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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
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
QFTIR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
UV
QFT
g = 1/ego
coupling constant
IR ??
Wilsonianprogram
(fermions: new degrees of
freedom)
UV
QFT
g = 1/ego
coupling constant
IR ??
gauge-gravity duality(Maldacena, 1997)
UV
QFTIR gravity
Wilsonianprogram
(fermions: new degrees of
freedom)
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
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?
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-thery)
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-thery)
RG equations
Landau-Wilson
Hamiltonian
long-wavelengths
ξt ∝ ξz
so what(currents,
symmetries)
black hole
gravitons
QFT
black hole
gravitons
QFT
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)
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)
transferof spectralweight to
high energiesbeyond any ordering
scale
Recall, eV = 104K
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)
transferof spectralweight to
high energiesbeyond any ordering
scale
Recall, eV = 104K
∆ = 0.6eV > ∆dimerization(Mott, 1976) ∆
Tcrit≈ 20
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)
transferof spectralweight to
high energiesbeyond any ordering
scale
Recall, eV = 104K
∆ = 0.6eV > ∆dimerization(Mott, 1976) ∆
Tcrit≈ 20
Can holography solve the Mott problem?
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
Charged system probe
JψOψ
UV
Jµ
RN-AdSds2, At
what has been done?
MIT, Leiden, McMaster,...
ψ Dirac Eq.
S0
Charged system probe
JψOψ
UV
Jµ
RN-AdSds2, At
what has been done?
MIT, Leiden, McMaster,...
ψ Dirac Eq.
S0
Charged system probe
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?
MIT, Leiden, McMaster,...
ψ Dirac Eq.
S0
Charged system probe
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?
MIT, Leiden, McMaster,...
a=0 defines FS
RN black hole
Dirac equation+... ina curved space-time
boundary physics:AdS2 ×R2
RN black hole
Dirac equation+... ina curved space-time
boundary physics:AdS2 ×R2
ds2 =L22
ζ2−dτ2 + dζ2
+
r20R2
dx2
x, τ, ζ → x,λτ,λζ
RN black hole
Dirac equation+... ina curved space-time
boundary physics:AdS2 ×R2
x → λ(1/z=∞)x = x
ds2 =L22
ζ2−dτ2 + dζ2
+
r20R2
dx2
x, τ, ζ → x,λτ,λζ
RN black hole
Dirac equation+... ina curved space-time
boundary physics:AdS2 ×R2
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
x, τ, ζ → x,λτ,λζ
√−giψ(D −m)ψ
AdS-RNMIT, Leiden group
G(ω, k) =Z
vF (k − kF )− ω − hω lnω
√−giψ(D −m)ψ
AdS-RNMIT, Leiden group
G(ω, k) =Z
vF (k − kF )− ω − hω lnω
marginal Fermi liquid
√−giψ(D −m)ψ
?? Mott Insulator
AdS-RNMIT, Leiden group
G(ω, k) =Z
vF (k − kF )− ω − hω lnω
marginal Fermi liquid
√−giψ(D −m)ψ
?? Mott Insulator
AdS-RNMIT, Leiden group
G(ω, k) =Z
vF (k − kF )− ω − hω lnω
How to Destroy the Fermi surface?
marginal Fermi liquid
√−giψ(D −m)ψ
?? Mott Insulator
AdS-RNMIT, Leiden group
G(ω, k) =Z
vF (k − kF )− ω − hω lnω
How to Destroy the Fermi surface?
decoherence
marginal Fermi liquid
√−giψ(D −m)ψ
?? Mott Insulator
AdS-RNMIT, Leiden group
G(ω, k) =Z
vF (k − kF )− ω − hω lnω
How to Destroy the Fermi surface?
decoherence
what if? ψ ∝ ar∆ + br−∆ ∆
not pole-likeImG = 0
is complex
marginal Fermi liquid
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
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
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
Finite Temperature Mott transition
∆
Tcrit≈ 20 vanadium oxide
T/µ = 5.15× 10−3 T/µ = 3.92× 10−2
Finite Temperature Mott transition
∆
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
Mottness
Hubbard
Holographydynamical
spectralweight
transfer
criticality in metals with a neutral condensate
charge ormagnetic order
gcharge ormagnetic order
no order
criticality in metals with a neutral condensate
charge ormagnetic order
gcharge ormagnetic order
no order
criticality in metals with a neutral condensate
AFM
nematic
SRO
Standard Account:Hertz-Millis `theory’
Standard Account:Hertz-Millis `theory’
Hubbard model
Standard Account:Hertz-Millis `theory’
Hubbard model bosonic order+gapless fermionsHS
Standard Account:Hertz-Millis `theory’
Hubbard model bosonic order+gapless fermionsHS
critical bosons
integrate out fermions
Standard Account:Hertz-Millis `theory’
Hubbard model bosonic order+gapless fermionsHS
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
holography to the rescue: requirements
not AdS in the far interior
1.) background with finite z
t → λzt x → λx
holography to the rescue: requirements
UV
QFTIR gravity
2.) neutral condensate:Φ = 0
Φφ?
classical field boundary operator
not AdS in the far interior
1.) background with finite z
t → λzt x → λx
holography to the rescue: requirements
3.) d=2+1
UV
QFTIR gravity
2.) neutral condensate:Φ = 0
Φφ?
classical field boundary operator
not AdS in the far interior
1.) background with finite z
t → λzt x → λx
1.) background with finite z
horizon
1.) background with finite z
horizon
1.) background with finite z
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
curvature<=matter fields from electron fluid
ds2 = L2
−f(r)dt2 +
dx2 + dy2
r2+ g(r)dr2
solve Einstein/Maxwell eom n(E) = βE
E2 −m2
f
curvature<=matter fields from electron fluid
ds2 = L2
−f(r)dt2 +
dx2 + dy2
r2+ g(r)dr2
solve Einstein/Maxwell eom n(E) = βE
E2 −m2
f
ds2 = L2
−dt2
r2z+
dx2
r2+ g
dr2
r2
Lifshitz geometry
z ≥ 1/(1−m2f ) ≥ 1 mf ∈ (0, 1)
curvature<=matter fields from electron fluid
ds2 = L2
−f(r)dt2 +
dx2 + dy2
r2+ g(r)dr2
solve Einstein/Maxwell eom n(E) = βE
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
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
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
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
d = 2 + 1 IRLifshitzd = 2 + 1 IRLifshitz
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)
Nature of T=0 Transition
1
gr3+z∂r
r−(1+z)∂rφ
= L2V (φ)
φ(r) = r(z+2)/2 sin
g (m2
cL2 −m2L2) log
r
rUV
Nature of T=0 Transition
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
Nature of T=0 Transition
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
Φ ∼ µ∆ exp
− (z + 2)π
2
g (m2cL
2 −m2L2)
Nature of T=0 Transition
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
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
z=4
m2cΦ = 0 Φ = 0
z = z1z = z2
d = 2 + 1 IRLifshitzd = 2 + 1 IRLifshitz
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
Dynamically generated gap(Mott Gap)
Hubbard Model
neutral scalar condensate in a metal
(with finite z and d=2+1)
Hertz/Millis
Dynamically generated gap(Mott Gap)
Hubbard Model
Holography
neutral scalar condensate in a metal
(with finite z and d=2+1)
Hertz/Millis
Dynamically generated gap(Mott Gap)
Hubbard Model
Holography
RN-AdS
neutral scalar condensate in a metal
(with finite z and d=2+1)
Hertz/Millis
Dynamically generated gap(Mott Gap)
Hubbard Model
Holography
electronstar
2+1 Lifshitz
RN-AdS
neutral scalar condensate in a metal
(with finite z and d=2+1)
Hertz/Millis
Dynamically generated gap(Mott Gap)
Hubbard Model
Holography
electronstar
2+1 Lifshitz
RN-AdSUV-IRmixing
neutral scalar condensate in a metal
(with finite z and d=2+1)
Hertz/Millis
51
thanks again to
M. Edalati R. G. LeighKa Wai Lo
NSF, EFRC