Quantum Monte Carlo

It is remarkable how the Froehlich polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field of ion displacements, has resisted full analytical or numerical solution at all coupling since 1950, when its Hamiltonian was first written. The field has been a testing ground for analytical, semi-analytical, and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational, exact diagonalisation (ED), and quantum Monte Carlo (QMC) techniques. This article reviews recent developments in the field of continuum and discrete (lattice) Froehlich (bi)polarons starting with the basics and covering a number of active directions of research.

A comprehensive theoretical and experimental study is presented of the magnetic susceptibility versus temperature \chi(T) of spin S = 1/2 two- and three-leg Heisenberg ladders and ladder oxide compounds. Extensive quantum Monte Carlo simulations of \chi(T) were carried out for both isolated and coupled two-leg ladders with spatially anisotropic intraladder exchange. Accurate fits to these and related literature QMC data were obtained. We have also calculated the one- and two-magnon dispersion relations and the dynamical spin structure factor for anisotropic isolated 2 x 12 ladders. The exchange constants in the two-leg ladder compound SrCu2O3 are estimated from LDA+U calculations. We report the detailed crystal structure of SrCu2O3 and of the three-leg ladder compound Sr2Cu3O5. New experimental \chi(T) data are reported for the two-leg ladder cuprates SrCu2O3 and LaCuO_{2.5}, and for the (nominally) two-leg ladder vanadates CaV2O5 and MgV2O5. The new and literature \chi(T) data for these compounds and for Sr2Cu3O5 are modeled using our QMC \chi(T) simulation fits, and the exchange coupling constants between the spins-1/2 are thereby estimated for each material. The surpisingly strong spatial anisotropy of the bilinear intraladder exchange constants in the cuprate compounds is discussed together with the results of other experiments sensitive to this anisotropy. Recent theoretical predictions are discussed including those which indicate that a four-spin cyclic exchange interaction within a Cu4 plaquette is important to determining the magnetic properties and which can significantly influence the exchange interactions estimated from \chi(T) data assuming the presence of only bilinear exchange.

This article is a heavily-revised, and double-blind peer-reviewed, version of the conference paper "Simulating Nature" listed below.

Purpose
To recuperate Heinz Von Foerster’s “Quantum Mechanical Theory of Memory” from Cybernetics: Circular, Causal, and Feedback Mechanisms in Biological and Social Systems and John von Neumann’s The Computer and the Brain for present-day, and future applications, on biophysics, theories of information and cognition, and quantum theories; the main objective is to ground cybernetic theory for a critical evaluation of the historical evolution of the Monte Carlo method, with potential for application to quantum computing.

Design/methodology/approach
Close-reading of selected texts, historiography, and case studies in current developments in the Monte Carlo method of high energy particle physics (HEP) for bridging seemingly incommensurable differences between the physical-mathematical and the biological sciences.

Findings
1. Usefulness of the cybernetic approach for historicizing the Monte Carlo method in relation to digital computing and quantum physics.
2.Development of an inter/trans-disciplinary approach to the hard sciences through a critical re-evaluation of the historical texts of von Foerster and von Neumann for application to quantum theory, biophysics, and computing.

Research Limitations
This work is largely theoretical and uses dialectical thought experiments to engage between sciences operating across different ontological scales.

Practical Implications
Consideration of developments of quantum computing and how that would change one's perception of information, data, and the way in which analysis is currently performed with big data.

Originality and Value
This is the first time that von Neumann and von Foerster have been contrasted and compared in relation to their epistemic compatibility, historical importance, and relevance for producing a creative approach to current scientific epistemology. This article hopes to change how we view transdisciplinary/interdisciplinary practices in the sciences and produce new vistas of thought in the history and philosophy of science.

Due to advances in computer hardware and new algorithms, it is now possible to perform highly accurate many-body simulations of realistic materials with all their intrinsic complications. The success of these simulations leaves us with a conundrum: how do we extract useful physical models and insight from these simulations? In this article, we present a formal theory of downfolding--extracting an effective Hamiltonian from first-principles calculations. The theory maps the downfolding problem into fitting information derived from wave functions sampled from a low-energy subspace of the full Hilbert space. Since this fitting process most commonly uses reduced density matrices, we term it density matrix downfolding (DMD).

We report large scale determinant quantum Monte Carlo calculations of the effective bandwidth, momentum distribution, and magnetic correlations of the square lattice fermion Hubbard Hamiltonian at half-filling. The sharp Fermi surface of the noninteracting limit is significantly broadened by the electronic correlations but retains signatures of the approach to the edges of the first Brillouin zone as the density increases. Finite-size scaling of simulations on large lattices allows us to extract the interaction dependence of the antiferromagnetic order parameter, exhibiting its evolution from weak-coupling to the strong-coupling Heisenberg limit. Our lattices provide improved resolution of the Green’s function in momentum space, allowing a more quantitative comparison with time-of-flight optical lattice experiments.

The interaction energy of a methane molecule encapsulated in a dodecahedral water cage is calculated using the MP2, MP2C, various dispersion-corrected DFT, and diffusion Monte Carlo (DMC) methods. The MP2, MP2C, and DMC methods give binding energies of −5.04, −4.60, and −5.3 ± 0.5 kcal/mol, respectively. In addition, the two- and three-body contributions are evaluated using the DFT, MP2, and CCSD(T) methods. All of the DFT methods considered appreciably overestimate the magnitude of the three-body contribution to the interaction energy. The two- and three-body energies are further analyzed by use of symmetry-adapted perturbation theory (SAPT) which allows decomposition into electrostatics, exchange, induction, and dispersion contributions. The SAPT calculations reveal that the induction, dispersion, and exchange three-body contributions to the methane-cage binding energy are all sizable, with the net three-body contribution to the binding energy being about 1 kcal/mol.

The first principles methods, density-functional theory and quantum Monte Carlo, have been used to examine the balance between van der Waals (vdW) forces and hydrogen bonding in ambient and high-pressure phases of ice. At higher pressure, the contribution to the lattice energy from vdW increases and that from hydrogen bonding decreases, leading vdW to have a substantial effect on the transition pressures between the crystalline ice phases. An important consequence, likely to be of relevance to molecular crystals in general, is that transition pressures obtained from density-functional theory exchange-correlation functionals which neglect vdW forces are greatly overestimated.

We reveal that local interactions in graphene allow novel spin liquids between the semimetal and antiferromagnetic Mott insulating phases, identified with algebraic spin liquid and Z2 spin liquid, respectively. We argue that the algebraic spin liquid can be regarded as the two-...