Black Holes for non-Fermi liquids and superconductivity

arousedpodunkUrban and Civil

Nov 15, 2013 (3 years and 9 months ago)

99 views

Black Holes for non-Fermi liquids and superconductivity
Sean Hartnoll
Harvard University
Work over the past two years and ongoing in collaboration with
Miranda Cheng,Frederik Denef,Tom Hartman,Chris Herzog,Diego Hofman,Gary
Horowitz,Cindy Keeler,Joe Polchinski,Subir Sachdev,Eva Silverstein,David Tong.
Feb.2010 { IPMU
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 1/23
Plan of talk
Context:non-Fermi liquids
Holographic approach I:with explicit fermions
Holographic approach II:without explicit fermions
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 2/23
Context:non-Fermi liquids
1
Invitation to the cuprates
2
Anomalous scalings
3
Fermi surface reconstruction
4
Absence of quasiparticles
5
Quantum critical scenario
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 3/23
Invitation to the cuprates

The most glamorous non-Fermi liquids are the cuprate high-T
c
superconductors.

What is a non-Fermi liquid?
 How does superconductivity emerge from a non-Fermi liquid?
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 4/23
Anomalous scalings

The`strange metal'regime is characterised by unconventional scaling
laws.Eg.

DC resistivity:  T,optical conductivity (!) !
0:65
.
[Plots from McKenzie et al.'97 and van der Marel et al.'03.]
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 5/23
Fermi surface reconstruction
 de Haas - van Alphen oscillations detect the size of Fermi surface.
 Underoped:Fermi pockets.,Overdoped:conventional Fermi surface
[Plots from Doiron-Leyraud et al.2007,Vignolle et al.2008]
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 6/23
Absence of quasiparticles
 Weakly interacting Fermi liquid has sharp quasiparticle excitations at
the Fermi surface (left below).
 The strange metallic region of the cuprates does not (right below).
 Some structure in ARPES,but broad.
[Plot from Ding et al.'96]
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 7/23
Possible scenario

Summary:
1
Scaling laws
2
Change in shape of Fermi surface
3
Absence of well dened quasiparticles

Consistent with a quantum critical point at T = 0 at a critical doping
controlling the strange metal region.
 Layered structure of cuprates suggests 2+1 dimensional critical
theory.

Such theories generically strongly coupled.Perturbative methods not
controlled.
)Turn to the holographic correspondence....
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 8/23
Holographic approach I:with explicit fermions
1
Renormalisation group and holography
2
Finite temperature and black holes
3
Fermi surfaces
4
Violation of Lifshitz-Kosevich scaling
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 9/23

Holographic correspondence:add the energy scale as an extra curved
spacetime dimension.
!
!"
#
!
$%
#
&'(
#
)"*%!(+
#

Curvature of the`holographic direction'contains the RG ow
information.
 Einstein's equations are the RG equations.
 Locality in spacetime and in energy on the same footing.
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 10/23
Finite temperature and density )black holes

A minimal dual framework to discuss charge and temperature is
Einstein-Maxwell theory
S
E
[A;g] =
Z
d
4
x
p
g


1
2
2

R +
6
L
2

+
1
4g
2
F
2

:

Unique solution with correct properties:dyonic black hole.
!
!"
#
!
$%
#
&
#
'(#)
#

Falling into the black hole )dissipation in quantum critical theory.
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 11/23
[Aside on T = 0 entropy]

These black holes have an entropy at zero temperature that may be
undesirable.
 Consider however the following plot of entropy over temperature as a
function of magnetic eld in Sr
3
Ru
2
O
7
 [Science,Sept.2009,Rost et al.]
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 12/23
Superconductivity without glue
[Gubser;Hartnoll,Herzog,Horowitz'08]

There is a natural instability of the black hole in the presence of
charged bulk matter.Charged scalars can lead to superconductivity.
!
!"
#
!
$%
#
&
#
'(#)
#
*
#
*
#
*
#
*
#
+
#+#+
#
+
#+#
#
+
#
+
#
!
!"
#
!
$%
#
&
#
'(#)
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
*
#
+,-
#
 Geometric instability.No`glue'or weakly coupled`pairing'required.
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 13/23
Criterion for instability
[Denef,Hartnoll'09]
 The critical temperature T
c
is determined by the charge and scaling
dimensions of operators in the quantum critical theory.
0.001
0.01
0.05
0.1
0.2
0.5
1.
2.
10.
0
1
2
3
4
5
6
0
1
2
3
4
5
6
g
q
D

Contour lines are T
c
=.[Cf.T
c
= 0:01T
F
?]
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 14/23
Fermi surfaces
[Sung-Sik Lee'09;Faulkner,Liu,McGreevy and Vegh'09]
 At zero temperature:peak in the fermion spectral function:
Imh i
R
(!;k).

Dispersion relation
!
v
F
+he
i 
!
2
= k k
F
:
 Looks like a (non-Landau for  <
1
2
) Fermi surface!
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 15/23
Saddle point (`large N') magnetic susceptibility
 Easy to compute   
@
2


0
@B
2

Plot result:
0
1
2
3
4
5
-
7
-
6
-
5
-
4
-
3
-
2
B
m
2
2
k
2
L
2
m
c
A
 Looks just like free (massive) bosons.
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 16/23
One loop free energy and quasinormal modes
[Denef,Hartnoll,Sachdev'09;Hartnoll,Hofman'09]

We derived formulae for the determinant as a sum over quasinormal
modes z
?
(`) of the black hole


1-loop,F
= 
jqBjAT
2
X
`
X
z
?
(`)
log

1
2






iz
?
(`)
2T
+
1
2




2
!
+UV:
 Using this formula exhibit quantum oscillations
 = A

T

;
ck
2
F
eB

cos
ck
2
F
eB
:
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 17/23
Violation of Lifshitz-Kosevich scaling
[Hartnoll,Hofman'09]

For free fermions,the amplitude of quantum oscillations at T B is
A  e
mT=B
:
 Seems to t data in underdoped and overdoped cuprates.

Strongly interacting fermions of the holographic correspondence give
A  e
T=B(T=)
21
:
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 18/23
Holographic approach II:without explicit fermions
1
Towards strange metallic holography
2
Lifshitz plus probe branes
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 19/23
Towards strange metals
[Hartnoll,Polchinski,Silverstein,Tong'09]

Model:Quantum critical sector coupled to gapped charge carriers.
Quantum critical
z
Charge carriers
E
g
self-
interaction
dissipation

Quantum critical sector has a dynamical scaling exponent z.

Results so far
 
T
2=z
J
t
;(!) 
J
t
!
2=z
:
z = 2 matches resistivity experiments and z = 3 (or maybe z = 2)
optical conductivity.
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 20/23
Probe brane setup

Quantum critical Liftshitz`bath'[Kachru,Liu,Mulligan]
ds
2
= L
2


dt
2
r
2z
+
dx
i
dx
i
r
2
+
dr
2
r
2

:
 Probe D brane bending into internal dimensions (charge carriers)
S = 
e
Z
dd
3
 V()
n
p
j
?
g +2
0
Fj;

Find D brane embedding and then compute conductivity etc.
 Conceptually similar to studies of mesons in Holographic QCD
[Karch-Katz,Sakai-Sugimoto...].
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 21/23
Compare and contrast

Fermions in charged black hole background
1
Maxwell eld interacts via gravity.
2
Eects of fermions are one loop (`1=N
0
).
3
Explicit fermions allows comparison with traditional experimental
probes.

Probe branes in Lifshitz background
1
Maxwell eld has self-interactions,no gravitational backreaction.
2
Eects of charge classical but suppressed by probe limit (`N
f
=N
0
c
).
3
Unclear to what extent charges are fermionic
(is this even well dened at strong coupling?).
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 22/23
Conclusions

Exotic but important materials cannot be described using weakly
interacting quasiparticles.
 The holographic correspondence is a nonperturbative method that
studies strongly interacting systems via`dual'classical description.

Finite temperature and density are dually described by black holes.
 Some initial results:
1
Onset of superconductivity without`glue'.
2
Non-Fermi liquid dispersion relations for fermionic excitations.
3
Quantum oscillations with potentially nonstandard temperature
dependence.
4
Scaling laws with similarities to those observed in`strange metals'.
Sean Hartnoll (Harvard U)
non-Fermi liquids and superconductivity
Feb.2010 23/23