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arXiv:1005.2235 · Physical Review C

Doubly periodic structure for the study of inhomogeneous bulk fermion matter with spatial localizations

Klaas Vantournhout, N. Jachowicz, J. Ryckebusch

Published Sep 20, 2011• 5 citations
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Abstract

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.

DOI
10.1103/physrevc.84.032801

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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.
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.
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.
I present a first-quantization deterministic algorithm for antisymmetrizing a spatially separated target-projectile system containing NTN_TNT​ and NpN_pNp​ identical fermions, respectively. The method constructs a fully antisymmetric wavefunction from the product of two independently antisymmetrized many-body states, each of which may be a superposition of Slater determinants. The algorithm uses a Dicke-state ancilla register that coherently encodes all one-particle exchange channels between the two subsystems, and, crucially, requires only single-particle swaps to generate the full antisymmetric structure. A total of O(NTNp)O(N_T N_p)O(NT​Np​) single-particle exchanges are needed, with up to NpN_pNp​ of them implemented in parallel, if an additional NpN_pNp​ ancillae are used. The correct fermionic phase is incorporated through application of ZZZ gates on NTN_TNT​ ancillae, after which the ancilla register is efficiently uncomputed using a compact sequence of controlled operations. This construction provides a nontrivial and scalable protocol for preparing fully antisymmetric states in reaction and scattering simulations, significantly expanding the range of systems that can be addressed with first-quantized quantum algorithms.
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.
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.
The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction n‾\overline{n}n, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.