Finite-Element Lattice Hamiltonian Matrix Eleents. Anharmonic Oscillators

TL;DR

This paper develops a finite-element lattice method to accurately compute matrix elements of the time evolution operator for anharmonic oscillators, preserving unitarity and achieving high precision without solving implicit equations.

Contribution

It introduces a finite-element approach for lattice field theory that accurately constructs matrix elements for anharmonic oscillators without solving implicit equations of motion.

Findings

Matrix elements are highly accurate, with errors less than 1% for ground state energies.

Two-state approximations reduce errors to below 0.1%.

Method preserves unitarity and can be extended to field theories in higher dimensions.

Abstract

The finite-element approach to lattice field theory is both highly accurate (relative errors $\sim 1/N^2$, where $N$ is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this paper we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is $H=p^2/2+\lambda q^{2k}/2k$. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the $k=2$ anharmonic oscillator ground state energy accurate to better than 1\%, while a two-state approximation reduces the error to less than 0.1\%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.