Variational Monte Carlo Approach Research Articles

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.

Low-lying quasiparticle excitations in strongly correlated superconductors: An ansatz from BCS quasiparticle excitations?

The question about the existence of Bogoliubov's quasiparticles in the BCS wave functions underneath Gutzwiller's projection is of importance to strongly correlated systems. We develop a method to examine the two-particle excitations of Gutzwiller-projected BCS wave functions by using the variational Monte Carlo approach. We find that the exact Gutzwiller-projected quasiparticle (GQP) dispersions are quantitatively reproduced by the Gutzwiller-projected Bogoliubov quasiparticles (GBQP) except the regions where d-wave Cooper pairing is strong. We believe GBQP provides a reasonable description to the low-energy excitations in strongly correlated superconducting systems because GBQP becomes more stable than GQP near the antinodes. Our framework offers a route to understand the intimate connection between Gutzwiller's projection and d-wave Cooper pairing.

Variational Monte Carlo approach to the two-dimensional Kondo lattice model

We study the phase diagram of the Kondo lattice model with nearest-neighbor hopping in the square lattice by means of the variational Monte Carlo technique. Specifically, we analyze a wide class of variational wave functions that allow magnetic and superconducting order parameters, so as to assess the possibility that superconductivity might emerge close to the magnetic instability, as is often observed in heavy-fermion systems. Indeed, we do find evidence of $d$-wave superconductivity in the paramagnetic sector, i.e., when magnetic order is not allowed in the variational wave function. However, when magnetism is allowed, it completely covers the superconducting region, which thus disappears from the phase diagram.

Stochastic extension of the Lanczos method for nuclear shell-model calculations with variational Monte Carlo method

We propose a new variational Monte Carlo (VMC) approach based on the Krylov subspace for large-scale shell-model calculations. A random walker in the VMC is formulated with the M-scheme representation, and samples a small number of configurations from a whole Hilbert space stochastically. This VMC framework is demonstrated in the shell-model calculations of 48Cr and 60Zn, and we discuss its relation to a small number of Lanczos iterations. By utilizing the wave function obtained by the conventional particle-hole-excitation truncation as an initial state, this VMC approach provides us with a sequence of systematically improved results.

Variational Monte Carlo with the multiscale entanglement renormalization ansatz

Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here we put forward a variational Monte Carlo approach for the multi-scale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with non-unitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just log(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two dimensional systems.

Variational Monte Carlo for spin-orbit interacting systems

Recently, a diffusion Monte Carlo algorithm was applied to the study of spin dependent interactions in condensed matter. Following some of the ideas presented therein, and applied to a Hamiltonian containing a Rashba-like interaction, a general variational Monte Carlo approach is here introduced that treats in an efficient and very accurate way the spin degrees of freedom in atoms when spin orbit effects are included in the Hamiltonian describing the electronic structure. We illustrate the algorithm on the evaluation of the spin-orbit splittings of isolated carbon and lead atoms. In the case of the carbon atom, we investigate the differences between the inclusion of spin-orbit in its realistic and effective spherically symmetrized forms. The method exhibits a very good accuracy in describing the small energy splittings, opening the way for a systematic quantum Monte Carlo studies of spin-orbit effects in atomic systems.

Nature of the Spin Liquid State of the Hubbard Model on a Honeycomb Lattice

Recent numerical work [Z. Y. Meng et al., Nature (London) 464, 847 (2010)] indicates the existence of a spin liquid (SL) phase that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum J(1)-J(2) spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model. We find three phases-antiferromagnetic, an unusual Z(2) SL state, and a dimerized state with spontaneously broken rotational symmetry. We identify the Z(2) SL state as the likely candidate for the SL phase of the Hubbard model.

Variational Monte Carlo Method with Dirichlet Boundary Conditions: Application to the Study of Confined Systems by Impenetrable Surfaces with Different Symmetries.

Variational Monte Carlo method is a powerful tool to determine approximate wave functions of atoms, molecules, and solids up to relatively large systems. In the present work, we extend the variational Monte Carlo approach to study confined systems. Important properties of the atoms, such as the spatial distribution of the electronic charge, the energy levels, or the filling of electronic shells, are modified under confinement. An expression of the energy very similar to the estimator used for free systems is derived. This opens the possibility to study confined systems with little changes in the solution of the corresponding free systems. This is illustrated by the study of helium atom in its ground state (1)S and the first (3)S excited state confined by spherical, cylindrical, and plane impenetrable surfaces. The average interelectronic distances are also calculated. They decrease in general when the confinement is stronger; however, it is seen that they present a minimum for excited states under confinement by open surfaces (cylindrical, planes) around the radii values corresponding to ionization. The ground (2)S and the first (2)P and (2)D excited states of the lithium atom are calculated under spherical constraints for different confinement radii. A crossing between the (2)S and (2)P states is observed around rc = 3 atomic units, illustrating the modification of the atomic energy level under confinement. Finally the carbon atom is studied in the spherical symmetry by using both variational and diffusion Monte Carlo methods. It is shown that the hybridized state sp(3) becomes lower in energy than the ground state (3)P due to a modification and a mixing of the atomic orbitals s, p under strong confinement. This result suggests a model, at least of pedagogical interest, to interpret the basic properties of carbon atom in chemistry.

Mechanism of superfluid-insulator transition in two-dimensional Bose Hubbard model

To understand the mechanism of Mott transitions in case of no magnetic influence, superfluid-insulator (Mott) transitions are studied for the S=0 Bose Hubbard model on the square lattice, using a variational Monte Carlo approach. In trial many-body wave functions, we introduce various types of attractive correlation factors between a doubly-occupied site (doublon, D) and an empty site (holon, H), which play a central role for the transition. We propose an improved picture of D–H binding; a Mott transition occurs when the D–H pair length becomes equivalent to the minimum D–D distance, which lengths are appropriately estimated. We confirm this picture is valid for all the wave functions with attractive D–H factors we consider, and point out it can be universal for nonmagnetic Mott transitions.

Knockout Reactions fromp-Shell Nuclei: Tests ofAb InitioStructure Models

Absolute cross sections have been determined following single neutron knockout reactions from 10Be and 10C at intermediate energy. Nucleon density distributions and bound-state wave function overlaps obtained from both variational Monte Carlo (VMC) and no core shell model (NCSM) ab initio calculations have been incorporated into the theoretical description of knockout reactions. Comparison to experimental cross sections demonstrates that the VMC approach, with the inclusion of 3-body forces, provides the best overall agreement while the NCSM and conventional shell-model calculations both overpredict the cross sections by 20% to 30% for 10Be and by 40% to 50% for 10C, respectively. This study gains new insight into the importance of 3-body forces and continuum effects in light nuclei and provides a sensitive technique to assess the accuracy of ab initio calculations for describing these effects.

Interaction-induced Fermi-surface renormalization in thet1−t2Hubbard model close to the Mott-Hubbard transition

We investigate the nature of the interaction-driven Mott-Hubbard transition of the half-filled $t_1{-}t_2$ Hubbard model in one dimension, using a full-fledged variational Monte Carlo approach including a distance-dependent Jastrow factor and backflow correlations. We present data for the evolution of the magnetic properties across the Mott-Hubbard transition and on the commensurate to incommensurate transition in the insulating state. Analyzing renormalized excitation spectra, we find that the Fermi surface renormalizes to perfect nesting right at the Mott-Hubbard transition in the insulating state, with a first-order reorganization when crossing into the conducting state.

Mechanism of formation of half-doped stripes in underdoped cuprates

Using a variational Monte-Carlo approach with a recently proposed stripe wave function, we showed that the strong correlation included in a t-J-type model has essentially all the necessary ingredients to form these stripes with modulations of charge density, spin magnetization, and pair field. If a perturbative effect of electron-phonon coupling to renormalize the effective mass or the hopping rate of holes is considered with the model, we find the half-doped stripes, which has on the average one half of a hole in one period of charge modulation, to be most stable, energetic wise in the underdoped region, $1/12\leq\delta\leq1/8$. This is in good agreement with the observation in the neutron scattering experiments. We also find long range Coulomb interaction to be less effective in the formation of half-doped stripes.

Variational study on the cluster glass wave function for the 2D extended t – J model

By using the variational Monte Carlo approach, we have demonstrated the existence of cluster glass states with a randomly-distributed short-ranged modulations is a natural outcome of the extended t – J models. As shown in our recent paper [1], these glassy states have almost the same energy as the uniform superconducting ground state if the modulation is moderate. Also we show that there are many possible degenerate states within a wide range of variational parameters. Because of the unusual degeneracy in the extended t – J models, random distribution of any defect or impurity in the materials could easily stabilize the glassy state. All these results obtained without introducing extra competing interactions or new order parameters seem to provide an appropriate framework to understand the experiments.

Coupled Quantum Dots as Two-Level Systems: A Variational Monte Carlo Approach

The electronic properties of two-dimensional double-quantum dots in the presence of external magnetic fields are investigated by a variational Monte Carlo method with s and s-p trial wavefunctions. We compute the exchange energy between two electrons as well as the two-electron total Coulomb energy for the singlet and triplet states in both strongly and weakly coupled quantum dots, and compare our data with the results of the numerically exact diagonalization of the Schrödinger Equation. In both systems, the singlet Coulomb energy decreases in magnetic fields as a consequence of magnetic localization, whereas the triplet Coulomb energy reaches a maximum value at intermediate magnetic fields before decreasing. Overall, good agreement between the two methods is obtained with s-p orbital trial wavefunction in strongly and weakly coupled quantum dots

Ionic dopants in He droplets: cluster energies from a variational and diffusion Monte Carlo approach

We present here an implementation of a combined variational and diffusion Monte Carlo (VMC and DMC) procedure that we have recently successfully used in describing the ‘solvation’ of molecular and atomic ions in small He clusters. We report the details of our numerical procedure which may use both pure Metropolis sampling and Biased (Langevin) sampling. We conclude by showing a series of test calculations we have performed both on atomic and molecular impurities.

Quantum Monte Carlo study of first-row atoms using transcorrelated variational Monte Carlo trial functions

The effect of using the transcorrelated variational Monte Carlo (TC-VMC) approach to construct a trial function for fixed node diffusion Monte Carlo (DMC) energy calculations has been investigated for the first-row atoms, Li to Ne. The computed energies are compared with fixed node DMC energies obtained using trial functions constructed from Hartree-Fock and density functional levels of theory. Despite major VMC energy improvement with TC-VMC trial functions, no improvement in DMC energy was observed using these trial functions for the first-row atoms studied. The implications of these results on the nodes of the trial wave functions are discussed.

A Variational Monte Carlo Approach to Atomic Structure

A simple version of the variational Monte Carlo method is applied to the electronic structure of atomic hydrogen, helium, lithium, and beryllium. For simplicity, the trial functions are taken as products of hydrogenic orbitals for which Z is treated as a variable parameter. The variationally optimized values of these parameters are interpreted as effective nuclear charges. These ab initio calculations are done using only spreadsheets that are straightforward enough for students to construct and use. The results are used to explicate several features of many electron atoms, including electron shielding in the ground state of helium, singlet�triplet splitting in the first excited state of helium, the difference in 2s and 2p penetration in lithium, and the trends in ionization energies for Be, Be+, and Be2+.