The Role of Charge Order in the Mechanism of High Temperature Superconductivity
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1 The Role of Charge Order in the Mechanism of High Temperature Superconductivity Talk at the Oak Ridge National Laboratory, March 28, 2008 Eduardo Fradkin Department of Physics University of Illinois at Urbana Champaign March 28, 2008
2 Collaborators Steven Kivelson (Stanford University) and Victor J. Emery (Brookhaven National Laboratory) Enrico Arrigoni (TU Graz, Austria), Vadim Oganesyan (Yale University), Erez Berg (Stanford), Eun-Ah Kim (Stanford), Shoucheng Zhang (Stanford), Michael Lawler (Toronto), Daniel Barci (UERJ, Brazil), Tom Lubensky (U. Penn),Erica Carlson (Purdue), Karin Dahmen (UIUC), Kai Sun (UIUC), E. Manousakis (FSU/NHMFL) John Tranquada (Brookhaven), Aharon Kapitulnik (Stanford), Peter Abbamonte (UIUC), Lance Cooper (UIUC)
3 References Steven A. Kivelson and Eduardo Fradkin, How optimal inhomogeneity produces high temperature superconductivity; chapter 15 of Treatise of High Temperature Superconductivity, J. Robert Schrieffer and J. Brooks, editors, Springer-Verlag (2007); arxiv:cond-mat/ Steven A. Kivelson, Eduardo Fradkin, Vadim Oganesyan, Ian Bindloss, John Tranquada, Aharon Kapitulnik and Craig Howald, How to detect fluctuating stripes in high temperature superconductors, Rev. Mod. Phys. 75, 1201 (2003); arxiv:cond-mat/ Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery, Electronic Liquid Crystal Phases of a Doped Mott Insulator, Nature 393, 550 (1998); arxiv:cond-mat/
4 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators
5 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems
6 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe?
7 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe? If inhomogeneous phases are part of the mechanism of high temperature superconductivity, is there an optimal degree of inhomogeneity?
8 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe? If inhomogeneous phases are part of the mechanism of high temperature superconductivity, is there an optimal degree of inhomogeneity? High T c Superconductivity in a Striped Hubbard Model
9 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe? If inhomogeneous phases are part of the mechanism of high temperature superconductivity, is there an optimal degree of inhomogeneity? High T c Superconductivity in a Striped Hubbard Model Isolated doped Hubbard Ladders as the prototype spin-gap systems
10 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe? If inhomogeneous phases are part of the mechanism of high temperature superconductivity, is there an optimal degree of inhomogeneity? High T c Superconductivity in a Striped Hubbard Model Isolated doped Hubbard Ladders as the prototype spin-gap systems Estimating T c : How high is high?
11 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe? If inhomogeneous phases are part of the mechanism of high temperature superconductivity, is there an optimal degree of inhomogeneity? High T c Superconductivity in a Striped Hubbard Model Isolated doped Hubbard Ladders as the prototype spin-gap systems Estimating T c : How high is high? Conclusions and Open Questions
12 Outline Electronic liquid crystal phases are unavoidable in doped Mott insulators Experimental evidence for electronic liquid crystal phases in the high T c superconductors, Sr 3 Ru 2 O 7, and quantum Hall systems Charge Order and Superconductivity: friend or foe? If inhomogeneous phases are part of the mechanism of high temperature superconductivity, is there an optimal degree of inhomogeneity? High T c Superconductivity in a Striped Hubbard Model Isolated doped Hubbard Ladders as the prototype spin-gap systems Estimating T c : How high is high? Conclusions and Open Questions
13 Phase Diagram of the High T c Superconductors T Mott Insulator pseudogap Superconductor x
14 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/
15 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise due to the competition between phase separation and strong correlations
16 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise due to the competition between phase separation and strong correlations Crystal Phases: break all continuous translation symmetries and rotations
17 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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
18 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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
19 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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 High Tc Superconductors : Lattice effects breaking of point group symmetries
21 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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 High Tc Superconductors : Lattice effects breaking of point group symmetries If lattice effects are weak (high T) continuous symmetries essentially recovered
22 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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 High Tc Superconductors : Lattice effects breaking of point group symmetries If lattice effects are weak (high T) continuous symmetries essentially recovered 2DEG in GaAs heterostructures continuous symmetries
23 Electron Liquid Crystal Phases Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery Electronic Liquid Crystal Phases of a Doped Mott Insulator Nature 393, 550 (1998); arxiv:cond-mat/ Doping a Mott insulator: inhomogeneous phases arise 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 High Tc Superconductors : Lattice effects breaking of point group symmetries If lattice effects are weak (high T) continuous symmetries essentially recovered 2DEG in GaAs heterostructures continuous symmetries
24 Electronic Liquid Crystal Phases in High T c Superconductors Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a superconductor
25 Electronic Liquid Crystal Phases in High T c Superconductors 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
26 Electronic Liquid Crystal Phases in High T c Superconductors 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
27 Electronic Liquid Crystal Phases in High T c Superconductors 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
28 Electronic Liquid Crystal Phases in High T c Superconductors 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
29 Electronic Liquid Crystal Phases in HTSC, cont d Soft Quantum Condensed Matter!
30 Schematic Phase Diagram of Doped Mott Insulators
31 Order Parameters for the Smectic (Stripe) State
32 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order
33 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe
34 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe if it coexists with spin order spin stripe
35 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ± Qch = ± 2π λ ch ê x
36 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ± Qch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Qs = (π, π) ± 1 2 Q ch
37 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ± Qch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Qs = (π, π) ± 1 2 Q ch Order Parameter: n Qch, Fourier component of the electron density at Q ch
38 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ± Qch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Qs = (π, π) ± 1 2 Q ch Order Parameter: n Qch, Fourier component of the electron density at Q ch Smectic (stripe) order couples linearly to disorder macroscopic glassy behavior
39 Order Parameters for the Smectic (Stripe) State Two-dimensionally ordered state with unidirectional CDW order charge modulation charge stripe if it coexists with spin order spin stripe stripe state new Bragg peaks of the electron density at k = ± Qch = ± 2π λ ch ê x spin stripe magnetic Bragg peaks at k = Qs = (π, π) ± 1 2 Q ch Order Parameter: n Qch, Fourier component of the electron density at Q ch Smectic (stripe) order couples linearly to disorder macroscopic glassy behavior
40 Order Parameters for the Nematic State
41 Order Parameters for the Nematic State Two-dimensionally translationally invariant metallic state with spontaneously broken rotational symmetry, i.e. spontaneous orthorhombicity
42 Order Parameters for the Nematic State Two-dimensionally translationally invariant metallic state with spontaneously broken rotational symmetry, i.e. spontaneous orthorhombicity Weak coupling picture: spontaneous quadrupolar distortion of the Fermi surface; a Pomeranchuk instability Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64, (2001); arxiv:cond-mat/
43 Order Parameters for the Nematic State Two-dimensionally translationally invariant metallic state with spontaneously broken rotational symmetry, i.e. spontaneous orthorhombicity Weak coupling picture: spontaneous quadrupolar distortion of the Fermi surface; a Pomeranchuk instability Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64, (2001); arxiv:cond-mat/ Strong coupling picture: a quantum melted stripe state. It has the same pattern of symmetry breaking as the Pomeranchuk picture. It has low energy dynamical stripe fluctuations, i.e. fluctuating stripes.
44 Order Parameters for the Nematic State Two-dimensionally translationally invariant metallic state with spontaneously broken rotational symmetry, i.e. spontaneous orthorhombicity Weak coupling picture: spontaneous quadrupolar distortion of the Fermi surface; a Pomeranchuk instability Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64, (2001); arxiv:cond-mat/ Strong coupling picture: a quantum melted stripe state. It has the same pattern of symmetry breaking as the Pomeranchuk picture. It has low energy dynamical stripe fluctuations, i.e. fluctuating stripes. Order Parameter: a 2 2 traceless symmetric tensor, which is odd under a π/2 rotation, e.g. extracted from the resistivity tensor Q = ρ xx ρ yy ρ xx + ρ yy
45 Order Parameters for the Nematic State Two-dimensionally translationally invariant metallic state with spontaneously broken rotational symmetry, i.e. spontaneous orthorhombicity Weak coupling picture: spontaneous quadrupolar distortion of the Fermi surface; a Pomeranchuk instability Vadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin, Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64, (2001); arxiv:cond-mat/ Strong coupling picture: a quantum melted stripe state. It has the same pattern of symmetry breaking as the Pomeranchuk picture. It has low energy dynamical stripe fluctuations, i.e. fluctuating stripes. Order Parameter: a 2 2 traceless symmetric tensor, which is odd under a π/2 rotation, e.g. extracted from the resistivity tensor Q = ρ xx ρ yy ρ xx + ρ yy Nematic charge order is fragile: charge nematic order parameter coupled to disorder 2D random field Ising model E. Carlson, K. Dahmen, E. Fradkin and S. Kivelson, PRL 2005
46 Stripe charge order in underdoped high T c superconductors La 2 xsr xcuo 4 La 2 xba xcuo 4 (non-sc) YBa 2Cu 3O 6+y Neutron scattering (Tranquada, Mook, Keimer) Transport (Ando) Resonant X-ray scattering (Abbamonte) P. Abbamonte et al, Nature Physics 1, 155 (2005)
47 Fluctuating stripe charge order and superconductivity 300 La1.86Sr0.14Cu0.988Zn0.012O4 La1.85Sr0.15CuO4 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) S. Kivelson et al, Rev. Mod. Phys. 75, 1201 (2003) (Tranquada, Mook) Coexistence of fluctuating stripe charge order and SC in La 2 x Sr xcuo 4 and YBa 2 Cu 3 O 6+y
48 Stripe charge order in underdoped high T c superconductors Underdoped La 2 x Sr xcuo 4, x = 5% with Zn (5%) Resonant X-ray scattering (P. Abbamonte et al, unpublished, 2007
49 Charge Order induced inside a SC vortex halo Neutron scattering in La 2 x Sr xcuo 4 in the mixed phase (B. Lake et al) STM in Bi 2 Sr 2 CaCu 2 O 8+δ (S. Davis et al); J. Hoffmanet al, Science 295, 466 (2002)
50 STM: Short range stripe order in Dy-Bi 2 Sr 2 CaCu 2 O 8+δ Kohsaka et al, Science 315, 1380 (2007); R( r,150 mv) = I( r, +150mV)/I( r, 150 mv) short range stripe order on scales ξ 0 (BSCCO and NaCCOC) (Kapitulnik, Davis, Yazdani)
51 Optimal Degree of Inhomogeneity in La 2 x Ba x CuO 4 Evidence of an optimal degree of inhomogeneity in high T c superconductors :The antinodal (pairing) gap is largest, even though T c is smallest, near x = 1 8 in La 2 x Ba xcuo 4 T. Valla et al, Science 314, 1914 (2006) (ARPES)
52 Why is T c so low for La 2 x Ba x CuO 4 at x = 1/8? Q. Li, M. Hücker, G.D. Gu, A.M. Tsvelik and J. M. Tranquada, Phys. Rev. Lett. 99, (2007)
53 Dynamical decoupling of the SC layers in La 2 x Ba x CuO
54 Dynamical decoupling of the SC layers in La 2 x Ba x CuO Striped state with magnetic antiphase domain walls and π shifted superconductivity
55 Dynamical decoupling of the SC layers in La 2 x Ba x CuO Striped state with magnetic antiphase domain walls and π shifted superconductivity The c-axis Josephson coupling vanishes by symmetry in the LTT structure
56 Dynamical decoupling of the SC layers in La 2 x Ba x CuO Striped state with magnetic antiphase domain walls and π shifted superconductivity The c-axis Josephson coupling vanishes by symmetry in the LTT structure This state has 2D superconductivity but the 3D superconductivity is absent!
57 Dynamical decoupling of the SC layers in La 2 x Ba x CuO Striped state with magnetic antiphase domain walls and π shifted superconductivity The c-axis Josephson coupling vanishes by symmetry in the LTT structure This state has 2D superconductivity but the 3D superconductivity is absent! E. Berg, E. Fradkin. E.-A. Kim, S. A. Kivelson, V. Oganesyan, J. M. Tranquada, and S.C. Zhang, Phys. Rev. Lett. 99, (2007)
58 Electron Nematic Order in 2DEG in high magnetic fields Electron nematic phase in the 2DEG in high magnetic fields J.P. Eisenstein et al, Phys. Rev. B 82, 394 (1999)
59 Electron Nematic Order in 2DEG in high magnetic fields K. B. Cooper et al, Phys. Rev. B 65, (2002) temperature dependent and tunable anisotropy
60 Electron Nematic Order in 2DEG in high magnetic fields K. B. Cooper et al, Phys. Rev. B 65, (2002) temperature dependent and tunable anisotropy linear I V curves (no pinning), no noise
61 Electron Nematic Order in 2DEG in high magnetic fields K. B. Cooper et al, Phys. Rev. B 65, (2002) temperature dependent and tunable anisotropy linear I V curves (no pinning), no noise A melted stripe state! Fradkin and Kivelson (1998), Fradkin, Kivelson, Manousakis and Nho (2000)
62 Electron Nematic Order in Sr 3 Ru 2 O 7 in magnetic fields Electron nematic phase in Sr 3Ru 2O 7 R. Borzi, S. Grigera, A. P. Mackenzie et al, Science 315, 214 (2007)
63 Electron Nematic Order in Sr 3 Ru 2 O 7 in magnetic fields A nematic phase preempts a metamagnetic quantum critical point in Sr 3Ru 2O 7 No lattice distortion has been detected
64 Electron Nematic Order in high T c superconductors Y. Ando et al, Phys. Rev. Lett. 88, (2002) Temperature-dependent transport anisotropy in underdoped La 2 x Sr xcuo 4 and YBa 2 Cu 3 O 6+y
65 Fluctuating Stripes as Nematic Order in underdoped YBa 2 Cu 3 O 6+y Fluctuating stripe order detected in inelastic neutron scattering, Mook et al (2000)
66 Fluctuating Stripes as Nematic Order in underdoped YBa 2 Cu 3 O 6+y Fluctuating stripe order detected in inelastic neutron scattering, Mook et al (2000) Temperature dependence of neutron inelastic scattering peaks reveal a nematic phase in YBa 2 Cu 3 O 6+y, y = 4.5 and T 150K, B. Keimer et al (2008)
67 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems
68 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that
69 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc Superconductors are never normal metals and don t have well defined quasiparticles in the normal state (linear resistivity, ARPES)
70 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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
71 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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
72 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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
73 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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
74 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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 Problem
75 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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 Problem BCS is so successful in conventional metals
76 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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 Problem BCS is so successful in conventional metals that the term mechanism naturally evokes the idea of a weak coupling instability
77 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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 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
78 Stripe Phases and the Mechanism of high T c superconductivity in Strongly Correlated Systems Since the discovery of high T c it has been clear that High Tc 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 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.
79 Superconductivity in a Doped Mott Insulator or How To Get Pairing from Repulsive Interactions
80 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
81 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:
82 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
83 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 Tc superconductivity
84 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 Tc superconductivity
85 Problems
86 Problems there is no real evidence that the simple 2D Hubbard model favors superconductivity (let alone high T c superconductivity)
87 Problems there is no real evidence that the simple 2D Hubbard model favors superconductivity (let alone high T c superconductivity) all evidence indicates that if anything it wants to be an insulator and to phase separate (finite size diagonalizations, various Monte Carlo simulations)
88 Problems there is no real evidence that the simple 2D Hubbard model favors superconductivity (let alone high T c 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
89 Problems there is no real evidence that the simple 2D Hubbard model favors superconductivity (let alone high T c 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
90 Problems there is no real evidence that the simple 2D Hubbard model favors superconductivity (let alone high T c 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
91 Why an Inhomogeneous State is Good for high T c SC
92 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly from strong repulsive interactions.
93 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly from strong repulsive interactions. Repulsive interactions lead to local superconductivity on mesoscale structures
94 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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
95 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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
96 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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 Strong local pairing does not guarantee a large critical temperature
97 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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 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
98 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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 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
99 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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 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
100 Why an Inhomogeneous State is Good for high T c SC An inhomogeneity induced pairing mechanism of high temperature superconductivity in which the pairing of electrons originates directly 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 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
101 Striped Hubbard Model: A Cartoon of the Strongly Correlated Stripe Phase
102 Striped Hubbard Model: A Cartoon of the Strongly Correlated Stripe Phase H = X < r, r >,σ t r, r h i c r,σ c r,σ + h.c. + X»ǫ r c r,σ c r,σ + U2 c r,σ c r, σ c r, σc r,σ r,σ t t t t ε ε ε ε δt t δt t δt A B Enrico Arrigoni, Eduardo Fradkin and Steven A. Kivelson, Mechanism of High Temperature Superconductivity in a striped Hubbard Model Phys. Rev. B 69, (2004); arxiv:cond-mat/
103 Physics of the 2-leg ladder t E t U E F V p p F1 p F2 p F2 p F1
104 Physics of the 2-leg ladder t E t U E F V p p F1 p F2 p F2 p F1 For U = V = 0 there are two bands of states
105 Physics of the 2-leg ladder t E t U E F V p p F1 p F2 p F2 p F1 For U = V = 0 there are two bands of states The bands have different Fermi wave vectors, p F1 p F2
106 Physics of the 2-leg ladder t E t U E F V p p F1 p F2 p F2 p F1 For U = V = 0 there are two bands of states The bands have different Fermi wave vectors, p F1 p F2 The only allowed scattering processes involve an even number of electrons (momentum conservation)
107 Physics of the 2-leg ladder t E t U E F V p p F1 p F2 p F2 p F1 For U = V = 0 there are two bands of states The bands have different Fermi wave vectors, p F1 p F2 The only allowed scattering processes involve an even number of electrons (momentum conservation) The coupling of CDW fluctuations with Q 1 = 2p F1 Q 2 = 2p F2 is suppressed
108 Physics of the 2-leg ladder t E t U E F V p p F1 p F2 p F2 p F1 For U = V = 0 there are two bands of states The bands have different Fermi wave vectors, p F1 p F2 The only allowed scattering processes involve an even number of electrons (momentum conservation) The coupling of CDW fluctuations with Q 1 = 2p F1 Q 2 = 2p F2 is suppressed
109 Why is there a Spin Gap
110 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
111 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
112 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
113 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
114 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 This is corroborated by extensive DMRG numerical calculations that find a spin gap for a broad range of parameters (doping and coupling constants) of doped 2-leg and 3-leg ladders (White, Noack and Scalapino), and analytically by bosonization methods Giamarchi and Schulz; Wu, Liu and Fradkin
115 What is it known about the 2-leg ladder
116 What is it known about the 2-leg ladder At x = 0 there is a unique fully gapped ground state ( C0S0 ); for U t, s J/2
117 What is it known about the 2-leg ladder At x = 0 there is a 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
118 What is it known about the 2-leg ladder At x = 0 there is a 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
119 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(y)] = iδ(y y )
120 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(y)] = iδ(y y ) x-dependence of s, K, v c, and µ depends on t /t and U/t
121 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(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
122 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(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
123 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(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
124 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(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
125 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(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!
126 What is it known about the 2-leg ladder At x = 0 there is a 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 H = dy v c 2 [K ( y θ) 2 + 1K ( xφ) 2 ] +... φ: CDW phase field; θ: SC phase field; [φ(y ), y θ(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!
127 Effects of Inter-ladder Couplings
128 Effects of Inter-ladder Couplings In the Luther-Emery phase, 0 < x < x c, there is a spin gap and single particle tunneling is irrelevant
129 Effects of Inter-ladder Couplings In the Luther-Emery phase, 0 < x < x c, there is a spin gap and single particle tunneling is irrelevant Second order processes in δt:
130 Effects of Inter-ladder Couplings In the Luther-Emery phase, 0 < x < x c, there is a spin gap and single particle tunneling is irrelevant Second order processes in δt: marginal (and small) forward scattering inter-ladder interactions
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