Quantum Monte Carlo Calculations Research Articles

Selected configuration interaction wave functions in phaseless auxiliary field quantum Monte Carlo

We present efficient algorithms for using selected configuration interaction (sCI) trial wave functions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC). These advances, geared toward optimizing computational performance for longer configuration interaction expansions, allow us to use up to a million configurations in the trial state for ph-AFQMC. In one example, we found the cost of ph-AFQMC per sample to increase only by a factor of about 3 for a calculation with 104 configurations compared to that with a single one, demonstrating the tiny computational overhead due to a longer expansion. This favorable scaling allows us to study the systematic convergence of the phaseless bias in auxiliary field quantum Monte Carlo calculations with an increasing number of configurations and provides a means to gauge the accuracy of ph-AFQMC with other trial states. We also show how the scalability issues of sCI trial states for large system sizes could be mitigated by restricting them to a moderately sized orbital active space and leveraging the near-cancellation of out of active space phaseless errors.

Weighted nodal domain averages of eigenstates for quantum Monte Carlo and beyond

We study the nodal properties of many-body eigenstates of stationary Schr\"odinger equation that affect the accuracy of real-space quantum Monte Carlo calculations. In particular, we introduce weighted nodal domain averages that provide a new probe of nodal surfaces beyond the usual expectations. Particular choices for the weight function reveal, for example, that the difference between two arbitrary fermionic eigenvalues is given by the nodal hypersurface integrals normalized by overlaps with the bosonic ground state of the given Hamiltonian. Noninteracting and fully interacting Be atom with corresponding almost exact and approximate wave functions are used to illustrate several aspects of these concepts. Variational formulations that employ different weights are proposed for prospective improvement of nodes in variational and fixed-node diffusion Monte Carlo calculations.

Atomic shell structure from Born probabilities: Comparison to other shell descriptors and persistence in molecules.

Real space chemical bonding descriptors, such as the electron localization function or the Laplacian of the electron density, have been widely used in electronic structure theory thanks to their power to provide chemically intuitive spatial images of bonded and non-bonded interactions. This capacity stems from their ability to display the shell structure of atoms and its distortion upon molecular formation. Here, we examine the spatial position of the N electrons of an atom at the maximum of the square of the wavefunction, the so-called Born maximum, as a shell structure descriptor for ground state atoms with Z = 1-36, comparing it to other available indices. The maximization is performed with the help of variational quantum Monte Carlo calculations. We show that many electron effects (mainly Pauli driven) are non-negligible, that Born shells are closer to the nucleus than any other of the examined descriptors, and that these shells are very well preserved in simple molecules.

How electrons Coulomb repulsion changes graphene band structure

Base on effective medium theory we introduce a multi sites method for calculation of realistic energy bands of strongly correlated systems. We found due to approximated self energy, the density of states that obtained directly by calculated local Green function does not reflects system energy bands truly. By using this method we investigated how electrons repulsion renormalizes graphene bands. Graphene realistic bands calculated in both the dynamical mean field theory (DMFT) and four sites beyond super cell approximation for different repulsions. Our calculated interacting graphene bands illustrate a semi metal to a Mott insulator anti ferromagnetic phase transition at repulsions u = 2.2 t and u = 0.6 t for DMFT and four sites beyond super cell approximation respectively. These values are much less than finite size quantum Monte Carlo calculation prediction. We showed that the graphene bands are very sensitive to electrons repulsions and this phase transition happens at low repulsions in comparison to graphene band width.

Unbiasing fermionic quantum Monte Carlo with a quantum computer

Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems1–4. Fermionic quantum Monte Carlo (QMC) methods5,6, which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed with the help of quantum computers, while achieving accuracy that is competitive with state-of-the-art classical methods without burdensome error mitigation. Compared with the popular variational quantum eigensolver7,8, our hybrid quantum-classical computational model offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the ground-state wavefunction.

π phase shift across stripes in a charge density wave system

Many strongly correlated materials are characterized by deeply intertwined charge and spin order. Besides their high superconducting transition temperatures, one of the central features of these complex patterns in cuprates is a phase shift which occurs across lines of decreased hole density. That is, when doped away from their antiferromagnetic phase, the additional charge is not distributed uniformly, but rather in ``stripes.'' The sublattices preferentially occupied by up and down spin are reversed across these stripes, a phenomenon referred to as a ``$\ensuremath{\pi}$ phase shift.'' Many of the spin-charge patterns, including the $\ensuremath{\pi}$ phase shift, are reproduced by density matrix renormalization group and quantum Monte Carlo calculations of simplified tight binding (repulsive Hubbard) models. In this paper we demonstrate that this sublattice reversal is generic by considering the corresponding phenomenon in the attractive Hubbard Hamiltonian, where a charge density wave phase forms at half filling. We introduce charge stripes via an appropriate local chemical potential; measurements of charge correlation across the resulting lines of lowered density reveal a clear $\ensuremath{\pi}$ phase.

Multi-scale simulation of the adsorption of lithium ion on graphite surface: From quantum Monte Carlo to molecular density functional theory.

The structure of the double-layer formed at the surface of carbon electrodes is governed by the interactions between the electrode and the electrolyte species. However, carbon is notoriously difficult to simulate accurately, even with well-established methods such as electronic density functional theory and molecular dynamics. Here, we focus on the important case of a lithium ion in contact with the surface of graphite, and we perform a series of reference quantum Monte Carlo calculations that allow us to benchmark various electronic density functional theory functionals. We then fit an accurate carbon-lithium pair potential, which is used in molecular density functional theory calculations to determine the free energy of the adsorption of the ion on the surface in the presence of water. The adsorption profile in aqueous solution differs markedly from the gas phase results, which emphasize the role of the solvent on the properties of the double-layer.

Ground-state properties of the narrowest zigzag graphene nanoribbon from quantum Monte Carlo and comparison with density functional theory.

By means of quantum Monte Carlo (QMC) calculations from first-principles, we study the ground-state properties of the narrowest zigzag graphene nanoribbon with an infinite linear acene structure. We show that this quasi-one-dimensional system is correlated and its ground state is made of localized π electrons whose spins are antiferromagnetically ordered. The antiferromagnetic (AFM) stabilization energy [36(3) meV per carbon atom] and the absolute magnetization [1.13(0.11) μB per unit cell] predicted by QMC are sizable, and they suggest the survival of antiferromagnetic correlations above room temperature. These values can be reproduced to some extent by density functional theory (DFT) within the DFT+U framework or by using hybrid functionals. Based on our QMC results, we then provide the strength of Hubbard repulsion in DFT+U suitable for this class of systems.

Momentum dependent optical lattice induced by an artificial gauge potential

We propose an experimentally feasible method to generate a one-dimensional optical lattice potential in an ultracold Bose gas system that depends on the transverse momentum of the atoms. The optical lattice is induced by the artificial gauge potential generated by a periodically driven multilaser Raman process. We study the many-body Bose-Hubbard model in an effective 1D case and show that the superfluid--Mott-insulator transition can be controlled via tuning the transverse momentum of the atomic gas. Such a feature enables us to control the phase of the quantum gas in the longitudinal direction by changing its transverse motional state. We examine our prediction via a strong-coupling expansion to an effective 1D Bose-Hubbard model and a quantum Monte Carlo calculation and discuss possible applications.

Quantum criticality at finite temperature for two-dimensional [formula omitted] models on the square and the honeycomb lattices

We study the quantum criticality at finite temperature for three two-dimensional (2D) JQ3 models using the first principles quantum Monte Carlo calculations (QMC). In particular, the associated universal quantities are obtained, and their inverse temperature dependence is investigated. The considered models are known to have quantum phase transitions from the Néel order to the valence-bond solid, and these transitions are shown to be possibly of second order for two of the studied models, with the remaining one being of first order. When the temperature dependence of the studied universal quantities is considered, we observe substantial differences between the two models likely possessing second-order phase transitions and the remaining model. Moreover, by using the associated data from both the models that may have continuous transitions, good data collapses are obtained for a number of the considered universal quantities. The findings presented here provide numerical evidence to support particular results established in the literature regarding the nature of the phase transitions of these JQ3 models, and can be employed as useful criterions to distinguish second-order phase transitions from first-order ones for the exotic criticalities of the JQ-type models.

Benchmarking exchange-correlation functionals in the spin-polarized inhomogeneous electron gas under warm dense conditions

Warm dense matter is a highly active research area both at the frontier and interface of material science and plasma physics. We assess the performance of commonly used exchange-correlation (XC) approximation (LDA, PBE, PBEsol, and AM05) in the spin-polarized inhomogeneous electron gas under warm dense conditions based on exact path-integral quantum Monte-Carlo calculations. This extends our recent analysis on the relevance of inhomogeneities in the spin-unpolarized warm dense electron gas [Z.~Moldabekov et al., J. Chem. Phys. 155, 124116 (2021)]. We demonstrate that the predictive accuracy of these XC functionals deteriorates with (1) a decrease in density (corresponding to an increase in the inter-electronic correlation strength) and (2) an increase of the characteristic wave number of the density perturbation. We provide recommendations for the applicability of the considered XC functionals at conditions typical for warm dense matter. Furthermore, we hint at future possibilities for constructing more accurate XC functionals under these conditions.

Enhancing d -wave superconductivity with nearest-neighbor attraction in the extended Hubbard model

Motivated by the recent discovery of the anomalously nearest-neighbor attraction arising from the electron-phonon coupling, we quantitatively investigate the enhancing effects of this additional attractive channel on the $d$-wave SC based on dynamic cluster quantum Monte Carlo calculations of doped two-dimensional extended Hubbard model with nearest-neighbor attraction $-V$. Focusing on the range of $0<-V/t \le 2$, our simulations indicate that the dynamics of $d$-wave projected pairing interaction is attractive at all frequencies and increases with $|V|$. Moreover, turning on $-V$ attraction enhances the $(\pi,\pi)$ spin fluctuations but only enhances (suppresses) the charge fluctuations for small (large) momentum transfer. Thus, at $V/t=-1$ relevant to ``holon folding branch'', the charge fluctuations are insufficient to compete with $d$-wave pairing interaction strengthened by enhanced spin fluctuations. Our work suggest the underlying rich interplay between the spin and charge fluctuations in giving rise to the superconducting properties.

Path Integral Estimates of the Quantum Fluctuations of the Relative Soliton-Soliton Velocity in a Gross-Pitaevskii Breather

In this paper, the quantum fluctuations of the relative velocity of constituent solitons in a Gross-Pitaevskii breather are studied. The breather is confined in a weak harmonic trap. These fluctuations are monitored, indirectly, using a two-body correlation function measured at a quarter of the harmonic period after the breather creation. The results of an ab initio quantum Monte Carlo calculation, based on the Feynman-Kac path integration method, are compared with the analytical predictions using the recently suggested approach within the Bogoliubov approximation, and a good agreement is obtained.

Ab initio calculations in atoms, molecules, and solids, treating spin-orbit coupling and electron interaction on an equal footing.

We incorporate explicit, non-perturbative treatment of spin-orbit coupling into ab initio auxiliary-field quantum Monte Carlo (AFQMC) calculations. The approach allows a general computational framework for molecular and bulk systems in which material specificity, electron correlation, and spin-orbit coupling effects can be captured accurately and on an equal footing, with favorable computational scaling vs system size. We adopt relativistic effective-core potentials that have been obtained by fitting to fully relativistic data and that have demonstrated a high degree of reliability and transferability in molecular systems. This results in a two-component spin-coupled Hamiltonian, which is then treated by generalizing the ab initio AFQMC approach. We demonstrate the method by computing the electron affinity in Pb, the bond dissociation energy in Br2 and I2, and solid Bi.

Enhancing the Accuracy of Ab Initio Molecular Dynamics by Fine Tuning of Effective Two-Body Interactions: Acetonitrile as a Test Case.

Grimme's dispersion-corrected density functional theory (DFT-D) methods have emerged among the most practical approaches to perform accurate quantum mechanical calculations on molecular systems ranging from small clusters to microscopic and mesoscopic samples, i.e., including hundreds or thousands of molecules. Moreover, DFT-D functionals can be easily integrated into popular ab initio molecular dynamics (MD) software packages to carry out first-principles condensed-phase simulations at an affordable computational cost. Here, starting from the well-established D3 version of the dispersion-correction term, we present a simple protocol to improve the accurate description of the intermolecular interactions of molecular clusters of growing size, considering acetonitrile as a test case. Optimization of the interaction energy was performed with reference to diffusion quantum Monte Carlo calculations, successfully reaching the same inherent accuracy of the latter (statistical error of ∼0.1 kcal/mol per molecule). The refined DFT-D3 model was then used to perform ab initio MD simulations of liquid acetonitrile, again showing significant improvements toward available experimental data with respect to the default correction.

Quantitative electron-electron interaction using local field factors from quantum Monte Carlo calculations

The effective electron-electron interaction in electron gas depends on both the density and spin local field factors. Variational diagrammatic quantum Monte Carlo calculations of the spin local field factor are reported. These are used together with the charge local field factor from previous diffusion quantum Monte Carlo calculations to quantitatively present the full effective spin-dependent electron-electron interaction in three-dimensional electron gas. Very simple quadratic formulas are presented for the local field factors that quantitatively produce all of the response functions of the electron gas at metallic densities. Exchange and correlation become increasingly important at low densities. At the compressibility divergence at ${r}_{s}=5.25$, both the direct (screened Coulomb) term and the charge-dependent exchange term in the electron-electron interaction at $q=0$ are separately divergent. However, due to large cancellations, their difference is finite, well behaved, and much smaller than either term separately. As a result, the spin contribution to the electron-electron interaction becomes an important factor. The static electron-electron interaction is repulsive as a function of density but is less repulsive for electrons with parallel spins. The effect of allowing a deformable, rather than rigid, positive background is shown to be as quantitatively important as exchange and correlation. As a simple concrete example, the electron-electron interaction is calculated using the measured bulk modulus of the alkali metals with a linear phonon dispersion. The net electron-electron interaction in lithium is attractive for wave vectors $0\ensuremath{-}2{k}_{F}$, which suggests superconductivity, and is mostly repulsive for the other alkali metals.

Collective excitations and quantum incompressibility in electron-hole bilayers

We apply quantum continuum mechanics to the calculation of the excitation spectrum of a coupled electron-hole bilayer. The theory expresses excitation energies in terms of ground-state intra- and inter-layer pair correlation functions, which are available from Quantum Monte Carlo calculations. The final formulas for the collective modes deduced from this approach coincide with the formulas obtained in the "quasi-localized particle approximation" by Kalman et al., and likewise, the theory predicts the existence of gapped excitations in the charged channels, with the gap arising from electron-hole correlation. An immediate consequence of the gap is that the static density-density response function of the charged channel vanishes as $q^2$ for wave vector $q \to 0$, rather than linearly in $q$, as commonly expected. In this sense, the system is {\it incompressible}. This feature, which has no analogue in the classical electron-hole plasma, is consistent with the existence of an excitonic ground state and implies the existence of a discontinuity in the chemical potential of electrons and holes when the numbers of electrons and holes are equal. It should be experimentally observable by monitoring the densities of electrons and holes in response to potentials that attempt to change these densities in opposite directions.

Superconducting Fluctuations in Overdoped Bi2Sr2CaCu2O8+δ

Fluctuating superconductivity - vestigial Cooper pairing in the resistive state of a material - is usually associated with low dimensionality, strong disorder or low carrier density. Here, we report single particle spectroscopic, thermodynamic and magnetic evidence for persistent superconducting fluctuations in heavily hole-doped cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ ($T_c$ = 66~K) despite the high carrier density. With a sign-problem free quantum Monte Carlo calculation, we show how a partially flat band at ($\pi$,0) can help enhance superconducting phase fluctuations. Finally, we discuss the implications of an anisotropic band structure on the phase-coherence-limited superconductivity in overdoped cuprates and other superconductors.

Assessment of the Accuracy of Density Functionals for Calculating Oxygen Reduction Reaction on Nitrogen-Doped Graphene.

Experimental studies of the oxygen reduction reaction (ORR) at nitrogen-doped graphene electrodes have reported a remarkably low overpotential, on the order of 0.5 V, similar to Pt-based electrodes. Theoretical calculations using density functional theory have lent support to this claim. However, other measurements have indicated that transition metal impurities are actually responsible for the ORR activity, thereby raising questions about the reliability of both the experiments and the calculations. To assess the accuracy of the theoretical calculations, various generalized gradient approximation (GGA), meta-GGA, and hybrid functionals are employed here and calibrated against high-level wave-function-based coupled-cluster calculations (CCSD(T)) of the overpotential as well as self-interaction corrected density functional calculations and published quantum Monte Carlo calculations of O adatom binding to graphene. The PBE0 and HSE06 hybrid functionals are found to give more accurate results than the GGA and meta-GGA functionals, as would be expected, and for a low dopant concentration, 3.1%, the overpotential is calculated to be 1.0 V. The GGA and meta-GGA functionals give a lower estimate by as much as 0.4 V. When the dopant concentration is doubled, the overpotential calculated with hybrid functionals decreases, while it increases in GGA functional calculations. The opposite trends result from different potential-determining steps, the *OOH species being of central importance in the hybrid functional calculations, while the reduction of *O determines the overpotential obtained in GGA and meta-GGA calculations. The results presented here are mainly based on calculations of periodic representations of the system, but a comparison is also made with molecular flake models that are found to give erratic results due to finite size effects and geometric distortions during energy minimization. The presence of the electrolyte has not been taken into account explicitly in the calculations presented here but is estimated to be important for definitive calculations of the overpotential.

Comparative study of He6 β -decay based on different similarity-renormalization-group evolved chiral interactions

We report on a study of the Gamow-Teller matrix element contributing to $^{6}\mathrm{He}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$ decay with similarity renormalization group (SRG) versions of momentum- and configuration-space two-nucleon interactions. These interactions are derived from two different formulations of chiral effective field theory ($\ensuremath{\chi}\mathrm{EFT})$---without and with the explicit inclusion of $\mathrm{\ensuremath{\Delta}}$ isobars. We consider evolution parameters ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{SRG}}$ in the range between 1.2 and 2.0 ${\mathrm{fm}}^{\ensuremath{-}1}$ and, for the $\mathrm{\ensuremath{\Delta}}$-less case, also the unevolved (bare) interaction. The axial current contains one- and two-body terms, consistently derived at tree level (no loops) in the two distinct $\ensuremath{\chi}\mathrm{EFT}$ formulations we have adopted here. The $^{6}\mathrm{He}$ and $^{6}\mathrm{Li}$ ground-state wave functions are obtained from hyperspherical-harmonics (HH) solutions of the nuclear many-body problem. In $A=6$ systems, the HH method is limited at present to treat only two-body interactions and non-SRG evolved currents. Our results exhibit a significant dependence on ${\mathrm{\ensuremath{\Lambda}}}_{\text{SRG}}$ of the contributions associated with two-body currents, suggesting that a consistent SRG-evolution of these is needed in order to obtain reliable estimates. We also show that the contributions from one-pion-exchange currents depend strongly on the model (chiral) interactions and on the momentum- or configuration-space cutoffs used to regularize them. These results might prove helpful in clarifying the origin of the sign difference recently found in no-core-shell-model and quantum Monte Carlo calculations of the $^{6}\mathrm{He}$ Gamow-Teller matrix element.