Quantum Monte Carlo Techniques Research Articles

Towards structural optimization of gold nanoclusters with quantum Monte Carlo.

We study the prospects of using quantum Monte Carlo techniques (QMC) to optimize the electronic wavefunctions and atomic geometries of gold compounds. Complex gold nanoclusters are widely studied for diverse biochemical applications, but the dynamic correlation and relativistic effects in gold set the bar high for reliable, predictive simulation methods. Here we study selected ground state properties of few-atom gold clusters by using density functional theory (DFT) and various implementations of the variational Monte Carlo (VMC) and diffusion Monte Carlo. We show that the QMC methods mitigate the exchange-correlation (XC) approximation made in the DFT approach: the average QMC results are more accurate and significantly more consistent than corresponding DFT results based on different XC functionals. Furthermore, we use demonstrate structural optimization of selected thiolated gold clusters with between 1 and 3 gold atoms using VMC forces. The optimization workflow is demonstrably consistent, robust, and its computational cost scales with nb, where b < 3 and n is the system size. We discuss the implications of these results while laying out steps for further developments.

Accelerated quantum Monte Carlo with probabilistic computers

Quantum Monte Carlo (QMC) techniques are widely used in a variety of scientific problems and much work has been dedicated to developing optimized algorithms that can accelerate QMC on standard processors (CPU). With the advent of various special purpose devices and domain specific hardware, it has become increasingly important to establish clear benchmarks of what improvements these technologies offer compared to existing technologies. In this paper, we demonstrate 2 to 3 orders of magnitude acceleration of a standard QMC algorithm using a specially designed digital processor, and a further 2 to 3 orders of magnitude by mapping it to a clockless analog processor. Our demonstration provides a roadmap for 5 to 6 orders of magnitude acceleration for a transverse field Ising model (TFIM) and could possibly be extended to other QMC models as well. The clockless analog hardware can be viewed as the classical counterpart of the quantum annealer and provides performance within a factor of < 10 of the latter. The convergence time for the clockless analog hardware scales with the number of qubits as ∼ N, improving the ∼ N2 scaling for CPU implementations, but appears worse than that reported for quantum annealers by D-Wave.

Solving 2D and 3D Lattice Models of Correlated Fermions—Combining Matrix Product States with Mean-Field Theory

Correlated electron states are at the root of many important phenomena including unconventional superconductivity (USC), where electron pairing arises from repulsive interactions. Computing the properties of correlated electrons, such as the critical temperature Tc for the onset of USC, efficiently and reliably from the microscopic physics with quantitative methods remains a major challenge for almost all models and materials. In this theoretical work, we combine matrix product states (MPS) with static mean field (MF) to provide a solution to this challenge for quasi-one-dimensional (Q1D) systems: two- and three-dimensional materials comprised of weakly coupled correlated 1D fermions. This MPS+MF framework for the ground state and thermal equilibrium properties of Q1D fermions is developed and validated for attractive Hubbard systems first, and further enhanced via analytical field theory. We then deploy it to compute Tc for superconductivity in 3D arrays of weakly coupled, doped, and repulsive Hubbard ladders. The MPS+MF framework thus enables the quantitative study of USC and high-Tc superconductivity—and potentially many more correlated phases—in fermionic Q1D systems based directly on their microscopic parameters, in ways inaccessible to previous methods. This approach further allows one to treat competing macroscopic orders, such as superconducting and insulating ones, on an equal footing. Benchmarks of the framework using auxiliary-field quantum Monte Carlo techniques show that the overestimation of, e.g., Tc due to its mean-field component, is near constant in microscopic parameters. These features of the MPS+MF approach to correlated fermions open up the possibility of designing deliberately optimized Q1D superconductors, from experiments in ultracold gases to synthesizing new materials.13 MoreReceived 8 July 2022Revised 7 December 2022Accepted 3 January 2023DOI:https://doi.org/10.1103/PhysRevX.13.011039Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFermionsSuperconductivityPhysical Systems3-dimensional systemsStrongly correlated systemsUnconventional superconductorsTechniquesBosonizationDensity matrix renormalization groupMean field theoryCondensed Matter, Materials & Applied Physics

Systematically improvable mean-field variational ansatz for strongly correlated systems: Application to the Hubbard model

A systematically improvable wave function is proposed for the numerical solution of strongly correlated systems. With a stochastic optimization method, based on the auxiliary field quantum Monte Carlo technique, an effective temperature ${T}_{\mathrm{eff}}$ is defined, probing the distance of the ground-state properties of the model in the thermodynamic limit from the ones of the proposed correlated mean-field ansatz. In this way, their uncertainties from the unbiased zero temperature limit may be estimated by simple and stable extrapolations well before the so-called sign problem gets prohibitive. At finite ${T}_{\mathrm{eff}}$, the convergence of the energy to the thermodynamic limit is indeed shown to already be possible in the Hubbard model for relatively small square lattices with linear dimension $L\ensuremath{\simeq}10$, thanks to appropriate averages over several twisted boundary conditions. Within the estimated energy accuracy of the proposed variational ansatz, two clear phases are identified, as the energy is lowered by spontaneously breaking some symmetries satisfied by the Hubbard Hamiltonian: (a) a stripe phase where both spin and translation symmetries are broken and (b) a strong coupling $d$-wave superconducting phase when the particle number is not conserved and global $U(1)$ symmetry is broken. On the other hand, the symmetric phase is stable in a wide region at large doping and small coupling.

S=1 dimer system K2Ni(MoO4)2 : A candidate for magnon Bose-Einstein condensation

Dimerized quantum magnets provide a unique possibility to investigate the Bose-Einstein condensation of magnetic excitations in crystalline systems at low temperature. Here, we model the low-temperature magnetic properties of the recently synthesized spin $S=1$ dimer system ${\mathrm{K}}_{2}\mathrm{Ni}{({\mathrm{MoO}}_{4})}_{2}$ and propose it as a candidate material for triplon and quintuplon condensation. Based on a first-principles analysis of its electronic structure, we derive an effective spin dimer model that we first solve within a mean-field approximation to refine its parameters in comparison to experiment. Finally, the model is solved by employing a numerically exact quantum Monte Carlo technique which leads to magnetic properties in good agreement with experimental magnetization and thermodynamic results. We discuss the emergent spin model of ${\mathrm{K}}_{2}\mathrm{Ni}{({\mathrm{MoO}}_{4})}_{2}$ in view of the condensation of magnetic excitations in a broad parameter regime. Finally, we comment on a geometrical peculiarity of the proposed model and discuss how it could host a supersolid phase upon structural distortions.

Correct and Accurate Polymorphic Energy Ordering of Transition-Metal Monoxides Obtained from Semilocal and Onsite-Hybrid Exchange-Correlation Approximations

The relative energetic stability of the structural phases of common antiferromagnetic transition-metal oxides (MnO, FeO, CoO, and NiO) within the semilocal and hybrid density functionals are fraught with difficulties. In particular, MnO is known to be the most difficult case for almost all common semilocal and hybrid density approximations. Here, we show that the meta-generalized gradient approximation (meta-GGA) constructed from the cuspless hydrogen model and Pauli kinetic energy density (MGGAC) can lead to the correct phase as the ground-state of MnO. The relative energy differences of zinc blende (zb) and rock salt (rs) structures as computed using MGGAC are found to be in nice agreement with those obtained from high-level correlation methods like the random phase approximation or quantum Monte Carlo techniques. Besides, we have also applied the onsite hybrid functionals (closely related to DFT+U) based on GGA and meta-GGA functionals, and it is shown that a relatively high amount of Hartree–Fock exchange is necessary to obtain rs as the ground-state phase. Our present investigation suggests the semilocal MGGAC and onsite hybrids, both being computationally cheap, as methods of choice for the calculation of the relative stability of antiferromagnetic transition-metal oxides having potential applications in solid-state physics and structural chemistry.

Stripes in the extended $t-t^\prime$ Hubbard model: A Variational Monte Carlo analysis

By using variational quantum Monte Carlo techniques, we investigate the instauration of stripes (i.e., charge and spin inhomogeneities) in the Hubbard model on the square lattice at hole doping \delta=1/8δ=1/8, with both nearest- (tt) and next-nearest-neighbor hopping (t^\primet′). Stripes with different wavelengths \lambdaλ (denoting the periodicity of the charge inhomogeneity) and character (bond- or site-centered) are stabilized for sufficiently large values of the electron-electron interaction U/tU/t. The general trend is that \lambdaλ increases going from negative to positive values of t^\prime/tt′/t and decreases by increasing U/tU/t. In particular, the \lambda=8λ=8 stripe obtained for t^\prime=0t′=0 and U/t=8U/t=8 [L.F. Tocchio, A. Montorsi, and F. Becca, SciPost Phys. 21 (2019)] shrinks to \lambda=6λ=6 for U/t\gtrsim 10U/t≳10. For t^\prime/t&lt;0t′/t&lt;0, the stripe with \lambda=5λ=5 is found to be remarkably stable, while for t^\prime/t&gt;0t′/t&gt;0, stripes with wavelength \lambda=12λ=12 and \lambda=16λ=16 are also obtained. In all these cases, pair-pair correlations are highly suppressed with respect to the uniform state (obtained for large values of |t^\prime/t||t′/t|), suggesting that striped states are not superconducting at \delta=1/8δ=1/8.

Comment on “Quasielastic lepton scattering and back-to-back nucleons in the short-time approximation”

The article of Pastore et al., whereas proposing an interesting and potentially useful approach for the generalization of quantum Monte Carlo techniques to the treatment of the nuclear electromagnetic response, features an incorrect and misleading discussion of $y$ scaling. The response to interactions with transversely polarized virtual photons receives sizable contributions from nonscaling processes in which the momentum transfer is shared between two nucleons. It follows that, contrary to what is stated by the the authors, $y$ scaling in the transverse channel is accidental.

A Continuous Time Quantum Monte Carlo as an Impurity Solver for Strongly Correlated System

We assess the continuous-time quantum Monte Carlo (CT-QMC) technique with hybridization expansion for solvingthe electronic structure of the strongly correlated system LaxSr1−xVO3 . The impurity solver method implemented here shows the fair agreement with the other available Monte Carlo techniques. From the study, it is found that the CT-QMC technique clearly distinguishes metallic phase, quasiparticle phase and insulating phases of the system depending upon the strength of the correlation. In case of La0.33Sr0.67VO3 system the metal-insulator transition is found to be at U = 4.5 eV for β = 6(eV)−1. The value of U depends with the value of β, and also the value of Hund’s coupling (J) and bandwidth (W). This technique allows the particle to exchange with the reservoir of the particles and the impurity sites, which is accounted numerically to treat the temporal fluctuation of the fermionic systems termed as dynamical mean field theory (DMFT). This theory is used to explain the phenomena of MottHubbard metal insulator transition of the materials which are applicable for designing the Mottronics, Neuromorphic computing, Quantum computing and resistive memory devices.

Self-energy method for time-dependent spectral functions of the Anderson impurity model within the time-dependent numerical renormalization group approach

The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation of motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time-evolution of spectral features at the different characteristic time-scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.

Finite-range effects in the unitary Fermi polaron

Quantum Monte Carlo techniques are employed to study the properties of polarons in an ultracold Fermi gas, at $T=0$, and in the unitary regime using both a zero-range model and a square-well potential. For a fixed density, the potential range is varied and results are extrapolated and compared against a zero-range model. A discussion regarding the choice of an interacting potential with a finite range is presented. We compute the polaron effective mass, the polaron binding energy, and the effective coupling between them. The latter is obtained using the Landau-Pomeranchuk's weakly interacting quasiparticle model. The contact parameter is estimated by fitting the pair distribution function of atoms in different spin states.

Quantum phase transition of many interacting spins coupled to a bosonic bath: Static and dynamical properties

By using worldline and diagrammatic quantum Monte Carlo techniques, matrix product state, and a variational approach \`a la Feynman, we investigate the equilibrium properties and relaxation features of a quantum system of $N$ spins antiferromagnetically interacting with each other, with strength $J$, and coupled to a common bath of bosonic oscillators, with strength $\ensuremath{\alpha}$. We show that, in the Ohmic regime, a Beretzinski-Thouless-Kosterlitz quantum phase transition occurs. While for $J=0$ the critical value of $\ensuremath{\alpha}$ decreases asymptotically with $1/N$ by increasing $N$, for nonvanishing $J$ it turns out to be practically independent on $N$, allowing to identify a finite range of values of $\ensuremath{\alpha}$ where spin phase coherence is preserved also for large $N$. Then, by using matrix product state simulations, and the Mori formalism and the variational approach \`a la Feynman jointly, we unveil the features of the relaxation, that, in particular, exhibits a nonmonotonic dependence on the temperature reminiscent of the Kondo effect. For the observed quantum phase transition we also establish a criterion analogous to that of the metal-insulator transition in solids.

The stability of e+H- 2.

The recently discovered positronic molecule e+H- 2 [J. Charry et al., Angew. Chem., Int. Ed. 57, 8859-8864 (2018)] has a new type of bond, the single-positron bond. We studied its stability using quantum Monte Carlo techniques. We computed an accurate potential energy curve of the reaction H- + PsH → e+H- 2 → H2 + Ps- to establish its global stability with respect to all possible dissociation channels and to define the range of its local stability. We showed that the e+H- 2 system is stable with respect to the dissociation into H- + PsH, with a binding energy of 23.5(1) mhartree. For R < 3.2 bohrs, the system is unstable, and it decays into H2 + Ps-. There are no other bound structures for R < 3.2 bohrs. We discuss possible routes to its experimental production.

Improved Accuracy on Noisy Devices by Nonunitary Variational Quantum Eigensolver for Chemistry Applications.

We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named nonunitary Variational Quantum Eigensolver (nu-VQE), in which a nonunitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wave function ansatz. In the present work, as nonunitary operator, we use the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for which we obtain accurate ground-state energies with shallower circuits, at the cost of an increased number of measurements. Finally, we also show that this method achieves an important error mitigation effect that drastically improves the quality of the results for VQE optimizations on today's noisy quantum computers. The absolute error in the calculated energy within our scheme is 1 order of magnitude smaller than the corresponding result using traditional VQE methods, with the same circuit depth.

Chiral bosonic topological insulator on the honeycomb lattice with anisotropic interactions

We study hard-core bosons on the honeycomb lattice in the presence of anisotropic nearest-neighbor repulsive interactions. Using a quantum Monte Carlo (QMC) technique, we extract the phase diagram of the model in terms of the filling and the anisotropy. At half-filling we find a dimer insulator phase near maximum anisotropy that is characterized by a finite topological entanglement entropy $\ln(2)/2$, indicative of a fractional quantum Hall state for bosons. We identify the presence of edge states and derive a QMC-based method to extract and verify their chirality. Remarkably, this phase arises in the absence of magnetic flux and without explicit lattice frustration.

Determining quantum Monte Carlo simulability with geometric phases

Although stoquastic Hamiltonians are known to be simulable via sign-problem-free quantum Monte Carlo (QMC) techniques, the non-stoquasticity of a Hamiltonian does not necessarily imply the existence of a QMC sign problem. We give a sufficient and necessary condition for the QMC-simulability of Hamiltonians in a fixed basis in terms of geometric phases associated with the chordless cycles of the weighted graphs whose adjacency matrices are the Hamiltonians. We use our findings to provide a construction for non-stoquastic, yet sign-problem-free and hence QMC-simulable, quantum many-body models. We also demonstrate why the simulation of truly sign-problematic models using the QMC weights of the stoquasticized Hamiltonian is generally sub-optimal. We offer a superior alternative.

A Modern View of the Equation of State in Nuclear and Neutron Star Matter

Background: We analyze several constraints on the nuclear equation of state (EOS) currently available from neutron star (NS) observations and laboratory experiments and study the existence of possible correlations among properties of nuclear matter at saturation density with NS observables. Methods: We use a set of different models that include several phenomenological EOSs based on Skyrme and relativistic mean field models as well as microscopic calculations based on different many-body approaches, i.e., the (Dirac–)Brueckner–Hartree–Fock theories, Quantum Monte Carlo techniques, and the variational method. Results: We find that almost all the models considered are compatible with the laboratory constraints of the nuclear matter properties as well as with the largest NS mass observed up to now, 2.14−0.09+0.10M⊙ for the object PSR J0740+6620, and with the upper limit of the maximum mass of about 2.3–2.5M⊙ deduced from the analysis of the GW170817 NS merger event. Conclusion: Our study shows that whereas no correlation exists between the tidal deformability and the value of the nuclear symmetry energy at saturation for any value of the NS mass, very weak correlations seem to exist with the derivative of the nuclear symmetry energy and with the nuclear incompressibility.

Effective mass and pairing gap in neutron matter

Microscopic simulations of strongly correlated systems, such as neutron matter (NM), is restricted to a small number of particles due to various complexities. Thus, to understand the physics of infinite systems one has to consider the finite size effects (FSE) arising from these limitations on system size. These conference proceedings are centered around different ways of treating FSE in order to study NM. Specifically, we first present a preliminary study, at the mean field level and beyond, of FSE in the pairing gap of neutron superfluids. We also present a systematic model-independent extraction of the effective mass at various densities in NM using Quantum Monte Carlo (QMC) techniques.

The nature of the chemical bond in the dicarbon molecule.

The molecular dissociation energy has often been explained and discussed in terms of singlet bonds, formed by bounded pairs of valence electrons. In this work, we use a highly correlated resonating valence bond ansatz, providing a consistent paradigm for the chemical bond, where spin fluctuations are shown to play a crucial role. Spin fluctuations are known to be important in magnetic systems and correspond to the zero point motion of the spin waves emerging from a magnetic broken symmetry state. Within our ansatz, a satisfactory description of the carbon dimer is determined by the magnetic interaction of two carbon atoms with antiferromagnetically ordered S = 1 magnetic moments. This is a first step that, thanks to the highly scalable and efficient quantum Monte Carlo techniques, may open the door for understanding challenging complex systems containing atoms with large spins (e.g., transition metals).

Long-range interaction in an open boundary-driven Heisenberg spin lattice: A far-from-equilibrium transition to ballistic transport

We study an open Heisenberg XXZ spin chain with long-range Ising-type interaction, which is incoherently driven at its boundaries and therefore a far-from-equilibrium steady state current is induced. Quantum Monte Carlo techniques make a regime accessible, where the system size is large enough to examine the transition from a diffusive to a ballistic transport regime. We find that the chain lengths for this transition are increasing with decreasing range of the Ising-type interactions between distant spins, while the strength of the incoherent driving does not have a relevant effect on the transport transition. The transition can be explained by the suppression of ferromagnetic domains at the edges of the chain.