Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy...

Phys. Rev. Lett. 1990

Saturday, April 3, 2010

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Ultracold 87Rbatoms - bosons

Superfluid-insulator transition

InsulatorSuperfluid

Quantumcritical

σ =4e2

Σ, a universal number.

Quantum critical transport

S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

Quantum “perfect fluid”with shortest possible

thermal equilibration time, τR

τR kBT

Quantum critical transport

S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

Quantum “perfect fluid”with shortest possible

thermal equilibration time, τR

τR kBT

The worst possible state for quantum computation

σ =4e2

K. Damle and S. Sachdev, 1997

σ =4e2

Collisionless

σ =4e2

Collision-dominated

InsulatorSuperfluid

Quantumcritical

ψ = 0 ψ = 0

d2rdτ|∂τψ|2 + v2|∇ψ|2 + (g − gc)|ψ|2 +

2|ψ|4

• Now add Dirac fermions, generalize the gaugegroup to SU(N), and allow maximal super-symmetry in 2+1 dimensions.

• Yields a model whose transport propertiescan be computed exactly in the large N limitvia the AdS/CFT correspondence.

• Most importantly, the large N limit exhibitshydrodynamic behavior, and the thermal equi-libration time remains finite as N → ∞: thisis a first for any solvable many body theory.

• Critical conductivity Σ =√2N3/2/3 (“self-

dual” value).

P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)

Imχ(k, ω)/k2 ImK√

k2 − ω2

Collisionless to hydrodynamic crossover of SYM3

Collisionless

P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)

Imχ(k, ω)/k2

ImDχc

Dk2 − iω

Collisionless to hydrodynamic crossover of SYM3

Collision-dominated

Field theories in D spacetime dimensions are char-

acterized by couplings g which obey the renormal-

ization group equation

du= β(g)

where u is the energy scale. The RG equation is

local in energy scale, i.e. the RHS does not depend

upon u.

Field theories in D spacetime dimensions are char-

acterized by couplings g which obey the renormal-

ization group equation

du= β(g)

where u is the energy scale. The RG equation is

local in energy scale, i.e. the RHS does not depend

upon u.

Key idea: ⇒ Implement u as an extra dimen-sion, and map to a local theory in D+1 dimensions.

At the RG fixed point, β(g) = 0, the D dimen-sional field theory is invariant under the scale trans-formation

xµ → xµ/b , u→ b u

At the RG fixed point, β(g) = 0, the D dimen-sional field theory is invariant under the scale trans-formation

xµ → xµ/b , u→ b u

This is an invariance of the metric of the theory in

D + 1 dimensions. The unique solution is

2dxµdxµ + L2 du2

Or, using the length scale z = L2/u

ds2= L2 dxµdxµ + dz2

This is the space AdSD+1, and L is the AdS radius.

J. McGreevy, arXiv0909.0518Saturday, April 3, 2010

Bonus: AdSD+1 is a solution of Einstein’s equations

with a negative cosmological constant, and is a sym-

metric space; the full group of symmetries of the

metric is SO(D + 1, 1) (in Euclidean signature)

Bonus: AdSD+1 is a solution of Einstein’s equations

with a negative cosmological constant, and is a sym-

metric space; the full group of symmetries of the

metric is SO(D + 1, 1) (in Euclidean signature)

SO(D+1, 1) is the group of conformal transforma-

tions in D dimensions, and relativistic field theo-

ries at the RG fixed point are conformally invari-

The quantum phase transitions of metals in two dimensions

HARVARDTalk online: sachdev.physics.harvard.edu

The quantum phase transitions of metals in two dimensions

HARVARDTalk online: sachdev.physics.harvard.edu

Field theories with singularities along lines in momentum space:

also being studied by the AdS/CFT correspondence

Max Metlitski, Harvard

HARVARD

arXiv:1001.1153,and to appear

Eun Gook Moon, Harvard

Phys. Rev. B 80, 035117 (2009),and to appear

Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and

change in Fermi surface

StrangeMetal

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

STM measurements of Z(r), the energy asymmetry

in density of states in Bi2Sr2CaCu2O8+δ.

Jhinhwan Lee,

A. R. Schmidt,

Kim, H. Eisaki,

J. P. Sethna, and

Jhinhwan Lee,

A. R. Schmidt,

Kim, H. Eisaki,

J. P. Sethna, and

Strong anisotropy of electronic states between

x and y directions:Electronic

“Ising-nematic” order

Phase diagram as a function of coupling r

φ = 0φ = 0

Quantum criticality of Ising-nematic order

Phase diagram as a function of T and r

φ = 0

Quantumcritical

φ = 0rrc

Quantum criticality of Ising-nematic ordering

T ∗ ?

φ = 0

Quantumcritical

φ = 0rrc

T ∗ ?

φ = 0

Quantumcritical

φ = 0rrc

StrangeMetal ?

dx⊥ψ

†na(x⊥)

∂τ− ivF (n)

∂x⊥

ψna(x⊥)

Fermi liquid theory

Infinite number of 1+1 dimensional chiral fermionsSaturday, April 3, 2010

dd−1y ψ†

∂τ− ivF

∂x− κ

S1 = u0

dd−1y ψ†

aψ†bψbψa.

Focus on one extended patch,and scale towards a fixedk0 while keeping the Fermivelocity vF and Fermi sur-face curvature, κ fixed.

Requires a 2+1 dimensionalfield theory at every k0.

Fermi liquid theory

dd−1y ψ†

∂τ− ivF

∂x− κ

S1 = u0

dd−1y ψ†

aψ†bψbψa.

x = x/s2

y = y/s

τ = τ/s2

ψ = ψs(d+1)/2

u0 = u0s

(1−d)

Fermi liquid theory

dd−1y ψ†

∂τ− ivF

∂x− κ

S1 = u0

dd−1y ψ†

aψ†bψbψa.

Fermi liquid theory

x = x/s2

y = y/s

τ = τ/s2

ψ = ψs(d+1)/2

u0 = u0s

(1−d)

dd−1y ψ†

∂τ− ivF

∂x− κ

S1 = u0

dd−1y ψ†

aψ†bψbψa.

Fermi liquid theory

ImΣ(k,ω) =

ω2 u20

4πv2FκΛd−2 , d > 2

ω2 u20

4πv2Fκln(|ω|) , d = 2

The theory has an exact “Galilean” symmetry:

ψ(x, y) → exp

θ · y + θ2

ψ(x, y + θ x)

This implies for the fermion Green’s function

G(qx, qy,ω) = G(εq,ω)

where εk = vF qx + κq2y/2.

Fermi liquid theory

PFermi liquid theory

(qx, qy)

Field theory defined at k0

Field theory defined at k1

P(qx, q

y)Fermi liquid theory

P(qx, q

y)Fermi liquid theory

εq = εq ensures compatibility

of redundant 2+1 dimensional field theories.

Quantum criticality of Ising-nematic transition

Sc =Nf

dτc†kα (∂τ + εk) ckα

Sφc = − γ

φq (cos kx− cos ky)c†k+q/2,αck−q/2,α

d2rdτ(∂τφ)2 + c2(∇φ)2 + (r − rc)φ2 + uφ4

A φ fluctuation at wavevector q couples most efficiently tofermions near ±k0.

Expand fermion kinetic energy at wavevectors about k0

L = ψ†+

ζ∂τ − i∂x − ∂2

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

− λφψ†

+ψ+ + ψ†−ψ−

(∂yφ)2 +r

Theory of Ising-nematic transition

L = ψ†+

ζ∂τ − i∂x − ∂2

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

− λφψ†

+ψ+ − ψ†−ψ−

(∂yφ)2 +r

Theory of a Fermi surface minimally coupledto an Abelian or non-Abelian gauge field with φ ∼ Ax

L = ψ†+

ζ∂τ − i∂x − ∂2

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

− λφψ†

+ψ+ + ψ†−ψ−

(∂yφ)2 +r

Theory of Ising-nematic transition

L = ψ†+

ζ∂τ − i∂x − ∂2

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

− λφψ†

+ψ+ + ψ†−ψ−

(∂yφ)2 +r

After tuning the single parameter r ∼ λ − λc, and sendingζ → 0, L describes a critical theory with no coupling constants.There is a separate copy of this critical theory for each directionq. This theory has 2 independent exponents z and η, and thecorrelation length and susceptibility exponents are given by

z − 1; γ = 1

The fermion and order parameter Green’s functions obey thescaling forms

G(q,ω) = ξ2−ηΦψ

(qx+q2y)ξ

2,ωξz

; D(q,ω) = ξz−1Φφ

qyξ,ωξ

We have computed the exponents to three loops, and find z = 3and η = 0.06824 at this order.

FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Underlying SDW ordering quantum critical pointin metal at x = xm; has associated Ising-nematic order

Theory of quantum criticality in the cuprates

Increasing SDW orderIncreasing SDW order

LargeFermi

surface

StrangeMetal

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

Onset of d-wave superconductivity

hides the SDW critical point x = xm

Fluctuating, paired Fermi

pockets

LargeFermi

surface

StrangeMetal

Competition between SDW order and superconductivitymoves the actual SDW quantum critical point to x = xs < xm.

pockets

V. Galitski andS. Sachdev, Phys.Rev. B 79, 134512(2009).

E. G. Moon andS. Sachdev, Phys.Rev. B 80, 035117(2009)

LargeFermi

surface

StrangeMetal

pockets

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gap

Thermallyfluctuating

pockets

surface

StrangeMetal

Magneticquantumcriticality

Spin gap

Thermallyfluctuating

pockets

Physics of competition: d-wave SC and SDW“eat up” same pieces of the large Fermi surface.

xs = xm − c|∆|

xs = xm − c|∆|2

∆ is the fermion pairing gap.

Shift of ordering critical point in a metal due to onset of superconductivity

SDW ordering:

Ising-nematic ordering:

E.G. Moon and S. Sachdev, to appearSaturday, April 3, 2010

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

surface

StrangeMetal

d-waveSC

pockets

E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).

E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

surface

StrangeMetal

d-waveSC

R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).

pockets

Onset of superconductivity

disrupts SDW order, but

VBS/CDW/Ising-nematic ordering can

survive

VBS/CDW and/orIsing-nematic order

G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223

Similar phase diagram for CeRhIn5

Similar phase diagram for the pnictides

Ishida, Nakai, and HosonoarXiv:0906.2045v1

0 0.02 0.04 0.06 0.08 0.10 0.12

AFM Ort/

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.

Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy...

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