Fermionic molecular dynamics
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1 Fermionic molecular dynamics Can it cook nuclear pasta? K. Vantournhout H. Feldmeier GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany NuSTAR seminar April 25, 2012 Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
2 Outline 1 Introduction 2 Recipes for pasta 3 Molecular dynamics for fermions 4 Periodic boundary conditions and FMD 5 Conclusion and outlook Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
3 Introduction : preface Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
4 Introduction : preface Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
5 Introduction : preface Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
6 Introduction : preface Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
7 Introduction : preface Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
8 Introduction : preface Eta Carinae and the bipolar Homunculus Nebula which surrounds the star. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
9 Introduction : preface March 16, 2012 SN2012aw in M95 Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
10 Introduction : preface Crab pulsar Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
11 Introduction : composition of a neutron star Coulomb Crystal Relativistic electron gas (e Z) Mantel Outer Crust Inner Crust Outer Core n+p+e+µ pasta phases Coulomb Crystal Relativistic electron gas dripped neutrons (e Z n)? Inner Core hyperons? -condensate? K-condensate? quark matter? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
12 Introduction : crustal matter and nuclear pastas meatballs spaghetti lasagne ziti Swiss cheese Inner Crust sauce Outer Core 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 the nuclear matter (Swiss cheese) transition to the neutron star core, uniform neutron matter (sauce) Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
13 Introduction : what is bucatini? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
14 Introduction : crustal matter and nuclear pastas meatballs spaghetti lasagne ziti Swiss cheese Inner Crust sauce Outer Core Where does the crustal and subnuclear density matter play a role? Supernova physics Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
15 Introduction : crustal matter and nuclear pastas meatballs spaghetti lasagne ziti Swiss cheese Inner Crust sauce Outer Core Where does the crustal and subnuclear density matter play a role? Supernova physics Neutron star cooling Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
16 Introduction : crustal matter and nuclear pastas meatballs spaghetti lasagne ziti Swiss cheese Inner Crust sauce Outer Core Where does the crustal and subnuclear density matter play a role? Supernova physics Neutron star cooling The r-process Neutron star glitches Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
17 Introduction : crustal matter and nuclear pastas meatballs spaghetti lasagne ziti Swiss cheese Inner Crust sauce Outer Core Where does the crustal and subnuclear density matter play a role? Supernova physics Neutron star cooling The r-process Neutron star glitches Gravitational waves Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
18 pastas : is there a relation to Turing instabilities? Tomographic reconstruction of the Belousov-Zhabotinsky reaction incorporated in an water-in-oil aerosol micro-emulsion. Turing patterns form in a reaction-diffusion medium under the condition of a long-ranged reaction inhibition and a short-ranged activation. T. Jr. Bánsági et al., Science 331, pp (2011) Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
19 pasta recipe : pasta alla liquid drop The liquid-drop model K. Nakazato et al., Phys. Rev. Lett. 103, (2009) Semi-empirical model mainly studies the canonical pasta phases does not allow dynamics the hard-coded geometry is studied computationally fairly simple Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
20 pasta recipe : pasta in scatola alla Hartree-Fock The Hartree-Fock model W.G. Newton et al., Phys. Rev. C 79, (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) Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
21 pasta recipe : pasta classica con dinamica molecolare The molecular-dynamics technique G. Watanabe et al., Phys. Rev. Lett. 103, (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 Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
22 pasta recipe : crosta croccante con dinamica molecolare The molecular-dynamics technique is awesome! C. Horowitz et al., Phys. Rev. Lett. 102, (2009) allows laboratory like simulations (e.g. viscosity or stress tensors) allows to study the effect of external probes time-dependent, out-of-equilibrium, thermodynamics, phase transitions,... lacks quantum features (mimics fermion behaviour by a Pauli potential) How can we use MD with correct quantum features? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
23 What is classical molecular dynamics I. Newton Molecular dynamics studies systems of interacting particles by numerically solving the equations of motion. These particles can be point-like, spherical or non-spherical rigid bodies with extra internal degrees of freedom. Time-dependent or static studies are done by solving Newton s or Hamilton s equations of motion (here for point-particles) : W.R. Hamilton q = H p, ṗ = H q. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
24 Fermionic molecular dynamics : the wave function J.C.F. Gauß The single particle fermion wave function is a parametrised quantum-dressed Gaussian wave packet q p = c kp A kp b kp χ kp ζ kp, k x Ab = exp 1 2 (x b) A 1 (x b) The antisymmetrised many-body fermion wave function Q is a Slater determinant of non-orthogonal fermion wave-packets : x 1,..., x A Q = 1 x 1 q 1... x 1 q A det. A! x A q 1... x A q A J.C. Slater Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
25 Fermionic molecular dynamics : the math Adam Ries FMD has a matrix formalism. All relevant fermion physics is contained in the overlap matrix n = [ q p q q ] A pq=1 and its inverse o = n 1. Expectation values are evaluated as : I = Q ˆB I Q Q Q = A q p ˆB I q q o qp pq=1 Time-dependent or static studies are done by means of variational principles on the variables z of trail state Q(z) : Lord Rayleigh ic ż = Q Ĥ Q z Q Q, Q Ĥ ˆT C M Q z = 0. Q Q H. Feldmeier et al., Rev. Mod. Phys. 72, pp (2000) Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
26 Fermionic molecular dynamics : its fascinating! Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
27 How to simulate bulk systems? Finite-sized simulations introduce unwanted surface effects. The more particles, the heavier the computation (N 2(4) ) The less particles, the larger the influence of surface effects. (488/1000 particles) Can we use a small number of particles and avoid surface effects? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
28 How to simulate bulk systems? Finite-sized simulations introduce unwanted surface effects. The more particles, the heavier the computation (N 2(4) ) The less particles, the larger the influence of surface effects. (488/1000 particles) Can we use a small number of particles and avoid surface effects? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
29 How to simulate bulk systems? Finite-sized simulations introduce unwanted surface effects. The more particles, the heavier the computation (N 2(4) ) The less particles, the larger the influence of surface effects. (488/1000 particles) Can we use a small number of particles and avoid surface effects? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
30 Fermionic molecular dynamics : bulk systems Periodic boundary conditions One unit cell contains A single particle states The unit cell is periodically replicated in all directions Which shape of unit cell? What kind of periodicity? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
31 Periodic boundary conditions : Bravais lattices A Bravais lattice is an infinite array of points periodically placed throughout space. a 2 a 1 The lattice appears identical from any point. Each lattice point is referred to through a Bravais lattice vector : R = n 1 a 1 + n 2 a 2 +. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
32 Periodic boundary conditions : Primitive cell A primitive cell tessellates space perfectly without overlap or voids. A primitive cell contains only a single lattice point. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
33 Periodic boundary conditions : Wigner-Seitz cell A Wigner-Seitz cell is a primitive cell preserving the lattice symmetry. It is a region of space that is closer to one lattice point than to any other. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
34 Periodic boundary conditions : Reciprocal lattice The reciprocal lattice is a dual lattice in the Fourier space of the Bravais lattice. Its lattice vectors K satisfy e i K R = 1. Its Wigner-Seitz cell is referred to as the first Brillouin zone. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
35 Periodic Boundary Conditions : Common lattices Square lattice Hexagonal lattice Simple cubic Face-centred cubic Body-centred cubic Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
36 Fermionic molecular dynamics : bulk systems Periodic boundary conditions One unit cell contains A single particle states The unit cell is periodically replicated in all directions Which Bravais lattice? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
37 Fermionic molecular dynamics : bulk systems What about FMD and PBC? Fermions require antisymmetrisation Antisymmetry does not stop at the border of the unit cell FMD must be applied to the infinite periodic system Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
38 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
39 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. N = Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
40 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. N = Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
41 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. N = Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
42 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. N = Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
43 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. N = Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
44 Fermionic molecular dynamics : the bulk overlap matrix The matrix formalism of FMD becomes infinite dimensional but has a multi-index block-toeplitz structure that can be exploited. N = Expectation values become finite-dimensional ρ,i = 1 A I,pq (k) qp (k)dk, V BZ BZ pq=1 = n P Q,pq = q p ˆT (Q P) q q where a volume-integral reflects the infinite system. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
45 ` ` Results : the one-dimensional free Fermi gas 4 (a) a/ ` = (b) a/ ` = ρ x 2 ρ kkf (c) a/ ` = (d) a/ ` = 1.0 k F = π/ ` ρ x 1.0 ρ kkf scaled position x/ ` scaled momentum k / k F For small overlap : the particles behave as distinguishable particles For large overlap : the particles reproduce free Fermi gas behaviour Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
46 Results : understanding antisymmetry and PBC Coordinate density ρv W Z Momentum density ρv BZ y/l k y /k l x/l k x /k l 0.0 Gaussian width : a = l/5 without periodicity without antisymmetry Densities in momentum and coordinate space are summed. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
47 Results : understanding antisymmetry and PBC Coordinate density ρv W Z 0.5 Momentum density ρv BZ y/l k y /k l x/l k x /k l 0.0 Gaussian width : a = l/5 without periodicity with antisymmetry Antisymmetry enables the Pauliexclusion principle. It seems like particles are pushed away. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
48 Results : understanding antisymmetry and PBC Coordinate density ρv W Z Momentum density ρv BZ y/l k y /k l x/l k x /k l 0.0 Gaussian width : a = l/5 without periodicity without antisymmetry Densities in momentum and coordinate space are summed. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
49 Results : understanding antisymmetry and PBC Coordinate density ρv W Z Momentum density ρv BZ y/l k y /k l x/l k x /k l 0.0 Gaussian width : a = l/5 with periodicity without antisymmetry Densities in momentum and coordinate space are summed and periodic corrections added in the spatial density. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
50 Results : understanding antisymmetry and PBC Coordinate density ρv W Z 0.5 Momentum density ρv BZ y/l k y /k l x/l k x /k l 0.0 Gaussian width : a = l/5 with periodicity with antisymmetry Antisymmetry is a long-range manybody correlation. Both densities tend to that of a free fermion system. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
51 Results : the two-dimensional free Fermi gas Minimal kinetic energy for N fermions per unit-cell. ρ = 1/fm 2. With increasing overlap, the system moves towards the results of the free Fermi gas, however there is a strong particle dependence. What is the origin of this seemingly erratic behaviour? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
52 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. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
53 Where is Wally? Where is my box? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
54 The effect of periodic boundary conditions Coordinate space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
55 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
56 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
57 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
58 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
59 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
60 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
61 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
62 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
63 The effect of periodic boundary conditions Momentum space Periodic boundary conditions introduce a discrete translation symmetry into the system as a result of the lattice (box). By construction, the simulation cannot leave this symmetry. To uphold the lattice symmetry, Bragg planes and Brillouin zones define the Fermi surface and replace the spherical Fermi distribution. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
64 Results : the two-dimensional free Fermi gas Spatial and momentum distribution for minimal kinetic energy y/l 0.0 k y /k l y/l 0.0 k y /k l y/l x/l ρ = 1/fm 2, square lattice x/l k y /k l k x /k l k x /k l Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
65 Results : the two-dimensional free Fermi gas Spatial and momentum distribution for minimal kinetic energy y/l y/l y/l x/l ρ = 1/fm 2, hexagonal lattice x/l k y /k l k y /k l k y /k l k x /k l k x /k l Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
66 Results : the two-dimensional free Fermi gas Brillouin zones are the limit for large overlap, and become more spherical with increasing N. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
67 Conclusion and outlook Conclusion : FMD can study bulk matter using PBC It reproduces free fermion behaviour The effect of PBC is well understood Outlook : We are finally on the stage where we can investigate the neutron star s crust. Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
68 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.
69 Introduction : composition of a neutron star Coulomb Crystal Relativistic electron gas (e Z) Mantel Outer Crust Inner Crust Outer Core n+p+e+µ pasta phases Coulomb Crystal Relativistic electron gas dripped neutrons (e Z n)? Inner Core hyperons? -condensate? K-condensate? quark matter? Klaas Vantournhout (GSI Darmstadt) Can we cook pasta with FMD? NuSTAR / 33
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