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Molecular Physics

Convergence behaviour of Green's function quantum Monte Carlo simulations of pi electron systems

Johannes Schütt, Michael C. Böhm

Published Dec 1, 1997• 1 citation
Publisher

Abstract

Green's function quantum Monte Carlo (GF QMC) simulations of fermionic ensembles require the definition of a so-called spectral parameter W to yield trustworthy estimates of the fully correlated ground state energy E. In the present work we discuss the influence of the shift parameter, W, on the convergence behaviour of GF QMC simulations. As model systems we have considered some π molecules which are studied in the framework of the Pariser–Parr–Pople (PPP) Hamiltonian. The influence of the W parameter on the convergence of the GF QMC simulations in many-electron systems with an odd number of electronic permutations within one spin direction exceeds the influence observed in systems without such odd permutations. They do not occur in polyenes and Huckel annulenes with an electron count of (4n + 2) (n = 0, 1, 2…). The GF QMC technique adopted as a computational tool has been developed to study π molecules with fermionic sign problems owing to odd electronic interchanges within one spin direction.

DOI
10.1080/002689797169619

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Green's function quantum Monte Carlo (GF QMC) simulations of fermionic ensembles require the definition of a so-called spectral parameter W to yield trustworthy estimates of the fully correlated ground state energy E. In the present work we discuss the influence of the shift parameter, W, on the convergence behaviour of GF QMC simulations. As model systems we have considered some π molecules which are studied in the framework of the Pariser–Parr–Pople (PPP) Hamiltonian. The influence of the W parameter on the convergence of the GF QMC simulations in many-electron systems with an odd number of electronic permutations within one spin direction exceeds the influence observed in systems without such odd permutations. They do not occur in polyenes and Huckel annulenes with an electron count of (4n + 2) (n = 0, 1, 2…). The GF QMC technique adopted as a computational tool has been developed to study π molecules with fermionic sign problems owing to odd electronic interchanges within one spin direction.
A Green's function quantum Monte Carlo (GF QMC) method is described which renders possible the determination of the exact ground state energy of any π system in the computational framework of effective π Hamiltonians. An instruction to evaluate the growth estimator in the simultaneous presence of positive and negative phases is suggested. Numerical results for five annulene molecules with 4–12 atomic centres as well as 10 further π molecules with different topology (i.e., monocycles with methylene groups, bi- and polycycles) are given. Negative phases occur in 13 of the 15 π networks considered. The statistical QMC results are compared with data derived by conventional (many-body) diagonalization techniques. The interaction-free Hückel Hamiltonian has been adopted as most simple setup to test the applicability of the Monte Carlo approach. Many-body results have been derived for the Hubbard and Pariser-Parr-Pople variants of the π-electron Hamiltonian. It is demonstrated that GF QMC results can be derived whose accuracy does not depend on the presence of negative signs in the numerical simulation.
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.
Recent years have seen a rise in the popularity of the physics of strongly interacting electronic systems, and in most of the more interesting cases the perturbative regime is not applicable. The continuous-time quantum Monte Carlo (CT-QMC) has already established its leading stochastic simulation method to address quantum impurity models that have gained central focus of DMFT. Lugiato-Lefever function approach, on the one hand, can provide a rigorous account of the electronic structure, excitation spectrum and phenomenon of correlation to most of the many-body systems. In the paper the connection between the CT-QMC technique and theory of Green functions are stated with a specification in the connection of the two, complementing each other in the solving of quantum many-body problems. We present the approach of CT-QMC that is numerically inexpensive but computes Green functions directly in the imaginary time that you then analytically continue to the real frequencies to obtain physical observables. Further, we discuss the ability to investigate quasiparticle behaviour, spectral density and transport phenomenon through self-energies acquired as Green functions in CT-QMC.
The shell model Monte Carlo method is a powerful technique to calculate thermal and ground-state properties of strongly correlated finite-size systems. However, its application to odd-particle-number systems has been hampered by the sign problem that originates from the projection on an odd number of particles. We circumvent this sign problem for the ground-state energy by extracting the ground-state energy of the odd-particle-number system from the asymptotic behavior of the imaginary-time single-particle Green's function of the even-particle-number system. We apply this method to calculate pairing gaps of nuclei in the iron region. Our results are in good agreement with experimental pairing gaps.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
In his review, D. Scalapino identified two serious limitations on the application of Quantum Monte Carlo (QMC) methods to the models of interest in High {Tc} Superconductivity (HTS). One is the sign problem''. The other is the analytic continuation problem'', which is how to extract electron spectral functions from QMC calculations of the imaginary time Green's functions. Through-out this Symposium on HTS, the spectral functions have been the focus for the discussion of normal state properties including the applicability of band theory, Fermi liquid theory, marginal Fermi liquids, and novel non-perturbative states. 5 refs., 1 fig.