVARD

4 Foundations of quantum many body theory: 1. Ground states connected adiabatically to independent electron states 2. Quasiparticle structure of excited states Metals E In Metal k

5 Foundations of quantum many body theory: 1. Ground states connected adiabatically to independent electron states 2. Quasiparticle structure of excited states Metals E In Metal k

6 Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle structure of excited states Famous examples: The fractional quantum Hall effect of electrons in two dimensions (e.g. in graphene) in the presence of a strong magnetic field. The ground state is described by Laughlin s wavefunction, and the excitations are quasiparticles which carry fractional charge.

7 Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle structure of excited states Famous examples: Electrons in one dimensional wires form the Luttinger liquid. The quanta of density oscillations ( phonons ) are a quasiparticle basis of the lowenergy Hilbert space. Similar comments apply to magnetic insulators in one dimension.

8 Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle 2. No quasiparticles structure of excited states

9 Modern phases of quantum matter: 1. Ground states disconnected from independent electron states: many-particle entanglement 2. Quasiparticle 2. No quasiparticles structure of excited states Only 2 examples: 1. Conformal field theories in spatial dimension d >1 2. Quantum critical metals in dimension d=2

10 Iron pnictides: a new class of high temperature superconductors Ishida, Nakai, and Hosono arxiv: v1

11 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) Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic 1.0 Superconductivity 0

12 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) Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " Neel (AF) and nematic order 2.0 AF SDW +nematic 1.0 Superconductivity 0

13 Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic 1.0 Superconductivity 0 Superconductor Bose condensate of pairs of electrons 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)

14 Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic 1.0 Superconductivity 0 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) Ordinary metal (Fermi liquid)

15 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) Resistivity 0 + AT Strange Metal no quasiparticles, Landau-Boltzmann theory does not apply BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic Superconductivity 1.0 0

16 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

17 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

18 Superfluid-insulator transition Ultracold 87 Rb atoms - bosons M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

19 T a complex field representing the Bose-Einstein condensate of the superfluid T KT Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g U/t

20 a h 0 /U Response to modulation of lattice depth: Observation of quasiparticle excitations across the superfluidinsulator transition of ultracold atoms in a 2-dimensional optical lattice j Mott Superfluid 1 1 Insulator j/j c 2 b Lattice depth (E r ) Lattice loading Modulation Hold time Ramp to atomic limit 0 T mod = k B T/U V T tot = 200 ms Time (ms) mod (Hz) Temperature measurement A = 0.03V 0 Figure 1 Illustration of the Higgs mode and experimental sequence. a, Classical energy b density 0.17V as a function of the order parameter Y. Within the ordered (superfluid) phase, Nambu Goldstone 1 and Higgs modes arise from phase and amplitude0.15 modulations (blue and red arrows in panel 1). As the coupling j 5 J/U (see main text) approaches 0 the critical value j c, the energy density transforms into a function with a minimum at Y 5 0 (panels 2 and 3) Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the 0.11 disordered (Mott insulating) phase, two gapped modes exist, respectively corresponding to particle and hole excitations in our case (red and blue arrow in panel 3). b, The Higgs 0.18 mode can 2 be excited with a periodic modulation of the coupling j, which amounts to a shaking of the classical energy density 0.16 potential. In the experimental sequence, 0 this is realized by a modulation of the optical lattice potential (see main text for details). t 5 1/n 0.14 mod ; E r, lattice recoil energy. V 0 = 8E r j/j c = 2.2 V 0 = 9E r j/j c = 1.6 system with a recently developed scheme based on single-atomresolved detection It is the high sensitivity of this method that 3 allowed us to reduce the modulation amplitude by almost an order of magnitude compared 0.16 with earlier experiments 20,21 and to stay well within the linear response regime 0 (Supplementary Information). V 0 = 10E r j/j c = 1.2 Macmillan Publishers Lim e Higgs Manuel mode. Endres, a, The Takeshi fittedfukuhara, gap valuesdavid hn 0 /UPekker, Marc region. Cheneau, b, Temperature Peter Schaub, response Christian to lattice Gross, modulati stic softening Eugene close Demler, to the Stefan critical Kuhr, point and inimmanuel quantitative Bloch, blue Nature line) 487, and454 fit with (2012). an error function (solid black

21 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T Quantum state with complex, many-body, long-range quantum Quantum entanglement critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g U/t

22 Long-range entanglement No quasiparticles - no simple description of excitations. The low energy excitations are described by a theory which has the same structure as Einstein s theory of special relativity, but with the sound velocity playing the role of the velocity of light. The theory of the critical point is strongly-coupled because the quartic-coupling u flows to a renormalization group fixed point (the Wilson-Fisher fixed point). This fixed point has an even larger symmetry corresponding to conformal transformations of spacetime: we refer to such a theory as a CFT3 Characteristics of quantum critical point

23 Long-range entanglement No quasiparticles - no simple description of excitations. The low energy excitations are described by a theory which has the same structure as Einstein s theory of special relativity, but with the sound velocity playing the role of the velocity of light. The theory of the critical point is strongly-coupled because the quartic-coupling u flows to a renormalization group fixed point (the Wilson-Fisher fixed point). This fixed point has an even larger symmetry corresponding to conformal transformations of spacetime: we refer to such a theory as a CFT3 Characteristics of quantum critical point

24 Long-range entanglement No quasiparticles - no simple description of excitations. The low energy excitations are described by a theory which has the same structure as Einstein s theory of special relativity, but with the sound velocity playing the role of the velocity of light. The theory of the critical point is strongly-coupled because the quartic-coupling u flows to a renormalization group fixed point (the Wilson-Fisher fixed point). This fixed point has an even larger symmetry corresponding to conformal transformations of spacetime: we refer to such a theory as a CFT3 Characteristics of quantum critical point

25 Long-range entanglement No quasiparticles - no simple description of excitations. The low energy excitations are described by a theory which has the same structure as Einstein s theory of special relativity, but with the sound velocity playing the role of the velocity of light. The theory of the critical point is strongly-coupled because the quartic-coupling u flows to a renormalization group fixed point (the Wilson-Fisher fixed point). This fixed point has an even larger symmetry corresponding to conformal transformations of spacetime: we refer to such a theory as a CFT3 Characteristics of quantum critical point

26 S = Z d 2 t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T A conformal field theory in 2+1 spacetime dimensions: T KT a CFT3 Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g U/t

27 T Quantum critical T KT Superfluid Insulator 0 g c c g U/t

28 T Boltzmann theory of Nambu- T KT Goldstone-Higgs excitations and vortices 0 Superfluid Quantum critical g c c Insulator Boltzmann theory of particles/holes g U/t

29 CFT3 at T>0 T Quantum critical T KT Superfluid Insulator 0 g c c g U/t

30 CFT3 at T>0 T T KT Quantum No quasiparticles: Boltzmann critical theory does not apply Superfluid Insulator 0 g c c g U/t

31 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

32 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

33 String theory Allows unification of the standard model of particle physics with gravity. Low-lying string modes correspond to gauge fields, gravitons, quarks...

34 A D-brane is a d-dimensional surface on which strings can end. The low-energy theory on a D-brane has no gravity, similar to theories of entangled electrons of interest to us: the strings connecting the particles on the D-brane encode the entanglement In d = 2, we obtain strongly-interacting CFT3s. These are dual to string theory on anti-de Sitter space: AdS4.

35 A D-brane is a d-dimensional surface on which strings can end. The low-energy theory on a D-brane has no gravity, similar to theories of entangled electrons of interest to us: the strings connecting the particles on the D-brane encode the entanglement In d = 2, we obtain strongly-interacting CFT3s. These are dual to string theory on anti-de Sitter space: AdS4.

36 A D-brane is a d-dimensional surface on which strings can end. The low-energy theory on a D-brane has no gravity, similar to theories of entangled electrons of interest to us: the strings connecting the particles on the D-brane encode the entanglement In d = 2, we obtain strongly-interacting CFT3s. These are dual to string theory on anti-de Sitter space: AdS4.

37 AdS/CFT correspondence at zero temperature AdS 4 R 2,1 The symmetry group of isometries of AdS 4 maps to the group of conformal symmetries of the CFT3 Minkowski CFT3 x i zr This emergent spacetime is a solution of Einstein gravity with a negative cosmological constant Z S E = d 4 x p apple R 1 g 2apple 2 + 6L 2

38 Gauge-gravity duality at non-zero temperatures r There is a family of r h solutions of Einstein s equations which are AdS 4 as r 0, but which have horizons at A horizon, similar to the surface of a black hole Z S E = d 4 x p g apple 1 2apple 2 R + 6L 2 r = r h.

39 Gauge-gravity duality at non-zero temperatures r r h A CFT3 at a temperature T 1/r h equal to the Hawking temperature of the horizon. A horizon, similar to the surface of a black hole Z S E = d 4 x p g apple 1 2apple 2 R + 6L 2

40 Gauge-gravity duality at non-zero temperatures r r h A CFT3 at a temperature T 1/r h equal to the Hawking temperature of the horizon. A horizon, similar to the surface of a black hole Dissipation and friction in the CFT3 = waves falling past the horizon

41 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures

42 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

43 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

44 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

45 Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

46 AdS4 theory of quantum criticality Most general e ective holographic theory for linear charge transport with 4 spatial derivatives: S bulk = 1 Z gm 2 d 4 x p apple 1 g 4 F abf ab + L 2 C abcd F ab F cd Z + d 4 x p apple R 1 g 2apple 2 + 6L 2, This action is characterized by 3 dimensionless parameters, which can be linked to data of the CFT (OPE coe cients): 2-point correlators of the conserved current J µ and the stress energy tensor T µ, and a 3-point T, J, J correlator. Constraints from both the CFT and the gravitational theory bound apple 1/12 = R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, (2013)

47 Holography+quantum Monte Carlo 0.50 shiwlêsq (i)/(e 2 /h) g = 0.08 g = wê2pt The holographic theory provides an excellent fit to imaginary-time quantum Monte Carlo on the boson Hubbard model with = 0.08, and we combine these methods to obtain absolute predictions for the frequency-dependent conductivity. W. Witczak-Krempa, E. Sorensen, and S. Sachdev, arxiv: See also K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof ev, arxiv:

48 Holography+quantum Monte Carlo shwlêsq ()/(e 2 /h) wê2pt The holographic theory provides an excellent fit to imaginary-time quantum Monte Carlo on the boson Hubbard model with = 0.08, and we combine these methods to obtain absolute predictions for the frequency-dependent conductivity. W. Witczak-Krempa, E. Sorensen, and S. Sachdev, arxiv: See also K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof ev, arxiv:

49 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

50 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

51 Resistivity 0 + AT Strange Metal no quasiparticles, Landau-Boltzmann theory does not apply BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW +nematic 1.0 Superconductivity 0 Ordinary 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) (Fermi liquid)

52 Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa 2 Cu 3 O y Tao Wu 1, Hadrien Mayaffre 1, Steffen Krämer 1, Mladen Horvatić 1, Claude Berthier 1, W. N. Hardy 2,3, Ruixing Liang 2,3, D. A. Bonn 2,3 & Marc-Henri Julien 1 8 S E P T E M B E R V O L N AT U R E erved Superconducting T (K) 40 Field-induced charge order Spin order p (hole/cu)

53 M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999) M. Vojta and O. Rosch, Physical Review B 77, (2008) M. Metlitski and S. Sachdev, Physical Review B 82, (2010) S. Sachdev and R. La Placa, Physical Review Letters 111, (2013) A. Allais, J. Bauer, and S. Sachdev, to appear , (2010) 80 Superconducting T (K) 40 d-wave bond Field-induced charge order Spin order order p (hole/cu)

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

55 Quantum phase transition with onset of antiferromagnetism in a metal ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

56 Quantum phase transition with onset of antiferromagnetism in a metal ncreasing SDW Find new instabilities h~' i6=0 upon approaching h~' i =0 critical point Metal with electron and hole pockets Metal with large Fermi surface r

57 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. Possible quantum critical point at optimal doping involving onset of Ising-nematic order = x 2 y 2 V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986) Ar. Abanov, A.V. Chubukov, and A.M. Finkel'stein, Europhys. Lett. 54, 488 (2001) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, (2010)

58 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond _ order, x, y X-ray6 scattering in underdoped YBCO is described by P ± (x max ) 10 x angular thermal fluctuations of a 6-component order 4 parameter,, x, y. 2 P + _ P _ r c L = 14 L = 10 L = 12 Possible 0 quantum critical point at optimal Science 338, doping 1606 (2012). involving 2 onset of Ising-nematic order = x 2 y r E. Berg, M. Metlitski, and S. Sachdev, Sign-problem-free Quantum Monte Carlo for antiferromagnetism in metals

59 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999) M. Metlitski and S. Sachdev, Physical Review B 82, (2010) S. Sachdev and R. La Placa, Physical Review Letters 111, (2013) Possible M. Vojta and O. quantum Rosch, Physical Review critical B 77, (2008) point at optimal doping involving onset of Ising-nematic order = x 2 y 2 A. Allais, J. Bauer, and S. Sachdev, to appear 1 1

60 1 1 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999) Possible M. Vojta and O. quantum Rosch, Physical Review critical B 77, (2008) point at optimal doping involving M. Metlitski and S. Sachdev, Physical Review B 82, (2010) S. Sachdev and R. onset La Placa, Physical of Review Ising-nematic Letters 111, (2013) order = x 2 y 2 A. Allais, J. Bauer, and S. Sachdev, to appear 1 1

61 An Intrinsic Bond-Centered Electronic Glass with Unidirectional Domains in Underdoped Cuprates Y. Kohsaka, 1 C. Taylor, 1 K. Fujita, 1,2 A. Schmidt, 1 C. Lupien, 3 T. Hanaguri, 4 M. Azuma, 5 M. Takano, 5 H. Eisaki, 6 H. Takagi, 2,4 S. Uchida, 2,7 J. C. Davis 1,8 * 9 MARCH 2007 VOL 315 SCIENCE Phys. Rev. Lett. 109, (2012). Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered YBa 2 Cu 3 O 6þ Superconductors Identified by Resonant Elastic X-ray Scattering A. J. Achkar, 1 R. Sutarto, 2,3 X. Mao, 1 F. He, 3 A. Frano, 4,5 S. Blanco-Canosa, 4 M. Le Tacon, 4 G. Ghiringhelli, 6 L. Braicovich, 6 M. Minola, 6 M. Moretti Sala, 7 C. Mazzoli, 6 Ruixing Liang, 2 D. A. Bonn, 2 W. N. Hardy, 2 B. Keimer, 4 G. A. Sawatzky, 2 and D. G. Hawthorn 1, * 1 also stripes in 214 cuprates). In such a case, the energy shifts may in fact be a signature of a novel electronic state, such as a valence bond solid [17]. Alternately, the energy

62 1 1 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999) Possible M. Vojta and O. quantum Rosch, Physical Review critical B 77, (2008) point at optimal doping involving M. Metlitski and S. Sachdev, Physical Review B 82, (2010) S. Sachdev and R. onset La Placa, Physical of Review Ising-nematic Letters 111, (2013) order = x 2 y 2 A. Allais, J. Bauer, and S. Sachdev, to appear 1 1

63 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. Possible quantum critical point at optimal doping involving onset of Ising-nematic order = x 2 y 2 L. E. Hayward, D. G. Hawthorn, R. G. Melko, and S. Sachdev, arxiv:

64 Comparison of Monte Carlo with experiments Z S x = d 2 r h 3.0 x (r) x (0)i Charge order structure factor S x S x /a YBCO ga 2 = 0.2 ga 2 = 0.25 ga 2 = 0.3 ga 2 = 0.35 ga 2 = L = 64, wa 2 =0.0, = T (K) For ga 2 =0.30 and wa 2 =0.0 wehave s = 160K. The height was also rescaled to make the peak heights match. L. E. Hayward, D. G. Hawthorn, R. G. Melko, and S. Sachdev, arxiv:

65 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. Possible quantum critical point at optimal doping involving onset of Ising-nematic order = x 2 y 2 L. E. Hayward, D. G. Hawthorn, R. G. Melko, and S. Sachdev, arxiv:

66 New physics in metals with antiferromagnetic correlation Weak-coupling instability to d-wave superconductivity,. Superconductivity survives at strong-coupling with critical antiferromagnetism Secondary instability to incommensurate d-wave bond order, x, y X-ray scattering in underdoped YBCO is described by angular thermal fluctuations of a 6-component order parameter,, x, y. Possible quantum critical point at optimal doping involving onset of Ising-nematic order = x 2 y 2

67 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

68 Outline 1. The simplest model without quasiparticles A. Superfluid-insulator transition of ultracold bosonic atoms in an optical lattice B. Conformal field theories in 2+1 dimensions, the AdS/CFT correspondence, and transport without quasiparticles. 2. Metals without quasiparticles A. The onset of antiferromagnetism in a metal B. Non-quasiparticle transport at the Ising-nematic quantum critical point

69 c T (K) S. Kasahara, H.J. Shi, K. Hashimoto, S. Tonegawa, Y. Mizukami, T. Shibauchi, K. Sugimoto, T. Fukuda, T. Terashima, A.H. Nevidomskyy, and d Y. Matsuda, Nature 486, 382 (2012) a M Paramagnetic (tetragonal) (010) T T * Electronic nematic H b (100) Fe As/P T T > T* T < T* (a.u.) 2 (a.u.) 35 K 6 Antiferromagnetic (orthorhombic) Superconducting x BaFe 2 (As 1 x P x ) 2 a Torque = 0 M H 0 a 4 (a.u.) c Single 0crystal 18 (deg.) (d M H b

70 Visualization of the emergence of the pseudogap state and the evolution to superconductivity in a lightly hole-doped Mott insulator Y. Kohsaka, T. Hanaguri, M. Azuma, M. Takano, J. C. Davis, and H. Takagi T > T Nature Physics, 8, 534 (2012). c c Y X 2 nm Cu 500 g 2 nm - 2 n Q xx (R), Ca 1.88 Na 0.12 CuO 2 Cl h O x 400 O y 500 Evidence for nematic order (i.e. breaking of 90 rotation symmetry) in Ca 1.88 Na 0.12 CuO 2 Cl 2. 2 nm O O(2) O

71 Quantum criticality of Ising-nematic ordering in a metal y Occupied states x Empty states A metal with a Fermi surface with full square lattice symmetry

72 Quantum criticality of Ising-nematic ordering in a metal or =0 =0 c Pomeranchuk instability as a function of coupling

73 Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n Fermi Fermi liquid =0 =0 liquid c Phase diagram as a function of T and

74 Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and

75 Quantum criticality of Ising-nematic ordering in a metal T Strange Metal TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and

76 Quantum criticality of Ising-nematic ordering in a metal T Strange Metal TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Common theoretical belief from an analysis of scattering of charged electronic quasiparticles o bosonic fluctuations: resistivity of strange metal (T ) T 4/3.

77 Quantum criticality of Ising-nematic ordering in a metal The standard model : Z S = d 2 rd (@ ) 2 + c 2 (r ) 2 +( c) 2 + u 4 S c = N f d c k ( + k ) c k Field theory of S c = =1 k Z g d N f X =1 X k,q bosonic order parameter q (cos k x cos k y )c k+q/2, c k q/2,

78 Quantum criticality of Ising-nematic ordering in a metal The standard model : Z S = d 2 rd (@ ) 2 + c 2 (r ) 2 +( c) 2 + u 4 S c = S c = N f =1 k Z g d N f X =1 d c k X k,q ( + k ) c k Electrons with a Fermi surface: " k = 2t(cos k x + cos k y ) µ... q (cos k x cos k y )c k+q/2, c k q/2,

79 Quantum criticality of Ising-nematic ordering in a metal The standard model : S = S c = S c = Z N f g Z d 2 rd (@ ) 2 + c 2 (r ) 2 Yukawa +( c) 2 + u 4 coupling =1 k d N f X =1 d c k X k,q ( + k ) c k between bosons and fermions q (cos k x cos k y )c k+q/2, c k q/2,

80 Quantum criticality of Ising-nematic ordering in a metal The standard model : Z S = d 2 rd (@ ) 2 + c 2 (r ) 2 +( c) 2 + u 4 S c = N f d c k ( + k ) c k S c = =1 k Z g d N f X =1 X k,q q (cos k x cos k y )c k+q/2, c k q/2,

81 Quantum criticality of Ising-nematic ordering in a metal The standard model : S = S c = S c = g Z N f X =1 Z d 2 rd (@ ) 2 + c 2 (r ) 2 +( c) 2 + u 4 Z d 2 rd d 2 rd c N f X r 2 2m µ This continuum theory has a conserved momentum P, and J,P 6= 0, and so the resistivity (T ) = 0. The resistivity of the strange metal is not determined by the scattering rate of charged excitations near the Fermi surface, but by the dominant rate of momentum loss by any excitation, whether neutral or charged, or fermionic or bosonic c c (@ 2 2 y)c + (@ 2 2 y)c c

82 Quantum criticality of Ising-nematic ordering in a metal The standard model : S = S c = S c = g Z N f X =1 Z d 2 rd (@ ) 2 + c 2 (r ) 2 +( c) 2 + u 4 Z d 2 rd d 2 rd c N f X r 2 2m µ c c (@ 2 2 y)c + (@ 2 2 y)c c This continuum theory has a conserved momentum P, and J,P 6= 0, and so the resistivity (T ) = 0. The resistivity of the strange metal is not determined by the scattering rate of charged excitations near the Fermi surface, but by the dominant rate of momentum loss by any excitation, whether neutral or charged, or fermionic or bosonic

83 Resistivity of strange metal In the presence of weak disorder of quenched Gaussian random fields Z S dis = d 2 rd V (r) c c + h(r), V (r) =0 ; V (r)v (r 0 )=V0 2 (r r 0 ), h(r) =0 ; h(r)h(r 0 )=h 2 0 (r r 0 ), we obtain the resistivity for current along angle # (T )= 1 Z 2 lim 0 J,P d 2 k (2 ) 2 k2 cos 2 ( k #) V0 2 Im R c c (, k) + h 2 0 Im D R (, k) S. A. Hartnoll, R. Mahajan, M. Punk and S. Sachdev, arxiv:

84 Resistivity of strange metal In the presence of weak disorder of quenched Gaussian random fields Z S dis = d 2 rd V (r) c c + h(r), V (r) =0 ; V (r)v (r 0 )=V0 2 (r r 0 ), h(r) =0 ; h(r)h(r 0 )=h 2 0 (r r 0 ), we obtain the resistivity for current along angle # (T )= 1 Z 2 lim 0 J,P d 2 k (2 ) 2 k2 cos 2 ( k #) V0 2 Im R c c (, k) + h 2 0 Im D R (, k) Fermi surface term: Obtain T -dependent corrections to residual resistivity similar to earlier work G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64, (2001) I. Paul, C. Pépin, B. N. Narozhny, and D. L. Maslov, Phys. Rev. Lett. 95, (2005). S. A. Hartnoll, R. Mahajan, M. Punk and S. Sachdev, arxiv:

85 Resistivity of strange metal In the presence of weak disorder of quenched Gaussian random fields Z S dis = d 2 rd V (r) c c + h(r), V (r) =0 ; V (r)v (r 0 )=V0 2 (r r 0 ), h(r) =0 ; h(r)h(r 0 )=h 2 0 (r r 0 ), we obtain the resistivity for current along angle # (T )= 1 Z 2 lim 0 J,P d 2 k (2 ) 2 k2 cos 2 ( k #) V0 2 Bosonic term: Dominant contribution: Im R c c (, k) + h 2 0 Im D R (, k) (T ) T (d z+ )/z Crosses over from the relativistic form (z = 1, 0) with (T ) T at higher T, to the Landau-damped form (z = 3, = 0) with (T ) (T ln(1/t )) 1/2 at lower T (subtle corrections to scaling specific to this field theory). S. A. Hartnoll, R. Mahajan, M. Punk and S. Sachdev, arxiv:

86 Resistivity of strange metal In the presence of weak disorder of quenched Gaussian random fields Z S dis = d 2 rd V (r) c c + h(r), V (r) =0 ; V (r)v (r 0 )=V0 2 (r r 0 ), h(r) =0 ; h(r)h(r 0 )=h 2 0 (r r 0 ), we obtain the resistivity for current along angle # (T )= 1 Z 2 lim 0 J,P d 2 k (2 ) 2 k2 cos 2 ( k #) V0 2 Bosonic term: Dominant contribution: Im R c c (, k) + h 2 0 Im D R (, k) (T ) T (d z+ )/z Also obtained in holographic theory of a generalized compressible quantum state (A. Lucas, S. Sachdev, and K. Schalm, arxiv: ). S. A. Hartnoll, R. Mahajan, M. Punk and S. Sachdev, arxiv:

87 Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. Quantitative predictions for transport by combining quantum Monte Carlo and holography. Metals with antiferromagnetic correlations have d-wave superconducting and bond order instabilities similar to observations. Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography.

88 Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. Quantitative predictions for transport by combining quantum Monte Carlo and holography. Metals with antiferromagnetic correlations have d-wave superconducting and bond order instabilities similar to observations. Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography.

89 Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. Quantitative predictions for transport by combining quantum Monte Carlo and holography. Metals with antiferromagnetic correlations have d-wave superconducting and bond order instabilities similar to observations. Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography.

90 Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. Quantitative predictions for transport by combining quantum Monte Carlo and holography. Metals with antiferromagnetic correlations have d-wave superconducting and bond order instabilities similar to observations. Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography.