The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher-order splitting will result in a higher-order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Green's function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically gives rise to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second-order forward propagators can have fourth-order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second-order propagators and their minimal correctors.
The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher-order splitting will result in a higher-order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Green's function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically gives rise to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second-order forward propagators can have fourth-order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second-order propagators and their minimal correctors.
We derive a fourth-order diffusion Monte Carlo algorithm for solving quantum many-body problems. The method uses a factorization of the imaginary time propagator in terms of the usual local energy E and Langevin operators L as well as an additional pseudo-potential consisting of the double commutator [EL, [L, EL]]. A new factorization of the propagator of the Fokker-Planck equation enables us to implement the Langevin algorithm to the necessary fourth order. We achieve this by the addition of correction terms to the drift steps and the use of a position-dependent Gaussian random walk. We show that in the case of bulk liquid helium the systematic step size errors are indeed fourth order over a wide range of step sizes.
Chau et al. [New J. Phys. 20, 073003 (2018)] presented a new and straight-forward derivation of a fourth-order approximation 'U7' of the time-evolution operator and hinted at its potential value as a symplectic integrator. U7 is based on the Suzuki-Trotter split-operator method and leads to an algorithm for numerical time propagation that is superior to established methods. We benchmark the performance of U7 and other algorithms, including a Runge-Kutta method and another recently developed Suzuki-Trotter-based scheme, that are exact up to fourth order in the evolution parameter, against various classical and quantum systems. We find U7 to deliver any given target accuracy with the lowest computational cost, across all systems and algorithms tested here. This study is accompanied by open-source numerical software that we hope will prove valuable in the classroom.
We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.
By decomposing the important sampled imaginary time Schr\"odinger evolution operator to fourth order with positive coefficients, we derived a number of distinct fourth-order diffusion Monte Carlo algorithms. These sophisticated algorithms require higher derivatives of the drift velocity and local energy and are more complicated to program. However, they allowed very large time steps to be used, converged faster with lesser correlations, and virtually eliminated the step size error. We demonstrated the effectiveness of these quartic algorithms by solving for the ground-state energy and radial density distribution of bulk liquid helium.
We consider the numerical propagation of models that combine both quantum and classical degrees of freedom, usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum-classical molecular dynamics models, one uses some propagation technique to deal with the quantum part and a different one for the classical equations. While the resulting procedure may, in principle, be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the geometrical structure of the equations. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular, one family of techniques that proves to be very efficient for the propagation of Schrödinger-like equations: the (quasi)-commutator free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrödinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.
Because the Schrodinger equation cannot be solved analytically for systems larger than hydrogen, approximate solution methods must be applied to study systems of chemical interest. Quantum Monte Carlo is one of these methods. Quantum Monte Carlo simulations involve the diffusion of walkers through configuration space and necessitate the use of one of two imaginary time propagators, 2nd or 4th order. The choice of the propagator implemented depends on a balance between accuracy and efficiency. Obtaining accurate solutions is the primary objective; once this is accomplished, finding ways to make the simulations efficient is the next intention. Using a model system of a multidimensional harmonic oscillator, this study intends to compare and contrast both 2nd and 4th order propagators' diminished accuracy as the dimensionality of the system is increased. Two phases of the propagation are sampled to determine accuracy: a single propagation step and the asymptotic limit. To test the accuracy after one step, we start a simulation by placing walkers at the origin, moving them forward one time step in imaginary time, creating normalized histograms of their positions and comparing these with the known result for the imaginary time propagation of the delta function centered at x=0. The convergence of the simulation is determined by observing the change in distribution from the known asymptotic limit distribution. This is done by distributing the walkers in the known asymptotic distribution and observing the change in average weight upon further propagation. A converged distribution’s average weight will not change.