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arXiv:1111.3221 · Journal of Physics Conference Series

Fully antisymmetrised dynamics for bulk fermion systems

Klaas Vantournhout, Hans Feldmeier

Published Feb 13, 2012• 2 citations
arXivHTMLPDF

Abstract

The neutron star's crust and mantel are typical examples of non-uniform bulk systems with spacial localisations. When modelling such systems at low temperatures, as is the case in the crust, one has to work with antisymmetrised many-body states to get the correct fermion behaviour. Fermionic molecular dynamics, which works with an antisymmetrised product of localised wave packets, should be an appropriate choice. Implementing periodic boundary conditions into the fermionic molecular dynamics formalism would allow the study of the neutron star's crust as a bulk quantum system. Unfortunately, the antisymmetrisation is a non-local entanglement which reaches far out of the periodically repeated unit cell. In this proceeding, we give a brief overview how periodic boundary conditions and fermionic molecular dynamics can be combined without truncating the long-range many-body correlation induced by the antisymmetry of the many-body state.

DOI
10.1088/1742-6596/342/1/012011

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The neutron star's crust and mantel are typical examples of non-uniform bulk systems with spacial localisations. When modelling such systems at low temperatures, as is the case in the crust, one has to work with antisymmetrised many-body states to get the correct fermion behaviour. Fermionic molecular dynamics, which works with an antisymmetrised product of localised wave packets, should be an appropriate choice. Implementing periodic boundary conditions into the fermionic molecular dynamics formalism would allow the study of the neutron star's crust as a bulk quantum system. Unfortunately, the antisymmetrisation is a non-local entanglement which reaches far out of the periodically repeated unit cell. In this proceeding, we give a brief overview how periodic boundary conditions and fermionic molecular dynamics can be combined without truncating the long-range many-body correlation induced by the antisymmetry of the many-body state.
We present a novel method to study the dynamics of bulk fermion systems such as the neutron-star crust.By introducing periodic boundary conditions into Fermionic Molecular Dynamics, it becomes possible to examine the long-range many-body correlations induced by antisymmetrisation in bulk fermion system.The presented technique treats the spins and the fermionic nature of the nucleons explicitly and permits investigating the dynamics of the system.Despite the increased complexity related to the periodic boundary conditions, the formalism remains computationally feasible.
Bulk fermionic matter, as can be notably found in supernova matter and neutrons stars, is subject to correlations of infinite range due to the antisymmetrization of the NNN-body wave function, which cannot be explicitly accounted for in a practical simulation. This problem is usually addressed in condensed matter physics by means of the so-called twist averaged boundary condition method. A different ansatz based on the localized Wannier representation has been proposed in the context of antisymmetrized molecular dynamics. In this paper we work out the formal relation between the two approaches. We show that, while the two coincide when working with exact eigenstates of the NNN-body Hamiltonian, differences appear in the case of variational approaches, which are currently used for the description of stellar matter. Some model applications with fermionic molecular dynamics are shown.
In cold atoms and in the crust of neutron stars, the pairing gap can reach values comparable with the Fermi energy. While in nuclei the neutron gap is smaller, it is still of the order of a few percent of the Fermi energy. The pairing mechanism in these systems is due to short range attractive interactions between fermions and the size of the Cooper pair is either comparable to the inter‐particle separation or it can be as big as a nucleus, which is still relatively small in size. Such a strong pairing gap is the result of the superposition of a very large number of particle–particle configurations, which contribute to the formation of the Cooper pairs. These systems have been shown to be the host of a large number of remarkable phenomena, in which the large magnitude of the pairing gap plays an essential role: quantum shock waves, quantum turbulence, Anderson–Bogoliubov–Higgs mode, vortex rings, domain walls, soliton vortices, vortex pinning in neutron star crust, unexpected dynamics of fragmented condensates and role of pairing correlations in collisions on heavy‐ions, Larkin–Ovchinnikov phase as an example of a Fermi supersolid, role of pairing correlations in controlling the dynamics of fissioning nuclei.
We present a method that offers perspectives to perform fully antisymmetrized simulations for inhomogeneous bulk fermion matter. The technique bears resemblance to classical periodic boundary conditions, using localized single-particle states. Such localized states are an ideal tool to discuss phenomena where spatial localization plays an important role. The antisymmetrization is obtained introducing a doubly periodic structure in the many-body fermion wave functions. This results in circulant matrices for the evaluation of expectation values, leading to a computationally tractable formalism to study fully antisymmetrized bulk fermion matter. We show that the proposed technique is able to reproduce essential fermion features in an elegant and computationally advantageous manner.
Fermionic Molecular Dynamics (FMD) models a system of fermions by means of many-body states which are composed of antisymmetrized products of single-particle states. These consist of one or several Gaussians localized in coordinate and momentum space. The parameters specifying them are the dynamical variables of the model. As the repulsive core of the nucleon-nucleon interaction induces short range correlations which cannot be accommodated by a Slater determinant, a novel approach, the unitary correlation operator method (UCOM), is applied. The unitary correlator moves two particles away from each other whenever their relative distance is within the repulsive core. The time-dependent variational principle yields the equations of motion for the variables. Energies of the stationary ground states are calculated and compared to exact many-body results for nuclei up to Ca 48. Time-dependent solutions are shown for collisions between nuclei.
An exact solution of an eight-site crystal model with periodic boundary conditions, a {001} thin film with face-centered-cubic crystal structure, is presented for the case of a heavy-fermion system. The object is to study modifications of the many-body electronic structure in the presence of surfaces. The model consists of (a) a single, fully symmetric orbital per site, with nearest-neighbor and second-nearest-neighbor hopping; (b) an infinite Coulomb repulsion between electrons on the same site; (c) antiferromagnetic superexchange interactions; and (d) a nearly half-filled band ((7/8 electron per site). Two structures are investigated: a two-layer and a four-layer fcc slab. The nature of the change from bulk to surface is studied by introducing a variable coupling between periodically extended slabs. The many-body spectrum, the one-electron occupancy, and the change in properties from bulk to surface are obtained and discussed.