Quantum phase transitions of insulators, superconductors and metals in two dimensions Talk online: sachdev.physics.harvard.edu HARVARD Friday, April 13, 2012 Outline 1. Phenomenology of the cuprate superconductors (and other compounds) 2. QPT of antiferromagnetic insulators (and bosons at rational filling) 3. QPT of d-wave superconductors: Fermi points of massless Dirac fermions 4. QPT of Fermi surfaces: A. Finite wavevector ordering (SDW/CDW): “Hot spots” on Fermi surfaces B. Zero wavevector ordering (Nematic): “Hot Fermi surfaces” Friday, April 13, 2012 Outline 1. Phenomenology of the cuprate superconductors (and other compounds) 2. QPT of antiferromagnetic insulators (and bosons at rational filling) 3. QPT of d-wave superconductors: Fermi points of massless Dirac fermions 4. QPT of Fermi surfaces: A. Finite wavevector ordering (SDW/CDW): “Hot spots” on Fermi surfaces B. Zero wavevector ordering (Nematic): “Hot Fermi surfaces” Friday, April 13, 2012 Max Metlitski Friday, April 13, 2012 Strategy 1. Write down local field theory for order parameter and fermions 2. Apply renormalization group to field theory Friday, April 13, 2012 Strategy 1. Write down local field theory for order parameter and fermions 2. Apply renormalization group to field theory Friday, April 13, 2012 Order parameter at a non- zero wavevector: “Hot spots” on the Fermi surface. Friday, April 13, 2012 T Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Magnetic quantum SC criticality Thermally fluctuating Spin gap SDW SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Friday, April 13, 2012 T Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Theory of SDW Magnetic quantum SC criticality quantum phase Thermally fluctuating Spin gap transition in SDW metal SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Friday, April 13, 2012 T Strange Fluctuating,Small Fermi Metal pocketspaired Fermi with Large pairingpockets fluctuations Fermi surface d-wave Theory of SDW Magnetic quantum SC criticality quantum phase Thermally fluctuating Spin gap transition in SDW metal SC+ SDW SC SDW (Small Fermi pockets) M "Normal" (Large Fermi H surface) Friday, April 13, 2012 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). Friday, April 13, 2012 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). Friday, April 13, 2012 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 Friday, April 13, 2012 =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 Friday, April 13, 2012 = 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 Friday, April 13, 2012 = 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 Friday, April 13, 2012 = 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 Friday, April 13, 2012 = 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 Friday, April 13, 2012 = 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-Moriya-Millis (HMM) 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) Friday, April 13, 2012 = 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-Moriya-Millis (HMM) 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) But, higher order terms contain an infinite number of marginal couplings . Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004). Friday, April 13, 2012 = 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. Friday, April 13, 2012 Order parameter at zero wavevector: “Hot Fermi surface”. Friday, April 13, 2012 T T* Strange Fluctuating,Fluctuating, Metal pairedpaired FermiFermi Large pocketspockets Fermi surface Tc d-wave Tsdw SC VBS and/or Fluctuating, Ising nematic order paired Fermi pockets SC+ SDW SC Hc2 SDW M Insulator SDW (Small Fermi "Normal" pockets) (Large Fermi H 22 surface) Friday, April 13, 2012 Broken rotational symmetry in the pseudogap phase of a high-Tc superconductor R. Daou, J. Chang, David LeBoeuf, Olivier Cyr- Choiniere, Francis Laliberte, Nicolas Doiron- Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer arXiv: 0909.4430, Nature, in press Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y x Fermi surface with full square lattice symmetry Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y x Spontaneous elongation along x direction: Ising order parameter φ > 0. Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y x Spontaneous elongation along y direction: Ising order parameter φ < 0. Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability φ =0 φ =0 λ λc Pomeranchuk instability as a function of coupling λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability T Quantum critical Tc φ =0 φ =0 λ λc Phase diagram as a function of T and λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability T Quantum critical Tc Classical d=2 Ising criticality φ =0 φ =0 λ λc Phase diagram as a function of T and λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability T Quantum critical Tc φ =0 φ =0 D=2+1 quantum λ criticalityλc ? Phase diagram as a function of T and λ Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Effective action for Ising order parameter = d2rdτ (∂ φ)2 + c2( φ)2 +(λ λ )φ2 + uφ4 Sφ τ ∇ − c Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Effective action for Ising order parameter = d2rdτ (∂ φ)2 + c2( φ)2 +(λ λ )φ2 + uφ4 Sφ τ ∇ − c Effective action for electrons: Nf = dτ c† ∂ c t c† c Sc iα τ iα − ij iα iα α=1 i i<j Nf dτc† (∂ + ε ) c ≡ kα τ k kα α=1 k Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Coupling between Ising order and electrons Nf = γ dτφ (cos k cos k )c† c Sφc − x − y kα kα α=1 k for spatially independent φ φ > 0 φ < 0 Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability Coupling between Ising order and electrons Nf φc = γ dτ φq (cos kx cos ky)c† ck q/2,α S − − k+q/2,α − α=1 k,q for spatially dependent φ φ > 0 φ < 0 Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability = d2rdτ (∂ φ)2 + c2( φ)2 +(λ λ )φ2 + uφ4 Sφ τ ∇ − c Nf = dτc† (∂ + ε ) c Sc kα τ k kα α=1 k Nf φc = γ dτ φq (cos kx cos ky)c† ck q/2,α S − − k+q/2,α − α=1 k,q Friday, April 13, 2012 A φ fluctuation at wavevector q couples most efficiently to fermions near k . ± 0 Expand fermion kinetic energy at wavevectors about k0 Friday, April 13, 2012 2 2 = ψ+† ζ∂τ i∂x ∂y ψ+ + ψ† ζ∂τ + i∂x ∂y ψ L − − − − − 1 2 r 2 λφ ψ+† ψ+ + ψ† ψ + (∂yφ) + φ − − − 2g 2 Friday, April 13, 2012 Emergent “Galilean invariance” at low energy (s = ): ± 2 is( θ y+ θ x) φ(x, y) φ(x, y + θx), ψ (x, y) e− 2 4 ψ (x, y + θx) → s → s which implies for the fermion Green’s function 2 G(qx,qy)=G(sqx + qy).