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1. Introduction to the Hubbard Model 2. Coupled Dimer Antiferromagnet 3 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<j i i t “hopping”. U local repulsion, µ chemical potential ij → → → Spin index α = , ↑ ↓ 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<j i i t “hopping”. U local repulsion, µ chemical potential ij → → → Spin index α = , ↑ ↓ 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<j k 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 .
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