Transport w/o
quasiparticles
Good metals, bad metals and insulators
Sean Hartnoll (Stanford) -- UCSB, Mar. 2013
1201.3917 w/ Diego Hofman
1212.2998 w/ Aristos Donos
Thursday, March 7, 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 t i =
6 0
Thursday, March 7, 13
Finite density transport
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:
s1
1
w HeVL
Only alternative to breaking momentum
conservation is to dilute the charge carriers. Makes
2
JP
spectral weight of delta function small:
⌧1
JJ
Thursday, March 7, 13
PP
Observed behaviors
Strange
metals
Conventional metals
(sharp Drude peak)
(unconventional
scalings)
Bad
metals
(no Drude peak,
violate MIR
bound)
Thursday, March 7, 13
Insulators
(vanishing dc
conductivity)
A conventional metal
Optical conductivity in graphene by
Li et al. 0807.3780
Thursday, March 7, 13
Metal-insulator transitions
Dramatic spectral
weight transfer from
Drude peak to
interband scales:
Itinerant to localized
charge
Uchida et al. ’91
Thursday, March 7, 13
Gapless insulators
Quantum spin liquid candidates show a power law
‘soft’ gap in the optical conductivity
Elsässer et al. 1208.1664
Thursday, March 7, 13
Herbertsmithite
by Pilon et al 1301.3501
A bad metal
La1.9Sr0.1CuO4 by Takenaka et al.’03,
from Hussey et al. ’04
Thursday, March 7, 13
Theory of sharp Drude peaks
Sharp 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:
O(kL ) =
Z
4
Y
i=1
d!i d2 ki
!
†
(k1 )
†
(k2 ) (k3 ) (k4 ) (k1 + k2
k3
In holographic models, often least irrelevant
operator is: J t (kL )
Thursday, March 7, 13
k4
kL )
Theory of sharp Drude peaks
Framework to treat ! perturbatively:
Memory matrix formalism. (cf. Rosch and Andrei, 1+1)
1
i! + M (!)
(!) =
M (!) =
Z
1/T
d
0
⌧
Ȧ(0)Q
!
1
,
i
QḂ(i )
QLQ
.
For case of scattering by a lattice
=
2
g 2 kL
~P
~
P
Im GR
OO (!, kL )
lim
!!0
!
.
g=0
(Hartnoll and Hofman, 1201.3917)
Thursday, March 7, 13
Semi-local criticality
(cf. Iqbal, Liu, Mezei)
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
r(T ) ⇠ T
2 (kL )
(Hartnoll and Hofman, 1201.3917)
(Confirmed numerically by Horowitz, Santos, Tong)
Theories with z < ∞ do not dissipate efficiently in
momentum-violating processes.
Thursday, March 7, 13
Making the lattice relevant
Claim: (at least some) metal-insulator transitions are
described by momentum-nonconserving operators
becoming relevant in the effective low energy theory.
(cf. Emery, Luther, Peschel, 1+1)
We found a holographic realization of this mechanism.
(Donos and Hartnoll, 1212.2998)
Simple theory
S=
Z
5
d x
p
✓
g R + 12
1
Fab F ab
4
1
Wab W ab
4
2
m
Ba B a
2
◆

2
Z
B^F ^W .
(Chern-Simons term not essential but
helps to find the IR geometries)
Thursday, March 7, 13
RG flow scenarios
The theory has (T=0) IR geometries both with
and without translation invariance
(Donos and Hartnoll, 1212.2998)
Thursday, March 7, 13
A technical simplification
To capture the physics without solving PDEs we use
a lattice that breaks translation invariance while
retaining homogeneity:
B (0) =
!2 + i!3 = e
ipx1
!2
(dx2 + idx3 )
(Invariant under Bianchi VII0 algebra,
cf. Nakamura-Ooguri-Park, Donos-Gauntlett, Kachru-Trivedi-....)
Beyond a simplification, realization of smectic metal
phases due to strong yet anisotropic lattice scattering
in the IR.
(cf. Emery, Fradkin, Kivelson, Lubensky;
Vishwanath, Carpentier)
Thursday, March 7, 13
Metal-insulator transition
(Donos and Hartnoll, 1212.2998)
The metallic and insulating IR geometries are:
2
ds =
2
dr
r2 dt2 + 2 + dx2R3 ,
r
p
A = 2 6 r dt ,
2
2
dr
dx
cr2 dt2 + 2 + 1/31 + r2/3 !22 + r1/3 !32 ,
cr
r
ds2 =
B = 0.
A = 0,
B = b !2 .
(cf. D’Hoker and Kraus)
0
0.15
sêH4pm3 L
WêHVm4 L
-10
-20
-30
0.05
-40
1
2
3
pêm
Thursday, March 7, 13
0.10
4
5
0.00
2.0
2.1 2.2
pêm
2.3
Spectral weight transfer
(Donos and Hartnoll, 1212.2998)
10
2.0
ReHsL HA.U.L
ReHsL HA.U.L
8
6
4
1.0
0.5
2
0
0.0
1.5
0.0
0.1
0.2
0.3
0.4
0.0
0.5
1.0
2.0
wê m
wêm
Metal
Insulator
(!) ⇠ !
Thursday, March 7, 13
1.5
4/3
at T = 0
2.5
3.0
Further comments
Can compute d.c. conductivities analytically:
metal:
insulator:
(T ) ⇠ T 2 (kL ) ,
(T ) ⇠ T 4/3 .
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?
Thursday, March 7, 13
Take home messages
Objective: non-quasiparticle language for transport.
Good metal: Effective low energy theory translation
invariant up to perturbative effects of momentumnonconserving 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.
Thursday, March 7, 13