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

63
Phys. Rev. Lett. 1990 Saturday, April 3, 2010

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

Page 1: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Phys. Rev. Lett. 1990

Saturday, April 3, 2010

Page 2: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Ultracold 87Rbatoms - bosons

Superfluid-insulator transition

Saturday, April 3, 2010

Page 3: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

σ =4e2

Σ, a universal number.

Saturday, April 3, 2010

Page 4: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Quantum critical transport

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

Quantum “perfect fluid”with shortest possible

thermal equilibration time, τR

τR kBT

Saturday, April 3, 2010

Page 5: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Saturday, April 3, 2010

Page 6: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

σ =4e2

ωkBT

ωkBT

K. Damle and S. Sachdev, 1997

1

Saturday, April 3, 2010

Page 7: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

σ =4e2

ωkBT

ωkBT

K. Damle and S. Sachdev, 1997

Collisionless

1

Saturday, April 3, 2010

Page 8: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

σ =4e2

ωkBT

ωkBT

K. Damle and S. Sachdev, 1997

Collision-dominated

1

Saturday, April 3, 2010

Page 9: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

CFT3

ψ = 0 ψ = 0

S =

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

u

2|ψ|4

Saturday, April 3, 2010

Page 10: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Saturday, April 3, 2010

Page 11: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Saturday, April 3, 2010

Page 12: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

• 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).

Saturday, April 3, 2010

Page 13: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

• 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).

Saturday, April 3, 2010

Page 14: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

• 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).

Saturday, April 3, 2010

Page 15: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

• 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).

Saturday, April 3, 2010

Page 16: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Saturday, April 3, 2010

Page 17: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Saturday, April 3, 2010

Page 18: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Field theories in D spacetime dimensions are char-

acterized by couplings g which obey the renormal-

ization group equation

udg

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.

Saturday, April 3, 2010

Page 19: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Field theories in D spacetime dimensions are char-

acterized by couplings g which obey the renormal-

ization group equation

udg

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.

Saturday, April 3, 2010

Page 20: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Saturday, April 3, 2010

Page 21: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

ds2=

u

L

2dxµdxµ + L2 du2

u2.

Or, using the length scale z = L2/u

ds2= L2 dxµdxµ + dz2

z2.

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

Saturday, April 3, 2010

Page 22: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Page 23: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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)

Saturday, April 3, 2010

Page 24: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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-

ant.

Saturday, April 3, 2010

Page 25: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Saturday, April 3, 2010

Page 26: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Saturday, April 3, 2010

Page 27: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

The quantum phase transitions of metals in two dimensions

HARVARDTalk online: sachdev.physics.harvard.edu

Saturday, April 3, 2010

Page 28: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Saturday, April 3, 2010

Page 29: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Max Metlitski, Harvard

HARVARD

arXiv:1001.1153,and to appear

Eun Gook Moon, Harvard

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

Saturday, April 3, 2010

Page 30: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Γ Γ

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

change in Fermi surface

StrangeMetal

Saturday, April 3, 2010

Page 31: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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.

Saturday, April 3, 2010

Page 32: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

B

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

AC

D

STM measurements of Z(r), the energy asymmetry

in density of states in Bi2Sr2CaCu2O8+δ.

Saturday, April 3, 2010

Page 33: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

B

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

AC

D

Strong anisotropy of electronic states between

x and y directions:Electronic

“Ising-nematic” order

STM measurements of Z(r), the energy asymmetry

in density of states in Bi2Sr2CaCu2O8+δ.

Saturday, April 3, 2010

Page 34: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

rrc

Phase diagram as a function of coupling r

φ = 0φ = 0

Quantum criticality of Ising-nematic order

Saturday, April 3, 2010

Page 35: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

T

Phase diagram as a function of T and r

φ = 0

Quantumcritical

φ = 0rrc

TI-n

Quantum criticality of Ising-nematic ordering

Saturday, April 3, 2010

Page 36: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

T

Phase diagram as a function of T and r

T ∗ ?

φ = 0

Quantumcritical

φ = 0rrc

TI-n

Quantum criticality of Ising-nematic ordering

Saturday, April 3, 2010

Page 37: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

T

Phase diagram as a function of T and r

T ∗ ?

φ = 0

Quantumcritical

φ = 0rrc

TI-n

StrangeMetal ?

Quantum criticality of Ising-nematic ordering

Saturday, April 3, 2010

Page 38: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

SFL =

dΩn

dx⊥ψ

†na(x⊥)

∂τ− ivF (n)

∂x⊥

ψna(x⊥)

Fermi liquid theory

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

Page 39: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

S0 =

dx

dd−1y ψ†

a

ζ∂

∂τ− ivF

∂x− κ

2∇2

y

ψa

S1 = u0

dx

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

Saturday, April 3, 2010

Page 40: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

S0 =

dx

dd−1y ψ†

a

ζ∂

∂τ− ivF

∂x− κ

2∇2

y

ψa

S1 = u0

dx

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

Saturday, April 3, 2010

Page 41: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

S0 =

dx

dd−1y ψ†

a

ζ∂

∂τ− ivF

∂x− κ

2∇2

y

ψa

S1 = u0

dx

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)

Saturday, April 3, 2010

Page 42: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

S0 =

dx

dd−1y ψ†

a

ζ∂

∂τ− ivF

∂x− κ

2∇2

y

ψa

S1 = u0

dx

dd−1y ψ†

aψ†bψbψa.

Fermi liquid theory

ImΣ(k,ω) =

ω2 u20

4πv2FκΛd−2 , d > 2

ω2 u20

4πv2Fκln(|ω|) , d = 2

Saturday, April 3, 2010

Page 43: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

The theory has an exact “Galilean” symmetry:

ψ(x, y) → exp

−i

vFκ

θ · y + θ2

2x

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

Saturday, April 3, 2010

Page 44: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

PFermi liquid theory

(qx, qy)

Field theory defined at k0

Saturday, April 3, 2010

Page 45: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Field theory defined at k1

P(qx, q

y)Fermi liquid theory

Saturday, April 3, 2010

Page 46: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

P(qx, q

y)Fermi liquid theory

εq = εq ensures compatibility

of redundant 2+1 dimensional field theories.

Saturday, April 3, 2010

Page 47: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Quantum criticality of Ising-nematic transition

Sc =Nf

α=1

k

dτc†kα (∂τ + εk) ckα

Sφc = − γ

Nf

α=1

k,q

φq (cos kx− cos ky)c†k+q/2,αck−q/2,α

Sφ =

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

Saturday, April 3, 2010

Page 48: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Expand fermion kinetic energy at wavevectors about k0

Saturday, April 3, 2010

Page 49: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

L = ψ†+

ζ∂τ − i∂x − ∂2

y

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

y

ψ−

− λφψ†

+ψ+ + ψ†−ψ−

+

12g

(∂yφ)2 +r

2φ2

Theory of Ising-nematic transition

Saturday, April 3, 2010

Page 50: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

L = ψ†+

ζ∂τ − i∂x − ∂2

y

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

y

ψ−

− λφψ†

+ψ+ − ψ†−ψ−

+

12g

(∂yφ)2 +r

2φ2

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

Saturday, April 3, 2010

Page 51: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

L = ψ†+

ζ∂τ − i∂x − ∂2

y

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

y

ψ−

− λφψ†

+ψ+ + ψ†−ψ−

+

12g

(∂yφ)2 +r

2φ2

Theory of Ising-nematic transition

Saturday, April 3, 2010

Page 52: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

L = ψ†+

ζ∂τ − i∂x − ∂2

y

ψ+ + ψ†

−ζ∂τ + i∂x − ∂2

y

ψ−

− λφψ†

+ψ+ + ψ†−ψ−

+

12g

(∂yφ)2 +r

2φ2

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

ν =1

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

z

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

Saturday, April 3, 2010

Page 53: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

T*

Saturday, April 3, 2010

Page 54: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

Onset of d-wave superconductivity

hides the SDW critical point x = xm

Theory of quantum criticality in the cuprates

Fluctuating, paired Fermi

pockets

T*

Saturday, April 3, 2010

Page 55: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

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

Theory of quantum criticality in the cuprates

Fluctuating, paired Fermi

pockets

T*

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

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

Saturday, April 3, 2010

Page 56: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

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

Theory of quantum criticality in the cuprates

Fluctuating, paired Fermi

pockets

T*

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

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

Saturday, April 3, 2010

Page 57: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin density wave (SDW)

Spin gap

Thermallyfluctuating

SDW

d-wavesuperconductor

Theory of quantum criticality in the cuprates

Fluctuating, paired Fermi

pockets

T*

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

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

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

Saturday, April 3, 2010

Page 58: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin density wave (SDW)

Spin gap

Thermallyfluctuating

SDW

d-wavesuperconductor

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

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

Theory of quantum criticality in the cuprates

Fluctuating, paired Fermi

pockets

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

T*

Saturday, April 3, 2010

Page 59: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Page 60: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

d-waveSC

T

Tsdw

Fluctuating, paired Fermi

pockets

T*

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

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

Saturday, April 3, 2010

Page 61: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

d-waveSC

T

Tsdw

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

Fluctuating, paired Fermi

pockets

Onset of superconductivity

disrupts SDW order, but

VBS/CDW/Ising-nematic ordering can

survive

VBS/CDW and/orIsing-nematic order

TI-n

Saturday, April 3, 2010

Page 62: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

Similar phase diagram for CeRhIn5

Saturday, April 3, 2010

Page 63: Phys. Rev. Lett. 1990qpt.physics.harvard.edu/talks/girvinfest.pdfThe RG equation is local in energy scale, i.e. the RHS does not depend upon u. Saturday, April 3, 2010. Field theories

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

150

100

50

0

SC

Ort

AFM Ort/

Tet

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.

Saturday, April 3, 2010