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Quantum transport & impurities in dense matter
Arnau Rios HuguetSTFC Advanced FellowDepartment of Physics
University of Surrey
XI HYPBarcelona, 4 October 2012
Impurity
Self-Consistent Green’s functions
Mean-free path
Transport
0 / 17
Quantum transport in NSsShear viscosity of neutron matter in CBF
Wambach, Ainsworth & Pines, NPA 555, 128 (1993)Benhar & Valli, PRL 99, 232501 (2007)
Shear viscosity: CBF vs BHF
Benhar, Polls, Valli & Vidaña, PRC 81, 024305 (2010)Benhar & Carbone, arxiv:0912.0129
• EoS not enough⇒ aim at complete NS models!
• Viscosity coefficient: CBF/BHF + Landau-Abrikosov-Khalatnikov
• Hybrid models, not really ab initio!
• Better if experimentally testable1 Mean-free path⇒ Optical potentials & scattering
1 / 17
What many-body technique?Self-consistent Green’s functions
Ramos, Polls & Dickhoff, NPA 503 1 (1989)Alm et al., PRC 53 2181 (1996)
Dewulf et al., PRL 90 152501 (2003)Frick & Müther, PRC 68 034310 (2003)Rios, PhD Thesis, U. Barcelona (2007)
Somà & Bozek, PRC 78 054003 (2008)
Ladder approximation within SCGF
In-medium interaction Ladder self-energy
Dyson equationpp & hh Pauli blocking
Spectral function
One-body properties Momentum distributionThermodynamics & EoS
Transport
• Self-consistency, pp+hh & full off-shell effects2 / 17
What many-body technique?Self-consistent Green’s functions
0 0.08 0.16 0.24 0.32! [fm-3]
10
20
30
40
Ener
gy, E
/A [M
eV]
T=10 MeV
T=20 MeV
SCGFBHFFPVirial
CDBONN
0 0.08 0.16 0.24 0.32! [fm-3]
20
30
40
Ener
gy, E
/A [M
eV]
T=10 MeV
T=20 MeV
Argonne V18
0 0.01 0.02 0.0310
20
30
0 0.08 0.16 0.24 0.32ρ [fm-3]
10-2
10-1
100
101
Pres
sure
, p [M
eV fm
-3]
T=5 MeVT=10 MeVT=15 MeVT=20 MeVFPVirial
CDBONN
0 0.08 0.16 0.24 0.32ρ [fm-3]
10-1
100
101
Pres
sure
, p [M
eV fm
-3]
Argonne V18
0 0.01 0.02 0.0310-2
10-1
100
0 0.5 1 1.5 2Momentum, k/kF
0
0.2
0.4
0.6
0.8
1
Mom
entu
m d
istri
butio
n, n
(k)
Av18SCHF
!=0.16 fm-3, T=4 MeV
0 0.5 1 1.5 2 2.5 3Momentum, k/kF
10-3
10-2
10-1
100
Mom
entu
m d
istri
butio
n, n
(k)
!=0.16 fm-3, T=4 MeV
0 5 10 15 20Temperature, T [MeV]
0
0,5
1
1,5
2
Entro
py p
er p
artic
le, S
/A
SDQ
m*w*k
SQP
SMF
SNK
Av18, ρ=0.16 fm-3
10-610-510-410-310-210-1
Spectral function, ρ=0.16 fm-3
10-610-510-410-310-210-1
(2π)
-1 A
(k,ω
) [M
eV-1
]
-500 -250 0 250 500ω−µ [MeV]
10-510-410-310-210-1
T=5 MeV
Argonne V18 k=0
k=kF
k=2kF
Spectral function Total Energy
Equation of State
Entropy
Microscopic properties Bulk properties
Ladder approximation
Momentum distribution
In-medium interaction
• Self-consistency, pp+hh & full off-shell effects2 / 17
What many-body technique?Self-consistent Green’s functions
0 0.08 0.16 0.24 0.32! [fm-3]
10
20
30
40
Ener
gy, E
/A [M
eV]
T=10 MeV
T=20 MeV
SCGFBHFFPVirial
CDBONN
0 0.08 0.16 0.24 0.32! [fm-3]
20
30
40
Ener
gy, E
/A [M
eV]
T=10 MeV
T=20 MeV
Argonne V18
0 0.01 0.02 0.0310
20
30
0 0.08 0.16 0.24 0.32ρ [fm-3]
10-2
10-1
100
101
Pres
sure
, p [M
eV fm
-3]
T=5 MeVT=10 MeVT=15 MeVT=20 MeVFPVirial
CDBONN
0 0.08 0.16 0.24 0.32ρ [fm-3]
10-1
100
101
Pres
sure
, p [M
eV fm
-3]
Argonne V18
0 0.01 0.02 0.0310-2
10-1
100
0 0.5 1 1.5 2Momentum, k/kF
0
0.2
0.4
0.6
0.8
1
Mom
entu
m d
istri
butio
n, n
(k)
Av18SCHF
!=0.16 fm-3, T=4 MeV
0 0.5 1 1.5 2 2.5 3Momentum, k/kF
10-3
10-2
10-1
100
Mom
entu
m d
istri
butio
n, n
(k)
!=0.16 fm-3, T=4 MeV
0 5 10 15 20Temperature, T [MeV]
0
0,5
1
1,5
2
Entro
py p
er p
artic
le, S
/A
SDQ
m*w*k
SQP
SMF
SNK
Av18, ρ=0.16 fm-3
10-610-510-410-310-210-1
Spectral function, ρ=0.16 fm-3
10-610-510-410-310-210-1
(2π)
-1 A
(k,ω
) [M
eV-1
]
-500 -250 0 250 500ω−µ [MeV]
10-510-410-310-210-1
T=5 MeV
Argonne V18 k=0
k=kF
k=2kF
Spectral function Total Energy
Equation of State
Entropy
Microscopic properties Bulk properties
Ladder approximation
Momentum distribution
In-medium interaction
Transport? Impurities?
• Self-consistency, pp+hh & full off-shell effects2 / 17
Why the mean-free path?Motivation
Glauber calculations(input parameter)
(p,A) scattering(absorption)
Shell model(verification)
Fermi Liquid Theory(validation)
Transport coefficients(calculation methods)
Transport simulations(in-medium cross sections)
Nucleon mean-free path
λk
3 / 17
Why the mean-free path?Motivation
Glauber calculations(input parameter)
(p,A) scattering(absorption)
Shell model(verification)
Fermi Liquid Theory(validation)
Transport coefficients(calculation methods)
Transport simulations(in-medium cross sections)
Nucleon mean-free path
λk
3 / 17
Damping and mean-free pathNaive optical potential model[
−∇2
2m+ Re Σ(εk) + iIm Σ(εk)
]ψ(r) = εkψ(r)
ψ(r) = Ne−i
{k− i
2λk
}r ⇒ p(r) = |ψ(r)|2 ∼ e−
rλk
λk = − k
2m
1
Im Σ(Ek)=
k
m
1
Γk=∂kεkΓk
• Mean-free path from quasi-particle properties
λk =vkΓk
• Fundamental asymptotic behavior for propagator in real time
4 / 17
Damping and mean-free pathNaive optical potential model[
−∇2
2m+ Re Σ(εk) + iIm Σ(εk)
]ψ(r) = εkψ(r)
ψ(r) = Ne−i
{k− i
2λk
}r ⇒ p(r) = |ψ(r)|2 ∼ e−
rλk
λk = − k
2m
1
Im Σ(Ek)=
k
m
1
Γk=∂kεkΓk
• Mean-free path from quasi-particle properties
λk =vkΓk
• Fundamental asymptotic behavior for propagator in real time
4 / 17
Damping and mean-free pathNaive optical potential model[
−∇2
2m+ Re Σ(εk) + iIm Σ(εk)
]ψ(r) = εkψ(r)
ψ(r) = Ne−i
{k− i
2λk
}r ⇒ p(r) = |ψ(r)|2 ∼ e−
rλk
λk = − k
2m
1
Im Σ(Ek)=
k
m
1
Γk=∂kεkΓk
• Mean-free path from quasi-particle properties
λk =vkΓk
• Fundamental asymptotic behavior for propagator in real time
GR(k, t)→ −iηke−iεkte−|Γk|t
4 / 17
Quasi-particle "pole"
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10
Energy, E [MeV]
-60
-50
-40
-30
-20
-10
0
10
20
Wid
th,
! [
Me
V]
0
0.5
1
1.5
2
2.5
3
C
Γk
εk
• Fundamental asymptotic behavior for propagator in real time
GR(k, t)→ −iηke−iεkte−|Γk|t ⇒ λk =∂kεkΓk
• Time-energy Fourier transform using retarded contour + Cauchy
GR(k, t) =
∫ ∞−∞
dω
2πe−iωtGR(k, ω) ∼
∫C′
dz
2πe−izt η(z)
z − (εk − i|Γk|)= −iηke−iεkte−|Γk|t
5 / 17
Quasi-particle "pole"
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10
Energy, E [MeV]
-60
-50
-40
-30
-20
-10
0
10
20
Wid
th,
! [
Me
V]
0
0.5
1
1.5
2
2.5
3
C’
Γk
εk
• Fundamental asymptotic behavior for propagator in real time
GR(k, t)→ −iηke−iεkte−|Γk|t ⇒ λk =∂kεkΓk
• Time-energy Fourier transform using retarded contour + Cauchy
GR(k, t) =
∫ ∞−∞
dω
2πe−iωtGR(k, ω) ∼
∫C′
dz
2πe−izt η(z)
z − (εk − i|Γk|)= −iηke−iεkte−|Γk|t
5 / 17
From the cut to the poleAnalytical continuation yields pole in propagator
-200-150
-100-50
0 50
-60 -40 -20 0 20 40
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Energy,
E [M
eV]
Width, Γ [MeV]
-200-150
-100-50
0 50
-60 -40 -20 0 20 40
-0.8-0.6-0.4-0.2
0 0.2 0.4 0.6 0.8
Energy,
E [M
eV]
Width, Γ [MeV]
a.c. self-energy by demanding analyticity across real axis
Σ(k, z) ≡{
Σ(k, z), Im z > 0Σ∗(k, z), Im z ≤ 0
G(k, z) =1
z − k2
2m− Σ(k, z)
6 / 17
From qp pole to mean-free path
1 Complex energy self-energy
Σ(k, z) =
∫ ∞−∞
dω
2π
γ(k, ω)
z − ω ⇐ γ(k, ω) = ImΣ(k, ω+)︸ ︷︷ ︸SCGF
2 Analytical continuation of self-energy
Σ(k, z) ≡{
Σ(k, z), Im z > 0Σ∗(k, z), Im z ≤ 0
3 Complex Dyson equation
G(k, z) =1
z − k2
2m− Σ(k, z)
4 Position of quasi-particle pole in complex plane
zk =k2
2m+ Re Σ(k, zk) + iIm Σ(k, zk) ⇒ zk = εk + iΓk
5 Mean-free pathλk =
1
Γk
∂εk∂k
7 / 17
Hunting the poleCDBonn, T = 0, ρ = 0.16 fm−3
0
0.05
0.1
0.15
0.2S
pe
ctr
al fu
nctio
n,
A [
Me
V-1
] k=0
0
0.05
0.1
0.15
0.2k=kF
0
0.05
0.1
0.15
0.2k=2kF
-80 -60 -40 -20 0Energy, ω−µ [MeV]
-50
-40
-30
-20
-10
0
10
Wid
th,
Γ [
Me
V]
0
1
2
-40 -20 0 20 40Energy, ω−µ [MeV]
-50
-40
-30
-20
-10
0
10
0
1
2
3
4
5
100 120 140 160 180Energy, ω−µ [MeV]
-50
-40
-30
-20
-10
0
10
0
1
2
3
• Cross: full pole position
• Circle: first renormalization (expansion on Im z to 1st order)
• Square: second renormalization (expansion on Im z to 2nd order)8 / 17
Model dependence
1
10
100
λ [
fm]
CDBonn+3BFT=0 MeV
λλ
0λ
2λ
2’
(ρ σnp
)-1
Symmetric matter, ρ=0.16 fm-3
-50 0 50 100 150 200 250 300 350ε−µ [MeV]
1
10
100
λ [
fm]
T=0 MeV, CDBonn+3BF
T=0 MeV, CDBonn
T=5 MeV, CDBonn
T=5 MeV, Av18
λk =1
Γk
∂εk∂k
• λ ∼ 4− 5 fm above 50 MeV
• Compatible with pA experiments
• Small model dependence• λ0 ⇒ no non-locality• λ2 ⇒ full non-locality• λ′2 ⇒ m∗k non-locality
• Classical approximation is incorrect!
• Little effect of 3BFs
A. Rios & V. Somà, PRL 108, 012501 (2012)9 / 17
Density & temperature dependence
1
10
100
λ [
fm]
ρ=0.16 fm-3
T=0 MeV
T=4 MeV
T=8 MeV
T=12 MeV
T=16 MeV
T=20 MeV
CD-Bonn
-50 0 50 100 150 200 250 300 350ε−µ [MeV]
1
10
100
λ [
fm]
T=5 MeV ρ=0.08 fm-3
ρ=0.12 fm-3
ρ=0.16 fm-3
ρ=0.20 fm-3
ρ=0.24 fm-3
ρ=0.28 fm-3
λk =1
Γk
∂εk∂k
• Effect of temperature close to FS
• At zero T, infinite λ at Fermi surface
• At finite T, finite λ at Fermi surface
• Density affects all energies
• Above 50 MeV, ρ increase lowers λ
A. Rios & V. Somà, in preparation10 / 17
Isospin asymmetric matterTuning correlations
Nuclear “trencadís”β=0
Symmetric matter
β=1Neutron matter
β≠0Asymmetric matter β≈1
Polaron
SR+Tensor correlations
SR correlations
Neutrons less correlatedProtons more correlated
Protons maximally correlatedHyper-impurities?
� = N � ZN + Z
Nuclei
Neutron stars
Frick, Rios et al. PRC 71, 014313 (2005)Rios et al. PRC 79, 064308 (2009)
Carbone et al. EPL 97 22001 (2012)
11 / 17
Asymmetric nuclear matterMomentum distribution
0 0.5 1 1.5 2 2.5 3
Momentum, k [fm-1
]
0
0.2
0.4
0.6
0.8
1
Mom
entu
m d
istr
ibut
ion,
n(k
)
β=0.0β=0.2β=0.4β=0.6β=0.8β=1.0
Free Fermi Gas
0 0.5 1 1.5 2 2.5 3
Momentum, k [fm-1
]
0
0.2
0.4
0.6
0.8
1
Mom
entu
m d
istr
ibut
ion,
n(k
)SCGF, Argonne v18ρ=0.16 fm
-3 T=5 MeV
Neutrons
Protons
• Correlations affect depletion⇒ non-perturbative effect
• Neutrons become less correlated
• Protons are more correlated A. Rios et al., PRC 79, 064308 (2009) 12 / 17
Asymmetric nuclear matterSpectral functions
10-5
10-4
10-3
10-2
10-1
β=0.2
β=0.92
Neutrons
10-5
10-4
10-3
10-2
10-1
(2π
)-1 A
(k,ω
) [M
eV-1
]
-500 -250 0 250 500
ω−µ [MeV]
10-5
10-4
10-3
10-2
10-1
10-5
10-4
10-3
10-2
10-1
Protons
10-5
10-4
10-3
10-2
10-1
(2π
)-1 A
(k,ω
) [M
eV-1
]
-500 -250 0 250 500
ω−µ [MeV]
10-5
10-4
10-3
10-2
10-1
β=0.2
β=0.92
ρ=0.16 fm-3
T=5 MeVk=0
k=kn
k=2kn
k=0
k=kp
k=2kp
13 / 17
mfp: isospin asymmetry
1
10
100
Neu
tro
n, λ
[fm
] ρ=0.16 fm-3
T=5 MeV
β=0.5
β=0.4
β=0.3
β=0.2
β=0.1
β=0.04
CD-Bonn
-50 0 50 100 150 200 250 300 350ε−µ [MeV]
1
10
100
Pro
ton
, λ
[fm
]
λqk =1
Γqk
∂εqk∂k
• Asymmetry dependence is weak
• Neutron λ increases slightly
• Proton λ decreases ∼ 2 fm
A. Rios & V. Somà, very preliminary!14 / 17
mfp: isospin asymmetry
1
10
100
Neu
tro
n, λ
[fm
] ρ=0.16 fm-3
T=5 MeV
β=0.5
β=0.4
β=0.3
β=0.2
β=0.1
β=0.04
CD-Bonn
-50 0 50 100 150 200 250 300 350ε−µ [MeV]
1
10
100
Pro
ton
, λ
[fm
]
Zuo et al., PRC 60 024605 (1999) 14 / 17
Impurity: further issues
• Many-body dynamics of a polaron?
• Treatment of a fragmented impurity?
• Definiton of particle threshold?
• Thermodynamical limit?
Gaudí’s impurity
Brueckner-like Λ polaron
Robertson & Dickhoff, PRC 70 044301 (2004)
15 / 17
Conclusions
• Ab initio description of nuclear mean-free path, λk
• Fully self-consistent & quantum mechanical calculation
• Agreement with empirical estimates
• Independent of temperature and model
• Density dependence is strong
• Working towards impurities with SCGF
• Correlation effects not seen in λk
• Adding three-body forces consistently!
A. Carbone
16 / 17