holography and mottness: a discrete marriage

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