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Kinetic theory of transport for inhomogeneous electron fluids
Andrew Lucas
Stanford Physics
Physics Next: From Quantum Fields to Condensed Matter; Hyatt Place Long Island
August 25, 2017
Collaborators 2
Sean HartnollStanford Physics
I A. Lucas and S. A. Hartnoll. “Kinetic theory of transport forinhomogeneous electron fluids”, 1706.04621
Introduction to Transport 3
Transport in Metals
I Ohm’s law – the “simplest” experiment...
E = ρJ (V = IR)
I ...yet ρ hard to compute in interesting systems:
transportgases
fluids
QFT
chaosstat.mech. black holes
I textbooks on canonical Boltzmann/kinetic transport:
Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)
57 years later, we have ‘completed’ this theory
Introduction to Transport 3
Transport in Metals
I Ohm’s law – the “simplest” experiment...
E = ρJ (V = IR)
I ...yet ρ hard to compute in interesting systems:
transportgases
fluids
QFT
chaosstat.mech. black holes
I textbooks on canonical Boltzmann/kinetic transport:
Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)
57 years later, we have ‘completed’ this theory
Introduction to Transport 3
Transport in Metals
I Ohm’s law – the “simplest” experiment...
E = ρJ (V = IR)
I ...yet ρ hard to compute in interesting systems:
transportgases
fluids
QFT
chaosstat.mech. black holes
I textbooks on canonical Boltzmann/kinetic transport:
Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)
57 years later, we have ‘completed’ this theory
Introduction to Transport 3
Transport in Metals
I Ohm’s law – the “simplest” experiment...
E = ρJ (V = IR)
I ...yet ρ hard to compute in interesting systems:
transportgases
fluids
QFT
chaosstat.mech. black holes
I textbooks on canonical Boltzmann/kinetic transport:
Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)
57 years later, we have ‘completed’ this theory
Old Boltzmann Transport 4
The Boltzmann Equation
I assume quasiparticles:
〈ψ†(k, ω)ψ(k, ω)〉 ∼ 1
ω − ε(k) + icω2 + · · · .
I non-equilibrium distribution function:
f(x,p) ∼ “particles of momentum p at position x”
I weak interactions + ∆x∆p� ~ =⇒ kinetic theory:
∂tf + v · ∂xf + F · ∂pf︸ ︷︷ ︸free-particle streaming
= C[f ]︸︷︷︸collisions
.
Old Boltzmann Transport 4
The Boltzmann Equation
I assume quasiparticles:
〈ψ†(k, ω)ψ(k, ω)〉 ∼ 1
ω − ε(k) + icω2 + · · · .
I non-equilibrium distribution function:
f(x,p) ∼ “particles of momentum p at position x”
I weak interactions + ∆x∆p� ~ =⇒ kinetic theory:
∂tf + v · ∂xf + F · ∂pf︸ ︷︷ ︸free-particle streaming
= C[f ]︸︷︷︸collisions
.
Old Boltzmann Transport 4
The Boltzmann Equation
I assume quasiparticles:
〈ψ†(k, ω)ψ(k, ω)〉 ∼ 1
ω − ε(k) + icω2 + · · · .
I non-equilibrium distribution function:
f(x,p) ∼ “particles of momentum p at position x”
I weak interactions + ∆x∆p� ~ =⇒ kinetic theory:
∂tf + v · ∂xf + F · ∂pf︸ ︷︷ ︸free-particle streaming
= C[f ]︸︷︷︸collisions
.
Old Boltzmann Transport 5
Computing Transport
I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable
I static linear response: f = feq + δf ,
v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms
+ eE · v∂feq
∂ε︸ ︷︷ ︸source term
=δCδf
∣∣∣∣eq
· δf︸ ︷︷ ︸collision operator
I BIG linear algebra problem...schematically:
L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq
∂εΦ︸ ︷︷ ︸
Φ not singular
I choose inner product 〈Jx|Φ〉 = Javgx :
σxx = 〈Jx|(W + L)−1|Jx〉.
Old Boltzmann Transport 5
Computing Transport
I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable
I static linear response: f = feq + δf ,
v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms
+ eE · v∂feq
∂ε︸ ︷︷ ︸source term
=δCδf
∣∣∣∣eq
· δf︸ ︷︷ ︸collision operator
I BIG linear algebra problem...schematically:
L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq
∂εΦ︸ ︷︷ ︸
Φ not singular
I choose inner product 〈Jx|Φ〉 = Javgx :
σxx = 〈Jx|(W + L)−1|Jx〉.
Old Boltzmann Transport 5
Computing Transport
I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable
I static linear response: f = feq + δf ,
v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms
+ eE · v∂feq
∂ε︸ ︷︷ ︸source term
=δCδf
∣∣∣∣eq
· δf︸ ︷︷ ︸collision operator
I BIG linear algebra problem...schematically:
L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq
∂εΦ︸ ︷︷ ︸
Φ not singular
I choose inner product 〈Jx|Φ〉 = Javgx :
σxx = 〈Jx|(W + L)−1|Jx〉.
Old Boltzmann Transport 5
Computing Transport
I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable
I static linear response: f = feq + δf ,
v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms
+ eE · v∂feq
∂ε︸ ︷︷ ︸source term
=δCδf
∣∣∣∣eq
· δf︸ ︷︷ ︸collision operator
I BIG linear algebra problem...schematically:
L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq
∂εΦ︸ ︷︷ ︸
Φ not singular
I choose inner product 〈Jx|Φ〉 = Javgx :
σxx = 〈Jx|(W + L)−1|Jx〉.
Old Boltzmann Transport 6
Conservation Laws
I exact solution hard. common approximations:
W =1
τ.
relaxation time approximation
L→ 0.
homogeneous fluid
I however there are conservation laws:
C[feq
(ε(p)− µ− δµ
T
)]= 0
I conservation of charge: W|Φ = 1〉 = 0
I and conservation of momentum: if density ρ 6= 0:
W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞
Old Boltzmann Transport 6
Conservation Laws
I exact solution hard. common approximations:
W =1
τ.
relaxation time approximation
L→ 0.
homogeneous fluid
I however there are conservation laws:
C[feq
(ε(p)− µ− δµ
T
)]= 0
I conservation of charge: W|Φ = 1〉 = 0
I and conservation of momentum: if density ρ 6= 0:
W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞
Old Boltzmann Transport 6
Conservation Laws
I exact solution hard. common approximations:
W =1
τ.
relaxation time approximation
L→ 0.
homogeneous fluid
I however there are conservation laws:
C[feq
(ε(p)− µ− δµ
T
)]= 0
I conservation of charge: W|Φ = 1〉 = 0
I and conservation of momentum: if density ρ 6= 0:
W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞
Old Boltzmann Transport 6
Conservation Laws
I exact solution hard. common approximations:
W =1
τ.
relaxation time approximation
L→ 0.
homogeneous fluid
I however there are conservation laws:
C[feq
(ε(p)− µ− δµ
T
)]= 0
I conservation of charge: W|Φ = 1〉 = 0
I and conservation of momentum: if density ρ 6= 0:
W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞
Strange Metals in Experiment 7
Viscous Flows in Constrictions
I at finite T , more scattering? ∂∂TW > 0 =⇒
∂∂T ρ = ∂
∂T (〈Jx|W−1|Jx〉)−1 > 0?
I however, ∂∂T ρ < 0 in graphene constrictions!
[Kumar et al; 1703.06672]
Strange Metals in Experiment 7
Viscous Flows in Constrictions
I at finite T , more scattering? ∂∂TW > 0 =⇒
∂∂T ρ = ∂
∂T (〈Jx|W−1|Jx〉)−1 > 0?I however, ∂
∂T ρ < 0 in graphene constrictions!
[Kumar et al; 1703.06672]
Strange Metals in Experiment 8
Electron-Electron Interaction Limited Resistivity in Fermi Liquids
I in a Fermi liquid:
τee ∼~µ
(kBT )2, ρ = AT 2 ∼ 1
τee. . .
I A depends on thermodynamics (not disorder?):[Jacko, Fjaerestad, Powell; 0805.4275]
Strange Metals in Experiment 8
Electron-Electron Interaction Limited Resistivity in Fermi Liquids
I in a Fermi liquid:
τee ∼~µ
(kBT )2, ρ = AT 2 ∼ 1
τee. . .
I A depends on thermodynamics (not disorder?):[Jacko, Fjaerestad, Powell; 0805.4275]
Strange Metals in Experiment 9
Linear Resistivity: A Challenge
I in a theory without quasiparticles:
τee &~kBT
.
I “Drude” ρ =m
ne2
1
τee∼ m
ne2
kBT
~:
[Bruin, Sakai, Perry, Mackenzie; (2013)]
Strange Metals in Experiment 9
Linear Resistivity: A Challenge
I in a theory without quasiparticles:
τee &~kBT
.
I “Drude” ρ =m
ne2
1
τee∼ m
ne2
kBT
~:
[Bruin, Sakai, Perry, Mackenzie; (2013)]
New Boltzmann Transport 10
Classical Disorder
I charge puddles: H = Hclean +
∫ddx n(x)Vimp(x):
⇠
�F
µ(x) = µ0 � Vimp(x)
I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical
algorithm to exactly solve the transport problem:
ρ = V 2imp × · · ·
New Boltzmann Transport 10
Classical Disorder
I charge puddles: H = Hclean +
∫ddx n(x)Vimp(x):
⇠
�F
µ(x) = µ0 � Vimp(x)
I for simplicity: neglect umklapp, phonons
I W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical
algorithm to exactly solve the transport problem:
ρ = V 2imp × · · ·
New Boltzmann Transport 10
Classical Disorder
I charge puddles: H = Hclean +
∫ddx n(x)Vimp(x):
⇠
�F
µ(x) = µ0 � Vimp(x)
I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentum
I must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical
algorithm to exactly solve the transport problem:
ρ = V 2imp × · · ·
New Boltzmann Transport 10
Classical Disorder
I charge puddles: H = Hclean +
∫ddx n(x)Vimp(x):
⇠
�F
µ(x) = µ0 � Vimp(x)
I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0
I Vimp perturbatively small – efficient analytical/numericalalgorithm to exactly solve the transport problem:
ρ = V 2imp × · · ·
New Boltzmann Transport 10
Classical Disorder
I charge puddles: H = Hclean +
∫ddx n(x)Vimp(x):
⇠
�F
µ(x) = µ0 � Vimp(x)
I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical
algorithm to exactly solve the transport problem:
ρ = V 2imp × · · ·
New Boltzmann Transport 11
Free Fermions (L 6= 0, W = 0)
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
I simple argument:
1
τDrude=vF
ξ×∆θ2
I formally:
ρ ∼ν(µ)V 2
imp
n2e2vFξ
I arrow of time =⇒ entropyproduction when W = 0!
New Boltzmann Transport 11
Free Fermions (L 6= 0, W = 0)
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
I simple argument:
1
τDrude=vF
ξ×∆θ2
I formally:
ρ ∼ν(µ)V 2
imp
n2e2vFξ
I arrow of time =⇒ entropyproduction when W = 0!
New Boltzmann Transport 11
Free Fermions (L 6= 0, W = 0)
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
I simple argument:
1
τDrude=vF
ξ×∆θ2
I formally:
ρ ∼ν(µ)V 2
imp
n2e2vFξ
I arrow of time =⇒ entropyproduction when W = 0!
New Boltzmann Transport 12
Toy Model: Single Fermi Surface
empty
filled
empty
filled
filled
kx
ky
kx
ky
I low T limit:
Φ ≈∑j∈Z
Φj(x)eijθ
I conserve charge (j = 0),momentum (j = ±1):
W ∼ vF
`ee
∑|j|≥2
|j〉〈j|
I analytically compute ρ!interactions alwaysdecrease ρ:
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ξ/`ee
ρ/ρ
res
kTFξ � 1kTFξ � 1
New Boltzmann Transport 12
Toy Model: Single Fermi Surface
empty
filled
empty
filled
filled
kx
ky
kx
ky
I low T limit:
Φ ≈∑j∈Z
Φj(x)eijθ
I conserve charge (j = 0),momentum (j = ±1):
W ∼ vF
`ee
∑|j|≥2
|j〉〈j|
I analytically compute ρ!interactions alwaysdecrease ρ:
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ξ/`ee
ρ/ρ
res
kTFξ � 1kTFξ � 1
New Boltzmann Transport 12
Toy Model: Single Fermi Surface
empty
filled
empty
filled
filled
kx
ky
kx
ky
I low T limit:
Φ ≈∑j∈Z
Φj(x)eijθ
I conserve charge (j = 0),momentum (j = ±1):
W ∼ vF
`ee
∑|j|≥2
|j〉〈j|
I analytically compute ρ!interactions alwaysdecrease ρ:
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ξ/`ee
ρ/ρ
res
kTFξ � 1kTFξ � 1
New Boltzmann Transport 12
Toy Model: Single Fermi Surface
empty
filled
empty
filled
filled
kx
ky
kx
ky
I low T limit:
Φ ≈∑j∈Z
Φj(x)eijθ
I conserve charge (j = 0),momentum (j = ±1):
W ∼ vF
`ee
∑|j|≥2
|j〉〈j|
I analytically compute ρ!interactions alwaysdecrease ρ:
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ξ/`ee
ρ/ρres
kTFξ � 1kTFξ � 1
New Boltzmann Transport 13
Comparison with Viscous Hydrodynamics
I `ee � ξ =⇒ viscoushydrodynamic transport
ρ ∼ η∫
d2x
V
(∇ 1
ne
)2
with η ∼ `ee
I QP random walks – seesVimp slower:
ρ ∼ 1
τ∼ vF`ee
ξ2
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
New Boltzmann Transport 13
Comparison with Viscous Hydrodynamics
I `ee � ξ =⇒ viscoushydrodynamic transport
ρ ∼ η∫
d2x
V
(∇ 1
ne
)2
with η ∼ `ee
I QP random walks – seesVimp slower:
ρ ∼ 1
τ∼ vF`ee
ξ2
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
New Boltzmann Transport 14
Toy Model: Two Fermi Surfaces
empty
filled
empty
filled
filled
kx
ky
kx
ky
I pockets have different vF
I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum
I numerical computationgives:
0 0.5 1 1.5 20
2
4
6
⇠/`ee
⇢/⇢
res
kTF⇠ ⌧ 1
vF,1/vF,2 = 0.3
vF,1/vF,2 = 0.5
vF,1/vF,2 = 0.7
vF,1/vF,2 = 1
0 0.5 1 1.5 2
0.5
1
1.5
⇠/`ee
⇢/⇢
res
kTF⇠ � 1
I similar to Baber scattering,but novel mechanism
New Boltzmann Transport 14
Toy Model: Two Fermi Surfaces
empty
filled
empty
filled
filled
kx
ky
kx
ky
I pockets have different vF
I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum
I numerical computationgives:
0 0.5 1 1.5 20
2
4
6
⇠/`ee
⇢/⇢
res
kTF⇠ ⌧ 1
vF,1/vF,2 = 0.3
vF,1/vF,2 = 0.5
vF,1/vF,2 = 0.7
vF,1/vF,2 = 1
0 0.5 1 1.5 2
0.5
1
1.5
⇠/`ee
⇢/⇢
res
kTF⇠ � 1
I similar to Baber scattering,but novel mechanism
New Boltzmann Transport 14
Toy Model: Two Fermi Surfaces
empty
filled
empty
filled
filled
kx
ky
kx
ky
I pockets have different vF
I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum
I numerical computationgives:
0 0.5 1 1.5 20
2
4
6
⇠/`ee
⇢/⇢
res
kTF⇠ ⌧ 1
vF,1/vF,2 = 0.3
vF,1/vF,2 = 0.5
vF,1/vF,2 = 0.7
vF,1/vF,2 = 1
0 0.5 1 1.5 2
0.5
1
1.5
⇠/`ee
⇢/⇢
res
kTF⇠ � 1
I similar to Baber scattering,but novel mechanism
New Boltzmann Transport 14
Toy Model: Two Fermi Surfaces
empty
filled
empty
filled
filled
kx
ky
kx
ky
I pockets have different vF
I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum
I numerical computationgives:
0 0.5 1 1.5 20
2
4
6
⇠/`ee⇢/⇢
res
kTF⇠ ⌧ 1
vF,1/vF,2 = 0.3
vF,1/vF,2 = 0.5
vF,1/vF,2 = 0.7
vF,1/vF,2 = 1
0 0.5 1 1.5 2
0.5
1
1.5
⇠/`ee
⇢/⇢
res
kTF⇠ � 1
I similar to Baber scattering,but novel mechanism
New Boltzmann Transport 14
Toy Model: Two Fermi Surfaces
empty
filled
empty
filled
filled
kx
ky
kx
ky
I pockets have different vF
I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum
I numerical computationgives:
0 0.5 1 1.5 20
2
4
6
⇠/`ee⇢/⇢
res
kTF⇠ ⌧ 1
vF,1/vF,2 = 0.3
vF,1/vF,2 = 0.5
vF,1/vF,2 = 0.7
vF,1/vF,2 = 1
0 0.5 1 1.5 2
0.5
1
1.5
⇠/`ee
⇢/⇢
res
kTF⇠ � 1
I similar to Baber scattering,but novel mechanism
New Boltzmann Transport 15
Why Do Interactions Increase the Resistivity?
I ∇ · J1 = ∇ · J2 = 0; QPspushed out of equilibrium:
J
Jimb
`ee
x
µimbalance
I all excited QPs relaxmomentum:
1
τ∼ `eevF
ξ2︸ ︷︷ ︸diffusion
× ξ2
`2ee︸︷︷︸imbalance
∼ vF
`ee
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
New Boltzmann Transport 15
Why Do Interactions Increase the Resistivity?
I ∇ · J1 = ∇ · J2 = 0; QPspushed out of equilibrium:
J
Jimb
`ee
x
µimbalance
I all excited QPs relaxmomentum:
1
τ∼ `eevF
ξ2︸ ︷︷ ︸diffusion
× ξ2
`2ee︸︷︷︸imbalance
∼ vF
`ee
Vimp ⇠
Vimp Vimp
x
y
x
y
x
y
New Boltzmann Transport 16
Bounds
I we proved a unified transport bound:
ρ ≤ T s
J2=〈Φodd|W|Φodd〉〈Φodd|E〉2
, subject to ∇ ·J [Φodd] = 0︸ ︷︷ ︸conservation of
charge, energy, imbalance...
upon integrating out non-conserved even modes:
W = Wodd + LTW−1evenL
I hydrodynamic limit:
T S ∼∫
ddx
(J − nv) ·Σ−1 · (J − nv)︸ ︷︷ ︸imbalance diffusion
+η(∇v)2
.I perturbative results from above become non-perturbative
bounds on ρ
New Boltzmann Transport 16
Bounds
I we proved a unified transport bound:
ρ ≤ T s
J2=〈Φodd|W|Φodd〉〈Φodd|E〉2
, subject to ∇ ·J [Φodd] = 0︸ ︷︷ ︸conservation of
charge, energy, imbalance...
upon integrating out non-conserved even modes:
W = Wodd + LTW−1evenL
I hydrodynamic limit:
T S ∼∫
ddx
(J − nv) ·Σ−1 · (J − nv)︸ ︷︷ ︸imbalance diffusion
+η(∇v)2
.
I perturbative results from above become non-perturbativebounds on ρ
New Boltzmann Transport 16
Bounds
I we proved a unified transport bound:
ρ ≤ T s
J2=〈Φodd|W|Φodd〉〈Φodd|E〉2
, subject to ∇ ·J [Φodd] = 0︸ ︷︷ ︸conservation of
charge, energy, imbalance...
upon integrating out non-conserved even modes:
W = Wodd + LTW−1evenL
I hydrodynamic limit:
T S ∼∫
ddx
(J − nv) ·Σ−1 · (J − nv)︸ ︷︷ ︸imbalance diffusion
+η(∇v)2
.I perturbative results from above become non-perturbative
bounds on ρ
Return to Experiments 17
Phenomenology: Enhanced Resistivity near Criticality
I sample phase diagram:
doping
Tρ ∼ T
ρ ∼ T 2
I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?
I layered materials with chemical doping =⇒ smootherpotentials...
I many ways to get this effect besides two Fermi surfaces...
Return to Experiments 17
Phenomenology: Enhanced Resistivity near Criticality
I sample phase diagram:
doping
Tρ ∼ T
ρ ∼ T 2
I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?
I layered materials with chemical doping =⇒ smootherpotentials...
I many ways to get this effect besides two Fermi surfaces...
Return to Experiments 17
Phenomenology: Enhanced Resistivity near Criticality
I sample phase diagram:
doping
Tρ ∼ T
ρ ∼ T 2
I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?
I layered materials with chemical doping =⇒ smootherpotentials...
I many ways to get this effect besides two Fermi surfaces...
Return to Experiments 17
Phenomenology: Enhanced Resistivity near Criticality
I sample phase diagram:
doping
Tρ ∼ T
ρ ∼ T 2
I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?
I layered materials with chemical doping =⇒ smootherpotentials...
I many ways to get this effect besides two Fermi surfaces...
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