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 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

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)

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

Fermi surface with full square lattice symmetry

Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y

Spontaneous elongation along x direction: Ising order parameter φ > 0.

Friday, April 13, 2012 Quantum criticality of Pomeranchuk instability y

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

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

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).

2 Every point on the Fermi surface sqx +qy = 0 has the same singularity: “Hot Fermi surface”.

Friday, April 13, 2012 “Hot” Fermi surfaces

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).

2 Every point on the Fermi surface sqx +qy = 0 has the same singularity: “Hot Fermi surface”.

Friday, April 13, 2012 Hertz-Moriya-Millis (HMM) theory

Integrate out fermions and obtain eﬀective action for φ

1 q2 ω = φ2 y + | | Lφ 2 g 4π q | y| Exponent z = 3 and mean-ﬁeld criticality ?

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 = 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 z Under the rescaling x = xe− , τ = τe− , the spatial gradients are ﬁxed if the ﬁelds transform as

(d+z 2)/2 (d+z 1)/2 ϕ = e − ϕ ;“ ψ = e − ψ.

Then the Yukawa coupling transforms as

(4 d z)/2 λ = e − − λ

For d = 2, with z = 2 the Yukawa coupling is invariant, and the bare time-derivative terms ζ, ζ are irrelevant.

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: = ϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β

With z = 2 scaling, ζ is irrelevant. So we take ζ 0 ( watch for dangerous→ irrelevancy).

Set ϕ wavefunction renormalization by keeping co-eﬃcient of ( ϕ )2 ﬁxed (as usual). ∇r

Set fermion wavefunction renormalization by keeping Yukawa coupling ﬁxed.

Y. Huh and S. Sachdev, Phys. Rev. B 78, 064512 (2008).

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: = ϕ ψ † σ ψ + ψ † σ ψ Lc − · 1α αβ 2β 2α αβ 1β We ﬁnd consistent two-loop RG factors, as ζ 0, for the → velocities vx, vy, and the wavefunction renormalizations.

Consistency check: the expression for the boson damp- 2 ing constant, γ = , is preserved under RG. πvxvy

Friday, April 13, 2012 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 = vy/vx whose ﬂow obeys dα 3 α2 = d −πN 1+α2

The velocities ﬂow as 1 dv (α)+ (α) 1 dv (α)+ (α) x = A B ; y = −A B vx d 2 vy d 2 3 α (α) A ≡ πN 1+α2

1 1 1 1 1 (α) α 1+ α tan− B ≡ 2πN α − α − α

The anomalous dimensions of ϕ and ψ are

1 1 1 2 1 1 η = α + + α tan− ϕ 2πN α − α2 α 1 1 1 1 1 η = α 1+ α tan− ψ −4πN α − α − α

Friday, April 13, 2012 RG-improved Migdal-Eliashberg theory α = v /v 0 logarithmically in the infrared. y x → Dynamical Nesting

v1 v2

Bare Fermi surface

Friday, April 13, 2012 RG-improved Migdal-Eliashberg theory α = v /v 0 logarithmically in the infrared. y x → Dynamical Nesting

Dressed Fermi surface

Friday, April 13, 2012 RG-improved Migdal-Eliashberg theory α = v /v 0 logarithmically in the infrared. y x → Dynamical Nesting

Bare Fermi surface

Friday, April 13, 2012 RG-improved Migdal-Eliashberg theory α = v /v 0 logarithmically in the infrared. y x → Dynamical Nesting

Dressed Fermi surface

Friday, April 13, 2012 RG-improved Migdal-Eliashberg theory α = v /v 0 logarithmically in the infrared. y x →

In ϕ SDW ﬂuctuations, characteristic q and ω scale as

3 ln(1/ω) 3 q ω1/2 exp . ∼ −64π2 N However, 1/N expansion cannot be trusted in the asymptotic regime.