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
Sunday, February 3, 13
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
Sunday, February 3, 13
Observed behaviors
Conventional metals(sharp Drude peak)
Insulators(vanishing dc conductivity)
Strangemetals
(unconventionalscalings)
Bad metals(no Drude peak,violate MIRbound)
Sunday, February 3, 13
A conventional metal
Optical conductivity in graphene byLi et al. 0807.3780
Sunday, February 3, 13
Metal-insulator transitionsDramatic spectral weight transfer from Drude peak
to interband scales: Itinerant to localized charge
Uchida et al. ’91Nicoletti et al ’11
Sunday, February 3, 13
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
Sunday, February 3, 13
A bad metal
La1.9Sr0.1CuO4 by Takenaka et al.’03,from Hussey et al. ’04
Sunday, February 3, 13
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)
Sunday, February 3, 13
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
.
Sunday, February 3, 13
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)
Sunday, February 3, 13
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)
Sunday, February 3, 13
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
(Donos and Hartnoll, 1212.2998)
Sunday, February 3, 13
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)
Sunday, February 3, 13
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 .
(Donos and Hartnoll, 1212.2998)
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)
Sunday, February 3, 13
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
(Donos and Hartnoll, 1212.2998)
Sunday, February 3, 13
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 .
Sunday, February 3, 13
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
Sunday, February 3, 13