CONAN—The cruncher of local exchange coefficients for strongly interacting confined systems in one dimension

Abstract

We consider a one-dimensional system of particles with strong zero-range interactions. This system can be mapped onto a spin chain of the Heisenberg type with exchange coefficients that depend on the external trap. In this paper, we present an algorithm that can be used to compute these exchange coefficients. We introduce an open source code CONAN (Coefficients of One-dimensional N-Atom Networks) which is based on this algorithm. CONAN works with arbitrary external potentials and we have tested its reliability for system sizes up to around 35 particles. As illustrative examples, we consider a harmonic trap and a box trap with a superimposed asymmetric tilted potential. For these examples, the computation time typically scales with the number of particles as O(N3.5±0.4). Computation times are around 10 s for N=10 particles and less than 10 min for N=20 particles. Program summaryProgram title: CONANProgram Files doi:http://dx.doi.org/10.17632/tw87vdy68b.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: CNature of problem: A system of N atoms (fermions or bosons) with a strong zero-range interaction confined in a one-dimensional potential V(x) can be described using a spin chain Heisenberg type Hamiltonian. This effective spin chain Hamiltonian is defined through N−1 exchange coefficients (also called geometric coefficients, αk). The exchange coefficients depend only on the integer N and the function V(x), but each coefficient is formally given as an (N−1)-dimensional integral. Given a number of particles N and a confining potential V(x), we wish to compute the exchange coefficients, but carrying out the (N−1)-dimensional integral numerically is not, to say the least, a method that scales well with the system size.Solution method: We wish to compute the exchange coefficients for a given system, but to do this we need to express them in a way that is more well-suited for a numerical implementation, i.e. in a way that does not involve an (N−1)-dimensional integral. In the submitted manuscript, we derive such an expression for the exchange coefficients. Our program, CONAN, is the numerical implementation of this formula for the exchange coefficients. Thus, CONAN takes as physical inputs the system size N and a smooth potential V(x), and returns the corresponding N−1 exchange coefficients appearing in the spin chain Hamiltonian.