A perfect Hashing function for exact diagonalization of many-body systems of identical particles

✍ Scribed by Shoudan Liang

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 355 KB
• Volume: 92
• Category: Article
• ISSN: 0010-4655
• DOI: 10.1016/0010-4655(95)00108-r

✦ Synopsis

In exact diagonalization studies of ground state properties of an interacting system using Lfinczos, it is crucial to find an efficient mapping from the configuration space to the index of the L~inczos vector. Here, we formulate a mapping (hashing |unction) that is simple to calculate for identical many-body systems of both fermions and bosons. A unique feature of our hashing functions is that they map many particle configurations to a sequence of consecutive integers. They are therefore )ptimal in using computer memory.

Exact diagonalization of Hamiltonians of interacting identical particles is a widely used technique in manybody physics and quantum chemistry. For example, exact diagonalization is one of most reliable technique in studying the effects of electron-electron correlations in systems ranging from high temperature superconductors [ 1 ] to conducting polymers [2]. Often the Hamiltonian matrix is sparse and only the ground state and a few excited states are needed. Therefore, it is ideal to use Lfinczos and related Arnoldi methods [3,4]. Compared to other techniques such as quantum Monte Carlo, the exact diagonalization gives exact answer and ~s relatively easy to implement.

The most computational intensive operation in exact diagonalization using a Lfinczos algorithm is multiplying ~he Hamiltonian matrix to the state vector. Although the Hamiltonian matrix is usually sparse, the storage of this matrix in the core memory of a computer is the limiting factor of how large a system that can be treated using this method.

In the second quantization formulation of quantum mechanics, the Hamiltonian scatters particles among allowed orbitals. Usually only two-body interactions are considered so that scattering involves only two particles. The Hamiltonian is therefore very easy to generate in particle occupation space. Therefore, in principle, the Hamiltonian matrix needs not to be stored; it can be generated when needed. However, it is only easy to generate the Hamiltonian matrix using the occupation space representation. We still need a method that converts the occupation configuration to the index of the eigenvector. To this end, one can use either a look-up table which requires large amount of memory, or a hashing technique. The hashing functions that have been proposed so far are not one-to-one mappings and complicated procedures are required to fix the resulting collisions problem.

We provide in this paper hashing functions that are one-to-one mappings and are therefore perfect. Furthermore, the configurations in occupation space of a fixed number of particles are mapped to a sequence of consecutive integers.