Quantum phase transitions of insulators, superconductors and metals in two dimensions

Quantum phase transitions of insulators, superconductors and metals in two dimensions Talk online: sachdev.physics.harvard.edu HARVARD

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

Max Metlitski

Strategy 1. Write down local field theory for order parameter and fermions 2. Apply renormalization group to field theory

Order parameter at a nonzero wavevector: Hot spots on the Fermi surface.

T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave SC SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Theory of SDW quantum phase transition in Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave SC metal SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron 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).

Hole-doped cuprates Increasing SDW order Γ Γ Γ ϕ Γ Hole pockets Electron 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).

Start from the spin-fermion model Z = Dc α Dϕ exp ( S) S = dτ k λ c kα dτ i τ ε k c kα c iα ϕ i σ αβ c iβ e ik r i + dτd 2 r 1 2 ( r ϕ ) 2 ζ + 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4

=3 =2 =4 =1 Low energy fermions ψ 1α, ψ 2α =1,...,4 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α v =1 1 =(v x,v y ), v =1 2 =( v x,v y )

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α v 1 v 2 ψ 1 fermions occupied ψ 2 fermions occupied

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α v 1 v 2 Hot spot Cold Fermi surfaces

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ Hertz-Moriya-Millis (HMM) theory 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Integrate out fermions and obtain non-local corrections to L ϕ L ϕ = 1 2 ϕ 2 q 2 + γ ω /2 ; γ = 2 πv x v y Exponent z = 2 and mean-field criticality (upto logarithms)

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ Hertz-Moriya-Millis (HMM) theory Integrate out fermions and obtain non-local corrections to L ϕ L ϕ = 1 2 ϕ 2 q 2 + γ ω /2 ; γ = 2 But, higher order terms contain an infinite number of marginal couplings...... 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 πv x v y Exponent z = 2 and mean-field criticality (upto logarithms) ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = λϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Perform RG on both fermions and ϕ, using a local field theory.

Order parameter at zero wavevector: Hot Fermi surface.

T Fluctuating, paired Fermi pockets T* Strange Metal Large Fermi surface T c Fluctuating, paired Fermi pockets T sdw d-wave SC VBS and/or Ising nematic order SC+ SDW SC H c2 H SDW Insulator M SDW (Small Fermi pockets) "Normal" (Large Fermi 22 surface)

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

Quantum criticality of Pomeranchuk instability y x Fermi surface with full square lattice symmetry

Quantum criticality of Pomeranchuk instability y x Spontaneous elongation along x direction: Ising order parameter φ > 0.

Quantum criticality of Pomeranchuk instability y x Spontaneous elongation along y direction: Ising order parameter φ < 0.

Quantum criticality of Pomeranchuk instability φ =0 φ =0 λ c λ Pomeranchuk instability as a function of coupling λ

Quantum criticality of Pomeranchuk instability T Quantum critical T c φ =0 φ =0 λ c λ Phase diagram as a function of T and λ

Quantum criticality of Pomeranchuk instability T Quantum critical T c Classical d=2 Ising criticality φ =0 φ =0 λ c λ Phase diagram as a function of T and λ

Quantum criticality of Pomeranchuk instability T Quantum critical T c φ =0 φ =0 D=2+1 quantum λ criticality c? λ Phase diagram as a function of T and λ

Quantum criticality of Pomeranchuk instability Effective action for Ising order parameter S φ = d 2 rdτ ( τ φ) 2 + c 2 ( φ) 2 +(λ λ c )φ 2 + uφ 4

Quantum criticality of Pomeranchuk instability Effective action for Ising order parameter S φ = d 2 rdτ ( τ φ) 2 + c 2 ( φ) 2 +(λ λ c )φ 2 + uφ 4 Effective action for electrons: S c = dτ N f i c iα τ c iα i<j t ij c iα c iα α=1 N f dτc kα ( τ + ε k ) c kα α=1 k

Quantum criticality of Pomeranchuk instability Coupling between Ising order and electrons S φc = γ dτ φ N f (cos k x cos k y )c kα c kα α=1 k for spatially independent φ φ > 0 φ < 0

Quantum criticality of Pomeranchuk instability Coupling between Ising order and electrons S φc = γ dτ N f φ q (cos k x cos k y )c k+q/2,α c k q/2,α α=1 k,q for spatially dependent φ φ > 0 φ < 0

Quantum criticality of Pomeranchuk instability S φ = d 2 rdτ ( τ φ) 2 + c 2 ( φ) 2 +(λ λ c )φ 2 + uφ 4 S c = S φc = γ N f α=1 k dτ N f α=1 dτc kα ( τ + ε k ) c kα k,q φ q (cos k x cos k y )c k+q/2,α c k q/2,α

A φ fluctuation at wavevector q couples most efficiently to fermions near ± k 0. Expand fermion kinetic energy at wavevectors about k 0

L = ψ y + ζ τ i x 2 ψ+ + ψ y ζ τ + i x 2 ψ ψ λφ +ψ + + ψ ψ + 1 2g ( yφ) 2 + r 2 φ2

Emergent Galilean invariance at low energy (s = ±): φ(x, y) φ(x, y + θx), ψ s (x, y) e is( θ 2 y+ θ2 4 x) ψ s (x, y + θx) which implies for the fermion Green s function G(q x,q y )=G(sq x + qy). 2 Every point on the Fermi surface sq x +qy 2 = 0 has the same singularity: Hot Fermi surface.

Hot Fermi surfaces Emergent Galilean invariance at low energy (s = ±): φ(x, y) φ(x, y + θx), ψ s (x, y) e is( θ 2 y+ θ2 4 x) ψ s (x, y + θx) which implies for the fermion Green s function G(q x,q y )=G(sq x + qy). 2 Every point on the Fermi surface sq x +qy 2 = 0 has the same singularity: Hot Fermi surface.

Hertz-Moriya-Millis (HMM) theory Integrate out fermions and obtain effective action for φ L φ = 1 q 2 y 2 φ2 g + ω 4π q y Exponent z = 3 and mean-field criticality?

Strategy 1. Write down local field theory for order parameter and fermions 2. Apply renormalization group to field theory

Order parameter at a nonzero wavevector: Hot spots on the Fermi surface.

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Under the rescaling x = xe, τ = τe z, the spatial gradients are fixed if the fields transform as ϕ = e (d+z 2)/2 ϕ ; ψ = e (d+z 1)/2 ψ. Then the Yukawa coupling transforms as λ = e (4 d z)/2 λ For d = 2, with z = 2 the Yukawa coupling is invariant, and the bare time-derivative terms ζ, ζ are irrelevant.

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β With z = 2 scaling, ζ is irrelevant. So we take ζ 0 ( watch for dangerous irrelevancy).

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Set ϕ wavefunction renormalization by keeping co-efficient of ( r ϕ ) 2 fixed (as usual).

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Set fermion wavefunction renormalization by keeping Yukawa coupling fixed. Y. Huh and S. Sachdev, Phys. Rev. B 78, 064512 (2008).

L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β We find consistent two-loop RG factors, as ζ 0, for the velocities v x, v y, and the wavefunction renormalizations. Consistency check: the expression for the boson damping constant, γ = 2 πv x v y, is preserved under RG.

RG-improved Migdal-Eliashberg theory RG flow can be computed a 1/N expansion (with N fermion species) in terms of a single dimensionless coupling α = v y /v x whose flow obeys dα d = 3 πn α 2 1+α 2 The velocities flow as 1 dv x v x d A(α) B(α) = A(α)+B(α) 2 3 α πn 1+α 2 1 1 2πN α α 1+ ; 1 dv y v y d = A(α)+B(α) 2 1 α α tan 1 1 α

RG-improved Migdal-Eliashberg theory RG flow can be computed a 1/N expansion (with N fermion species) in terms of a single dimensionless coupling α = v y /v x whose flow obeys dα d = 3 πn α 2 1+α 2 The anomalous dimensions of ϕ and ψ are 1 1 1 η ϕ = 2πN α α + α 2 + α2 tan 1 1 α η ψ = 1 1 1 4πN α α 1+ α α tan 1 1 α

RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting v 1 v 2 Bare Fermi surface

RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting Dressed Fermi surface

RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting Bare Fermi surface

RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting Dressed Fermi surface

RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. In ϕ SDW fluctuations, characteristic q and ω scale as q ω 1/2 exp 3 64π 2 ln(1/ω) N 3 However, 1/N expansion cannot be trusted in the asymptotic regime..

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) ϕ propagator 1 N 1 (q 2 + γ ω ) fermion propagator v q + iζω + i 1 1 N γv ωf v 2 q 2 ω

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) ϕ propagator 1 N 1 (q 2 + γ ω ) fermion propagator v q + iζω + i 1 1 N γv ωf v 2 q 2 ω Dangerous

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) Ignoring fermion self energy: 1 N 2 1 ζ 2 1 ω

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) Ignoring fermion self energy: 1 N 2 1 Actual order 1 N 0 ζ 2 1 ω

Double line representation A way to compute the order of a diagram. Extra powers of N come from the Fermi-surface What are the conditions for all propagators to be on the Fermi surface? Concentrate on diagrams involving a single pair of hot-spots Any bosonic momentum may be (uniquely) written as R. Shankar, Rev. Mod. Phys. 66, 129 (1994). S. W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Phys. Rev. B 72, 054531 (2005).

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) = Singularities as ζ 0 appear when fermions in closed blue and red line loops are exactly on the Fermi surface Actual order 1 N 0

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) = Actual order 1 N 0 Graph is planar after turning fermion propagators also into double lines by drawing additional dotted single line loops for each fermion loop Sung-Sik Lee, arxiv:0905.4532

New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) = Actual order 1 N 0 A consistent analysis requires resummation of all planar graphs

Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions