Outline 1. Introduction to the Hubbard model Superexchange and antiferromagnetism
2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point
3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model
4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover
5. Hubbard model as a SU(2) gauge theory Spin liquids, valence bond solids: analogies with SQED and SYM
Monday, June 14, 2010 Outline 6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Monday, June 14, 2010 Outline 6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Monday, June 14, 2010 The Hubbard Model
1 1 H = tijci†αcjα + U ni ni µ ci†αciα − ↑ − 2 ↓ − 2 − i niα = ci†αciα ci†αcjβ + cjβci†α = δijδαβ ciαcjβ + cjβciα =0 Will study on the honeycomb and square lattices Monday, June 14, 2010 The Hubbard Model 1 1 H = tijci†αcjα + U ni ni µ ci†αciα − ↑ − 2 ↓ − 2 − i niα = ci†αciα ci†αcjβ + cjβci†α = δijδαβ ciαcjβ + cjβciα =0 Will study on the honeycomb and square lattices Monday, June 14, 2010 Fermi surface+antiferromagnetism Hole states occupied Γ Electron states occupied Γ + The electron spin polarization obeys iK r S(r, τ) = ϕ (r, τ)e · where K is the ordering wavevector. Monday, June 14, 2010 Fermi surfaces in electron- and hole-doped cuprates Hole states occupied Γ Electron states occupied Γ Effective Hamiltonian for quasiparticles: H = t c† c ε c† c 0 − ij iα iα ≡ k kα kα i with tij non-zero for first, second and third neighbor, leads to satisfactory agree- ment with experiments. The area of the occupied electron states, e, from Luttinger’s theory is A 2π2(1 p) for hole-doping p e = − A 2π2(1 + x) for electron-doping x The area of the occupied hole states, h, which form a closed Fermi surface and so appear in quantum oscillation experimentsA is =4π2 . Ah − Ae Monday, June 14, 2010 Spin density wave theory In the presence of spin density wave order, ϕ at wavevector K = (π,π), we have an additional term which mixes electron states with momentum separated by K H = ϕ c† σ c sdw · k,α αβ k+K,β k,α,β where σ are the Pauli matrices. The electron dispersions obtained by diagonalizing H + H for ϕ (0, 0, 1) are 0 sdw ∝ εk + εk+K εk εk+K 2 Ek = − + ϕ ± 2 ± 2 This leads to the Fermi surfaces shown in the following slides for electron and hole doping. Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole Electron pockets pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole Electron pockets pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole Electron Hot spots pockets pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Hole Γ pockets Γ Γ Γ Hole Electron Hot spots pockets pockets Fermi surface breaks up at hot spots into electron and hole “pockets” S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole Electron Hot spots pockets pockets Fermi surface breaks up at hot spots into electron and hole “pockets” S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Evidence for small Fermi pockets Fermi liquid behaviour in an underdoped high Tc superconductor Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich arXiv:0912.3022 Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole Electron pockets pockets Large Fermi surface breaks up into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Hole-doped cuprates Increasing SDW order Γ Γ Γ ϕ Γ Hole Electron pockets pockets ϕ fluctuations act on the large Fermi surface S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Monday, June 14, 2010 Start from the “spin-fermion” model = c ϕ exp ( ) Z D αD −S ∂ = dτ c† ε c S kα ∂τ − k kα k iK r λ dτ c† ϕ σ c e · i − iα i · αβ iβ i 2 1 2 ζ 2 s 2 u 4 + dτd r ( r ϕ ) + (∂τ ϕ ) + ϕ + ϕ 2 ∇ 2 2 4 Monday, June 14, 2010 =3 =2 =4 =1 Low energy fermions ψ1α, ψ2α =1,...,4 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α v=1 =(v ,v ), v=1 =( v ,v ) 1 x y 2 − x y Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α v1 v2 ψ1 fermions ψ2 fermions occupied occupied Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α v1 v2 “Hot spot” “Cold” Fermi surfaces Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Hertz theory Integrate out fermions and obtain non-local corrections to Lϕ 1 2 2 2 ϕ = ϕ q + γ ω /2; γ = L 2 | | πvxvy Exponent z = 2 and mean-field criticality (upto logarithms) Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Hertz theory Integrate out fermions and obtain non-local corrections to Lϕ 1 2 2 2 ϕ = ϕ q + γ ω /2; γ = L 2 | | πvxvy Exponent z = 2 and mean-field criticality (upto logarithms) OK in d =3, but higher order terms contain an infinite number of marginal couplings in d =2 Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Monday, June 14, 2010 = ψ † ζ∂ iv ψ + ψ † ζ∂ iv ψ Lf 1α τ − 1 · ∇r 1α 2α τ − 2 · ∇r 2α 1 ζ s u Order parameter: = ( ϕ )2 + (∂ ϕ )2 + ϕ 2 + ϕ 4 Lϕ 2 ∇r 2 τ 2 4 “Yukawa” coupling: = λϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β Perform RG on both fermions and ϕ , using a local field theory. Monday, June 14, 2010 Outline 6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations 7. Instabilities near the SDW critical point d-wave superconductivity and other orders 8. Global phase diagram of the cuprates Competition for the Fermi surface Monday, June 14, 2010 Outline 6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations 7. Instabilities near the SDW critical point d-wave superconductivity and other orders 8. Global phase diagram of the cuprates Competition for the Fermi surface Monday, June 14, 2010 Superconductivity by SDW fluctuation exchange Monday, June 14, 2010 Physical Review B 34, 8190 (1986) Monday, June 14, 2010 Spin density wave theory in hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Monday, June 14, 2010 Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates Increasing SDW order Γ Γ Γ ϕ Γ Fermions at the large Fermi surface exchange fluctuations of the SDW order parameter ϕ . David Pines, Douglas Scalapino Monday, June 14, 2010 Pairing by SDW fluctuation exchange We now allow the SDW field ϕ to be dynamical, coupling to elec- trons as H = ϕ c† σ c . sdw − q · k,α αβ k+K+q,β k,q,α,β Exchange of a ϕ quantum leads to the effective interaction 1 Hee = Vαβ,γδ(q)c† ck+q,βcp† ,γ cp q,δ, −2 k,α − q p,γ,δ k,α,β where the pairing interaction is χ V (q)=σ σ 0 , αβ,γδ αβ · γδ ξ 2 +(q K)2 − − 2 with χ0ξ the SDW susceptibility and ξ the SDW correlation length. Monday, June 14, 2010 BCS Gap equation In BCS theory, this interaction leads to the ‘gap equation’ for the pairing gap ∆k ck c k . ∝ ↑ − ↓ 3χ0 ∆p ∆k = 2 2 − ξ +(p k K) 2 2 p − 2 ε + ∆ − − p p Non-zero solutions of this equation require that ∆ and ∆ have opposite signs when p k K. k p − ≈ Monday, June 14, 2010 Increasing SDW order d-wave pairing of the large Fermi surface ϕ _ K + Γ + _ ck c k ∆k = ∆0(cos(kx) cos(ky)) ↑ − ↓∝ − Monday, June 14, 2010 d-wave pairing in the theory of hotspots =3 =2 =4 =1 Low energy fermions ψ1α, ψ2α =1,...,4 Monday, June 14, 2010 d-wave pairing in the theory of hotspots Hot spots have ψ3 2 strong instability to d-wave pairing near SDW critical point. This instability is stronger than the BCS instability of a 3 ψ1 Fermi liquid. Pairing order parameter: εαβ ψ3 ψ1 ψ3 ψ1 1α 1β − 2α 2β Monday, June 14, 2010 ϕ + - + - + 3 3 3 3 3 ψ1 ψ2 ψ1 ψ2 ψ1 d-wave Cooper pairing instability in particle-particle channel Monday, June 14, 2010 2¯ 4 1 3¯ ky Γ 3 1¯ 4¯ 2 Q =(π, 0) kx Similar theory applies to the pnictides, and leads to s pairing. ± Monday, June 14, 2010 Emergent Pseudospin symmetry Continuum theory of hotspots in invariant under: ψ ψ ↑ U ↑ ψ † → ψ † ↓ ↓ where U are arbitrary SU(2) matrices which can be different on different hotspots . Monday, June 14, 2010 ϕ + - + - + 3 3 3 3 3 ψ1 ψ2 ψ1 ψ2 ψ1 d-wave Cooper pairing instability in particle-particle channel Monday, June 14, 2010 ϕ + - + - + 3 3 3 3 3 ψ1 ψ2 ψ1 ψ2 ψ1 Bond density wave (with local Ising-nematic order) instability in particle-hole channel Monday, June 14, 2010 3 ψ2 d-wave pairing has a partner instability in the particle-hole channel 3 ψ1 Density-wave order parameter: 3 1 3 1 ψ † ψ ψ † ψ 1α 1α − 2α 2α Monday, June 14, 2010 3 ψ2 3 ψ1 Monday, June 14, 2010 3 ψ2 Q 3 3 ψ1 ψ1 Q 3 ψ2 Single ordering wavevector Q: ck† Q/2,αck+Q/2,α = − Φ(cos kx cosky) − Monday, June 14, 2010 "1 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond: VBS order !1 No modulations on sites. Modulated bond-density wave with local Ising-nematic ordering: ck† Q/2,αck+Q/2,α =Φ(cos kx cos ky) − − Monday, June 14, 2010 "1 “Bond density” measures amplitude for electrons to be in spin-singlet valence bond: VBS order !1 No modulations on sites. Modulated bond-density wave with local Ising-nematic ordering: ck† Q/2,αck+Q/2,α =Φ(cos kx cos ky) − − Monday, June 14, 2010 STM measurements of Z(r), the energy asymmetry in density of states in Bi2Sr2CaCu2O8+δ. M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and Eun-Ah Kim, preprint Monday, June 14, 2010 STM measurements of Z(r), the energy asymmetry in density of states in Bi2Sr2CaCu2O8+δ. M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and Eun-Ah Kim, preprint C A B D Monday, June 14, 2010 STM measurements of Z(r), the energy asymmetry in density of states in Bi2Sr2CaCu2O8+δ. M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and Eun-Ah Kim, preprint C Strong anisotropy of A B electronic states between D x and y directions: Electronic “Ising-nematic” order Monday, June 14, 2010 Outline 6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations 7. Instabilities near the SDW critical point d-wave superconductivity and other orders 8. Global phase diagram of the cuprates Competition for the Fermi surface Monday, June 14, 2010 Outline 6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations 7. Instabilities near the SDW critical point d-wave superconductivity and other orders 8. Global phase diagram of the cuprates Competition for the Fermi surface Monday, June 14, 2010 d-wave Antiferro- supercon- magnetism ductivity Fermi surface Monday, June 14, 2010 d-wave Antiferro- supercon- magnetism ductivity Fermi surface Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Fluctuating Metal Fermi Large pockets Fermi surface IncreasingIncreasing SDW SDW order order Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = xm Monday, June 14, 2010 d-wave Antiferro- supercon- magnetism ductivity Fermi surface Monday, June 14, 2010 Spin d-wave density supercon- wave ductivity Fermi surface Monday, June 14, 2010 Spin d-wave density supercon- wave ductivity Fermi surface Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Fluctuating Metal Fermi Large pockets Fermi surface IncreasingIncreasing SDW SDW order order Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = xm Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Fluctuating,Small Fermi Metal pairedpockets Fermi with Large pairingpockets fluctuations Fermi surface d-wave superconductor Spin density wave (SDW) Onset of d-wave superconductivity hides the critical point x = xm Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Fluctuating,Small Fermi Metal E. Demler, S. Sachdev pairedpockets Fermi with Large and Y. Zhang, Phys. pairingpockets fluctuations Fermi Rev. Lett. 87, surface 067202 (2001). E. G. Moon and S. Sachdev, Phy. d-wave Rev. B 80, 035117 superconductor (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = xs Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Fluctuating,Small Fermi Metal E. Demler, S. Sachdev pairedpockets Fermi with Large and Y. Zhang, Phys. pairingpockets fluctuations Fermi Rev. Lett. 87, surface 067202 (2001). E. G. Moon and S. Sachdev, Phy. d-wave Rev. B 80, 035117 superconductor (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = xs Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Fluctuating,Small Fermi Metal E. Demler, S. Sachdev pairedpockets Fermi with Large and Y. Zhang, Phys. pairingpockets fluctuations Fermi Rev. Lett. 87, surface 067202 (2001). E. G. Moon and S. Sachdev, Phy. d-wave Rev. B 80, 035117 Magnetic (2009) quantum superconductor Thermally criticality fluctuating SDW Spin gap Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = xs Monday, June 14, 2010 Theory of quantum criticality in the cuprates T* Strange Metal Fluctuating,Small Fermi V. Galitski and pairedpockets Fermi with Large S. Sachdev, Phys. pairingpockets fluctuations Fermi Rev. B 79, 134512 surface (2009). E. G. Moon and d-wave S. Sachdev, Phys. Magnetic Rev. B 80, 035117 quantum superconductor (2009) Thermally criticality fluctuating SDW Spin gap Spin density wave (SDW) Physics of competition: d-wave SC and SDW “eat up” same pieces of the large Fermi surface. Monday, June 14, 2010 T T* Strange Fluctuating,Small Fermi Metal pairedpockets Fermi with Large pairingpockets fluctuations Fermi surface d-wave Magnetic quantum SC criticality Thermally fluctuating Spin gap SDW Monday, June 14, 2010 T T* Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Tsdw SC SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Monday, June 14, 2010 T T* Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Tsdw SC SC+ SDW SC SDW (Small Fermi Quantum oscillations pockets) M "Normal" (Large Fermi H surface) Monday, June 14, 2010 T T* Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Tsdw SC SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Monday, June 14, 2010 J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow, D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono, M. Oda, M. Ido, and J. Mesot, Physical Review B 78, 104525 (2008). J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann, H. M. Roennow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess, M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda, M. Ido, and J. Mesot, Phys. Rev. Lett. 102, 177006 (2009). Monday, June 14, 2010 T T* Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Tsdw SC SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Monday, June 14, 2010 Similar phase diagram for CeRhIn5 G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223 Monday, June 14, 2010 Similar phase diagram for the pnictides 150 100 Tet Ort 50 Ishida, Nakai, and Hosono arXiv:0906.2045v1 AFM /Ort SC 0 0 0.02 0.04 0.06 0.08 0.10 0.12 S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, arXiv:0911.3136. Monday, June 14, 2010