Quantum Monte Carlo Results Research Articles

X marks the spot: Accurate energies from intersecting extrapolations of continuum quantum Monte Carlo data.

We explore the application of an extrapolative method that yields very accurate total and relative energies from variational and diffusion quantum Monte Carlo (VMC and DMC) results. For a trial wave function consisting of a small configuration interaction (CI) wave function obtained from full CI quantum Monte Carlo and reoptimized in the presence of a Jastrow factor and an optional backflow transformation, we find that the VMC and DMC energies are smooth functions of the sum of the squared coefficients of the initial CI wave function and that quadratic extrapolations of the non-backflow VMC and backflow DMC energies intersect within uncertainty of the exact total energy. With adequate statistical treatment of quasi-random fluctuations, the extrapolate and intersect with polynomials of order two method is shown to yield results in agreement with benchmark-quality total and relative energies for the C2, N2, CO2, and H2O molecules, as well as for the C2 molecule in its first electronic singlet excited state, using only small CI expansion sizes.

Production of positronium chloride: A study of the charge exchange reaction between Ps and Cl.

We present cross sections for the formation of positronium chloride (PsCl) in its ground state from the charge exchange between positronium (Ps) and chloride (Cl-) in the range of 10 meV-100eV Ps energy. We have used theoretical models based on the first Born approximation in its three-body formulation. We simulated the collisions between Ps and Cl- using abinitio binding energies and positronic wave functions at both the mean-field and correlated levels extrapolated to the complete basis set limit. The accuracy of these abinitio data was benchmarked on the PsF system with the existing highly accurate results, including the very recent quantum Monte Carlo results. We have investigated Ps excited states up to n = 4. The results suggest that the channel Ps(n = 2) is of particular interest for the production of PsCl in the ground state and shows that an accurate treatment of correlation effects (i.e., electron-electron and electron-positron correlations) leads to a significant change in the magnitude of the PsCl production cross section with respect to the mean-field level.

Metal-Insulator Transition in a Semiconductor Heterobilayer Model.

Transition metal dichalcogenide superlattices provide an exciting new platform for exploring and understanding a variety of phases of matter. The moiré continuum Hamiltonian, of two-dimensional jellium in a modulating potential, provides a fundamental model for such systems. Accurate computations with this model are essential for interpreting experimental observations and making predictions for future explorations. In this work, we combine two complementary quantum MonteCarlo (QMC) methods, phaseless auxiliary field quantum MonteCarlo and fixed-phase diffusion MonteCarlo, to study the ground state of this Hamiltonian. We observe a metal-insulator transition between a paramagnet and a 120° Néel ordered state as the moiré potential depth and the interaction strength are varied. We find significant differences from existing results by Hartree-Fock and exact diagonalization studies. In addition, we benchmark density-functional theory, and suggest an optimal hybrid functional which best approximates our QMC results.

Straintronics with single-layer MoS2 : A quantum Monte Carlo study

Using state-of-the-art quantum Monte Carlo (QMC) methods, we study straintronic properties of a single MoS2 monolayer. 2D MoS2 is a quintessential straintronic material for which many experiments have been performed. First, we determine the equilibrium atomic structure which is not known experimentally and is strictly needed to correctly determine the straintronic properties. That enables us to precisely analyze the quasiparticle band gaps for any applied biaxial strain, which we describe by a bivariate paraboloid function of lattice constant and internal structural parameter. Using the fixed-node QMC calculations fitted by analytical formulas, we localize the following excited state crossings between the direct, K→K, and indirect Γ→X and K→K/2 excitations. Based on this highly accurate many-body treatment, we predict a gauge factor of 136 meV/% for the K→K transition and a fairly narrow window of ≈2.8% from compressive to mildly tensile strains, accounting for only ≈0.3 eV band gap change maintaining the direct character of the gap. Consequently, we suggest that, compared to other 2D materials, such as phosphorene, there is only a limited straintronic tuneability in this material often studied for its straintronic properties. QMC results are compared to results of standard DFT modeling, which reveal insights into the corresponding inaccuracies and therefore open a window for educated use of rapid DFT approaches. Published by the American Physical Society 2024

Off-diagonal long-range order in arrays of dipolar droplets

We report quantum Monte Carlo results of harmonically confined quantum Bose dipoles within a range of interactions covering the evolution from a gas phase to the formation of an array of droplets. Scaling the experimental setup to a computationally accessible domain we characterize that evolution in qualitative agreement with experiments. Our microscopic approach generates ground-state results free from approximations, albeit with some controlled statistical noise. The simultaneous estimation of the static structure factor and the one-body density matrix allows for a better knowledge of the quantum coherence between droplets. Our results show a narrow window of interaction strengths where diagonal and off-diagonal long-range order can coexist. This domain, which is the key signal of a supersolid state, is reduced with respect to the one predicted by the extended Gross–Pitaevskii equation. Differences are probably due to an increase of attraction in our model, observed previously in the calculation of critical atom numbers for single dipolar drops.

Toward accurate modeling of structure and energetics of bulk hexagonal boron nitride.

Materials that exhibit both strong covalent and weak van der Waals interactions pose a considerable challenge to many computational methods, such as DFT. This makes assessing the accuracy of calculated properties, such as exfoliation energies in layered materials like hexagonal boron nitride (h-BN) problematic, when experimental data are not available. In this paper, we investigate the accuracy of equilibrium lattice constants and exfoliation energy calculation for various DFT-based computational approaches in bulk h-BN. We contrast these results with available experiments and reference fixed-node diffusion quantum Monte Carlo (QMC) results. From our reference QMC calculation, we obtained an exfoliation energy of 2 meV/atom (-0.38 0.02 J/m ).

Efficient Tensor Network Algorithm for Layered Systems.

Strongly correlated layered 2D systems are of central importance in condensed matter physics, but their numerical study is very challenging. Motivated by the enormous successes of tensor networks for 1D and 2D systems, we develop an efficient tensor network approach based on infinite projected entangled-pair states for layered 2D systems. Starting from an anisotropic 3D infinite projected entangled-pair state ansatz, we propose a contraction scheme in which the weakly interacting layers are effectively decoupled away from the center of the layers, such that they can be efficiently contracted using 2D contraction methods while keeping the center of the layers connected in order to capture the most relevant interlayer correlations. We present benchmark data for the anisotropic 3D Heisenberg model on a cubic lattice, which shows close agreement with quantum MonteCarlo and full 3D contraction results. Finally, we study the dimer to Néel phase transition in the Shastry-Sutherland model with interlayer coupling, a frustrated spin model that is out of reach of quantum MonteCarlo due to the negative sign problem.

Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons.

We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.

Spectral Functions of the Holstein Polaron: Exact and Approximate Solutions.

It is generally accepted that the dynamical mean field theory gives a good solution of the Holstein model, but only in dimensions greater than two. Here, we show that this theory, which becomes exact in the weak coupling and in the atomic limit, provides an excellent, numerically cheap, approximate solution for the spectral function of the Holstein model in the whole range of parameters, even in one dimension. To establish this, we make a detailed comparison with the spectral functions that we obtain using the newly developed momentum-space numerically exact hierarchical equations of motion method, which yields electronic correlation functions directly in real time. We crosscheck these conclusions with our path integral quantum MonteCarlo and exact diagonalization results, as well as with the available numerically exact results from the literature.

Assessing the accuracy of compound formation energies with quantum Monte Carlo

Accurately predicting the formation energy of a compound, which describes its thermodynamic stability, is a key challenge in materials physics. Here, we employ many-body quantum Monte Carlo (QMC) with single-reference trial functions to compute the formation energy of two electronically disparate compounds, the intermetallic VPt$_2$ and the semiconductor CuI, for which standard density functional theory (DFT) predictions using both the Perdew-Burke Ernzerhof (PBE) and the strongly constrained and appropriately normed (SCAN) density functional approximations deviate markedly from available experimental values. For VPt$_2$, we find an agreement between QMC, SCAN, and PBE0 estimates, which therefore remain in disagreement with the much less exothermic experimental value. For CuI, the QMC result agrees with neither SCAN nor PBE pointing towards DFT exchange-correlation biases, likely related to the localized Cu $3d$ electrons. Compared to the behavior of some density functional approximations within DFT, spin-averaged QMC exhibits a smaller but still appreciable deviation when compared to experiment. The QMC result is slightly improved by incorporating spin-orbit corrections for CuI and solid I$_2$, so that experiment and theory are brought into imperfect but reasonable agreement within about 120~meV/atom.

Quantitative functional renormalization for three-dimensional quantum Heisenberg models

We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by \sim5\%∼5% and we confirm consistency of our data with established critical exponents \nuν and \etaη of the three-dimensional Heisenberg universality class. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T\rightarrow0T→0. While extensions towards higher loop orders \ellℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing \ellℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.

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.

Polaron Interactions and Bipolarons in One-Dimensional Bose Gases in the Strong Coupling Regime

Bose polarons, quasiparticles composed of mobile impurities surrounded by cold Bose gas, can experience strong interactions mediated by the many-body environment and form bipolaron bound states. Here we present a detailed study of heavy polarons in a one-dimensional Bose gas by formulating a nonperturbative theory and complementing it with exact numerical simulations. We develop an analytic approach for weak boson-boson interactions and arbitrarily strong impurity-boson couplings. Our approach is based on a mean-field theory that accounts for deformations of the superfluid by the impurities and in this way minimizes quantum fluctuations. The mean-field equations are solved exactly in the Born-Oppenheimer approximation, leading to an analytic expression for the interaction potential of heavy polarons, which is found to be in excellent agreement with quantum MonteCarlo (QMC) results. In the strong coupling limit, the potential substantially deviates from the exponential form valid for weak coupling and has a linear shape at short distances. Taking into account the leading-order Born-Huang corrections, we calculate bipolaron binding energies for impurity-boson mass ratios as low as 3 and find excellent agreement with QMC results.

Optical spectral weight, phase stiffness, and Tc bounds for trivial and topological flat band superconductors

We present exact results that give insight into how interactions lead to transport and superconductivity in a flat band where the electrons have no kinetic energy. We obtain bounds for the optical spectral weight for flat-band superconductors that lead to upper bounds for the superfluid stiffness and the two-dimensional (2D) [Formula: see text] We focus on on-site attraction [Formula: see text] on the Lieb lattice with trivial flat bands and on the π-flux model with topological flat bands. For trivial flat bands, the low-energy optical spectral weight [Formula: see text] with [Formula: see text] , where n is the flat-band density and Ω is the Marzari-Vanderbilt spread of the Wannier functions (WFs). We also obtain a lower bound involving the quantum metric. For topological flat bands, with an obstruction to localized WFs respecting all symmetries, we again obtain an upper bound for [Formula: see text] linear in [Formula: see text] We discuss the insights obtained from our bounds by comparing them with mean-field and quantum Monte Carlo results.

Towards efficient and accurate ab initio solutions to periodic systems via transcorrelation and coupled cluster theory

We propose a streamlined combination scheme of the transcorrelation (TC) and coupled cluster (CC) theory, which not only increases the convergence rate with respect to the basis set, but also extends the applicability of the lowest order CC approximations to strongly correlated regimes in the three dimensional uniform electron gas (3D UEG). With the correct physical insights built into the correlator used in TC, highly accurate ground state energies with errors $\leq 0.001 $ a.u./electron relative to the state-of-the-art quantum Monte Carlo results can be obtained across a wide range of densities. The greatly improved efficiency and accuracy of our methods hold great promise for strongly correlated solids where many other methods fail.

Three-dimensional isometric tensor networks

Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor network, an operation needed to calculate quantum expectation values. Here we introduce a method for the time-evolution of three-dimensional isometric tensor networks which respects the isometric structure and therefore renders contraction simple through a special canonical form. Our method involves a tetrahedral site-splitting which allows to move the orthogonality center of an embedded tree tensor network in a simple cubic lattice to any position. Using imaginary time-evolution to find an isometric tensor network representation of the ground state of the 3D transverse field Ising model across the entire phase diagram, we perform a systematic benchmark study of this method in comparison with exact Lanczos and quantum Monte Carlo results. We show that the obtained energy matches the exact groundstate result accurately deep in the ferromagnetic and polarized phases, while the regime close to the critical point requires larger bond dimensions. This behavior is in close analogy with the two-dimensional case, which we also discuss for comparison.

Convergence to the fixed-node limit in deep variational Monte Carlo.

Variational quantum Monte Carlo (QMC) is an ab initio method for solving the electronic Schrödinger equation that is exact in principle, but limited by the flexibility of the available Ansätze in practice. The recently introduced deep QMC approach, specifically two deep-neural-network Ansätze PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such Ansätze. Here, we analyze how deep variational QMC approaches the fixed-node limit with increasing network size. First, we demonstrate that a deep neural network can overcome the limitations of a small basis set and reach the mean-field (MF) complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H4 and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark MF and many-body Ansätze on H2O, increasing the fraction of recovered fixed-node correlation energy of single-determinant Slater-Jastrow-type Ansätze by half an order of magnitude compared to previous variational QMC results, and demonstrate that a single-determinant Slater-Jastrow-backflow version of the Ansatz overcomes the fixed-node limitations. This analysis helps understand the superb accuracy of deep variational Ansätze in comparison to the traditional trial wavefunctions at the respective level of theory and will guide future improvements of the neural-network architectures in deep QMC.

Shell model for superfluids in rough-walled nanopores

Recent experiments on the flow of helium-4 fluid through nanopores with tunable pore radius provide a platform for studying the quasi-one-dimensional (quasi-1D) superfluid behaviors. In the extreme 1D limit, the helium atoms are localized by disordered small variations in the substrate potential provided by the pore walls. In the limit of wide pore radius, a solid layer of helium-4 is expected to coat the pore walls smoothing out the substrate potential, and superfluidity is observed in the central region. Building on earlier quantum Monte Carlo results, we propose a scenario for this crossover using a shell model of coupled Luttinger liquids. We find that a small radius pore will always localize the helium atoms, but above a critical radius, a single 1D channel flows through the pore and can be described by Luttinger liquid theory.

Low-dimensional fluctuations and pseudogap in Gaudin-Yang Fermi gases

Pseudogap is a ubiquitous phenomenon in strongly correlated systems such as high-$T_{\rm c}$ superconductors, ultracold atoms and nuclear physics. While pairing fluctuations inducing the pseudogap are known to be enhanced in low-dimensional systems, such effects have not been explored well in one of the most fundamental 1D models, that is, Gaudin-Yang model. In this work, we show that the pseudogap effect can be visible in the single-particle excitation in this system using a diagrammatic approach. Fermionic single-particle spectra exhibit a unique crossover from the double-particle dispersion to pseudogap state with increasing the attractive interaction and the number density at finite temperature. Surprisingly, our results of thermodynamic quantities in unpolarized and polarized gases show an excellent agreement with the recent quantum Monte Carlo and complex Langevin results, even in the region where the pseudogap appears.

Simple and Accurate Exchange Energy for Density Functional Theory.

A non-empirical exchange functional based on an interpolation between two limits of electron density, slowly varying limit and asymptotic limit, is proposed. In the slowly varying limit, we follow the study by Kleinman from 1984 which considered the response of a free-electron gas to an external periodic potential, but further assume that the perturbing potential also induces Bragg diffraction of the Fermi electrons. The interpolation function is motivated by the exact exchange functional of a hydrogen atom. Combined with our recently proposed correlation functional, tests on 56 small molecules show that, for the first-row molecules, the exchange-correlation combo predicts the total energies four times more accurately than the presently available Quantum Monte Carlo results. For the second-row molecules, errors of the core electrons exchange energies can be corrected, leading to the most accurate first- and second-row molecular total energy predictions reported to date despite minimal computational efforts. The calculated bond energies, zero point energies, and dipole moments are also presented, which do not outperform other methods.