Session Q24: Focus Session: Quantum Monte Carlo Simulations of Fermion and Boson Systems II

Path Integral representation of quantum particles in fluids: Convergence of observables

In previous work the Path Integral Monte Carlo (PIMC) technique was used to simulate a low mass quantum particle (qp) in a dense Lennard-Jones 6-12 fluid having the thermodynamic properties of Xenon. Because of the difference in thermal wavelengths between the qp and the fluid molecules, the fluid molecules can be treated classically. This combination of using quantum mechanics for the qp and classical mechanics for the fluid molecules is known as a hybrid model. In the path integral formulation the qp is represented as a closed chain of P classical particles where the quantum uncertainty in the position of the qp is manifested by the finite spread of the polymer chain. The PIMC technique allows standard classical Monte Carlo techniques to be used to compute quantum mechanical equilibrium values like the ortho-Positronium pick-off decay rate. Here we compare the convergence of PIMC for different thermodynamic states, including one near the liquid-vapor critical point of the fluid. We employ the correlation function of the iterated quantum observables to estimate the number of statistically independent configurations in a run and provide an estimate of the standard error.   

Converged Nuclear Quantum Statistics from Semi-Classical Path Integrals

The quantum nature of nuclear motions plays a vital role in the structure, stability, and thermodynamics of molecular systems. The standard approach to take nuclear quantum effects (NQE) into account is the Feynman-Kac imaginary-time path-integral molecular dynamics (PIMD). Conventional PIMD simulations require exceedingly large number of classical subsystems (beads) to accurately capture NQE, resulting in considerable computational cost even at room temperature due to the rather high internal vibrational frequencies of many molecules of interest. We propose a novel parameter-free form for the PI partition function and estimators to calculate converged thermodynamic averages. Our approach requires the same ingredients as the conventional PIMD simulations, but decreases the number of required beads by roughly an order of magnitude. This greatly extends the applicability of {\it ab initio} PIMD for realistic molecular systems. The developed method has been applied to study the thermodynamics of N$_2$, H$_2$O, CO$_2$, and C$_6$H$_6$ molecules. For all of the considered systems at room temperature, 4 to 8 beads are enough to recover the NQE contribution to the total energy within 2\% of the fully converged quantum result.   

Destruction of spin-nematic order on randomly depleted triangular lattices

We consider the spin-1 Heisenberg model with biquadratic interactions on a 2-dimensional triangular lattice with random site dilution. It has been shown for this model that the ground state on a clean lattice exhibits spin nematic order. Using the stochastic series expansion (SSE) quantum Monte Carlo (QMC) algorithm, we study the nature of the order-disorder transition in the thermodynamic limit by extrapolating the ground state nematic order averaged over disorder realizations.   

Quantum Monte Carlo of atomic and molecular systems with heavy elements

We carry out quantum Monte Carlo calculations of atomic and molecular systems with several heavy atoms such as Mo, W and Bi. In particular, we compare the correlation energies vs their lighter counterparts in the same column of the periodic table in order to reveal trends with regard to the atomic number Z. One of the observations is that the correlation energy for the isoelectronic valence space/states is mildly decreasing with increasing Z. Similar observation applies also to the fixed-node errors, supporting thus our recent observation that the fixed-node error increases with electronic density for the same (or similar) complexity of the wave function and bonding. In addition, for Bi systems we study the impact of the spin-orbit on the electronic structure, in particular, on binding, correlation and excitation energies.   

Flat histogram quantum Monte Carlo for analytic continuation to real time

We study a recently developed technique based on the application of flat histogram ideas to quantum Monte Carlo. We use this technique to make the histogram of the single-particle Green's function flat as a function of the imaginary time, and the stochastic analytical inference technique to obtain the spectral functions for the $t-J$ model. We find that this application of the flat histogram idea to the quantum Monte Carlo method dramatically improves the quality of the results of the analytic continuation.   

Pseudopotentials for quantum Monte Carlo calculations of transition metal oxides

Quantum Monte Carlo calculations of transition metal oxides are partially limited by the availability of high quality pseudopotentials that are both accurate in QMC and compatible with major electronic structure codes, e.g. by not being overly hard in the standard planewave basis. Following insight gained from recent GW calculations, a set of neon core pseudopotentials with small cutoff radii have been created for the early transition metal elements Sc to Zn within the local density approximation of DFT. The pseudopotentials have been tested for energy consistency within QMC by calculating the first through fourth ionization potentials of the isolated transition metal (TM) atoms and the binding curve of each TM-O dimer. The vast majority of the ionization potentials fall within 0.3 eV of the experimental values, with exceptions occurring mainly for atoms with multiple unpaired d electrons where multireference effects are the strongest. The equilibrium bond lengths of the dimers are within 1\% of experimental values and the binding energy errors are typically less than 0.3 eV. Given the uniform treatment of the core, the larger deviations occasionally observed may primarily reflect the limitations of a Slater-Jastrow trial wavefunction.   

First-principles quantum Monte Carlo study of the density-density response in free-standing graphene

Electrons in graphene behave like 2D massless Dirac fermions in low energy. According to RPA, the usual screening in a metal should be absent; however, experiments on free-standing graphene suggest that the electron interactions are screened and it is a weakly correlated semimetal. However, it is still unclear whether the screening effect is due to exciton effects, the sigma electrons or the core electrons. We will report progress on first-principles quantum Monte Carlo calculations of the density-density response of free-standing graphene. We will use this quantity to investigate the screening in graphene.   

Studies of materials from simple metal atoms by quantum Monte Carlo methods

We carry out quantum Monte Carlo (QMC) calculations of systems from simple metal elements such as Li and Na with the goal of studying the cohesive/binding energies, structural characteristics as well as the accuracy of QMC methods for these elements. For Na we use test small-core pseudo potentials vs large-core pseudopotentials with the core polarization and relaxation correction potentials. We test orbital sets from several DFT functionals in order to assess the accuracy of the corresponding wave functions and fixed-node biases. It turns out that the valence correlations energies are very accurate, typically, 97\% or higher in most of the tested systems. This provides a validation framework for further QMC studies of these systems in non-equilibrium conformations and at high pressures.   

Generalized Moment Method for Gap Estimation and Quantum Monte Carlo Level Spectroscopy

We formulate a convergent sequence for the gap estimation in the worldline quantum Monte Carlo method. The ambiguity left in the conventional gap calculation for quantum systems is eliminated. The level spectroscopy from quantum Monte Carlo data is developed as an application of the unbiased gap estimation. From the spectrum analysis, we precisely determine the Kosterlitz-Thouless type quantum phase transition point in the spin-Peierls model as $\lambda_{\rm c} = 0.2245 \pm 0.0017$ for phonon frequency $\omega=1/4$. We demonstrate that the criticality at the transition point is described by the $k=1$ $SU(2)$ Wess-Zumino-Witten model. The detailed comparison to the previous approach to the gap estimation is shown and discussed. Reference: H. Suwa and S. Todo, arXiv:1402.0847.   

Numerical investigations of the sign problem in full configuration interaction quantum Monte Carlo

We explore the sign problem in full configuration interaction quantum Monte Carlo in a systematic manner. A large data set generated for the 1D k-space Hubbard model is used to demonstrate that, under certain conditions, the amount of information required to store the exact-on-average wave function has sub-linear cost with the number of Hilbert space states. It is our hope that this data set will be of use to people looking to improve both FCIQMC and its initiator adaptation. The sign problem is then discussed in connection with seniority zero configuration interaction and pair coupled cluster theories. Reference: Phys. Rev. B 90, 155130 (2014).   

Overcoming the fermion sign problem in homogeneous systems

Explicit treatment of many-body Fermi statistics in path integral Monte Carlo results in exponentially scaling computational cost due to the near cancellation of contributions to observables from even and odd permutations. Through direct analysis of exchange statistics we find that individual exchange probabilities in homogeneous systems are, barring known combinatorial factors, independent of the configuration of other permutations present. For two representative systems, 3-He and the homogeneous electron gas, we find that this allows the entire antisymmetrized density matrix to be generated from a simple model depending on only a few parameters obtainable directly from a standard PIMC simulation. Finally, we show this model may be extended to arbitrary order, resulting in a polynomial scaling algorithm for measuring fermionic observables.   

Infinite Variance in Fermion Quantum Monte Carlo Calculations Without the Sign Problem

For several important classes of fermion problems, for example the half-filled repulsive Hubbard model and the spin-balanced attractive Hubbard model, quantum Monte Carlo methods using auxiliary-fields allow exact calculations without the sign problem. We show, however, that in most commonly employed algorithms the variance diverges, leading to unreliable estimate of the Monte Carlo statistical error. An approach is proposed to solve the problem. Illustrative results in Hubbard model will be presented.   

Spin-Liquid Behavior of a Simple Spin Model on the Triangular Lattice

I will report on numerical studies of phase transition between competing magnetic (M) and valence bond solid states (VBS) using unbiased quantum Monte Carlo methods in sign-problem free models on non-bipartite lattices in two dimensions. On bipartite lattices the transition between these two phases is a direct second order critical point - consistent with various aspects of the ``deconfined'' criticality scenario. In contrast, on non-bipartite lattices an intermediate phase appears between M and VBS. We present evidence that this new phase is a quantum spin liquid.