holography and mottness: a discrete marriage
TRANSCRIPT
Holography and Mottness: a Discrete Marriage
Thanks to: NSF, EFRC (DOE)
M. Edalati R. G. LeighKa Wai Lo
Mott Problem
emergentgravity
Mott Problem
What interacting problemscan we solve in quantum mechanics?
#of solvable interactingQMproblems
#of interactingQMproblems= 0
#of solvable interactingQMproblems
#of interactingQMproblems= 0
interacting systems H = T + V
interacting systems H = T + V
easy
interacting systems H = T + V
impossibleeasy
interacting systems H = T + V
rs = �V �/�T �
rs < 1
impossibleeasy
interacting systems H = T + V
rs = �V �/�T �
rs < 1
`free’ systems(perturbative interactions)
metals,Fermi liquids,...
impossibleeasy
interacting systems H = T + V
rs = �V �/�T �
rs < 1
`free’ systems(perturbative interactions)
metals,Fermi liquids,...
impossibleeasy
p
p
1
2
Figure 1: Fermi sea (shaded) with two low-lying excitations, an electron atp1 and a hole at p2.
that something very different might emerge. All we can do here is to check
the guess for consistency (naturalness), and compare it with experiment.
Begin by examining the free action∫
dt d3p{
iψ†σ(p)∂tψσ(p) − (ε(p) − εF)ψ†
σ(p)ψσ(p)}
. (12)
Here σ is a spin index and εF is the Fermi energy. The single-electron energy
ε(p) would be p2/2m for a free electron, but in the spirit of writing down the
most general possible action we make no assumption about its form.5 The
ground state of this theory is the Fermi sea, with all states ε(p) < εF filled
and all states ε(p) > εF empty. The Fermi surface is defined by ε(p) = εF.
Low lying excitations are obtained by adding an electron just above the Fermi
surface, or removing one (producing a hole) just below, as shown in figure 1.
Now we need to ask how the fields behave as we scale all energies by a
factor s < 1. In the relativistic case, the momentum scaled with the energy,5A possible p-dependent coefficient in the time-derivative term has been absorbed into
the normalization of ψσ(p).
13
short-rangerepulsions are irrelevant
interacting systems H = T + V
rs = �V �/�T �
rs < 1
`free’ systems(perturbative interactions)
metals,Fermi liquids,...
impossibleeasy
p
p
1
2
Figure 1: Fermi sea (shaded) with two low-lying excitations, an electron atp1 and a hole at p2.
that something very different might emerge. All we can do here is to check
the guess for consistency (naturalness), and compare it with experiment.
Begin by examining the free action∫
dt d3p{
iψ†σ(p)∂tψσ(p) − (ε(p) − εF)ψ†
σ(p)ψσ(p)}
. (12)
Here σ is a spin index and εF is the Fermi energy. The single-electron energy
ε(p) would be p2/2m for a free electron, but in the spirit of writing down the
most general possible action we make no assumption about its form.5 The
ground state of this theory is the Fermi sea, with all states ε(p) < εF filled
and all states ε(p) > εF empty. The Fermi surface is defined by ε(p) = εF.
Low lying excitations are obtained by adding an electron just above the Fermi
surface, or removing one (producing a hole) just below, as shown in figure 1.
Now we need to ask how the fields behave as we scale all energies by a
factor s < 1. In the relativistic case, the momentum scaled with the energy,5A possible p-dependent coefficient in the time-derivative term has been absorbed into
the normalization of ψσ(p).
13
short-rangerepulsions are irrelevant
strongly coupled systems(non-perturbative)
QCD, high-temperaturesuperconductivity,...
rs � 1
why is strong coupling hard?
UV
why is strong coupling hard?
UV
IR
why is strong coupling hard?
emergentlow-energy physics
(new degrees of freedom not in UV)
UV
IR
why is strong coupling hard?
emergentlow-energy physics
(new degrees of freedom not in UV)
UV
IR
why is strong coupling hard?
emergentlow-energy physics
(new degrees of freedom not in UV)
UV
IR
why is strong coupling hard?
emergentlow-energy physics
(new degrees of freedom not in UV)
UV
IR
why is strong coupling hard?
emergentlow-energy physics
(new degrees of freedom not in UV)
UV
IR
?
What computational tools do we have forstrongly correlated electron matter?
Is strong coupling hard because we are in the wrong dimension?
Yes
UV
QFT
L = T − gϕ4 · · ·
UV
QFTIR ??
L = T − gϕ4 · · ·
UV
QFTIR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
L = T − gϕ4 · · ·
UV
QFTIR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
L = T − gϕ4 · · ·
UV
QFT
g = 1/ego
coupling constant
IR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
L = T − gϕ4 · · ·
UV
QFT
g = 1/ego
coupling constant
IR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
dg(E)
dlnE= β(g(E))
locality in energy
L = T − gϕ4 · · ·
UV
QFT
g = 1/ego
coupling constant
IR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
dg(E)
dlnE= β(g(E))
locality in energy
implement E-scaling with an extra dimension
L = T − gϕ4 · · ·
UV
QFT
g = 1/ego
coupling constant
IR ??
gauge-gravity duality(Maldacena, 1997)
UV
QFTIR gravity
Wilsonianprogram
(fermions: new degrees of
freedom)
dg(E)
dlnE= β(g(E))
locality in energy
implement E-scaling with an extra dimension
L = T − gϕ4 · · ·
what’s the geometry?
what’s the geometry?
dg(E)
dlnE= β(g(E))
scale invariance(continuous)
= 0
what’s the geometry?
E → λExµ → xµ/λ
dg(E)
dlnE= β(g(E))
scale invariance(continuous)
= 0
what’s the geometry?
E → λExµ → xµ/λ
solve Einstein equations
dg(E)
dlnE= β(g(E))
scale invariance(continuous)
= 0
what’s the geometry?
E → λExµ → xµ/λ
solve Einstein equations
ds2 =� u
L
�2ηµνdx
µdxν +
�L
E
�2
dE2
anti de-Sitter space
dg(E)
dlnE= β(g(E))
scale invariance(continuous)
= 0
Holography
Λ1Λ2Λ3Λ4
Λ5Λ6
ds2 = −dt2 + d�x2
Holography
Holography
AdSd+1
ds2 =L2
r2�−dt2 + d�x2 + dr2
�
r = L2/E
r → 0
Holography
AdSd+1
ds2 =L2
r2�−dt2 + d�x2 + dr2
�
r = L2/E
r → 0
geometrize RG flowβ(g) is local
{t, �x, r} → {λt,λ�x,λr}
symmetry:
Holography
AdSd+1
ds2 =L2
r2�−dt2 + d�x2 + dr2
�
r = L2/E
r → 0
geometrize RG flowβ(g) is local
{t, �x, r} → {λt,λ�x,λr}
symmetry:
QFT in d-dimensions
weakly-coupledclassical gravity
in d+1
UV
QFTIR
dual construction
UV
QFTIR
operatorsO
dual construction
UV
QFTIR
operatorsOφ
fields
dual construction
UV
QFTIR
operatorsOφ
fields duality
dual construction
UV
QFTIR
operatorsOφ
fields
e�ddxφOO
duality
dual construction
ZQFT = e−Son−shellADS (φ(φ∂ADS=JO
))Claim:
UV
QFTIR
operatorsOφ
fields
e�ddxφOO
duality
dual construction
UV
QFTIR
some fermionicsystem
How does it work?
O
UV
QFTIR
some fermionicsystem
How does it work?
Ofieldsψ
duality
UV
QFTIR
Dirac equation
some fermionicsystem
How does it work?
Ofieldsψ
duality
UV
QFTIR
Dirac equation
some fermionicsystem
How does it work?
Ofieldsψ
duality
Atµ
UV
QFTIR
Dirac equation
some fermionicsystem
ψ ≈ arm + br−m
source response
How does it work?
Ofieldsψ
duality
Atµ
UV
QFTIR
Dirac equation
some fermionicsystem
ψ ≈ arm + br−m
source response
How does it work?
Green function: =f(UV,IR)GO =b
a
Ofieldsψ
duality
Atµ
Why is gravity holographic?
Why is gravity holographic?
Why is gravity holographic?
Why is gravity holographic?
Entropy scales with the area not the volume: gravity is naturally holographic
Why is gravity holographic?
black hole
gravitons
QFT
black hole
gravitons
QFT
ZQFT = e−Son−shellADS (φ(φ∂ADS=JO
))
black hole
gravitons
QFT
reality
ZQFT = e−Son−shellADS (φ(φ∂ADS=JO
))
What holography does for you?
What holography does for you?
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
What holography does for you?
holography
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
What holography does for you?
holography
RG=GR
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
What holography does for you?
holography
RG=GR
strong-coupling is easy
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
What holography does for you?
holography
RG=GR
strong-coupling is easy
microscopic UV model not easy(need M-theory)
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
What holography does for you?
holography
RG=GR
strong-coupling is easy
microscopic UV model not easy(need M-theory)
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
so what(currents,
symmetries)
Can holography solve the Mott problem?
NiO insulates?
What is a Mott Insulator?
d8
NiO insulates?
EMPTY STATES=METAL
What is a Mott Insulator?
d8
NiO insulates?
EMPTY STATES=METAL
band theory fails!
What is a Mott Insulator?
d8
NiO insulates?
EMPTY STATES=METAL
band theory fails!
What is a Mott Insulator?
d8
perhaps thiscosts energy
(N rooms N occupants)
Mott Problem: NiO (Band theory
failure)
t
U � t
Half-filledband
Free electrons
Half-filledband
Free electrons
U
ε
µ
gap with no symmetrybreaking!!
charge gap
U � t
Mott Insulator-Ordering
=Mottness Mott Insulator-Ordering
=Mottness Mott Insulator-Ordering
Slater
=Mottness Mott Insulator-Ordering
Slater
its all about order.No Mottness
=Mottness Mott Insulator-Ordering
Slater
its all about order.No Mottness
of course it does!!order is secondary
Anderson
=Mottness Mott Insulator-Ordering
Slater
its all about order.No Mottness
of course it does!!order is secondary
Anderson
=Mottness Mott Insulator-Ordering
Slater
its all about order.No Mottness
of course it does!!order is secondary
Anderson
Why is the Mott problem important?
Why is the Mott problem important?
Why is the Mott problem important?
interactions dominate:Strong Coupling Physics
U/t = 10� 1
Experimental facts: Mottness
cm-1
T=360 K
T=295 K
σ(ω) Ω-1cm-1
VO2
M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, 205118 (2006)
Experimental facts: Mottness
cm-1
T=360 K
T=295 K
σ(ω) Ω-1cm-1
VO2
M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, 205118 (2006)
transferof spectralweight to
high energiesbeyond any ordering
scale
Recall, eV = 104K
Experimental facts: Mottness
cm-1
T=360 K
T=295 K
σ(ω) Ω-1cm-1
VO2
M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, 205118 (2006)
transferof spectralweight to
high energiesbeyond any ordering
scale
Recall, eV = 104K
∆ = 0.6eV > ∆dimerization(Mott, 1976) ∆
Tcrit≈ 20
Experimental facts: Mottness
cm-1
T=360 K
T=295 K
σ(ω) Ω-1cm-1
VO2
M. M. Qazilbash, K. S. Burch, D. Whisler, D. Shrekenhamer, B. G. Chae, H. T. Kim, and D. N. Basov PRB 74, 205118 (2006)
transferof spectralweight to
high energiesbeyond any ordering
scale
Recall, eV = 104K
∆ = 0.6eV > ∆dimerization(Mott, 1976) ∆
Tcrit≈ 20
Experimental facts: Mottness
What gravitational theory gives rise to a gap in ImG without spontaneous symmetry breaking?
dynamically generated gap: Mott gap (for probe fermions)
What gravitational theory gives rise to a gap in ImG without spontaneous symmetry breaking?
√−giψ̄(D −m)ψ
AdS-RNMIT, Leiden group
What has been done?
√−giψ̄(D −m)ψ
AdS-RNMIT, Leiden group
What has been done?
√−giψ̄(D −m)ψ
AdS-RNMIT, Leiden group
What has been done?
Fermi peak
G(ω, k) =Z
vF (k − kF )− ω − Σ
√−giψ̄(D −m)ψ
AdS-RNMIT, Leiden group
What has been done?
Fermi peak
G(ω, k) =Z
vF (k − kF )− ω − Σ
marginal Fermi liquid
√−giψ̄(D −m)ψ
AdS-RNMIT, Leiden group
What has been done?
Fermi peak
G(ω, k) =Z
vF (k − kF )− ω − Σ
marginal Fermi liquid
?? Mott Insulator
decoherence
?? Mott Insulator
Sprobe(ψ, ψ̄) =
�ddx
√−giψ̄(ΓMDM −m+ · · · )ψ
Sprobe(ψ, ψ̄) =
�ddx
√−giψ̄(ΓMDM −m+ · · · )ψ
what is hidden here?
√−giψ̄(D −m− ipF )ψconsider
FµνΓµν
Sprobe(ψ, ψ̄) =
�ddx
√−giψ̄(ΓMDM −m+ · · · )ψ
what is hidden here?
QED anomalous magnetic moment of an electron (Schwinger 1949)
√−giψ̄(D −m− ipF )ψconsider
FµνΓµν
Sprobe(ψ, ψ̄) =
�ddx
√−giψ̄(ΓMDM −m+ · · · )ψ
what is hidden here?
P=0
How is the spectrum modified?
P=0
Fermi surfacepeak
How is the spectrum modified?
P=0
Fermi surfacepeak
How is the spectrum modified?
P > 4.2
P=0
Fermi surfacepeak
How is the spectrum modified?
P > 4.2
P=0
Fermi surfacepeak
dynamically generated gap:
How is the spectrum modified?
P > 4.2
P=0
Fermi surfacepeak
dynamically generated gap:
spectralweight transfer
How is the spectrum modified?
P > 4.2
P=0
Fermi surfacepeak
dynamically generated gap:
spectralweight transfer
How is the spectrum modified?
P > 4.2
confirmed by Gubser, Gauntlett, 2011
Gauntlett, 2011
Mechanism?
UV
QFTIR
Mechanism?
UV
QFTIR
AdS2
emergentspacetime symmetry
Mechanism?
UV
QFTIR
AdS2
emergentspacetime symmetry
O±ψ
Mechanism?
UV
QFTIR
AdS2
emergentspacetime symmetry
O±ψ
operatorsO
Mechanism?
UV
QFTIR
AdS2
emergentspacetime symmetry
O±ψ
holography withinholography
operatorsO
Mechanism?
UV
QFTIR
where is k_F?
AdS2
emergentspacetime symmetry
O±ψ
holography withinholography
operatorsO
k_F moves into log-oscillatory region: IRacquires a complex dimensionO±
Mechanism?
UV
QFTIR
where is k_F?
AdS2
emergentspacetime symmetry
O±ψ
holography withinholography
operatorsO
What does a complex scaling dimension mean?
O = µ(λ)O(λr)
continuousscale invariance
O = µ(λ)O(λr)
continuousscale invariance
1 = µ(λ)λ∆
∆ = − lnµ
lnλ
O = µ(λ)O(λr)
continuousscale invariance
1 = µ(λ)λ∆
∆ = − lnµ
lnλ
is real, independent of
scale
∆
what about complex ∆
O = µ(λ)O(λr)
continuousscale invariance
1 = µ(λ)λ∆
∆ = − lnµ
lnλ
is real, independent of
scale
∆
1 = µλ∆
1 = µλ∆e2πin =
1 = µλ∆e2πin =
Discrete scale invariance (DSI)
n=0:CSI
∆ = − lnµ
lnλ+
2πin
lnλ
scaling dimension depends on scale
λn = λn
1 = µλ∆e2πin =
Discrete scale invariance (DSI)
n=0:CSI
∆ = − lnµ
lnλ+
2πin
lnλ
scaling dimension depends on scale
λn = λn
1 = µλ∆e2πin =
Discrete scale invariance (DSI)
n=0:CSI
∆ = − lnµ
lnλ+
2πin
lnλ
magnification
scaling dimension depends on scale
λn = λn
1 = µλ∆e2πin =
Discrete scale invariance (DSI)
n=0:CSI
∆ = − lnµ
lnλ+
2πin
lnλ
magnification
preferred scale
liquid
solid
analogy
liquid
solid
continuous translationalinvariance
discrete translationalinvariance
analogy
example
example
example
example
n iterations
number of segments
2n3−n
length
example
n iterations
number of segments
2n3−n
length
scale invariance onlyfor λp = 3p
example
n iterations
number of segments
2n3−n
length
scale invariance onlyfor λp = 3p
discrete scale invariance: D = − ln 2
ln 3+
2πin
ln 3
more examples
DLAturbulence
stock market crashes
percolationearthquakes fractals TB Law
discrete scale invariance
hidden scale(length, energy,...)
G =β−(0, k)
α−(0, k)
p=-0.4 p=0.0
p=0.2 p=1.0
no poles outside log-oscillatory region for p > 1/√6
G =β−(0, k)
α−(0, k)
p=-0.4 p=0.0
p=0.2 p=1.0
no poles outside log-oscillatory region for p > 1/√6
Mott physics and DSI are linked!
is there an instability?
x
y
is there an instability?
x
y
is there an instability?
E → E + iγ γ > 0
x
y
is there an instability?
E → E + iγ γ > 0
x
y �φ� �= 0
condensate(bosonic)
is there an instability?
E → E + iγ γ > 0
x
y �φ� �= 0
condensate(bosonic)
does this happen for fermions?
quasi-normal modes
Imω < 0 no instability
quasi-normal modes
Imω < 0 no instability
quasi-particle residue
quasi-normal modes
Imω < 0 no instability
quasi-particle residue
but the residue drops to zero:
opening of a gap
G(ω, k) = G(ωλn, k)
discrete scale invariancein energy
G(ω, k) = G(ωλn, k)
discrete scale invariancein energy
G(ω, k) = G(ωλn, k)
emergent IR scale
discrete scale invariancein energy
G(ω, k) = G(ωλn, k)
emergent IR scale
no condensate
discrete scale invariancein energy
G(ω, k) = G(ωλn, k)
emergent IR scale
energy gap: Mott gap (Mottness)
no condensate
continuous scale invariance
discrete scale invariancein energy
continuous scale invariance
discrete scale invariancein energy
is this the symmetry that is ultimatelybroken in the Mott
problem?
a.) yes
b.) no
a.) yes
b.) no
holography
a.) yes
b.) no
holography
VO_2, cuprates,...
a.) yes
b.) no
holography
VO_2, cuprates,...
if yes: holography has solved the Mott problem
what else can holography do?
Finite Temperature Mott transition from Holography
T/µ = 5.15× 10−3 T/µ = 3.92× 10−2
Finite Temperature Mott transition from Holography
∆
Tcrit≈ 20 vanadium oxide
T/µ = 5.15× 10−3 T/µ = 3.92× 10−2
Finite Temperature Mott transition from Holography
∆
Tcrit≈ 20 vanadium oxide
T/µ = 5.15× 10−3 T/µ = 3.92× 10−2
∆
Tcrit≈ 10
looks just like the experiments
Mottness
looks just like the experiments
toy model: merging of UV and IR fixed points
β = (α− α∗)− (g − g∗)2
g± = g∗ ±√α− α∗
toy model: merging of UV and IR fixed points
β = (α− α∗)− (g − g∗)2
g± = g∗ ±√α− α∗
β
gUV ≡ g+gIR ≡ g−
α > α∗
toy model: merging of UV and IR fixed points
β = (α− α∗)− (g − g∗)2
g± = g∗ ±√α− α∗
β
gUV ≡ g+gIR ≡ g−
α > α∗
α = α∗
g∗
toy model: merging of UV and IR fixed points
β = (α− α∗)− (g − g∗)2
g± = g∗ ±√α− α∗
β
gUV ≡ g+gIR ≡ g−
α > α∗
α = α∗
g∗
α < α∗
ΛIR = ΛUVe−π/
√α∗−α
toy model: merging of UV and IR fixed points
β = (α− α∗)− (g − g∗)2
g± = g∗ ±√α− α∗
β
gUV ≡ g+gIR ≡ g−
α > α∗
α = α∗
g∗
α < α∗
ΛIR = ΛUVe−π/
√α∗−α
BKT transition Kaplan, arxiv:0905.4752
are complex(conformality lost)g±
DSI