What can we learn about finite density
systems from holography?
HOLOGRAPHIC THEORY relates one set of physical laws acting in a volume with
a different set of physical laws acting on a boundary surface, as represented here by the
juggler and her colorful two-dimensional image. The surface laws involve quantum
particles that have “color” charges and interact very like the quarks and gluons of standard
particle physics. The interior laws are a form of string theory and include the force of
gravity (experienced by the juggler), which is hard to describe in terms of quantum
mechanics. Nevertheless, the physics on the surface and in the interior are completely
equivalent, despite their radically different descriptions.
Mark Mezei (MIT)
Collaborators: Nabil Iqbal, Hong Liu,
Qimiao Si, David Vegh, Raghu Mahajan
Outline of the talk
Introduction and motivation
•  Why do we use holography?
•  What are we aiming at?
The role of AdS2
•  AdS2 as semi-local quantum liquid
•  Universal intermediate energy phase
Bifurcating criticality
•  Conformality lost
•  The spiral
Conclusion and outlook
Outline of the talk
Introduction and motivation
•  Why do we use holography?
•  What are we aiming at?
The role of AdS2
•  AdS2 as semi-local quantum liquid
•  Universal intermediate energy phase
Bifurcating criticality
•  Conformality lost
•  The spiral
Conclusion and outlook
Introduction and motivation
•  Gauge/gravity duality can be used to infer new dynamical phenomena in
strongly coupled theories
•  The bottom-up approach provides us with flexibility for model building and
enables the scan a family of theories
•  UV/IR connection maps the whole RG-flow
Strongly
correlated
fermionic
•  Intinerant electronic CM
systems without
quasi-particles
are an enormous
theoretical challenge
systems at finite density
•  Is it possible to identify the dynamics behind these materials from the study of
“Photoemission
During the experiments”
last two decades,on
these pillars are challenged at
our toy models?
S-S Lee
bothblack
experimental
theoretical level.
holes and
HL, McGreevy, Vegh
Cubrovic, Zaanen, Schalm
Solving Dirac
equation for !,
extracting
boundary values
Universality of 2point functions:
(controlled by Dirac
equation)
do not depend on which specific theory and operator we use. Quantum phase transitions
High Tc cuprates
Results will only depend on charge q and dimension m .
of heavy fermion metals
Will now use q and m as input parameters
Outline of the talk
Introduction and motivation
•  Why do we use holography?
•  What are we aiming at?
The role of AdS2
•  AdS2 as semi-local quantum liquid
•  Universal intermediate energy phase
Bifurcating criticality
•  Conformality lost
•  The spiral
Conclusion and outlook
RN black brane for finite density
Einstein gravity with U(1) gauge field:
•  Simplest dual system that gives
finite density in the boundary
theory
•  Extremal RNBH geometry:
�
�
2
2
R
dz
ds2 = 2 −f dt2 + d�x2 +
z
f
f = 1 + 3z 4 − 4z 3 At = µ(1 − z)
x
z
UV
IR
CF Td +
•  Near horizon geometry: AdS2 × R2
f = 6(z − 1)2 + . . .
AdS4
AdS2 × R2
�
UV
dd xµJ t
(z → 1)
•  In the IR we get emergent
conformal symmetry
•  Finite µ =⇒ finite ρ
can be read off the fall-off of the
gauge profile
SLQL
IR
FL
SC
AFM
AdS2 as semi-local quantum liquid
•  With the method of matched
asymptotic series we calculate
the low-energy Green’s function
for a neutral scalar operator:
GR =
b+ + b − G
a+ + a− G
G = c(νk ) ω
2νk
νk =
�
1
(m2 + k 2 )R22 +
4
•  Fourier transformation shows
GE ∼
1
τ 2δ
GE ∼ ex/ξ
1
+ν
2
(x � ξ)
δ=
ξ∼
1
µ∗
1
G(t) ∼ 2δ
t
(x � ξ)
•  Can also be seen from geodesics
•  Finite entropy density motivates
the interpretation as SLQL
Iqbal, Liu, MM: [arXiv:1105.4621]
Instabilities of AdS2
•  Violates the Third Law of Thermodynamics
•  BF bound violation: m2 R2 > −d2 /4
–  AdS4 BF bound: m2 R2 > −9/4
–  AdS2 BF bound: m2 R22 = m2 R2 /6 > −1/4 =⇒ m2 R2 > −3/2
•  Analysis of GR:
–  Scalars:
•  a+ (k) = 0 , pole in UHP
•  νk imaginary, bifurcating susceptibility
–  Fermions:
•  a+ (k) = 0 , Fermi surface
•  νk imaginary, back reaction of bulk fermions (instability)
•  Plethora of endpoints:
–  Holographic superconductors (superfluids)
–  Antiferromagnet
Iqbal, Liu, MM, Si: [arXiv:1003.0010]
–  Fermi liquid
Iqbal, Liu, MM: [arXiv:1105.4621], Hartnoll et al.: [arXiv:1105.3197,
1011.6469, 1011.2502, 1008.2828]
–  Lifshitz fixpoint
Karchu et al.: [arXiv:1007.2490, 0911.3586]
–  Spatially modulated phase
Ooguri et al.: [arXiv:1011.4144, 1007.2490]
Summary of SLQL
•  Intermediate energy phase from
gravity
•  Scaling symmetry
CF Td +
–  Infinite correlation time
–  Semi-local in the spatial directions
•  Shares similarities with cuprates
and heavy fermion metals (and their
DMFT description)
•  Many instabilities, endpoints can be
obtained from gravity
�
UV
dd xµJ t
SLQL
AFM
FL
SC
IR
Outline of the talk
Introduction and motivation
•  Why do we use holography?
•  What are we aiming at?
The role of AdS2
•  AdS2 as semi-local quantum liquid
•  Universal intermediate energy phase
Bifurcating criticality
•  Conformality lost
•  The spiral
Conclusion and outlook
Bifurcating criticality
•  Consider ν → 0 for neutral scalar:
�
u + k 2 /6
b+ + b− G
�
GR =
−−−→ GR =
a+ + a− G ω=0
α + α̃ u + k 2 /6
β
#
−−−→ GR = +
u=0
α log ω
β + β̃
(u → 0)
•  From the condensed side we get:
�2 × R2
AdS
AdS2 × R2
AdS4
IR
UV
ψ
z
Iqbal, Liu, MM: to appear
On the condensed side
•  The entropy density decreases, some of the deconfined excitations
of SLQL get confined
•  Had we started with a different bulk field (and consider ν → 0):
–  Charged scalar → AdS4 or Lifshitz in the IR
–  Charged scalar → electron star in the IR
–  No residual T = 0 entropy
Ising ferromagnet
�O�
Holographic system
�O�
T > Tc
0.2
H
0.1
�O�
�O1 �
T = Tc
�0.3
�0.2
�0.1
0.1
H
�0.1
�O�
T < Tc
�0.2
H
u<0
0.2
0.3
H
Conformality lost
•  The condensate is exponentially small, strongly reminiscent of the
Efimov effect and BKT scaling:
�
�
nπ
�On � ∼ exp − √
2 u
•  The physics is the same: conformality is lost:
tIR
Β�g, Α�
g�
g�
g�
Α�Α�
Α�Α�
Α�Α�
(a)
tUV
g
g
g�
gUV
t
(b)
FIG. 1: (a) A toy β-function. For α > α∗ there are fixed points at g± which are UV- and IR-stable
respectively; these fixed points merge at g∗ for α = α∗ , and disappear for α <Son
α∗ ; et
(b)al.:[arXiv:0905.4752]
The RG flow
√
of the coupling g as a function of t = ln µ in the non-conformal phase, with (tUV −tIR ) ∝ 1/ α∗ − α.
Outline of the talk
Introduction and motivation
•  Why do we use holography?
•  What are we aiming at?
The role of AdS2
•  AdS2 as semi-local quantum liquid
•  Universal intermediate energy phase
Bifurcating criticality
•  Conformality lost
•  The spiral
Conclusion and outlook
Conclusion and outlook
•  We argued that an AdS2 factor in the geometry diagnoses an
intermediate energy SLQL phase
–  Universal in holographic finite density system for some parameter range
–  Momentum dependence indicates that it the system separates into cluster with
some number of d.o.f.’s
•  We showed one non-trivial dynamical phenomenon in this phase,
bifurcating criticality
–  The susceptibility does not diverge
–  Spiral with a tower of “Efimov states”
•  It is of interest to explore the dynamics in this phase
–
–
–
–
How instabilities can be understood from AdS2 point of view?
NFL/FL transition
Iqbal, Liu, MM, Vegh: to appear
EFT description
Map the phase structure and study the free energy in detail
Iqbal, Liu, MM, Mahajan: to appear
Iqbal, Liu, MM: TASI lectures