transport w/o quasiparticlesqpt.physics.harvard.edu/simons/hartnoll.pdftransport w/o quasiparticles...

Transport w/o quasiparticles

Good metals, bad metals and insulators

Sean Hartnoll (Stanford) -- Caneel Bay, Jan. 2013

Based on:1201.3917 w/ Diego Hofman1212.2998 w/ Aristos Donos

(also work in progress with Barkeshli & Mahajan)

Sunday, February 3, 13

Black holes and dissipation

The single most important reason that holography is useful is the unique way that black holes act as classical dissipative systems.

E.g. Witczak-Krempa and Sachdev: holographic transport at quantum criticality satisfies sum rules that are obscured in a Boltzmann approach.

This talk will be about transport in strongly correlated finite density systems: hJ ti 6= 0


Finite density transportIf the total momentum (or any other operator that overlaps with the total current) is conserved, the d.c. conductivity is infinite.

The optical conductivity of a perfect metal:

Only alternative to breaking momentum conservation is to dilute the charge carriers. Makes spectral weight of delta function small:

If the total momentum (or any other operator that overlaps with the total current) is conserved, the d.c. conductivity is infinite.

The optical conductivity of a perfect metal:

1 w HeVL

s1

�2JP

�JJ�PP⌧ 1


Observed behaviors

Conventional metals(sharp Drude peak)

Insulators(vanishing dc conductivity)

Strangemetals

(unconventionalscalings)

Bad metals(no Drude peak,violate MIRbound)


A conventional metal

Optical conductivity in graphene byLi et al. 0807.3780


Metal-insulator transitionsDramatic spectral weight transfer from Drude peak

to interband scales: Itinerant to localized charge

Uchida et al. ’91Nicoletti et al ’11


Gapless insulatorsQuantum spin liquid candidates show a power law

‘soft’ gap in the optical conductivity

Elsässer et al. 1208.1664

Herbertsmithiteby Pilon et al 1301.3501


A bad metal

La1.9Sr0.1CuO4 by Takenaka et al.’03,from Hussey et al. ’04


Theory of sharp Drude peaksSharp Drude peak ⇔ Momentum relaxation rate ! small

Can treat momentum-nonconserving operators (remnant of UV lattice) as perturbations of a translationally invariant effective IR theory.

E.g. Umklapp scattering in a Fermi liquid:

In holographic models, often least irrelevant operator is:

J t(kL)

O(kL) =

Z 4Y

i=1

d!id2ki

! †(k1)

†(k2) (k3) (k4)�(k1 + k2 � k3 � k4 � kL)


Theory of sharp Drude peaksFramework to treat ! perturbatively: Memory matrix formalism. (cf. Rosch and Andrei, 1+1)

For case of scattering by a lattice

M(!) =

Z 1/T

0d�

⌧A(0)Q i

! �QLQQB(i�)

�.

�(!) =1

�i! +M(!)��1� ,

(Hartnoll and Hofman, 1201.3917)

� =g2k2L�~P ~P

lim!!0

ImGROO(!, kL)!

��g=0

.


Semi-local criticality

Common in holography that IR geometries have z = ∞ (with or without ground state entropy).

Scaling of time but not space ⇒ efficient low energy dissipation in momentum-violating processes.

Find e.g. dc resistivity

(Hartnoll and Hofman, 1201.3917)

(cf. Iqbal, Liu, Mezei)

(Confirmed numerically by Horowitz, Santos, Tong)

Theories with z < ∞ do not dissipate efficiently in momentum-violating processes.

r(T ) ⇠ T 2�(kL)


Making the lattice relevantClaim: (at least some) metal-insulator transitions are described by momentum-nonconserving operators becoming relevant in the effective low energy theory.

We found a holographic realization of this mechanism.

Simple theory

(cf. Emery, Luther, Peschel, 1+1)

(Donos and Hartnoll, 1212.2998)

S =

Zd

5x

p�g

✓R+ 12� 1

4FabF

ab � 1

4WabW

ab � m

2

2BaB

a

◆�

2

ZB ^ F ^W .

(Chern-Simons term not essential but helps to find the IR geometries)


RG flow scenarios

89

,QVXODWRU8QVWDEOHIL[HG�SRLQW

0HWDOOLF$G6��[�5�

�

89

,QVXODWRU

0HWDOOLF$G6��[�5�

�

The theory has (T=0) IR geometries both with and without translation invariance



To capture the physics without solving PDEs we use a lattice that breaks translation invariance while retaining homogeneity:

Beyond a simplification, realization of smectic metal phases due to strong yet anisotropic lattice scattering in the IR.

A technical simplification

(Invariant under Bianchi VII0 algebra,cf. Nakamura-Ooguri-Park, Donos-Gauntlett, Kachru-Trivedi-....)

B(0) = �!2

!2 + i!3 = e

ipx1 (dx2 + idx3)

(cf. Emery, Fradkin, Kivelson, Lubensky)


Metal-insulator transition

The metallic and insulating IR geometries are:

ds

2 = �r

2dt

2 +dr

2

r

2+ dx

2R3 , A = 2

p6 r dt , B = 0 .

ds

2 = �cr

2dt

2 +dr

2

cr

2+

dx

21

r

1/3+ r

2/3!

22 + r

1/3!

23 , A = 0 , B = b!2 .


2.0 2.1 2.2 2.30.00

0.05

0.10

0.15

pêm

sêH4pm

3 L1 2 3 4 5

-40

-30

-20

-10

0

pêm

WêHVm

4 L

(cf. D’Hoker and Kraus)


Spectral weight transfer

0.0 0.1 0.2 0.3 0.40

2

4

6

8

10

wêm

ReHsLHA.U

.L

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

wêm

ReHsLHA

.U.L

Metal Insulator

�(!) ⇠ !4/3 at T = 0



Can compute d.c. conductivities analytically:

In between the metallic and insulating phases we found bad metals with no Drude peak and large resistivities.

‘Mid-infrared peak’ in insulating phase.

No immediate connection to commensurability. Insulators without underlying Mott physics?

Further comments

metal: �(T ) ⇠ T�2�(kL) ,insulator: �(T ) ⇠ T 4/3 .


Objective: non-quasiparticle language for transport.

Good metal: Effective low energy theory translation invariant up to perturbative effects of momentum-nonconserving operators.

Effects become relevant: metal-insulator transition.

Holography precisely realizes this scenario.

Simple model exhibits experimental features that are difficult to otherwise describe in a controlled way: major spectral weight transfer, bad metals, insulators with power law gaps.

Take home messages


transport w/o quasiparticlesqpt.physics.harvard.edu/simons/hartnoll.pdftransport w/o quasiparticles...

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