k. vantournhout h. feldmeier nustar seminar...nustar seminar april 25, 2012 klaas vantournhout (gsi...

Introduction Recipes for pasta Molecular dynamics for fermions Periodic boundary conditions and FMD Conclusion and outlook

Fermionic molecular dynamicsCan it cook nuclear pasta?

K. Vantournhout H. Feldmeier

GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany

NuSTAR seminarApril 25, 2012

Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 1 / 33


Outline

1 Introduction

2 Recipes for pasta

3 Molecular dynamics for fermions

4 Periodic boundary conditions and FMD

5 Conclusion and outlook



Introduction : preface




Eta Carinae and thebipolar HomunculusNebula which surroundsthe star.




March 16, 2012SN2012aw in M95




Crab pulsar



Introduction : composition of a neutron star

Outer Core

Inner Crust

Outer Crust

Inner Core

pasta phasesMantel

?K-condensate?-condensate?

quark matter?

hyperons?

n+p+e+µ

Coulomb CrystalRelativistic electron gasdripped neutrons

(e Z)

(e Z n)

Coulomb CrystalRelativistic electron gas



Introduction : crustal matter and nuclear pastas

spaghettimeatballs lasagne ziti Swiss cheese

sauce

OuterCore

InnerCrust

Starting from the inner crust, with increasing density

three-dimensional spherical nuclear clusters (meatballs)

two-dimensional cylindrical tubes of dense matter (spaghetti)

one-dimensional slabs interlaid with planar voids (lasagne)

two-dimensional cylindrical neutron liquids within the nuclear matter(bucatini)

three-dimensional spherical neutron-liquid-bubbles embedded in thenuclear matter (Swiss cheese)

transition to the neutron star core, uniform neutron matter (sauce)



Introduction : what is bucatini?



Introduction : crustal matter and nuclear pastas

spaghettimeatballs lasagne ziti Swiss cheese

sauce

OuterCore

InnerCrust

Where does the crustal and subnuclear density matter play a role?

Supernova physics

Neutron star cooling

The r-process

Neutron star glitches

Gravitational waves



pastas : is there a relation to Turing instabilities?

T. Jr. Bánsági et al., Science 331, pp. 1309–1312 (2011)

Tomographic reconstruction ofthe Belousov-Zhabotinskyreaction incorporated in anwater-in-oil aerosolmicro-emulsion.

Turing patterns form in areaction-diffusion mediumunder the condition of along-ranged reaction inhibitionand a short-ranged activation.



pasta recipe : pasta alla liquid drop

The liquid-drop model

K. Nakazato et al., Phys. Rev. Lett. 103, 132501 (2009)

Semi-empirical model

mainly studies the canonical pasta phases

does not allow dynamics

the hard-coded geometry is studied

computationally fairly simple



pasta recipe : pasta in scatola alla Hartree-Fock

The Hartree-Fock model

W.G. Newton et al., Phys. Rev. C 79, 055801 (2009)

mean-field model

studies the canonical pasta phases

does not allow dynamics

the studied geometry strongly depends on the box-geometry

computationally very intensive (large configuration space)



pasta recipe : pasta classica con dinamica molecolare

The molecular-dynamics technique

G. Watanabe et al., Phys. Rev. Lett. 103, 121101 (2009)

a quantum-dressed classical many-body technique

studies the canonical and intermediate pasta phases

allows dynamics

the studied geometry is independent of the box-geometry

computationally fairly intensive



pasta recipe : crosta croccante con dinamica molecolare

The molecular-dynamics technique is awesome!

C. Horowitz et al., Phys. Rev. Lett. 102, 191102 (2009)

allows laboratory like simulations (e.g. viscosity or stress tensors)allows to study the effect of external probestime-dependent, out-of-equilibrium, thermodynamics, phasetransitions, . . .lacks quantum features (mimics fermion behaviour by a Paulipotential)

How can we use MD with correct quantum features?



What is classical molecular dynamics

I. Newton

Molecular dynamics studies systems ofinteracting particles by numerically solvingthe equations of motion. These particles canbe point-like, spherical or non-spherical rigidbodies with extra internal degrees of freedom.

Time-dependent or static studies are done bysolving Newton’s or Hamilton’s equations ofmotion (here for point-particles) :

q =∂ H

∂ p, p =−∂ H

∂ q.

W.R. Hamilton



Fermionic molecular dynamics : the wave function

J.C.F. Gauß The single particle fermion wave function is aparametrised quantum-dressed Gaussian wavepacket

|qp⟩=∑

k

ckp|Akpbkp⟩ ⊗ |χkp⟩ ⊗ |ζkp⟩,

⟨x |Ab⟩= exp�

−1

2(x − b) ·A−1 · (x − b)

�

The antisymmetrised many-body fermionwave function |Q⟩ is a Slater determinant ofnon-orthogonal fermion wave-packets :

⟨x1, . . . , xA|Q⟩=1pA!

det

⟨x1|q1⟩ . . . ⟨x1|qA⟩...

. . ....

⟨xA|q1⟩ . . . ⟨xA|qA⟩

J.C. Slater



Fermionic molecular dynamics : the math

Adam Ries FMD has a matrix formalism. All relevantfermion physics is contained in the overlapmatrix n= [⟨qp|qq⟩]Apq=1 and its inverseo= n−1. Expectation values are evaluated as :

BI =⟨Q|BI |Q⟩⟨Q|Q⟩ =

A∑

pq=1

⟨qp|BI |qq⟩oqp

Time-dependent or static studies are done bymeans of variational principles on thevariables z of trail state |Q(z)⟩ :

iC·z = ∂

∂ z?⟨Q|H|Q⟩⟨Q|Q⟩ ,

∂

∂ z?⟨Q|H− TC M |Q⟩

⟨Q|Q⟩ = 0.

H. Feldmeier et al., Rev. Mod. Phys. 72, pp. 655–688 (2000)

Lord Rayleigh



Fermionic molecular dynamics : its fascinating!



How to simulate bulk systems?

Finite-sized simulations introduceunwanted surface effects.

The more particles, the heavierthe computation (N2(4))

The less particles, the largerthe influence of surface effects.(488/1000 particles)

Can we use a small number ofparticles and avoid surfaceeffects?



Fermionic molecular dynamics : bulk systems

Periodic boundary conditions

One unit cell contains A singleparticle states

The unit cell is periodicallyreplicated in all directions

Which shape of unit cell?What kind of periodicity?



Periodic boundary conditions : Bravais lattices

a1

a2

A Bravais lattice is an infinite arrayof points periodically placedthroughout space.

The lattice appears identicalfrom any point.

Each lattice point is referred tothrough a Bravais latticevector :

R = n1a1 + n2a2 + · · · .



Periodic boundary conditions : Primitive cell

A primitive cell tessellates spaceperfectly without overlap or voids.

A primitive cell contains only asingle lattice point.



Periodic boundary conditions : Wigner-Seitz cell

A Wigner-Seitz cell is a primitivecell preserving the lattice symmetry.

It is a region of space that is closerto one lattice point than to anyother.



Periodic boundary conditions : Reciprocal lattice

The reciprocal lattice is a duallattice in the Fourier space of theBravais lattice.

Its lattice vectors K satisfy

eiK ·R = 1.

Its Wigner-Seitz cell is referredto as the first Brillouin zone.



Periodic Boundary Conditions : Common lattices

Square lattice Hexagonal lattice

Simple cubic Face-centred cubic Body-centred cubic




Periodic boundary conditions

One unit cell contains A singleparticle states

The unit cell is periodicallyreplicated in all directions

Which Bravais lattice?




What about FMD and PBC?

Fermions requireantisymmetrisation

Antisymmetry does not stop atthe border of the unit cell

FMD must be applied to theinfinite periodic system



Fermionic molecular dynamics : the bulk overlap matrix

The matrix formalism of FMD becomes infinite dimensional but has amulti-index block-Toeplitz structure that can be exploited.





N=





N=

= nP−Q,pq = ⟨qp|T (Q− P)|qq⟩

Expectation values becomefinite-dimensional

Bρ,I =1

VBZ

∫

BZ

A∑

pq=1

B I ,pq(k)O qp(k)dk,

where a volume-integral reflects theinfinite system.



Results : the one-dimensional free Fermi gas

−2 −1 0 1 2

0

1

2

3

4

ρx

a/ = 0.2(a)

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

ρkk

F

a/ = 0.2(b)

−2 −1 0 1 2scaled position x/

0.0

0.5

1.0

1.5

2.0

ρx

a/ = 1.0(c)

0.0 0.5 1.0 1.5 2.0scaled momentum |k|/ kF

0.0

0.5

1.0

1.5

2.0

ρkk

F

a/ = 1.0kF = π/

(d)

` `

` `

`

`

``

For small overlap : the particles behave as distinguishable particles

For large overlap : the particles reproduce free Fermi gas behaviour



Results : understanding antisymmetry and PBC

−0.5

0.0

0.5

y/`

−0.5 0.0 0.5x/`

Coordinate density ρVW Z

0.0

1.0

2.0

−2

−1

0

1

2

k y/k `

−2 −1 0 1 2kx/k`

Momentum density ρVBZ

0.0

5.0

10.0

Gaussian width :p

a = `/5without periodicity

without antisymmetryDensities in momentum and coordinatespace are summed.




−0.5

0.0

0.5

y/`

−0.5 0.0 0.5x/`


0.0

0.5

1.0

−2

−1

0

1

2

k y/k `

−2 −1 0 1 2kx/k`


0.0

0.5

1.0

1.5

2.0

2.5

Gaussian width :p


with antisymmetry

Antisymmetry enables the Pauli-exclusion principle. It seems likeparticles are pushed away.




−0.5

0.0

0.5

y/`

−0.5 0.0 0.5x/`


0.0

1.0

2.0

−2

−1

0

1

2

k y/k `

−2 −1 0 1 2kx/k`


0.0

5.0

10.0

Gaussian width :p


without antisymmetryDensities in momentum and coordinatespace are summed.




−0.5

0.0

0.5

y/`

−0.5 0.0 0.5x/`


0.0

1.0

2.0

−2

−1

0

1

2

k y/k `

−2 −1 0 1 2kx/k`


0.0

5.0

10.0

Gaussian width :p

a = `/5with periodicity

without antisymmetry

Densities in momentum and coordinatespace are summed and periodic correc-tions added in the spatial density.




−0.5

0.0

0.5

y/`

−0.5 0.0 0.5x/`


0.0

0.5

1.0

−2

−1

0

1

2

k y/k `

−2 −1 0 1 2kx/k`


0.0

0.5

1.0

1.5

2.0

2.5

Gaussian width :p

a = `/5with periodicity

with antisymmetry

Antisymmetry is a long-range many-body correlation. Both densities tend tothat of a free fermion system.



Results : the two-dimensional free Fermi gas

With increasing overlap, the system moves towardsthe results of the free Fermi gas, however there is astrong particle dependence.

Minimal kinetic energyfor N fermions perunit-cell.ρ = 1/fm2.

What is the origin ofthis seemingly erraticbehaviour?



Where is Wally?

Simulations often introduce constraints to overcome certain problems.But how do these constraints influence the physics of the system under

study? Periodic boundary conditions are such a restriction.



Where is Wally?

Where is my box?



The effect of periodic boundary conditions

Coordinate space

Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.

To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.



The effect of periodic boundary conditions

Momentum space

Periodic boundary conditionsintroduce a discrete translationsymmetry into the system as aresult of the lattice (box). Byconstruction, the simulation cannotleave this symmetry.

To uphold the lattice symmetry,Bragg planes and Brillouin zonesdefine the Fermi surface and replacethe spherical Fermi distribution.




Spatial and momentum distribution for minimal kinetic energy

−0.4

−0.2

0.0

0.2

0.4

y/`

−0.4

−0.2

0.0

0.2

0.4

y/`

−0.4

−0.2

0.0

0.2

0.4

y/`

−0.4−0.2 0.0 0.2 0.4x/`

−0.4−0.2 0.0 0.2 0.4x/`

0.8 0.9 1.0 1.1 1.2 1.3

−2

−1

0

1

2

k y/k `

−2

−1

0

1

2

k y/k `

−2

−1

0

1

2k y/k `

−2 −1 0 1 2kx/k`

−2 −1 0 1 2kx/k`

0.0 0.5 1.0

ρ = 1/fm2, square latticeKlaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 28 / 33



Spatial and momentum distribution for minimal kinetic energy

−0.4

−0.2

0.0

0.2

0.4

y/`

−0.4

−0.2

0.0

0.2

0.4

y/`

−0.4

−0.2

0.0

0.2

0.4

y/`

−0.4−0.20.0 0.2 0.4x/`

−0.4−0.20.0 0.2 0.4x/`

0.7 0.8 0.9 1.0 1.1 1.2

−2

−1

0

1

2

k y/k `

−2

−1

0

1

2

k y/k `

−2

−1

0

1

2k y/k `

−2 −1 0 1 2kx/k`

−2 −1 0 1 2kx/k`

0.0 0.5 1.0

ρ = 1/fm2, hexagonal latticeKlaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR 2012 28 / 33



Brillouin zones are the limit for large overlap, and become morespherical with increasing N .



Conclusion and outlook

Conclusion :

FMD can study bulk matter usingPBC

It reproduces free fermionbehaviour

The effect of PBC is wellunderstood

Outlook : We are finally on the stagewhere we can investigate the neutronstar’s crust.


THE ANTISYMMETRY OF A WAVE FUNCTION IS A LONG-RANGE EFFECT, TREAT IT

WITH RESPECT!

THE GEOMETRY OF THE LATTICE INFLUENCES THE PHYSICS AND SHOULD BE

TREATED AS AN ACTIVE PARAMETER IN SIMULATIONS.


Introduction : composition of a neutron star

Outer Core

Inner Crust

Outer Crust

Inner Core

pasta phasesMantel

?K-condensate?-condensate?

quark matter?

hyperons?

n+p+e+µ

Coulomb CrystalRelativistic electron gasdripped neutrons

(e Z)

(e Z n)

Coulomb CrystalRelativistic electron gas


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k. vantournhout h. feldmeier nustar seminar...nustar seminar april 25, 2012 klaas vantournhout (gsi...

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