mottness and metallic criticality from...

Mottness and Metallic Criticality from Holography

Thanks to: NSF, EFRC (DOE)

M. Edalati R. G. LeighKa Wai Lo

Unsolved condensed matter problems

magnetismin metals

nematicsheavy fermions

turbulence

quantum criticality

high T_c (cuprate, organics,...

Unsolved condensed matter problems

physics at strong coupling

magnetismin metals

nematicsheavy fermions

turbulence

quantum criticality

high T_c (cuprate, organics,...

composite or bound states not in UV theory

Strong Coupling

QCD

emergentlow-energy physics

turbulence?

vulcanization

Mott problem,

magnetism andnematicsin metals

emergentgravity

Mott problem,

magnetism andnematicsin metals

What computational tools do we have forstrongly correlated electron systems?

DMFT

the only differencebetween this and

a theory is that this is not a theory

UV

QFT

UV

QFTIR ??

UV

QFTIR ??

Wilsonianprogram

(fermions: new degrees of

freedom)

UV

QFT

g = 1/ego

coupling constant

IR ??

Wilsonianprogram


freedom)

UV

QFT

g = 1/ego

coupling constant

IR ??

gauge-gravity duality(Maldacena, 1997)

UV

QFTIR gravity

Wilsonianprogram


freedom)

UV

QFT

g = 1/ego

coupling constant

IR ??

gauge-gravity duality(Maldacena, 1997)

UV

QFTIR gravity

Wilsonianprogram


freedom)

dg(E)

dlnE= β(g(E))

locality in energy

AdSd+1

geometrize RG flowβ(g) is local

`Holography’

RN black hole

t, x, r → λt,λx,λr

symmetry:

r → 0

ds2 =L2

r2−dt2 + dx2 + dr2

UV

QFTIR

UV

QFTIR

operatorsO

UV

QFTIR

operatorsOφ

fields

ZQFT = e−Son−shellADS (φ(φ∂ADS=JO

))Claim:

UV

QFTIR

operatorsOφ

fields

What holography does for you?


RG equations

Landau-Wilson

Hamiltonian

long-wavelengths

ξt ∝ ξz


holography

RG equations

Landau-Wilson

Hamiltonian

long-wavelengths

ξt ∝ ξz


holography

RG=GR

RG equations

Landau-Wilson

Hamiltonian

long-wavelengths

ξt ∝ ξz


holography

RG=GR

strong-coupling is easy

RG equations

Landau-Wilson

Hamiltonian

long-wavelengths

ξt ∝ ξz


holography

RG=GR


microscopic UV model not easy(need M-thery)

RG equations

Landau-Wilson

Hamiltonian

long-wavelengths

ξt ∝ ξz


holography

RG=GR


microscopic UV model not easy(need M-thery)

RG equations

Landau-Wilson

Hamiltonian

long-wavelengths

ξt ∝ ξz

so what(currents,

symmetries)

black hole

gravitons

QFT

black hole

gravitons

QFT

reality

Can holography solve the Mott problem?

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)


cm-1

T=360 K

T=295 K

σ(ω) Ω-1cm-1

VO2


transferof spectralweight to

high energiesbeyond any ordering

scale

Recall, eV = 104K


cm-1

T=360 K

T=295 K

σ(ω) Ω-1cm-1

VO2


transferof spectralweight to

high energiesbeyond any ordering

scale

Recall, eV = 104K

∆ = 0.6eV > ∆dimerization(Mott, 1976) ∆

Tcrit≈ 20


Wilsonian program

densityof

states

d(UHB)

Wilsonian program

densityof

states

d(UHB)

Wilsonian program

?

densityof

states

d(UHB)

Wilsonian program

?

Charge 2e stuff is transferred down?

densityof

states

d(UHB)

Wilsonian program

?

Charge 2e stuff is transferred down?

physics of non-rigid bands

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?

S0

Charged system probe

JψOψ

UV

Jµ

what has been done?

MIT, Leiden, McMaster,...

S0


JψOψ

UV

Jµ

RN-AdSds2, At

what has been done?


ψ Dirac Eq.

S0


JψOψ

UV

Jµ

RN-AdSds2, At

what has been done?


ψ Dirac Eq.

S0


JψOψ

UV

in-falling boundary conditions

Retarded Green function:G =b

a

ψ(r → ∞) ≈ arm + br−m

=f(UV,IR)

Jµ

RN-AdSds2, At

what has been done?


ψ Dirac Eq.

S0


JψOψ

UV

in-falling boundary conditions

Retarded Green function:G =b

a

ψ(r → ∞) ≈ arm + br−m

=f(UV,IR)

Jµ

RN-AdSds2, At

what has been done?


a=0 defines FS

RN black hole

Dirac equation+... ina curved space-time

boundary physics:AdS2 ×R2

RN black hole



ds2 =L22

ζ2−dτ2 + dζ2

+

r20R2

dx2

x, τ, ζ → x,λτ,λζ

RN black hole



x → λ(1/z=∞)x = x

ds2 =L22

ζ2−dτ2 + dζ2

+

r20R2

dx2


RN black hole



x → λ(1/z=∞)x = x

S ∝ T 2/z = 0 T → 0

finite T=0 entropy(oops!)

ds2 =L22

ζ2−dτ2 + dζ2

+

r20R2

dx2


√−giψ(D −m)ψ

AdS-RNMIT, Leiden group

G(ω, k) =Z

vF (k − kF )− ω − hω lnω



G(ω, k) =Z


marginal Fermi liquid


?? Mott Insulator


G(ω, k) =Z




?? Mott Insulator


G(ω, k) =Z


How to Destroy the Fermi surface?



?? Mott Insulator


G(ω, k) =Z



decoherence



?? Mott Insulator


G(ω, k) =Z



decoherence

what if? ψ ∝ ar∆ + br−∆ ∆

not pole-likeImG = 0

is complex


fermions in RN Ads_d+1 coupled to a gauge fieldthrough a dipole interaction

√−giψ(D −m− ipF )ψconsider

P=0

How is the spectrum modified?

P=0

Fermi surfacepeak


P=0

Fermi surfacepeak


P > 4.2

P=0

Fermi surfacepeak

dynamically generated gap:


P > 4.2

P=0

Fermi surfacepeak


spectralweight transfer


P > 4.2

P=0

Fermi surfacepeak


spectralweight transfer


P > 4.2

confirmed by Gubser, Gauntlett, 2011

Gauntlett, 2011

Mechanism

log-oscillatory

k_F moves into log-oscillatory region:acquires a complex dimensionO±

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

near horizon

p: time-reversal breaking mass term (in bulk)

ψI±(ζ) = ψ(0)I±(ζ) + ω ψ(1)

I±(ζ) + ω2ψ(2)I±(ζ) + · · ·

−ψ(0)I± (ζ) = iσ2

1 +

qedζ

− L2

ζ

mσ3 +

ped ±

kL

r0

σ1

ψ(0)I±(ζ),

m2k = m2 +

ped ±

kL

r0

2

ed = 1/2d(d− 1)

radial Dirac Equation

p=0.0 p=0.1 p=1.8

special points

−1.54 < p < −0.53

1 > νkF > 1/2

ω ∝ k − kFω ∝ (k − kF )

2νkF

`Fermi Liquid’

p = −0.53

νkF = 1/2

−0.53 < p < 1/√6

MFL

1/2 > νkF > 0

ω = ω ∝ (k − kF )1/(2νkF

)

NFL

quasi-normal modes

Imω < 0 no instability

quasi-normal modes

Imω < 0 no instability

quasi-particle residue

Finite Temperature Mott transition

T/µ = 5.15× 10−3 T/µ = 3.92× 10−2


∆

Tcrit≈ 20 vanadium oxide

T/µ = 5.15× 10−3 T/µ = 3.92× 10−2


∆

Tcrit≈ 20 vanadium oxide

T/µ = 5.15× 10−3 T/µ = 3.92× 10−2

∆

Tcrit≈ 10

spectral weighttransfer

UV-IR mixing

T ↑

p1

p3

σ(ω) Ω-1cm-1

VO2

Hubbard

Holographydynamical

spectralweight

transfer

Mottness

Hubbard

Holographydynamical

spectralweight

transfer

criticality in metals with a neutral condensate

charge ormagnetic order

gcharge ormagnetic order

no order


charge ormagnetic order

gcharge ormagnetic order

no order


AFM

nematic

SRO

Standard Account:Hertz-Millis `theory’


Hubbard model


Hubbard model bosonic order+gapless fermionsHS



critical bosons

integrate out fermions



critical bosons

integrate out fermions

no true high-energy scale

Belitz,Kirkpatrick, T. Vojta( FM) Abanov/Cubukov,Sachdev/

Melitskii,S. Lee (AFM) (d=2+1)

holography to the rescue: requirements


not AdS in the far interior

1.) background with finite z

t → λzt x → λx


UV

QFTIR gravity

2.) neutral condensate:Φ = 0

Φφ?

classical field boundary operator





3.) d=2+1

UV

QFTIR gravity

2.) neutral condensate:Φ = 0

Φφ?

classical field boundary operator





horizon


AdS4 + Φ

rsfermionsdeform interior

geometry

fermion flux

charge density

horizon

curvature<=matter fields from electron fluid

ds2 = L2

−f(r)dt2 +

dx2 + dy2

r2+ g(r)dr2


ds2 = L2

−f(r)dt2 +

dx2 + dy2

r2+ g(r)dr2

solve Einstein/Maxwell eom n(E) = βE

E2 −m2

f


ds2 = L2

−f(r)dt2 +

dx2 + dy2

r2+ g(r)dr2


E2 −m2

f

ds2 = L2

−dt2

r2z+

dx2

r2+ g

dr2

r2

Lifshitz geometry

z ≥ 1/(1−m2f ) ≥ 1 mf ∈ (0, 1)


ds2 = L2

−f(r)dt2 +

dx2 + dy2

r2+ g(r)dr2


E2 −m2

f

ds2 = L2

−dt2

r2z+

dx2

r2+ g

dr2

r2

Lifshitz geometry

z ≥ 1/(1−m2f ) ≥ 1 mf ∈ (0, 1)

no entropy at T=0 (excellent)

Lifshitz/AdS: entropy~

RN-AdS

Tc

T 2/z

temperature dependence

2.) Neutral Scalar Potential

Sφ = − 1

2κ2λ

d4x

√−g

1

2(∂φ)2 + V (φ)

,

V (φ) =1

4

φ2 +m2

2 −m4.

expand around φ = 0

r → 0

φ = Ar3−∆(1 + · · · ) +B r∆(1 + · · · )

source response:condensate

φ = Ar3−∆(1 + · · · ) +B r∆(1 + · · · )

Are there solutions in which A=0 but B is nonzero?

spontaneous symmetry breaking

φ = Ar3−∆(1 + · · · ) +B r∆(1 + · · · )

Are there solutions in which A=0 but B is nonzero?

such solutions are guaranteed when IR

anomalous dimension is complex: violation of

BF bound

spontaneous symmetry breaking

AdS4 + Φ

rs ∆IR =1

2

(2 + z) +

(2 + z)2 + 4gm2

∆ =3

2+

m2 + 9/4

exterior BF boundm2 = −9/4

interior BF bound

m2 = −(2 + z)2/(4g) > −9/4

we want solutions in which m^2>-9/4! (no violation of

exterior BF bound: SSB)

z=2

z=3

yes

Φ = 0

but

m2 > −9/4

yes

Φ = 0

but

m2 > −9/4

spontaneous condensation of neutral scalar at boundary

what happens for larger values of z?

what happens for larger values of z? AdS2 × R2

BF bound=-3/2

m2cΦ = 0 Φ = 0

z = z1

d = 2 + 1 IRLifshitzd = 2 + 1 IRLifshitz

z = z1

Thus far

ΛIR = ΛUV exp(−c2/

m2c −m2)

kaplan, et al. PRD 80,125005 (2009)

m2cΦ = 0 Φ = 0

z = z1


z = z1

Thus far

ΛIR = ΛUV exp(−c2/

m2c −m2)

kaplan, et al. PRD 80,125005 (2009)

m2cΦ = 0 Φ = 0

z = z1


z = z1

Thus far

does BKTTransition

Φ ∼ µ∆ exp(−c1/m2

c −m2)

Nature of T=0 Transition

1

gr3+z∂r

r−(1+z)∂rφ

= L2V (φ)


1

gr3+z∂r

r−(1+z)∂rφ

= L2V (φ)

φ(r) = r(z+2)/2 sin

g (m2

cL2 −m2L2) log

r

rUV


1

gr3+z∂r

r−(1+z)∂rφ

= L2V (φ)

φ(r) = r(z+2)/2 sin

g (m2

cL2 −m2L2) log

r

rUV

demand: φ(rUV) = φ(rIR) = 0


1

gr3+z∂r

r−(1+z)∂rφ

= L2V (φ)

φ(r) = r(z+2)/2 sin

g (m2

cL2 −m2L2) log

r

rUV


Φ ∼ µ∆ exp

− (z + 2)π

2

g (m2cL

2 −m2L2)


1

gr3+z∂r

r−(1+z)∂rφ

= L2V (φ)

φ(r) = r(z+2)/2 sin

g (m2

cL2 −m2L2) log

r

rUV


Φ ∼ µ∆ exp

− (z + 2)π

2

g (m2cL

2 −m2L2)

BKT Transition

∆Ω ∝ e−κ

(m−mc)α

Free energy

not power-law

back reaction

AdS4 + Φ

rs ∆IR =1

2

(2 + z) +

(2 + z)2 + 4gm2

∆ =3

2+

m2 + 9/4

Φ = 0

back reaction

AdS4 + Φ

rs ∆IR =1

2

(2 + z) +

(2 + z)2 + 4gm2

∆ =3

2+

m2 + 9/4

Φ = 0

use eom to determine new metric

back reaction

AdS4 + Φ

rs ∆IR =1

2

(2 + z) +

(2 + z)2 + 4gm2

∆ =3

2+

m2 + 9/4

Φ = 0

use eom to determine new metric

z(mf ,β) → z(mf ,β,mL,λ)

z=4

m2cΦ = 0 Φ = 0

z = z1z = z2


key conclusion: z changes across the transition

finite T solution

no back reaction

λ = 10λ = 100

critical exponents

B ∼ (1− T/Tc)βc dB/dA|A=0 ∼ (1− T/Tc)

−γc

∆Ω ∼ (1− T/Tc)νc

βc ∼ 0.5

B ∼ A1/δc

γc ∼ 1.015

δc ∼ 3.07 νc ∼ 2.06

critical exponents

B ∼ (1− T/Tc)βc dB/dA|A=0 ∼ (1− T/Tc)

−γc

∆Ω ∼ (1− T/Tc)νc

βc ∼ 0.5

B ∼ A1/δc

γc ∼ 1.015

δc ∼ 3.07 νc ∼ 2.06

mean-field transition

Hubbard Model

Dynamically generated gap(Mott Gap)

Hubbard Model


Hubbard Model

neutral scalar condensate in a metal

(with finite z and d=2+1)

Hertz/Millis


Hubbard Model

Holography



Hertz/Millis


Hubbard Model

Holography

RN-AdS



Hertz/Millis


Hubbard Model

Holography

electronstar

2+1 Lifshitz

RN-AdS



Hertz/Millis


Hubbard Model

Holography

electronstar

2+1 Lifshitz

RN-AdSUV-IRmixing



Hertz/Millis

51

thanks again to

M. Edalati R. G. LeighKa Wai Lo

NSF, EFRC

mottness and metallic criticality from...

Documents