Strongly coupled metals and insulators Sean Hartnoll (Stanford) - - PowerPoint PPT Presentation

Strongly coupled metals and insulators

Sean Hartnoll (Stanford) Gauge/gravity duality 2013 @ MPI, Munich

Saturday, July 27, 13

Take home buzzwords

Memory matrix. Wiedemann-Franz law. Insulators.

Saturday, July 27, 13

Charge transport at strong coupling

• Most computations of conductivities
• etc. use the Boltzmann equation:

− ~ E · @fk @~ k = −Iei[fk] − Iee[fk]

• Assumes long lived ‘quasiparticles’,

not useful at strong coupling.

• First objective: effective field theory

framework for strongly coupled transport.

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• Theorem (1960s, easy):

If there exists a conserved quantity P that overlaps with the electrical current operator J, i.e. Then the d.c. conductivity is infinite:

• Example: absence of lattice and

impurities ⇒momentum conserved

χP J 6= 0

σ ∼ χ2

P J

χP P δ(ω)

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• Consequence:

Suppose conservation of P is violated

• nly by an irrelevant operator O in the

low energy effective theory. Then the d.c. conductivity is large:

σ = χ2

P J

χP P 1 Γ

Relaxation rate

• At low temperatures, dominant T

dependence is from Γ. Thus, resistivity:

ρ ∼ Γ

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Γ EF Γ ≪ EF ω σ1

There is a sum rule that: is given by fixed UV data.

Z ∞ Re σ(ω)dω

• The small scale Γ furthermore gives

a Drude peak:

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Almost conserved quantities

• (From Andrei-Shimshoni-Rosch ’03,

studying Luttinger liquids)

D = 1 2 χ2

JP

χP P

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Memory matrix

• It is a theorem that

σ = lim

ω→0

X

AB

χJAM −1

AB(ω)χBJ

• With

MAB(ω) = Z 1/T dλ ⌧ ˙ A(0)Q i ω − QLQQ ˙ B(iλ)

• Almost-conserved quantities
• Where L is the `Liouville operator’ LA =[H,A]

and Q projects onto the space of operators

• rthogonal to the set A,B,... we kept.

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• Remnant of UV lattice in IR is a

momentum-carrying operator O(kL).

• If O is irrelevant, Γ can be computed

perturbatively in the IR coupling g of O:

Γ = g2k2

L

χP P lim

ω→0

Im GR

OO(ω, kL)

ω

• g=0

Hartnoll-Hofman @ 1201.3917

(case of impurities: Hartnoll-Kovtun-Muller-Sachdev @ 0706.3215)

Saturday, July 27, 13

• Results quoted so far are all derived

using the ‘‘memory matrix formalism’’.

• This formalism builds around almost-

conserved quantities, and is the correct way to think about charge transport in strongly correlated metallic systems. Suggested reading: Hartnoll-Hofman @ 1201.3917 Mahajan-Barkeshli-Hartnoll @ 1304.4249 Andrei-Shimshoni-Rosch @ cond-mat/0307578

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Fermi liquids:

The physics is at nonzero momentum

• Famously (1930s!): a clean Fermi

liquid has a low T electrical resistivity

ρ ∼ T 2

• An effective field theory derivation of

this result reveals nontrivial physics.

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• Recall the formula for relaxation rate:

Γ = g2k2

L

χP P lim

ω→0

Im GR

OO(ω, kL)

ω

• g=0
• Significant relaxation requires low

energy spectral weight (i.e. on shell excitations) at nonzero momentum kL.

• Clearly, such excitations do not exist in

e.g. a Lorentz invariant theory:

ω ∼ k ⇒ Γ ∼ e−kL/T

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• In a Fermi Liquid, low energy

excitations live on the Fermi surface.

• Leading irrelevant operator with finite

momentum is the umklapp operator:

O(kL) = Z 4 Y

i=1

dωid2ki ! ψ†(k1)ψ†(k2)ψ(k3)ψ(k4)δ(k1 + k2 − k3 − k4 − kL)

Saturday, July 27, 13

• Using the RG flow for Fermi surfaces
• f Polchinski (hep-th/9210046), the

umklapp operator O(w,k) has scaling dimension 𝚬=1. It is irrelevant.

• Dimensional analysis then gives

ρ ∼ Γ ∼ T 2

• Lesson: resistivity of a Fermi liquid

depends upon the interplay of two momentum scales: kL and kF.

[Hartnoll-Hofman (1201.3917)]

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Holography at nonzero density 101

• Density ⇒Electric flux at boundary.
• IR physics determined by near

horizon geometry.

• Saturday, July 27, 13

• An interesting class of IR geometries

is AdS2 x R2 or conformal to AdS2 x R2

• Via formula for Γ, power law resistivity:

ρ ∼ lim

ω→0

Im GR

JtJt(ω, kL)

ω ∼ T 2∆(kL)

Hartnoll-Hofman (1201.3917), Hartnoll-Shaghoulian (1203.4236) Anantua-Hartnoll-Martin-Ramirez (1210.1590) Verified with numerical lattice by Horowitz-Santos-Tong (1204.0519)

(space does not scale!)

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Wiedemann-Franz law

• Electrical and thermal conductivity:

L ≡  T = ⇡2 3 ≡ L0 = 1 T 2 R d✏✏2f 0

FD(✏)

R d✏f 0

FD(✏)

• Ratio of conductivities in a Fermi liquid:
• The WF law requires:

(i) long lived electronic quasiparticles. (ii) no additional heat carriers. (iii) elastic scattering.

✓ Jx Qx ◆ = ✓ σ αT αT ¯ κT ◆ ✓ Ex (rxT)/T ◆

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Lots of conserved quantities in a Fermi liquid

• Low energy description: Patchwise excitations
• f a Fermi surface (Shankar, Polchinski).
• Infinitely many conserved quantities:

θ

kF (θ)

k

• Each patch theory

is a free fermion:

Hθ = Z dk ✏θk c†

θkcθk

δnθk ≡ c†

θkcθk

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Ratio in Non-Fermi liquids

• Strong interactions to arbitrarily low energies.
• Simplest expectation: only almost-conserved

quantity in effective low energy theory is total momentum P.

σ = χ2

JP

• M −1

P P ,

κ = 1 T χ2

QP

• M −1

P P

κ σT = 1 T 2 χ2

QP

χ2

JP

Universal ratio of thermodynamic susceptilibites

• Implying that:

(Mahajan-Barkeshli-Hartnoll @ 1304.4249)

Saturday, July 27, 13

Some NFLs violate WF

• YbRh2Si2,

YbAgGe, c-axis CeCoIn5, YBCO.

• All suggestive
• f strong

interactions?

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Some “NFLs” satisfy WF

a-axis CeCoIn5, CeRhIn5, Sr3Ru2O7 .

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Quick lessons from data

• Not all linear in temperature

resistivities are the same.

• Some are suggestive of strong

coupling physics (violate WF).

• Others necessarily have a

quasiparticle description (satisfy WF).

Mahajan-Barkeshli-Hartnoll @ 1304.4249

Saturday, July 27, 13

• So far, have described good metals:

momentum non-conservation described by irrelevant operators in IR. Physics captured by (i) formula for Γ and (ii) knowledge of IR kinematics.

Holographic insulators

Momentum space becomes relevant

• If the lattice operators becomes relevant

in the IR, we might expect to obtain insulators or perhaps incoherent metals.

(Donos-Hartnoll @ 1212.2998)

Saturday, July 27, 13

Localization transitions 101

• Metal-insulator transitions are dramatic

phenomena: re-arrangement of degrees

• f freedom from itinerant to localized.
• Spectral weight

transfer from the Drude peak to the UV scale.

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Theories of localization

• Band insulators
• Anderson localization (impurities):

Free electrons in random potential have localized wavefunctions.

• Mott transition (charge commensurability):

Electrons ‘jam’ at half filling. Low energy excitations particle-hole symmetric:

review: Dobrosavljevic @ 1112.6166

χP J = 0

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Holographic insulator

S = Z d5x√−g ✓ R + 12 − 1 4FabF ab − 1 4WabW ab − m2 2 BaBa ◆ − κ 2 Z B ∧ F ∧ W .

• Bulk action:
• Used a helical lattice to avoid solving PDEs:

B(0) = λ ω2

ω2 + iω3 = eipx1 (dx2 + idx3)

• cf. Ooguri-Park (1007.3737), Donos-Gauntlett (1109.3866),

Iizuka-Kachru-Kundu-Narayan-Sircar-Trivedi (1201.4861)

• Objective: realize a new type of localization.

Main input from holography is that operators get O(1) anomalous dimensions.

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• As a function of UV

parameters, IR geometry undergoes a phase transition when lattice becomes relevant.

ds2 = −cr2dt2 + dr2 cr2 + dx2

1

r1/3 + r2/3ω2

2 + r1/3ω2 3 ,

A = 0 , B = b ω2 .

• Zero temperature IR geometry

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• Spectral weight transfer!

0.0 0.1 0.2 0.3 0.4 2 4 6 8 10 ΩêΜ ReHΣL HA.U.L

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 ΩêΜ ReHΣL HA.U.L

Metal Insulator

σ(ω) ∼ ω4/3 at T = 0

(Donos-Hartnoll @ 1212.2998)

Saturday, July 27, 13

Conclusions

• Strongly coupled metallic transport

should be organized around almost conserved quantities: Memory matrix.

• The Wiedemann-Franz law differentiates

non-Fermi liquids with and without quasiparticles.

• Holography enables the realization of a

new, strong coupling, mechanism of charge localization -- lattice scattering becomes relevant in the IR.

Saturday, July 27, 13