Electronic Liquid Crystal Phases in Strongly Correlated Systems

Transcription

1 Electronic Liquid Crystal Phases in Strongly Correlated Systems Lectures at the Les Houches Summer School, May 2009 Eduardo Fradkin Department of Physics University of Illinois at Urbana Champaign May 29, 2009

2 Les Houches, July 1982: Volver...Que veinte años no es nada... (C. Gardel et al, 1930)

3 Outline

4 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics

5 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors

6 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations)

7 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations) The Electron nematic/smectic quantum phase transition

8 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations) The Electron nematic/smectic quantum phase transition Nematic order and time reversal symmetry breaking

9 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations) The Electron nematic/smectic quantum phase transition Nematic order and time reversal symmetry breaking ELCs in Microscopic Models

10 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations) The Electron nematic/smectic quantum phase transition Nematic order and time reversal symmetry breaking ELCs in Microscopic Models ELC phases and the mechanism of high temperature superconductivity: Optimal Inhomogeneity

11 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations) The Electron nematic/smectic quantum phase transition Nematic order and time reversal symmetry breaking ELCs in Microscopic Models ELC phases and the mechanism of high temperature superconductivity: Optimal Inhomogeneity The Pair Density Wave phase

12 Outline Electronic Liquid Crystals Phases: symmetries and order parameters, strong coupling vs weak coupling physics Experimental Evidence in 2DEG, Sr 3 Ru 2 O 7, High Temperature Superconductors Theories of nematic phases in Fermi systems (and generalizations) The Electron nematic/smectic quantum phase transition Nematic order and time reversal symmetry breaking ELCs in Microscopic Models ELC phases and the mechanism of high temperature superconductivity: Optimal Inhomogeneity The Pair Density Wave phase Outlook

13 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998)

14 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998) Doping a Mott insulator: inhomogeneous phases due to the competition between phase separation and strong correlations

15 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998) Doping a Mott insulator: inhomogeneous phases due to the competition between phase separation and strong correlations Crystal Phases: break all continuous translation symmetries and rotations

16 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998) Doping a Mott insulator: inhomogeneous phases due to the competition between phase separation and strong correlations Crystal Phases: break all continuous translation symmetries and rotations Smectic (Stripe) phases: break one translation symmetry and rotations

17 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998) Doping a Mott insulator: inhomogeneous phases due to the competition between phase separation and strong correlations Crystal Phases: break all continuous translation symmetries and rotations Smectic (Stripe) phases: break one translation symmetry and rotations Nematic and Hexatic Phases: are uniform and anisotropic

18 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998) Doping a Mott insulator: inhomogeneous phases due to the competition between phase separation and strong correlations Crystal Phases: break all continuous translation symmetries and rotations Smectic (Stripe) phases: break one translation symmetry and rotations Nematic and Hexatic Phases: are uniform and anisotropic Uniform fluids: break no spatial symmetries

19 Electron Liquid Crystal Phases S. Kivelson, E. Fradkin, V. Emery, Nature 393, 550 (1998) Doping a Mott insulator: inhomogeneous phases due to the competition between phase separation and strong correlations Crystal Phases: break all continuous translation symmetries and rotations Smectic (Stripe) phases: break one translation symmetry and rotations Nematic and Hexatic Phases: are uniform and anisotropic Uniform fluids: break no spatial symmetries

20 Electronic Liquid Crystal Phases in Strongly Correlated Systems

21 Electronic Liquid Crystal Phases in Strongly Correlated Systems Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a superconductor

22 Electronic Liquid Crystal Phases in Strongly Correlated Systems Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a superconductor Nematic: Lattice effects reduce the symmetry to a rotations by π/2 ( Ising ); translation and reflection symmetries are unbroken; it is an anisotropic liquid with a preferred axis

23 Electronic Liquid Crystal Phases in Strongly Correlated Systems Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a superconductor Nematic: Lattice effects reduce the symmetry to a rotations by π/2 ( Ising ); translation and reflection symmetries are unbroken; it is an anisotropic liquid with a preferred axis Smectic: breaks translation symmetry only in one direction but liquid-like on the other; Stripe phase; (infinite) anisotropy of conductivity tensor

24 Electronic Liquid Crystal Phases in Strongly Correlated Systems Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a superconductor Nematic: Lattice effects reduce the symmetry to a rotations by π/2 ( Ising ); translation and reflection symmetries are unbroken; it is an anisotropic liquid with a preferred axis Smectic: breaks translation symmetry only in one direction but liquid-like on the other; Stripe phase; (infinite) anisotropy of conductivity tensor Crystal(s): electron solids ( CDW ); insulating states.

25 Electronic Liquid Crystal Phases in Strongly Correlated Systems Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a superconductor Nematic: Lattice effects reduce the symmetry to a rotations by π/2 ( Ising ); translation and reflection symmetries are unbroken; it is an anisotropic liquid with a preferred axis Smectic: breaks translation symmetry only in one direction but liquid-like on the other; Stripe phase; (infinite) anisotropy of conductivity tensor Crystal(s): electron solids ( CDW ); insulating states.

26 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors

27 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates

28 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates Anisotropic transport in 2DEG in large magnetic fields and in Sr 3Ru 2O 7

29 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates Anisotropic transport in 2DEG in large magnetic fields and in Sr 3Ru 2O 7 Common underlying physical mechanism:

30 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates Anisotropic transport in 2DEG in large magnetic fields and in Sr 3Ru 2O 7 Common underlying physical mechanism: { effective short range attractive forces Competition long(er) range repulsive (Coulomb) interactions Frustrated phase separation spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al)

31 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates Anisotropic transport in 2DEG in large magnetic fields and in Sr 3Ru 2O 7 Common underlying physical mechanism: { effective short range attractive forces Competition long(er) range repulsive (Coulomb) interactions Frustrated phase separation spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al) Examples in classical systems: blockcopolymers, ferrofluids, etc.

32 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates Anisotropic transport in 2DEG in large magnetic fields and in Sr 3Ru 2O 7 Common underlying physical mechanism: { effective short range attractive forces Competition long(er) range repulsive (Coulomb) interactions Frustrated phase separation spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al) Examples in classical systems: blockcopolymers, ferrofluids, etc. Astrophysical examples: Pasta Phases (meatballs, spaghetti and lasagna!) of neutron stars lightly doped with protons (G. Ravenhall et al,1983)

33 Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. Stripe and Nematic phases of cuprate superconductors Stripe phases of manganites and nickelates Anisotropic transport in 2DEG in large magnetic fields and in Sr 3Ru 2O 7 Common underlying physical mechanism: { effective short range attractive forces Competition long(er) range repulsive (Coulomb) interactions Frustrated phase separation spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al) Examples in classical systems: blockcopolymers, ferrofluids, etc. Astrophysical examples: Pasta Phases (meatballs, spaghetti and lasagna!) of neutron stars lightly doped with protons (G. Ravenhall et al,1983) Analogues in lipid bilayers intercalated with DNA (Lubensky et al, 2000)

34 Soft Quantum Matter or Quantum Soft Matter

35 Electron Liquid Crystal Phases Crystal Nematic Smectic Isotropic

36 Schematic Phase Diagram of Doped Mott Insulators Isotropic (Disordered) Temperature Nematic Crystal Superconducting Smectic C 3 hω C 2 C 1 ω measures transverse zero-point stripe fluctuations of the stripes. Systems with large coupling to lattice displacements (e. g. manganites) are more classical than systems with primarily electronic correlations (e. g. cuprates); nickelates lie in-between.

37 Phase Diagram of the High T c Superconductors T antiferromagnet pseudogap bad metal superconductor 1 8 x Full lines: phase boundaries for the antiferromagnetic and superconducting phases. Broken line: phase boundary for a system with static stripe order and a 1/8 anomaly Dotted line: crossover between the bad metal and pseudogap regimes

38 Order Parameter for Charge Smectic (Stripe) Ordered States

39 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW)

40 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe

41 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe if it coexists with spin order spin stripe

42 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ±Q ch = ± 2π λ ch ê x

43 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ±Q ch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Q spin = (π, π) ± 1 2 Q ch

44 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ±Q ch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Q spin = (π, π) ± 1 2 Q ch Charge Order Parameter: n Qch, Fourier component of the electron density at Q ch.

45 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ±Q ch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Q spin = (π, π) ± 1 2 Q ch Charge Order Parameter: n Qch, Fourier component of the electron density at Q ch. Spin Order Parameter: S Qspin, Fourier component of the electron density at Q spin.

46 Order Parameter for Charge Smectic (Stripe) Ordered States unidirectional charge density wave (CDW) charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ±Q ch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Q spin = (π, π) ± 1 2 Q ch Charge Order Parameter: n Qch, Fourier component of the electron density at Q ch. Spin Order Parameter: S Qspin, Fourier component of the electron density at Q spin.

47 Nematic Order

48 Nematic Order Translationally invariant state with broken rotational symmetry

49 Nematic Order Translationally invariant state with broken rotational symmetry Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor

50 Nematic Order Translationally invariant state with broken rotational symmetry Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor Order parameter: a director, a headless vector

51 Nematic Order Translationally invariant state with broken rotational symmetry Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor Order parameter: a director, a headless vector In D = 2 one can use the static structure factor

52 Nematic Order Translationally invariant state with broken rotational symmetry Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor Order parameter: a director, a headless vector In D = 2 one can use the static structure factor to construct S( k) = Z dω 2π S( k, ω)

53 Nematic Order Translationally invariant state with broken rotational symmetry Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor Order parameter: a director, a headless vector In D = 2 one can use the static structure factor to construct S( k) = Z dω 2π S( k, ω) Q k = S( k) S(R k) S( k) + S(R k) where S(k, ω) is the dynamic structure factor, the dynamic (charge-density) correlation function, and R = rotation by π/2.

54 Nematic Order Translationally invariant state with broken rotational symmetry Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor Order parameter: a director, a headless vector In D = 2 one can use the static structure factor to construct S( k) = Z dω 2π S( k, ω) Q k = S( k) S(R k) S( k) + S(R k) where S(k, ω) is the dynamic structure factor, the dynamic (charge-density) correlation function, and R = rotation by π/2.

55 Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q

56 Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q ( ) ρxx ρ Q ij = yy ρ xy ρ xy ρ yy ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N,

57 Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q ( ) ρxx ρ Q ij = yy ρ xy ρ xy ρ yy ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N, N = Q xx + iq xy = N e iϕ Under a rotation by a fixed angle θ, N transforms as

58 Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q ( ) ρxx ρ Q ij = yy ρ xy ρ xy ρ yy ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N, N = Q xx + iq xy = N e iϕ Under a rotation by a fixed angle θ, N transforms as N N e i2θ Hence, it changes sign under a rotation by π/2 and it is invariant under a rotation by π. On the other hand, it is invariant under uniform translations by R.

59 Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q ( ) ρxx ρ Q ij = yy ρ xy ρ xy ρ yy ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N, N = Q xx + iq xy = N e iϕ Under a rotation by a fixed angle θ, N transforms as N N e i2θ Hence, it changes sign under a rotation by π/2 and it is invariant under a rotation by π. On the other hand, it is invariant under uniform translations by R.

60 Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 hω c B 3/2 hω c 1/2 hωc E F Angular Momentum bulk edge Al As Ga As heterostructure Electrons in Landau levels are strongly correlated systems, K.E.= 0

61 Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 hω c B 3/2 hω c 1/2 hωc E F Angular Momentum bulk edge Al As Ga As heterostructure Electrons in Landau levels are strongly correlated systems, K.E.= 0 For N = 0, 1 Fractional (and integer) Quantum Hall Effects

62 Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 hω c B 3/2 hω c 1/2 hωc E F Angular Momentum bulk edge Al As Ga As heterostructure Electrons in Landau levels are strongly correlated systems, K.E.= 0 For N = 0, 1 Fractional (and integer) Quantum Hall Effects Integer QH states for N 2

63 Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 hω c B 3/2 hω c 1/2 hωc E F Angular Momentum bulk edge Al As Ga As heterostructure Electrons in Landau levels are strongly correlated systems, K.E.= 0 For N = 0, 1 Fractional (and integer) Quantum Hall Effects Integer QH states for N 2 Hartree-Fock predicts stripe phases for large N (Koulakov et al, Moessner and Chalker (1996))

64 Transport Anisotropy in the 2DEG M. P. Lilly et al (1999), R. R. Du et al (1999)

65 Transport Anisotropy in the 2DEG M. P. Lilly et al (1999), R. R. Du et al (1999)

66 Transport Anisotropy in the 2DEG K. B. Cooper et al (2002)

67 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition!

68 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state?

69 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect

70 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect The anisotropy is finite as T 0

71 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect The anisotropy is finite as T 0 The anisotropy has a pronounced temperature dependence it is an ordering effect

72 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect The anisotropy is finite as T 0 The anisotropy has a pronounced temperature dependence it is an ordering effect I V curves are linear (at low V) and no threshold electric field no pinning

73 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect The anisotropy is finite as T 0 The anisotropy has a pronounced temperature dependence it is an ordering effect I V curves are linear (at low V) and no threshold electric field no pinning No broad-band noise is observed in the peak region

74 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect The anisotropy is finite as T 0 The anisotropy has a pronounced temperature dependence it is an ordering effect I V curves are linear (at low V) and no threshold electric field no pinning No broad-band noise is observed in the peak region The 2DEG behaves as a uniform anisotropic fluid: it is a nematic charged fluid

75 Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? The effects gets bigger in cleaner systems it is a correlation effect The anisotropy is finite as T 0 The anisotropy has a pronounced temperature dependence it is an ordering effect I V curves are linear (at low V) and no threshold electric field no pinning No broad-band noise is observed in the peak region The 2DEG behaves as a uniform anisotropic fluid: it is a nematic charged fluid

76 Transport Anisotropy in the 2DEG The 2DEG behaves like a Nematic fluid! Classical Monte Carlo simulation of a classical 2D XY model for nematic order with coupling J and external field h, on a lattice Fit of the order parameter to the data of M. Lilly and coworkers, at ν = 9/2 (after deconvoluting the effects of the geometry.) Best fit: J = 73mK and h = 0.05J = 3.5mK and T c = 65mK. E. Fradkin, S. A. Kivelson, E. Manousakis and K. Nho, Phys. Rev. Lett. 84, 1982 (2000). K. B. Cooper et al., Phys. Rev. B 65, (2002)

77 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide

78 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide Sr 2 Ru 1 O 4 is a quasi-2d single layer correlated oxide and a low T c superconductor (p x + ip y?)

79 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide Sr 2 Ru 1 O 4 is a quasi-2d single layer correlated oxide and a low T c superconductor (p x + ip y?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at low fields

80 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide Sr 2 Ru 1 O 4 is a quasi-2d single layer correlated oxide and a low T c superconductor (p x + ip y?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at low fields It is a bad metal (linear resistivity over a large temperature range) except at the lowest temperatures

81 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide Sr 2 Ru 1 O 4 is a quasi-2d single layer correlated oxide and a low T c superconductor (p x + ip y?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at low fields It is a bad metal (linear resistivity over a large temperature range) except at the lowest temperatures Clean samples seemed to suggest a field tuned quantum critical end-point

82 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide Sr 2 Ru 1 O 4 is a quasi-2d single layer correlated oxide and a low T c superconductor (p x + ip y?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at low fields It is a bad metal (linear resistivity over a large temperature range) except at the lowest temperatures Clean samples seemed to suggest a field tuned quantum critical end-point Ultra-clean samples find instead a new phase with spontaneous transport anisotropy for a narrow range of fields

83 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Sr 3 Ru 2 O 7 is a strongly correlated quasi-2d bilayer oxide Sr 2 Ru 1 O 4 is a quasi-2d single layer correlated oxide and a low T c superconductor (p x + ip y?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at low fields It is a bad metal (linear resistivity over a large temperature range) except at the lowest temperatures Clean samples seemed to suggest a field tuned quantum critical end-point Ultra-clean samples find instead a new phase with spontaneous transport anisotropy for a narrow range of fields

84 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Phase diagram of Sr 3Ru 2O 7 in the temperature-magnetic field plane. (from Grigera et al (2004).

85 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields R. Borzi et al (2007)

86 Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields R. Borzi et al (2007)

87 Charge and Spin Order in the Cuprate Superconductors Stripe charge order in underdoped high temperature superconductors(la 2 x Sr x CuO 4, La 1.6 x Nd 0.4 Sr x CuO 4 and YBa 2 Cu 3 O 6+x ) (Tranquada, Ando, Mook, Keimer) Coexistence of fluctuating stripe charge order and superconductivity in La 2 x Sr x CuO 4 and YBa 2 Cu 3 O 6+x (Mook, Tranquada) and nematic order (Keimer). Dynamical layer decoupling in stripe ordered La 2 x Ba x CuO 4 and in La 2 x Sr x CuO 4 at finite fields (transport, (Tranquada et al (2007)), Josephson resonance (Basov et al (2009))) Induced charge order in the SC phase in vortex halos in La 2 x Sr x CuO 4 and underdoped YBa 2 Cu 3 O 6+x (neutrons: B. Lake, Keimer; STM: Davis) STM Experiments: short range stripe order (on scales long compared to ξ 0 ), possible broken rotational symmetry (Bi 2 Sr 2 CaCu 2 O 8+δ ) (Kapitulnik, Davis, Yazdani) Transport experiments give evidence for charge domain switching in YBa 2 Cu 3 O 6+x wires (Van Harlingen/Weissmann)

88 Charge and Spin Order in the Cuprate Superconductors Static spin stripe order in La 2 x Ba x CuO 4 near x = 1/8 in neutron scattering (Fujita et al (2004))

89 Charge and Spin Order in the Cuprate Superconductors Static charge stripe order in La 2 x Ba x CuO 4 near x = 1/8 in resonant X-ray scattering (Abbamonte et. al.(2005))

90 Induced stripe order in La 2 x Sr x CuO 4 by Zn impurities 300 La 1.86Sr 0.14Cu 0.988Zn 0.012O 4 La 1.85Sr 0.15CuO 4 I 7K (arb. units) (a) E = 1.5 mev (c) E = 2 mev Intensity (arb. units) I 7K -I 80K (arb. units) (b) E = 0 (d) E = (0.5+h,0.5,0) T = 1.5 K T = 50 K (0.5+h,0.5,0) Intensity (arb. units) Magnetic neutron scattering with and without Zn (Kivelson et al (2003))

91 Electron Nematic Order in High Temperature Superconductors (i) (j) Temperature-dependent transport anisotropy in underdoped La 2 x Sr xcuo 4 and YBa 2 Cu 3 O 6+x ; Ando et al (2002)

92 Charge Nematic Order in underdoped YBa 2 Cu 3 O 6+x (y = 6.45) Intensity maps of the spin-excitation spectrum at 3, 7,and 50 mev, respectively. Colormap of the intensity at 3 mev, as it would be observed in a crystal consisting of two perpendicular twin domains with equal population. Scans along a and b through Q AF. (from Hinkov et al (2007).

93 Charge Nematic Order in underdoped YBa 2 Cu 3 O 6+x (y = 6.45) a) Incommensurability δ (red symbols), half-width-at-half-maximum of the incommensurate peaks along a (ξa 1, black symbols) and along b (ξ 1 b, open blue symbols) in reciprocal lattice units. (from Hinkov et al 2007)).

94 Static stripe order in underdoped YBa 2 Cu 3 O 6+x at finite fields Hinkov et al (2008)

95 Charge Order induced inside a SC vortex halo Induced charge order in the SC phase in vortex halos: neutrons in La 2 x Sr x CuO 4 (B. Lake et al, 2002), STM in optimally doped Bi 2 Sr 2 CaCu 2 O 8+δ (S. Davis et al, 2004)

96 STM: Short range stripe order in Dy-Bi 2 Sr 2 CaCu 2 O 8+δ (k) Left (l) Right Left: STM R-maps in Dy-Bi 2 Sr 2 CaCu 2 O 8+δ at high bias: R( r, 150 mv) = I( r, +150mV)/I( r, 150 mv) Right: Short range nematic order with ξ 0 ; Kohsaka et al (2007)

97 Optimal Degree of Inhomogeneity in La 2 x Ba x CuO 4 ARPES in La 2 x Ba xcuo 4 : the antinodal (pairing) gap is largest, even though T c is lowest, near x = 1 ; T. Valla et al (2006), ZX Shen et al (2008) 8

98 Dynamical Layer Decoupling in La 2 x Ba x CuO 4 near x = 1/8 Dynamical layer decoupling in transport (Li et al (2008))

99 Dynamical Layer Decoupling in La 2 x Ba x CuO 4 near x = 1/8 Kosterlitz-Thouless resistive transition (Li et al (2008)

100 Dynamical Layer Decoupling in La 2 x Sr x CuO 4 in a magnetic field Layer decoupling seen in Josephson resonance spectroscopy: c-axis penetration depth λ c vs c-axis conductivity σ 1c (Basov et al, 2009)

101 Theory of the Nematic Fermi Fluid V. Oganesyan, S. A. Kivelson, and E. Fradkin (2001) The nematic order parameter for two-dimensional Fermi fluid is the 2 2 symmetric traceless tensor ˆQ(x) 1 «Ψ 2 ( r) x y 2 2 x y kf 2 2 x y y 2 x 2 Ψ( r), It can also be represented by a complex valued field Q whose expectation value is the nematic phase is Q Ψ ( x + i y) 2 Ψ = Q e 2iθ = Q 11 + iq 12 0 Q carries angular momentum l = 2. Q 0 the Fermi surface spontaneously distorts and becomes an ellipse with eccentricity Q This state breaks rotational invariance mod π

102 Theory of the Nematic Fermi Fluid A Fermi liquid model: Landau on Landau H = d r Ψ ( r)ǫ( )Ψ( r) d r d r F 2 ( r r )Tr[ˆQ( r)ˆq( r )] ǫ( k) = v F q[1 + a( q k F ) 2 ], q k k F F 2 ( r) = (2π) 2 d ke i q r F 2 /[1 + κf 2 q 2 ] F 2 is a Landau parameter. Landau energy density functional: V[Q] = E(Q) κ κ Tr[QDQ] 4 4 Tr[Q2 DQ] +... E(Q) = E(0) + A 4 Tr[Q2 ] + B 8 Tr[Q4 ] +... where A = 1 2N F + F 2, N F is the density of states at the Fermi surface, B = 3aN F F 2 3, and E 8EF 2 F v F k F is the Fermi energy. If A < 0 nematic phase

103 Theory of the Nematic Fermi Fluid This model has two phases: an isotropic Fermi liquid phase a nematic non-fermi liquid phase separated by a quantum critical point at 2N F F 2 = 1 Quantum Critical Behavior Parametrize the distance to the Pomeranchuk QCP by δ = 1/2 1/N F F 2 and define s = ω/qv F The transverse collective nematic modes have Landau damping at the QCP. Their effective action has a kernel κq 2 ω + δ i qv F The dynamic critical exponent is z = 3.

104 Theory of the Nematic Fermi Fluid Physics of the Nematic Phase: Transverse Goldstone boson which is generically overdamped except for φ = 0, ±π/4, ±π/2 (symmetry directions) where it is underdamped Anisotropic (Drude) Transport ρ xx ρ yy = 1 m y m x = Re Q + O(Q 3 ) ρ xx + ρ yy 2 m y + m x E F Quasiparticle scattering rate (one loop); In general Σ (ǫ, k) = π (κkf) 2 1/3 kxky 3 κn F Along a symmetry direction: Σ (ǫ) = π 3N F κ 1 k 2 F (κk 2 F )1/4 The Nematic Phase is a non-fermi Liquid! 4/3 ǫ v F k F ǫ 2v F k F 3/2 2/

105 Local Quantum Criticality at the Nematic QCP Since Σ (ω) Σ (ω) (as ω 0), we need to asses the validity of these results as they signal a failure of perturbation theory We use higher dimensional bosonization as a non-perturbative tool (Haldane 1993, Castro Neto and Fradkin 1993, Houghton and Marston 1993) Higher dimensional bosonization reproduces the collective modes found in Hartree-Fock+ RPA and is consistent with the Hertz-Millis analysis of quantum criticality: d eff = d + z = 5. (Lawler et al, 2006) The fermion propagator takes the form G F (x,t) = G 0(x,t)Z(x,t)

106 Local Quantum Criticality at the Nematic QCP At the Nematic-FL QCP: G F (x,0) = G 0(x,0) e const. x 1/3, G F (0,t) = G 0(0,t) e const. t 2/3 ln t, Quasiparticle residue: Z = lim x Z(x,0) = 0! DOS: N(ω) = N(0) 1 const. ω 2/3 lnω Lawler et al, 2006 n(k) Z k F k

107 Local Quantum Criticality at the Nematic QCP Behavior near the QCP: for T = 0 and δ 1 (FL side), Z e const./ δ At the QCP ((δ = 0), T F T T κ) Z(x,0) e const. Tx2 ln(l/x) 0 but, Z(0,t) is finite as L! Local quantum criticality Irrelevant quartic interactions of strength u lead to a renormalization that smears the QCP at T > 0 (Millis 1993) δ δ(t) = ut ln ut 1/3 Z(x,0) e const. Tx2 ln(ξ/x), where ξ = δ(t) 1/2

108 Generalizations: Charge Order in Higher Angular Momentum Channels Higher angular momentum particle-hole condensates Q l = Ψ ( x + i y) l For l odd: breaks rotational invariance (mod 2π/l). It also breaks parity P and time reversal T but PT is invariant; e.g. For l = 3 ( Varma loop state ) l even: Hexatic (l = 6), etc.

109 Nematic Order in the Triplet Channel Wu, Sun, Fradkin and Zhang (2007) Order Parameters in the Spin Triplet Channel (α, β =, ) Q a l(r) = Ψ α(r)σ a αβ ( x + i y) l Ψ β (r) n a 1 + in a 2 l 0 Broken rotational invariance in space and in spin space; particle-hole condensate analog of He 3A and He 3B Time Reversal: T Q a lt 1 = ( 1) l+1 Q a l Parity: PQ a lp 1 = ( 1) l Q a l Q a l rotates under an SO spin(3) transformation, and Q a l Q a le ilθ under a rotation in space by θ Q a l is invariant under a rotation by π/l and a spin flip

110 Nematic Order in the Triplet Channel Landau theory Free Energy F[n] = r( n n 2 2 ) + v 1( n n 2 2 ) 2 + v 2 n 1 n 2 2 Pomeranchuk: r < 0, (F A l < 2, l 1) v2 > 0, n 1 n 2 = 0 ( α phase) v2 > 0, n 1 n 2 = 0 and n 1 = n 2, ( β phase) l = 2 α phase: nematic-spin-nematic ; in this phase the spin up and spin down FS have an l = 2 nematic distortion but are rotated by π/2 In the β phases there are two isotropic FS bur spin is not a good quantum number: there is an effective spin orbit interaction! Define a d vector: d(k) = (cos(lθ k ),sin(lθ k ),0); In the β phases it winds about the undistorted FS: for l = 1, w = 1 Rashba, w = 1 Dresselhaus

111 Nematic States in the Strongly Coupled Emery Model of a CuO plane S.A. Kivelson, E.Fradkin and T. Geballe (2004) U d V pd V pp t pd U p ǫ t pp Energetics of the 2D Cu O model in the strong coupling limit: t pd /U p, t pd /U d,t pd /V pd,t pd /V pp 0 U d > U p V pd > V pp and t pp/t pd 0 as a function of hole doping x > 0 (x = 0 half-filling) Energy to add one hole: µ 2V pd + ǫ Energy of two holes on nearby O sites: µ + V pp + ǫ

112 Nematic States in the Strongly Coupled Emery Model of a CuO plane Effective One-Dimensional Dynamics at Strong Coupling In the strong coupling limit, and at t pp = 0, the motion of an extra hole is strongly constrained. The following is an allowed move which takes two steps. The final and initial states are degenerate, and their energy is E 0 + µ

113 Nematic States in the Strongly Coupled Emery Model a) b) a) Intermediate state for the hole to turn a corner; it has energy E 0 + µ + V pp t eff = t2 pd V pp t pd b) Intermediate state for the hole to continue on the same row; it has energy E 0 + µ + ǫ t eff = t2 pd ǫ

114 Nematic States in the Strongly Coupled Emery Model The ground state at x = 0 is an antiferromagnetic insulator Doped holes behave like one-dimensional spinless fermions H c = t X j [c j c j+1 + h.c.] + X j [ǫ jˆn j + V pdˆn jˆn j+1 ] at x = 1 it is a Nematic insulator the ground state for x 0 and x 1 is a uniform array of 1D Luttinger liquids it is an Ising Nematic Phase. This result follows since for x 0 the ground state energy is E nematic = E(x = 0) + c x + W x 3 + O(x 5 ) where c = 2V pd + ǫ +... and W = π 2 2 /6m, while the energy of the isotropic state is E isotropic = E(x = 0) + c x + (1/4)W x 3 + V eff x 2 (V eff : effective coupling for holes on intersecting rows and columns) E nematic < E isotropic A similar argument holds for x 1. the density of mobile charge x but k F = (1 x)π/2 For t pp 0 this 1D state crosses over (most likely) to a 2D (Ising) Nematic Fermi liquid state.

115 Nematic States in the Strongly Coupled Emery Model Phase diagram in the strong coupling limit T Isotropic N two phase coexistence x Nematic 1 In the Classical" Regime, ǫ/t pd, with U d > ǫ, the doped holes are distributed on O sites at an energy cost µ per doped hole and an interaction J = V pp/4 per neighboring holes on the O sub-lattice This is a classical lattice gas equivalent to a 2D classical Ising antiferromagnet with exchange J in a uniform field" µ, and an effective magnetization (per O site) m = 1 x The classical Ising antiferromagnet at temperature T and magnetization m = 1 x has the phase diagram of the figure. Quantum fluctuations lead to a similar phase diagram, except for the extra nematic phase.

116 The Quantum Nematic-Smectic Phase Transition

117 The Quantum Nematic-Smectic Phase Transition

118 The Quantum Nematic-Smectic Phase Transition

119 The Quantum Nematic-Smectic Phase Transition

120 The Quantum Nematic-Smectic Phase Transition

121 The Quantum Nematic-Smectic Phase Transition

122 Stripe Phases and the Mechanism of high temperature superconductivity in Strongly Correlated Systems Since the discovery of high temperature superconductivity it has been clear that High Temperature Superconductors are never normal metals and don t have well defined quasiparticles in the normal state (linear resistivity, ARPES) the parent compounds are strongly correlated Mott insulators repulsive interactions dominate the quasiparticles are an emergent low-energy property of the superconducting state whatever the mechanism is has to account for these facts

123 Stripe Phases and the Mechanism of high temperature superconductivity in Strongly Correlated Systems Problem BCS is so successful in conventional metals that the term mechanism naturally evokes the idea of a weak coupling instability with (write here your favorite boson) mediating an attractive interaction between well defined quasiparticles. The basic assumptions of BCS theory are not satisfied in these systems.

124 Stripe Phases and the Mechanism of HTSC Superconductivity in a Doped Mott Insulator or How To Get Pairing from Repulsive Interactions Universal assumption: 2D Hubbard-like models should contain the essential physics RVB mechanism: Mott insulator: spins are bound in singlet valence bonds; it is a strongly correlated spin liquid, essentially a pre-paired insulating state spin-charge separation in the doped state leads to high temperature superconductivity

125 Stripe Phases and the Mechanism of HTSC Problems there is no real evidence that the simple 2D Hubbard model favors superconductivity (let alone high temperature superconductivity ) all evidence indicates that if anything it wants to be an insulator and to phase separate (finite size diagonalizations, various Monte Carlo simulations) strong tendency for the ground states to be inhomogeneous and possibly anisotropic no evidence (yet) for a spin liquid in 2D Hubbard-type models

126 Stripe Phases and the Mechanism of HTSC Why an Inhomogeneous State is Good for high T c SC Inhomogeneity induced pairing mechanism: pairing from strong repulsive interactions. Repulsive interactions lead to local superconductivity on mesoscale structures The strength of this pairing tendency decreases as the size of the structures increases above an optimal size The physics responsible for the pairing within a structure Coulomb frustrated phase separation mesoscale electronic structures

127 Stripe Phases and the Mechanism of HTSC Pairing and Coherence Strong local pairing does not guarantee a large critical temperature In an isolated system, the phase ordering (condensation) temperature is suppressed by phase fluctuations, often to T = 0 The highest possible Tc is obtained with an intermediate degree of inhomogeneity The optimal Tc always occurs at a point of crossover from a pairing dominated regime when the degree of inhomogeneity is suboptimal, to a phase ordering regime with a pseudo-gap when the system is too granular

128 Stripe Phases and the Mechanism of HTSC H = A Cartoon of the Strongly Correlated Stripe Phase [ ] t r, r c r,σ c r,σ + h.c. + [ǫ r c r,σ c r,σ + U2 ] c r,σ c r, σ c r, σc r,σ r,σ < r, r >,σ t t t t ε ε ε ε δt t δt t δt A B Arrigoni, Fradkin and Kivelson (2004)

129 Stripe Phases and the Mechanism of HTSC Physics of the 2-leg ladder t t E U V E F p p F1 p F2 p F2 p F1 U = V = 0: two bands with different Fermi wave vectors, p F1 p F2 The only allowed processes involve an even number of electrons Coupling of CDW fluctuations with Q 1 = 2p F1 Q 2 = 2p F2 is suppressed

130 Stripe Phases and the Mechanism of HTSC Why is there a Spin Gap Scattering of electron pairs with zero center of mass momentum from one system to the other is peturbatively relevant The electrons can gain zero-point energy by delocalizing between the two bands. The electrons need to pair, which may cost some energy. When the energy gained by delocalizing between the two bands exceeds the energy cost of pairing, the system is driven to a spin-gap phase.

131 Stripe Phases and the Mechanism of HTSC What is it known about the 2-leg ladder x = 0: unique fully gapped ground state ( C0S0 ); for U t, s J/2 For 0 < x < x c 0.3, Luther-Emery liquid: no charge gap and large spin gap ( C1S0 ); spin gap s as x, and s 0 as x x c Effective Hamiltonian for the charge degrees of freedom Z H = dy»k vc ( yθ) 2 + 1K 2 ( xφ) φ: CDW phase field; θ: SC phase field; [φ(y ), yθ(y)] = iδ(y y ) x-dependence of s, K, v c, and µ depends on t /t and U/t... represent cosine potentials: Mott gap M at x = 0 K 2 as x 0; K 1 for x 0.1, and K 1/2 for x x c 1 2 K χ SC s/t χ CDW s/t 2 K χ CDW(T) and χ SC(T) for 0 < x < x c For x 0.1, χ SC χ CDW!

132 Stripe Phases and the Mechanism of HTSC Effects of Inter-ladder Couplings Luther-Emery phase: spin gap and no single particle tunneling Second order processes in δt: marginal (and small) forward scattering inter-ladder interactions possibly relevant couplings: Josephson and CDW Relevant Perturbations H = X Z dy J h J cos 2π θj ) + V cos P J y + i 2π φ J ) J: ladder index; P J = 2πx J, φ J = φ J+1 φ J, etc. J and V are effective couplings which must be computed from microscopics; estimate: J V (δt) 2 /J

133 Stripe Phases and the Mechanism of HTSC Period 2 works for x 1 If all the ladders are equivalent: period 2 stripe (columnar) state (Sachdev and Vojta) Isolated ladder: T c = 0 J 0 and V = 0, T C > 0 x 0.1: CDW couplings are irrelevant (1 < K < 2) Inter-ladder Josephson coupling leads to a SC state in a small x with low T c. T c δt x 2J χ SC(T c) = 1 For larger x, K < 1 and χ CDW is more strongly divergent than χ SC CDW couplings become more relevant Insulating, incommensurate CDW state with ordering wave number P = 2πx.

134 Stripe Phases and the Mechanism of HTSC Period 4 works! Alternating array of inequivalent A and B type ladders in the LE regime SC T c: CDW T c: (2J ) 2 χ A SC(T c)χ B SC(T c) = 1 (2V) 2 χ A CDW(P,T c)χ B CDW(P,T c) = 1 2D CDW order is greatly suppressed due to the mismatch between ordering vectors, P A and P B, on neighboring ladders

135 Stripe Phases and the Mechanism of HTSC Period 4 works! For inequivalent ladders SC beats CDW if 2 > K 1 A + K 1 B K A ; 2 > K 1 A T c s J f W «α ; α = + K 1 B K B 2K A K B [4K A K B K A K B ] J δt 2 /J and f W J; T c is (power law) small for small J! (α 1).

136 Stripe Phases and the Mechanism of HTSC How reliable are these estimates? This is a mean-field estimate for T c and it is an upper bound to the actual T c. T c should be suppressed by phase fluctuations by up to a factor of 2. Indeed, perturbative RG studies for small J yield the same power law dependence. This result is asymptotically exact for J << f W. Since T c is a smooth function of δt/j, it is reasonable to extrapolate for δt J. T max c s high T c. This is in contrast to the exponentially small T c as obtained in a BCS-like mechanism.

137 Stripe Phases and the Mechanism of HTSC Tc J 2 s (x) SC CDW 0 xc (2) xc (4) xc x The broken line is the spin gap s(x) as a function of doping x x c(2) and x c(4): SC-CDW QPT for period 2 and period 4 For x x c the isolated ladders do not have a spin gap

138 The Pair Density Wave and Dynamical Layer Decoupling

139 The Pair Density Wave and Dynamical Layer Decoupling

140 The Pair Density Wave and Dynamical Layer Decoupling

141 The Pair Density Wave and Dynamical Layer Decoupling

142 The Pair Density Wave and Dynamical Layer Decoupling

143 The Pair Density Wave and Dynamical Layer Decoupling

144 The Pair Density Wave and Dynamical Layer Decoupling

145 Quasiparticle Spectral Function of the Striped Superconductor

146 The Pair Density Wave and Dynamical Layer Decoupling

147 The Pair Density Wave and Dynamical Layer Decoupling

148 The Pair Density Wave and Dynamical Layer Decoupling

149 The Pair Density Wave and Dynamical Layer Decoupling

150 The Pair Density Wave and Dynamical Layer Decoupling

151 The Pair Density Wave and Dynamical Layer Decoupling

152 The Pair Density Wave and Dynamical Layer Decoupling

153 The Pair Density Wave and Dynamical Layer Decoupling

154 The Pair Density Wave and Dynamical Layer Decoupling

155 The Pair Density Wave and Dynamical Layer Decoupling

156 The Pair Density Wave and Dynamical Layer Decoupling

157 The Pair Density Wave and Dynamical Layer Decoupling

158 The Pair Density Wave and Dynamical Layer Decoupling

159 The Pair Density Wave and Dynamical Layer Decoupling

160 The Pair Density Wave and Dynamical Layer Decoupling