Session V21: General Theory: Computational Quantum Monte Carlo Methods

Efficiency of Rejection-Free Monte Carlo Algorithms for Particles Systems

We calculated the efficiency of rejection-free methods for dynamic Monte Carlo studies of off-lattice systems. Following the methodology of Watanabe et al\footnote{\label{watanabe}H. Watanabe, S. Yukawa, M.A. Novotny and N. Ito, \textit{Efficiency of Rejection-free dynamic Monte Carlo methods for homogenous spin models, hard disk systems, and hard sphere system}, Phys. Rew. E, \textbf{74}, 026707 (2006)}, we studied $d=1$ particles models including the hard rod model, and models with both harmonic and Lennard-Jones potentials. The hard-rod results are in agreement with [1], namely the efficiency near the close-packing density $\rho_{_{cp}}$ is proportional to $(\rho_{_{cp}}-\rho)^{-d}$, where $d$ is the dimension of the system and $\rho$ the system density. We also report on the algorithmic efficiency for cases with heterogenous particles. Some results in $d=2$ will also be presented.   

Faster QMC with Lagrange and splines

Computing the wave function can be the most time-intensive part of a quantum Monte Carlo calculation. Orbitals represented by extended basis functions scale in evaluation time as O($N^3$) while localized basis functions scale as O($N^2$). Two methods of localizing the orbital representation are: piecewise-polynomial (pp) interpolation and transformation to a localized basis. The Lagrange form of the pp-interpolant is simple but has discontinuous derivatives at sampling points. The pp-spline is continuous in certain low derivatives. The B-spline shares the pp-spline's derivative continuity but is a transformation not an interpolant. These methods have O($N^2$) scaling at all $N$ tested (up to $N=864$). While increasing the number of sampling points of the original orbital, the total QMC energy converges to the value calculated using plane waves at similar sampling point numbers for Lagrange, pp-splines and B-splines. At fixed sampling density, the three are of comparable speed. pp-splines use $8$ memory words per sampling point, Lagrange use $5$ and B-splines use $1$. Due to smaller memory usage, B-splines are the best choice.   

Approach toward Linear Scaling QMC

Quantum Monte Carlo simulations of fermions are currently done for relatively small system sizes, e.g., fewer than one thousand fermions. The most time-consuming part of the code for larger systems depends critically on the speed with which the ratio of a wavefunction for two different configurations can be evaluated. Most of the time goes into calculating the ratio of two determinants; this scales naively as O($n^3$) operations. Work by Williamson, et al. (2) have improved the procedure for evaluating the elements of the Slater matrix, so it can be done in linear time. Our work involves developing methods to evaluate the ratio of these Slater determinants quickly. We compare a number of methods including work involving iterative techniques, sparse approximate inverses, and faster matrix updating.\\ (2) A. J. Williamson, R.Q. Hood and J.C. Grossman, Phys. Rev. Lett. 87, 246406 (2001)   

Spinor path integral Quantum Monte Carlo for fermions

We have developed a continuous-space path integral method for spin 1/2 fermions with fixed-phase approximation. The internal spin degrees of freedom of each particle is represented by four extra dimensions. This effectively maps each spinor onto two of the excited states of a four dimensional harmonic oscillator. The phases that appear in the problem can be treated within the fixed-phase approximation. This mapping preserves rotational invariance and allows us to treat spin interactions and fermionic exchange on equal footing, which may lead to new theoretical insights. The technique is illustrated for a few simple models, including a spin in a magnetic field and interacting electrons in a quantum dot in a magnetic field at finite temperature. We will discuss possible extensions of the method to molecules and solids using variational and diffusion Quantum Monte Carlo.   

Accuracy of the fixed-node and pseudopotential approximations in diffusion Monte Carlo

Diffusion Monte Carlo is one of the most accurate methods for molecules and solids. Its accuracy is controlled by the fixed-node and pseudopotential approximations. For atoms and small molecular systems, efficient energy optimization methods enable the optimization of all parameters of many-body wave functions and systematically eliminate the fixed error. This enables our pseudopotential tests. Calculations for Si and C atoms and dimers demonstrate the importance of optimized multi-determinant trial-wave functions for chemical accuracy. The fixed-node error of the two seemingly similar dimers, Si$_2$ and C$_2$, differs dramatically with values of 0.1~eV and 1~eV, respectively. Calculations of the ionization energies and electron affinities of Si and C as well as binding energies and bond lengths for the Si$_2$ and C$_2$ for relativistic LDA, PBE and HF pseudopotentials of the Troullier-Martins, Vanderbilt and Gaussian form assess their accuracy and efficiency. The results are compared to experimental and quantum chemistry data. Reduced non-locality by larger cutoff distances and the Vanderbilt construction improve the efficiency. PBE and HF pseudopotentials result in accurate energies and HF pseudopotential are the most accurate for the dimer geometries.   

Correction of finite-size errors in many-body electronic structure

Quantum Monte Carlo (QMC) calculations using simulation cells with periodic boundary conditions are subject to finite-size errors. Often, such errors are corrected or reduced by extrapolation using increasingly larger simulation cells, combined with size corrections from less accurate calculations, such as from Hartree Fock (HF) or density functional theory (DFT). Direct extrapolation is computationally costly. Size-corrections from standard HF and DFT calculations introduce additional errors and are less reliable. This has led to several recent attempts at improved methods to correct for finite-size errors. Here we develop a scheme which uses modified density functionals to estimate the finite size errors. Tests on simple solids and molecules using plane-wave auxiliary-field QMC calculations show encouraging results.   

Wang-Landau integration --- The application of Wang-Landau sampling in numerical integration

Wang-Landau sampling was first introduced to simulate the density of states in energy space for various physical systems. This technique can be extended to numerical integrations due to certain similarities in nature of these two problems. It can be further applied to study quantum many-body systems. We report the feasibility of this application by discussing the correspondence between Wang-Landau integration and Wang-Landau sampling for Ising model. Numerical results for 1D and 2D integrations are shown. In particular, the utilization of this algorithm in the periodic lattice Anderson model is discussed as an illustrative example.   

Quantum Monte Carlo with short directed loops

We introduce a new type of directed loop algorithm with short-loop generation for the stochastic series expansion quantum Monte Carlo method[1]. Short-loop algorithms have been shown to greatly improve the dynamics at low temperature in studies of classical spin ice models[2]. We will discuss the framework of this algorithm and make comparisons to the conventional directed loop algorithm in a specific quantum spin model. \newline [1]O.Suljuasen and A. W. Sandvik, Phys. Rev. E66, 046701 (2002). \newline [2]R. Melko et al., Phys. Rev. Lett. 87, 067203 (2001).   

Optimization of quantum Monte Carlo wave functions by energy minimization

We present a simple, robust and highly efficient method for optimizing all parameters of many-body wave functions by energy minimization in quantum Monte Carlo calculations. Using a strong zero-variance principle, the optimal parameters are determined by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its derivatives [1-2]. We discuss the connection with previously-proposed energy minimization methods, namely the modified Newton method [3] and the perturbative energy fluctuation potential method [4]. Application of the method to electronic atomic and molecular systems show that it systematically reduces the diffusion Monte Carlo fixed-node error. [1] J. Toulouse and C. J. Umrigar, submitted to J. Chem. Phys. [2] C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, cond-mat/0611094. [3] C. J. Umrigar and C. Filippi, Phys. Rev. Lett. 94, 150201 (2005). [4] A. Scemama and C. Filippi, Phys. Rev. B 73, 241101 (2006).   

Systematic improvement of trial wavefunctions for Constrained Path Quantum Monte Carlo

Constrained Path Monte Carlo (CPMC) provides an approximate solution to the Fermion sign problem for lattice models such as the Hubbard model. In the zero-temperature CPMC algorithm, a trial wavefunction is used to eliminate random walkers when their overlap with the trial function becomes zero. CPMC often produces surprisingly good results for ground state energy and correlation functions, even when a simple trial function is used. However, there is no reason to expect that simple wavefunctions (free electron or Hartree Fock) will have any overlap with complex correlated ground states. We therefore describe a method to improve CPMC results by optimizing the trial wavefunction. The trial function we use is a sum of Slater determinants that is optimized by the Path Integral Renormalization Group (PIRG) procedure. The wavefunction produced by PIRG is a sum of L determinants, with an energy that is variational. We show CPMC+PIRG data for a system where CPMC with a free electron trial function fails, the Hubbard model on an anisotropic triangular lattice.   

Generalized pairing wave functions in electronic structure quantum Monte Carlo

We investigate several types of trial wave functions with pairing orbitals in fixed-node quantum Monte Carlo. Following upon our previous study[1], we explore the possibilities of expanding the wave function in linear combinations of pfaffians. We observe that molecular systems require much larger expansions than atomic systems and linear combinations of a few pfaffians lead to rather small gains in correlation energy. Further, we test the wave function based on fully-antisymmetrized product of independent pair orbitals. Despite its seemingly large variational potential, we do not observe significant gains in correlation energy. Finally, we combine these developments with the recently proposed inhomogeneous backflow transformations[2]. The trade-offs between computational efficiency and amounts correlation energy recovered will be discussed. \newline [1] M. Bajdich et al. Phys. Rev. Lett. 96, 130201 (2006). \newline [2] N. D. Drummond et. al., J. Chem. Phys. 124, 224104 (2006).   

Bond breaking in auxiliary-field quantum Monte Carlo

Bond stretching mimics different levels of electron correlations in the system and provides a challenging testbed for all approximate many-body computational methods. Using the recently developed phaseless auxiliary-field quantum Monte Carlo (AF QMC) method, we study the potential-energy curves of several well-known molecules --- BH, N$_2$, and F$_2$ --- and of the H$_{50}$ chain. To control the sign/phase problem, the phaseless AF QMC method constrains the random walks with an approximate phase condition that depends on a trial wave function. With single-determinant unrestricted Hartree-Fock trial wave functions, the phaseless AF QMC method generally gives better overall accuracy and a more uniform behavior than the coupled cluster CCSD(T) method in mapping the potential-energy curve. In the molecules, the use of multiple-determinant trial wave functions from multi-configuration self-consistent-field calculations is also explored. The increase in computational cost versus the gains in statistical and systematic accuracy are examined. With such trial wave functions, excellent results are obtained across the entire region between equilibrium and the dissociation limit.   

Bond Breaking of Simple Molecules in Auxiliary-Field Quantum Monte Carlo with GVB Wave Functions

Accurate potential energy curves are an essential ingredient in understanding chemical reactions. This problem spans a wide range of correlations, with correlation effects being the most important in the bond-breaking regime. We report potential energy curves of simple molecules, including water and the carbon dimer, within the framework of the auxiliary-field quantum Monte Carlo (AFQMC) method. AFQMC projects the many-body ground-state from a trial wave function, which is also used to control the sign/phase problem. A previous calculation\footnote{Al-Saidi, Zhang, Krakauer, J. Chem. Phys. \textbf{124}, 224101 (2006)} showed that AFQMC could provide a fairly uniform description of the bond stretching of a water molecule, even with a simple unrestricted Hartree-Fock (UHF) trial wave function. We investigate the use of Generalized Valence Bond (GVB). GVB gives a better description of the molecule than UHF; so it is a simple yet efficient alternative to using a single Slater determinant trial wave function. We will compare AFQMC results with other correlated methods and the exact configuration interaction calculations.   

Monte-Carlo Simulations of High-q Anti-ferromagnetic Potts Models

Using a highly efficient cluster-flip Monte-Carlo algorithm$^{1}$, we have investigated the ordering in the five- and the six-state anti-ferromagnetic Potts models on a simple cubic lattice. Using a method developed previously$^{2 }$to examine in detail the distribution of spins on sublattices, we show that the five-state case has a phase transition only at zero temperature, and that the six-state case is disordered at all temperatures. $^{1}$ R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987). $^{2}$ S. Rahman, E. Rush and R.H. Swendsen, Phys. Rev. \underline {B58}, 9125 (1998). * present address: Dept. of Materials Science and Engineering, Virginia Tech, VA, USA