Variational Monte Carlo Approach Research Articles

Variational Monte Carlo approach to the SU(4) spin-orbital model on the triangular lattice

Variational Monte Carlo approach to the SU(4) spin-orbital model on the triangular lattice

Evaluation of the excitation spectra with diffusion Monte Carlo on an auxiliary bosonic ground state.

We aim to improve upon the variational Monte Carlo (VMC) approach for excitations replacing the Jastrow factor by an auxiliary bosonic (AB) ground state and multiplying it by a fermionic component factor. The instantaneous change in imaginary time of an arbitrary excitation in the original interacting fermionic system is obtained by measuring observables via the ground-state distribution of walkers of an AB system that is subject to an auxiliary effective potential. The effective potential is used to (i) drive the AB system's ground-state configuration space toward the configuration space of the excitations of the original fermionic system and (ii) subtract from a diffusion Monte Carlo (DMC) calculation contributions that can be included in conventional approximations, such as mean-field and configuration interaction (CI) methods. In this novel approach, the AB ground state is treated statistically in DMC, whereas the fermionic component of the original system is expanded in a basis. The excitation energies of the fermionic eigenstates are obtained by sampling a fermion-boson coupling term on the AB ground state. We show that this approach can take advantage of and correct for approximate eigenstates obtained via mean-field calculations or truncated interactions. We demonstrate that the AB ground-state factor incorporates the correlations missed by standard Jastrow factors, further reducing basis truncation errors. Relevant parts of the theory have been tested in soluble model systems and exhibit excellent agreement with exact analytical data and CI and VMC approaches. In particular, for limited basis set expansions and sufficient statistics, AB approaches outperform CI and VMC in terms of basis size for the same systems. The implementation of this method in current codes, despite being demanding, will be facilitated by reusing procedures already developed for calculating ground-state properties with DMC and excitations with VMC.

Variational Monte Carlo approach to partial differential equations with neural networks

The accurate numerical solution of partial differential equations (PDEs) is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here, we develop a variational approach for solving PDEs governing the evolution of high dimensional probability distributions. Our approach naturally works on the unbounded continuous domain and encodes the full probability density function through its variational parameters, which are adapted dynamically during the evolution to optimally reflect the dynamics of the density. In contrast to previous works, this dynamical adaptation of the parameters is carried out using an explicit prescription avoiding iterative gradient descent. For the considered benchmark cases we observe excellent agreement with numerical solutions as well as analytical solutions for tasks that are challenging for traditional computational approaches.

Autoregressive neural-network wavefunctions for ab initio quantum chemistry

In recent years, neural-network quantum states have emerged as powerful tools for the study of quantum many-body systems. Electronic structure calculations are one such canonical many-body problem that have attracted sustained research efforts spanning multiple decades, whilst only recently being attempted with neural-network quantum states. However, the complex non-local interactions and high sample complexity are substantial challenges that call for bespoke solutions. Here, we parameterize the electronic wavefunction with an autoregressive neural network that permits highly efficient and scalable sampling, whilst also embedding physical priors reflecting the structure of molecular systems without sacrificing expressibility. This allows us to perform electronic structure calculations on molecules with up to 30 spin orbitals—at least an order of magnitude more Slater determinants than previous applications of conventional neural-network quantum states—and we find that our ansatz can outperform the de facto gold-standard coupled-cluster methods even in the presence of strong quantum correlations. With a highly expressive neural network for which sampling is no longer a computational bottleneck, we conclude that the barriers to further scaling are not associated with the wavefunction ansatz itself, but rather are inherent to any variational Monte Carlo approach. To perform electronic structure calculations in quantum chemistry systems, methods are needed that are both accurate and scalable as the size of the molecule of interest increases. Barrett and colleagues employ an autoregressive neural-network ansatz that allows them to study larger molecules than previously attempted with neural-network quantum state approaches.

Two-orbital model for possible superconductivity pairing mechanism in nickelates

The newly synthesized strontium doped $R{\mathrm{NiO}}_{2}$ ($R$=Nd, Pr, and La) superconductors have stimulated extensive interests in understanding their pairing mechanism and pairing nature. Here we study the pairing mechanism in this family from a two-orbital model comprising the Ni- $3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and $3{d}_{xy}$ orbitals, equipped with extended Hubbard interactions and induced low-energy effective superexchange interactions. We then study the pairing symmetry in this system by using large scale variational Monte Carlo approach. Our results yield the intraorbital ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave singlet pairing as the leading pairing symmetry in the nickelates, which is analogous to the cuprates. However, there exist two important differences between the physical properties of the two families due to the fact that at the low Sr-doping regime, while the Ni-$3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbitals remain half-filled and singlyins occupied to form a Mott-insulating background, the Ni-$3{d}_{xy}$ orbitals accommodate nearly all the extra doped holes, which move freely on this background. The first difference lies in the single-particle aspect: while the $3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ degree of freedom remains Mott insulating with spectra weight pinned down at zero at low dopings, the $3{d}_{xy}$ one behaves as Fermi liquid with spectra weight near 1. The second difference lies in the pairing aspect: while the huge intra-$3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-orbital pairing gap is actually a pseudogap which has nothing to do with the SC, the small intra-$3{d}_{xy}$-orbital pairing gap serves as the true superconducting pairing gap, which is related to the ${T}_{c}$ via the BCS relation. Both differences can be verified by the angle-resolved photoemission spectrum.

Optimization of semi-empirical mass formula co-efficients using least square minimization and variational Monte-Carlo approaches

In this paper, both least squares minimization (LSM) and variational Monte-Carlo techniques have been implemented to determine the co-efficients of semi-empirical mass formula (SEMF). First, the experimental binding energies (BEs) are determined for all the available nuclei from atomic mass evaluation (AME2016) data. Then, LSM technique is implemented in Gnumeric worksheet to minimize relative mean squared error (RMSE) to obtain the SEMF co-efficients by considering only the first three co-efficients which are deduced from liquid drop model. The mean percentage error (MPE) value, between obtained BEs from the optimized co-efficients and the experimental BEs, has been determined. Then, to emphasize the relevance of empirical terms, they have been introduced successively one after other and the procedure has been repeated. A reduction in MPE-value has been observed after each iteration. This same procedure has also been employed using Monte-Carlo approach to obtain SEMF co-efficients by minimizing RMSE-value as in variational principle.

A variational Monte Carlo approach for core excitations.

We present a systematically improvable approach to core excitations in variational Monte Carlo. Building on recent work in excited-state-specific Monte Carlo, we show how a straightforward protocol, starting from a quantum chemistry guess, is able to capture core state's strong orbital relaxations, maintain accuracy in the near-nuclear region during these relaxations, and explicitly balance accuracy between ground and core excited states. In water, ammonia, and methane, which serve as prototypical representatives for oxygen, nitrogen, and carbon core states, respectively, this approach predicts core excitation energies within 0.3 eV of experiment and core excitation peak separations within 0.1 eV of experiment.

Helium-like atoms in magnetic fields: A variational Monte Carlo approach using compact trial functions

In the present manuscript the electronic structure of the helium-like atoms (nuclear charge Z = 2 - 5 ) is explored in the presence of a magnetic field B ≤ 7 a.u. The states of interest are the lowest states 1 1 ( 0 ) + and 1 3 ( - 1 ) + in this range of magnetic fields. A compact trial function with only eight variational parameters is designed following the arguments of physical relevance and the variational energy is calculated by employing the variational Monte Carlo method. The results are in good agreement compared to the previous literature. The critical magnetic field where the lowest states cross each other is estimated and it is found that they exhibit a quadratic behavior as a function of the nuclear charge.

Dynamical Structure Factor of the J1−J2 Heisenberg Model on the Triangular Lattice: Magnons, Spinons, and Gauge Fields

Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials. However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering. In this work, we investigate the low-energy properties of the $S=1/2$ Heisenberg model on the triangular lattice, including both nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ super-exchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models. For $J_2=0$, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at $K$ points (i.e., at the corners of the Brillouin zone). The strong renormalization of the magnon branch (also including roton-like minima around the $M$ points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a non-trivial magnetic $\pi$-flux threading half of the triangular plaquettes. When increasing the frustrating ratio $J_2/J_1$, we detect a progessive softening of the magnon branch at $M$, which eventually becomes gapless within the spin-liquid phase. This feature is captured by the band structure of the unprojected wave function (with $2$ Dirac points for each spin component). In addition, we observe an intense signal at low energies around the $K$ points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.

Variational and diffusion Monte Carlo simulations of a hydrogen molecular ion in a spherical box**Project supported by the National Key R&D Program of China (Grant No. 2018YFA0305900), the National Natural Science Foundation of China (Grant Nos. 51632002, 51572108, 91745203, 11634004, 11174102, and 1174121), the National Key Research and Development Program of China (Grant No. 2016YFB0201204), the Program for Changjiang Scholars

The variational and diffusion Monte Carlo approaches are used to study the ground-state properties of a hydrogen molecular ion in a spheroidal box. In this work, we successfully treat the zero-point motion of protons in the same formalism with as of electrons and avoid the Born–Oppenheimer approximation in density function theory. The study shows that the total energy increases with the decrease in volume, and that the distance between protons decreases as the pressure increases. Considering the motion of protons, the kinetic energy of the electron is higher than that of the fixed model under the same conditions and increases by 5%. The kinetic energy of the proton is found to be small under high pressure, which is only a fraction of the kinetic energy of the electron.

Two-hole ground state wavefunction: Non-BCS pairing in a t−J two-leg ladder

Superconductivity is usually described in the framework of the Bardeen-Cooper-Schrieffer (BCS) wavefunction, which even includes the resonating-valence-bond (RVB) wavefunction proposed for the high-temperature superconductivity in the cuprate. A natural question is \emph{if} any fundamental physics could be possibly missed by applying such a scheme to strongly correlated systems. Here we study the pairing wavefunction of two holes injected into a Mott insulator/antiferromagnet in a two-leg ladder using variational Monte Carlo (VMC) approach. By comparing with density matrix renormalization group (DMRG) calculation, we show that a conventional BCS or RVB pairing of the doped holes makes qualitatively wrong predictions and is incompatible with the fundamental pairing force in the $t$-$J$ model, which is kinetic-energy-driven by nature. By contrast, a non-BCS-like wavefunction incorporating such novel effect will result in a substantially enhanced pairing strength and improved ground state energy as compared to the DMRG results. We argue that the non-BCS form of such a new ground state wavefunction is essential to describe a doped Mott antiferromagnet at finite doping.

Magic v.3: An integrated software package for systematic structure-based coarse-graining

Molecular simulations of many phenomena related to biomolecular systems, soft matter and nanomaterials require consideration of length scales above 10 nm and time scales longer than 1μs, which necessitates the use of coarse-grained (low resolution) models, where each site of the model represents a group of atoms, and where the solvent is often omitted. Our software package MagiC is designed to perform systematic structure-based coarse-graining of molecular models, in which the effective pairwise potentials between coarse-grained sites of low-resolution molecular models are constructed to reproduce structural distribution functions obtained from modeling of systems in a high resolution (atomistic) description. The software takes as input atomistic trajectories generated by an external molecular dynamics package, and produce as an output interaction potentials for coarse-grained models which can be directly used in a coarse-grained simulations package. Here we present a major update (v.3) of the software with substantially improved functionality, compatibility with several major atomistic and coarse-grained simulations packages (GROMACS, LAMMPS, GALAMOST), analysis suite with graphical possibilities, diagnostics, documentation. We describe briefly the coarse-graining methodology, the structure of the software, describe users actions, and illustrate the whole process with two complex examples: cholesterol containing lipid bilayers and condensation of DNA caused by multivalent ions. Program summaryProgram Title: MagiCProgram Files doi:http://dx.doi.org/10.17632/9gnfxyshj8.1Licensing provisions: GPLv3Programming language: Fortran, PythonNature of problem: Systematic bottom-up multiscale modeling is a complex multi-stage process, in which results of simulations of a high-resolution (atomistic) model are used to construct a low resolution (coarse-grained) model, providing the same structural properties for the coarse-grained system as for the high-resolution system. Within the approach, structural properties of the high-resolution model are computed in terms of radial distribution functions and distributions of intramolecular degrees of freedom. Then the inverse problem is solved, in which the interaction potentials for the low-resolution model are determined which provide distribution functions coinciding with those obtained in the high-resolution simulations. The low-resolution model can be then used for simulations of the same system on larger length and time-scale.Solution method: The presented software package implements all stages of the systematic structure-based coarse-graining. It works as an integrated pipeline, giving the user ability to easily derive a coarse-grained model for a multicomponent complex molecular system and then use it for large-scale simulations. MagiC implements two approaches to solve the inverse problem: (i) the Inverse Monte Carlo method in which the inverse problem is solved using the Newton–Raphson method, with inversion of the Jacobian for the discretized relationship between interaction potentials and structural distribution functions; and (ii) the Iterative Boltzmann approach in which the inverse problem is solved using approximative relationships neglecting correlations between different degrees of freedom. The inverse solver includes also variational Inverse Monte Carlo approach when some of the coarse-grained potentials are fixed while others vary in order to fit the whole set of reference distribution functions.Additional comments: The MagiC main module can be also used for conventional Monte Carlo simulations of molecular systems described by tabulated pairwise potentials in the canonical ensemble.

Spectral signatures of fractionalization in the frustrated Heisenberg model on the square lattice

We employ a variational Monte Carlo approach to efficiently obtain the dynamical structure factor for the spin-1/2 $J_1-J_2$ Heisenberg model on the square lattice. Upon increasing the frustrating ratio $J_2/J_1$, the ground state undergoes a continuous transition from a N\'eel antiferromagnet to a $\mathbb{Z}_{2}$ gapless spin liquid. We identify the characteristic spectral features in both phases and highlight the existence of a broad continuum of excitations in the proximity of the spin-liquid phase. The magnon branch, which dominates the spectrum of the unfrustrated Heisenberg model, gradually loses its spectral weight, thus releasing nearly-deconfined spinons, whose signatures are visible even in the magnetically ordered state. Our results show how free spinons emerge across a quantum critical point, providing evidence for the fractionalization of magnons into deconfined spinons.

Atomization of correlated molecular-hydrogen chain: A fully microscopic variational Monte Carlo solution

We discuss electronic properties and their evolution for the linear chain of $H_2$ molecules in the presence of a uniform external force $f$ acting along the chain. The system is described by an extended Hubbard model within a fully microscopic approach. Explicitly, the microscopic parameters describing the intra- and inter-site Coulomb interactions are determined together with the hopping integrals by optimizing the system ground state energy and the single-particle wave functions in the correlated state. The many-body wave function is taken in the Jastrow form and the Variational Monte-Carlo (VMC) method is used in combination with an ab initio approach to determine the energy. Both the effective Bohr radii of the renormalized single-particle wave functions and the many-body wave function parameters are determined for each $f$. Hence, the evolution of the system can be analyzed in detail as a function of the equilibrium intermolecular distance, which in turn is determined for each $f$ value. The transition to the atomic state, including the Peierls distortion stability, can thus be studied in a systematic manner, particularly near the threshold of the dissociation of the molecular into atomic chain. The computational reliability of VMC approach is also estimated.

Unifying neural-network quantum states and correlator product states via tensor networks

Correlator product states (CPS) are a powerful and very broad class of states for quantum lattice systems whose (unnormalised) amplitudes in a fixed basis can be sampled exactly and efficiently. They work by gluing together states of overlapping clusters of sites on the lattice, called correlators. Recently Carleo and Troyer (2017 Science 355 602) introduced a new type sampleable ansatz called neural-network quantum states (NQS) that are inspired by the restricted Boltzmann model used in machine learning. By employing the formalism of tensor networks we show that NQS are a special form of CPS with novel properties. Diagramatically a number of simple observations become transparent. Namely, that NQS are CPS built from extensively sized GHZ-form correlators making them uniquely unbiased geometrically. The appearance of GHZ correlators also relates NQS to canonical polyadic decompositions of tensors. Another immediate implication of the NQS equivalence to CPS is that we are able to formulate exact NQS representations for a wide range of paradigmatic states, including superpositions of weighed-graph states, the Laughlin state, toric code states, and the resonating valence bond state. These examples reveal the potential of using higher dimensional hidden units and a second hidden layer in NQS. The major outlook of this study is the elevation of NQS to correlator operators allowing them to enhance conventional well-established variational Monte Carlo approaches for strongly correlated fermions.

Variational cluster approach to superconductivity and magnetism in the Kondo lattice model

We investigate in detail antiferromagnetic (AF) and superconducting (SC) phases as well as their coexistence in the two-dimensional Kondo lattice model on a square lattice, which is a paradigmatic model for heavy fermion materials. The results presented are mainly obtained using the variational cluster approximation (VCA) and are complemented by analytical findings for the equations of motion of pairing susceptibilities. A particularly interesting aspect is the possibility to have s-wave SC near half filling as reported by Bodensiek \textit{et al.} [Phys. Rev. Lett. \textbf{110}, 146406 (2013)]. When doping the system, we identify three regions which correspond to an AF metallic phase with small Fermi surface at weak coupling, an AF metal with a different Fermi surface topology at intermediate coupling, and a paramagnetic metal with a large Fermi surface at strong coupling. The transition between these two AF phases is found to be discontinuous at lower fillings, but turns to a continuous one when approaching half-filling. In the quest for s-wave superconductivity, only solutions are found which possess mean-field character. No true superconducting solutions caused by correlation effects are found in the s-wave channel. In contrast, we clearly identify robust d-wave pairing away from half-filling. However, we show that only by treating antiferromagnetism and superconductivity on equal footing artificial superconducting solutions at half-filling can be avoided. Our VCA findings support scenarios previously identified by variational Monte Carlo approaches and are a starting point for future investigations with VCA and further approaches such as cluster-embedding methods.

Projector Quantum MonteCarlo Method for Nonlinear Wave Functions.

We reformulate the projected imaginary-time evolution of the full configuration interaction quantum MonteCarlo method in terms of a Lagrangian minimization. This naturally leads to the admission of polynomial complex wave function parametrizations, circumventing the exponential scaling of the approach. While previously these functions have traditionally inhabited the domain of variational MonteCarlo approaches, we consider recent developments for the identification of deep-learning neural networks to optimize this Lagrangian, which can be written as a modification of the propagator for the wave function dynamics. We demonstrate this approach with a form of tensor network state, and use it to find solutions to the strongly correlated Hubbard model, as well as its application to a fully periodic abinitio graphene sheet. The number of variables which can be simultaneously optimized greatly exceeds alternative formulations of variational MonteCarlo methods, allowing for systematic improvability of the wave function flexibility towards exactness for a number of different forms, while blurring the line between traditional variational and projector quantum MonteCarlo approaches.

Spin liquid nature in the HeisenbergJ1−J2triangular antiferromagnet

We investigate the spin-$\frac{1}{2}$ Heisenberg model on the triangular lattice in the presence of nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ antiferromagnetic couplings. Motivated by recent findings from density-matrix renormalization group (DMRG) claiming the existence of a gapped spin liquid with signatures of spontaneously broken lattice point group symmetry [Zhu and White, Phys. Rev. B 92, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403 (2015)], we employ the variational Monte Carlo (VMC) approach to analyze the model from an alternative perspective that considers both magnetically ordered and paramagnetic trial states. We find a quantum paramagnet in the regime $0.08\lesssim J_2/J_1\lesssim 0.16$, framed by $120^{\circ}$ coplanar (stripe collinear) antiferromagnetic order for smaller (larger) $J_2/J_1$. By considering the optimization of spin-liquid wave functions of a different gauge group and lattice point group content as derived from Abrikosov mean-field theory, we obtain the gapless $U(1)$ Dirac spin liquid as the energetically most preferable state in comparison to all symmetric or nematic gapped $\mathbb{Z}_{2}$ spin liquids so far advocated by DMRG. Moreover, by the application of few Lanczos iterations, we find the energy to be the same as the DMRG result within error-bars. To further resolve the intriguing disagreement between VMC and DMRG, we complement our methodological approach by the pseudofermion functional renormalization group (PFFRG) to compare the spin structure factors for the paramagnetic regime calculated by VMC, DMRG, and PFFRG. This model promises to be an ideal test-bed for future numerical refinements in tracking the long-range correlations in frustrated magnets.

High-temperature superconductivity in the two-dimensional t–J model: Gutzwiller wavefunction solution

A systematic diagrammatic expansion for Gutzwiller wavefunctions (DE-GWFs) proposed very recently is used for the description of the superconducting (SC) ground state in the two-dimensional square-lattice t–J model with the hopping electron amplitudes t (and ) between nearest (and next-nearest) neighbors. For the example of the SC state analysis we provide a detailed comparison of the methodʼs results with those of other approaches. Namely, (i) the truncated DE-GWF method reproduces the variational Monte Carlo (VMC) results and (ii) in the lowest (zeroth) order of the expansion the method can reproduce the analytical results of the standard Gutzwiller approximation (GA), as well as of the recently proposed ‘grand-canonical Gutzwiller approximation’ (called either GCGA or SGA). We obtain important features of the SC state. First, the SC gap at the Fermi surface resembles a wave only for optimally and overdoped systems, being diminished in the antinodal regions for the underdoped case in a qualitative agreement with experiment. Corrections to the gap structure are shown to arise from the longer range of the real-space pairing. Second, the nodal Fermi velocity is almost constant as a function of doping and agrees semi-quantitatively with experimental results. Third, we compare the doping dependence of the gap magnitude with experimental data. Fourth, we analyze the k-space properties of the model: Fermi surface topology and effective dispersion. The DE-GWF method opens up new perspectives for studying strongly correlated systems, as it (i) works in the thermodynamic limit, (ii) is comparable in accuracy to VMC, and (iii) has numerical complexity comparable to that of the GA (i.e., it provides the results much faster than the VMC approach).

Tetramerization in a SU(4) Heisenberg model on the honeycomb lattice

The SU(4) Heisenberg model can serve as a low-energy model of the Mott insulating state in materials where the spins and orbitals are highly symmetric or in systems of alkaline-earth atoms on an optical lattice. Recently, it has been argued that on the honeycomb lattice, the model exhibits a unique spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the algebraic spin-orbital liquid toward the spontaneous formation of SU(4) singlet plaquettes (tetramerization). Using a variational Monte Carlo approach to evaluate the projected wave function of fermions with $\ensuremath{\pi}$-flux state, we find that the algebraic liquid is robust and that a small finite value of the next-nearest exchange is needed to induce tetramerization. We also studied the phase diagram of a model which interpolates between the nearest-neighbor Heisenberg model and a Hamiltonian for which the singlet-plaquette-product state is an exact ground state.