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

Gràcies!

V. Somà

T. U. Darmstadt

W. H. Dickhoff

Wash. U. St. Louis

A. Polls, A. Carbone

University of Barcelona

UNIVERSITAT DE BARCELONA

U

B

17 / 17