A general spectral method for the numerical simulation of one-dimensional interacting fermions

This work introduces a general framework for the direct numerical simulation of systems of interacting fermions in one spatial dimension. The approach is based on a specially adapted nodal spectral Galerkin method, where the basis functions are constructed to obey the antisymmetry relations of fermionic wave functions. An efficient Matlab program for the assembly of the stiffness and potential matrices is presented, which exploits the combinatorial structure of the sparsity pattern arising from this discretization to achieve optimal run-time complexity. This program allows the accurate discretization of systems with multiple fermions subject to arbitrary potentials, e.g., for verifying the accuracy of multi-particle approximations such as Hartree–Fock in the few-particle limit. It can be used for eigenvalue computations or numerical solutions of the time-dependent Schrödinger equation.Program summaryProgram title: assembleFermiMatrixCatalogue identifier: AEKO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEKO_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 102No. of bytes in distributed program, including test data, etc.: 2294Distribution format: tar.gzProgramming language: MATLABComputer: Any architecture supported by MATLABOperating system: Any supported by MATLAB; tested under Linux (x86-64) and Mac OS X (10.6)RAM: Depends on the dataClassification: 4.3, 2.2Nature of problem: The direct numerical solution of the multi-particle one-dimensional Schrödinger equation in a quantum well is challenging due to the exponential growth in the number of degrees of freedom with increasing particles.Solution method: A nodal spectral Galerkin scheme is used where the basis functions are constructed to obey the antisymmetry relations of the fermionic wave function. The assembly of these matrices is performed efficiently by exploiting the combinatorial structure of the sparsity patterns.Restrictions: Only one-dimensional computational domains with homogeneous Dirichlet or periodic boundary conditions are supported.Running time: Seconds to minutes