This chapter describes the range of techniques available for the stochastic evaluation of path integrals and how this flexibility may be effectively exploited to incorporate one's understanding of the underlying physics in the method. It provides precise calculations of physical observables which are exact within controllable systematic and stochastic sampling errors. A special problem frequently arising in stochastic solutions of many-body problems is sampling of the inverse of a matrix. The principal intellectual challenge is to exploit the freedom and to utilize physical insight to incorporate as much of the essential physics of the problem as possible into the stochastic algorithm. It is natural to ask whether any of the functional integral representations of the many-body evolution operator using fields are preferable to the particle coordinate path integral for stochastic calculations. One alternative to evaluate the evolution operator for spin systems is to use the Trotter formula for other path integrals.
This chapter describes the range of techniques available for the stochastic evaluation of path integrals and how this flexibility may be effectively exploited to incorporate one's understanding of the underlying physics in the method. It provides precise calculations of physical observables which are exact within controllable systematic and stochastic sampling errors. A special problem frequently arising in stochastic solutions of many-body problems is sampling of the inverse of a matrix. The principal intellectual challenge is to exploit the freedom and to utilize physical insight to incorporate as much of the essential physics of the problem as possible into the stochastic algorithm. It is natural to ask whether any of the functional integral representations of the many-body evolution operator using fields are preferable to the particle coordinate path integral for stochastic calculations. One alternative to evaluate the evolution operator for spin systems is to use the Trotter formula for other path integrals.
Abstract The focus of Chapter 12 is the Feynman path integral formulations of quantum mechanics and quantum statistical mechanics. The connection between real-time quantum propagator and the density matrix is established via the Wick rotation, and a Trotter expansion is used to derive discrete path integral expressions for both. The continuous limit of the discrete path integral is derived to yield the functional integral formulation of the path integral. The construction of estimators and their averages for computing quantum observables is discussed. Next, the path integral is extended to incorporate Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, and a useful recursive formula for the partition function is derived that accelerates convergence by building up N-body partition functions from those of smaller systems. Quantum free energies are discussed, and algorithms for the evaluation of path integrals using molecular dynamics or Monte Carlo are presented. The chapter concludes with a discussion of higher-order factorizations and ring-polymer contraction methods to reduce the computational overhead of path-integral calculations
Abstract The path integral discussed in the preceding chapters can be exactly calculated only in a few restricted systems such as harmonic oscillators. The WKB method and perturbational expansions based on the above path integral method are in fact applicable to systems close to classical ones and weakly interacting systems, respectively. However, in order to study interesting phenomena such as high Tc superconductivity, we have to treat strongly correlated systems in which the quantum effect plays an important role. It is effective in these systems to extract some physically crucial contributions numerically. This will be possible if we use the quantum Monte Carlo method. Although there are many different types of quantum Monte Carlo methods, we explain here mainly the quantum statistical Monte Carlo method based on a Trotter-like formula and on the important sampling method applied to transformed classical systems.
… a general method, suitable for electronic computing machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. N. Metropolis et al. , 1953 Summary Quantum Monte Carlo methods have been very useful in providing exact results, or, at least, exact constraints on properties of electronic systems, in particular for the homogeneous electron gas. The results are, in many cases, more accurate than those from other quantum many-body methods, and provide unique capabilities and insights. In this chapter we introduce the general properties of stochastic methods and motivate their use on the quantum many-body problem. In particular, we discuss Markov chains and the computation of error estimates. The methods that we introduce in the next four chapters are quite different from those in Parts II and III: stochastic or quantum Monte Carlo methods. In stochastic methods, instead of solving deterministically for properties of the quantum many-body system, one sets up a random walk that samples for the properties. Historically the most important role of QMC for the electronic structure field has been to provide input into the other methods, most notably the QMC calculation of the HEG [109], used for the exchange– correlation functional in DFT. A second important role has been as benchmarks for other methods such asGW. There are systems for which QMC is uniquely suited, for example the Wigner transition in the low-density electron gas, see Sec. 3.1. In this chapter we introduce general properties of simulations, in particular Markov chains, and error estimates. In the following chapters we will apply this theory to three general classes of quantum Monte Carlo algorithms, namely variational (Ch. 23), projector (Ch. 24), and path-integral Monte Carlo (Ch. 25); Ch. 18 already introduced the QMC calculation of the impurity Green's function used in the dynamical mean-field method.We note that there are a variety of other QMC methods not covered in this book. Simulations First let us define what we mean by a simulation, since the word has other meanings in applied science. The dimensionality of phase space (i.e., the Hilbert space for a quantum system) is large or infinite. Even a classical system requires the positions and momenta of all particles, and, hence, the phase space for N classical particles has dimensionality 6 N .
Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem.
This chapter addresses a range of problems for which functional integrals provide unique insights and approximations. It shows how to generate a variety of alternative functional integrals for the evolution operator which gives rise to different physical approximations in the stationary-phase approximation. The chapter aims to perform the general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental vibrational excitations of the system. It also shows how time-dependent mean field solutions may be used to calculate transition amplitudes between specific states, approximate quantum eigenstates of large amplitude collective motion, study barrier penetration and spontaneous fission. The chapter evaluates the asymptotic behavior of large orders of perturbation theory and explores the freedom available in formulating functional integrals for the evolution operator and how to exploit this freedom in making physical approximations.
Path integrals are a powerful method for calculating real time, finite temperature, and ground state properties of quantum systems. By exploiting some remarkable properties of the symmetric Trotter formula and the discrete Fourier transform, we arrive at a high-accuracy method for removing ``time slice'' errors in Trotter-approximated propagators. We provide an explicit demonstration of the method applied to the two-body density matrix of 4He. Our method is simultaneously fast, high precision, and computationally simple and can be applied to a wide variety of quantum propagators.