Convergence Characteristics of the Cumulant Expansion for Fourier Path Integrals

The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables $N$ included. It is proved that when the cumulant expansion is truncated at order $p$, the asymptotic convergence rate of the density matrix behaves like $N^{-(2p+1)}$. The complex algebra associated with the proof is simplified by introducing a diagrammatic representation of the contributing terms along with an associated linked-cluster theorem. The cumulant terms at each order are expanded in a series such that the the asymptotic convergence rate is maintained without the need to calculate the full cumulant at order $p$. Using this truncated expansion of each cumulant at order $p$, the numerical cost in developing Fourier path integral expressions having convergence order $N^{-(2p+1)}$ is shown to be approximately linear in the number of required potential energy evaluations making the method promising for actual numerical implementation.Comment: 47 pages, 2 figures, submitted to PR

The Spin Density Matrix II: Application to a system of two quantum dots

This work is a sequel to our work "The Spin Density Matrix I: General Theory and Exact Master Equations" (eprint arXiv:0708.0644 [cond-mat]). Here we compare pure- and pseudo-spin dynamics using as an example a system of two quantum dots, a pair of localized conduction-band electrons in an n-doped GaAs semiconductor. Pure-spin dynamics is obtained by tracing out the orbital degrees of freedom, whereas pseudo-spin dynamics retains (as is conventional) an implicit coordinate dependence. We show that magnetic field inhomogeneity and spin-orbit interaction result in a non-unitary evolution in pure-spin dynamics, whereas these interactions contribute to the effective pseudo-spin Hamiltonian via terms that are asymmetric in spin permutations, in particular, the Dzyaloshinskii-Moriya (DM) spin-orbit interaction. We numerically investigate the non-unitary effects in the dynamics of the triplet states population, purity, and Lamb energy shift, as a function of interdot distance and magnetic field difference. The spin-orbit interaction is found to produce effects of roughly four orders of magnitude smaller than those due to magnetic field difference in the pure-spin model. We estimate the spin-orbit interaction magnitude in the DM-interaction term. Our estimate gives a smaller value than that recently obtained by Kavokin [Phys. Rev. B 64, 075305 (2001)], who did not include double occupancy effects. We show that a necessary and sufficient condition for obtaining a universal set of quantum logic gates, involving only two spins, in both pure- and pseudo-spin models is that the magnetic field inhomogeneity and the Heisenberg interaction are both non-vanishing. We also briefly analyze pure-spin dynamics in the electron on liquid helium system recently proposed by Lyon [Phys. Rev. A 74, 052338 (2006)].Comment: 16 pages including 12 figures. Sequel to "The Spin Density Matrix I: General Theory and Exact Master Equations", arXiv:0708.064

Cumulant expansion and analytic continuation in Monte Carlo simulation of classical Lennard-Jones clusters

The Monte Carlo (MC) estimates of thermal averages are usually functions of system control parameters., such as temperature, volume, and interaction couplings. Given the MC average at a set of prescribed control parameters lambda(0), the problem of analytic continuation of the MC data to lambda values in the neighborhood of lambda(0) is considered in both classic and quantum domains. The key result is the theorem that links the differential properties of thermal averages to the higher order cumulants. The theorem and analytic continuation formulas expressed via higher order cumulants are numerically tested on the classical Lennard-Jones cluster system of N = 13, 55, and 147 neon particles.open110sciescopu

A numerical study of the asymptotic convergence characteristics of partial averaged and reweighted Fourier path integral methods

The asymptotic convergence characteristics with respect to the number of included path variables of the partial average and reweighted Fourier path integral methods are numerically compared using a Gaussian fit to the one-dimensional Lennard-Jones potential. Using harmonic inversion to determine the parameters of the Gaussian fit potential appropriate for neon, the energy eigenvalues and thermodynamic properties of the Gaussian fit and the Lennard-Jones interaction agree to better than 1%. Using Monte Carlo methods to study the Gaussian fit potential, the systematic error associated with the truncation of the number of path variables is found to be larger for the reweighted method than the partial average method for the same number of path variables actually used. © 2009 Wiley Periodicals, Inc