Variational Monte Carlo Techniques Research Articles

Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation

Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. We employ time-dependent Jastrow factors and backflow transformations, enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we employ the time-dependent variational Monte Carlo technique and introduce a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately describe the time evolution, providing insight into quantum dynamical effects in interacting electronic systems, beyond the capabilities of mean-field.

Variational MonteCarlo Study of the 1/9-Magnetization Plateau in Kagome Antiferromagnets.

Motivated by very recent experimental observations of the 1/9-magnetization plateaus in YCu_{3}(OH)_{6+x}Br_{3-x} and YCu_{3}(OD)_{6+x}Br_{3-x}, our study delves into the magnetic-field-induced phase transitions in the nearest-neighbor antiferromagnetic Heisenberg model on the kagome lattice using the variational MonteCarlo technique. We uncover a phase transition from a zero-field Dirac spin liquid to a field-induced magnetically disordered phase that exhibits the 1/9-magnetization plateau. Through a comprehensive analysis encompassing the magnetization distribution, spin correlations, chiral order parameter, topological entanglement entropy, ground-state degeneracy, Chern number, and excitation spectrum, we pinpoint the phase associated with this magnetization plateau as a chiral Z_{3} topological quantum spin liquid and elucidate its diverse physical properties.

Efficient solutions of fermionic systems using artificial neural networks

In this study, we explore the similarities and differences between variational Monte Carlo techniques that employ conventional and artificial neural network representations of the ground-state wave function for fermionic systems. Our primary focus is on shallow neural network architectures, specifically the restricted Boltzmann machine, and we examine unsupervised learning algorithms that are appropriate for modeling complex many-body correlations. We assess the advantages and drawbacks of conventional and neural network wave functions by applying them to a range of circular quantum dot systems. Our findings, which include results for systems containing up to 90 electrons, emphasize the efficient implementation of these methods on both homogeneous and heterogeneous high-performance computing facilities.

Static and dynamical signatures of Dzyaloshinskii-Moriya interactions in the Heisenberg model on the kagome lattice

Motivated by recent experiments on Cs_22Cu_33SnF_{12}12 and YCu_{3}3(OH)_{6}6Cl_{3}3, we consider the {S=1/2}S=1/2 Heisenberg model on the kagome lattice with nearest-neighbor super-exchange JJ and (out-of-plane) Dzyaloshinskii-Moriya interaction J_DJD, which favors (in-plane) {Q=(0,0)}Q=(0,0) magnetic order. By using both variational Monte Carlo and tensor-network approaches, we show that the ground state develops a finite magnetization for J_D/J \gtrsim 0.03 \mathrm{-} 0.04JD/J≳0.03−0.04; instead, for smaller values of the Dzyaloshinskii-Moriya interaction, the ground state has no magnetic order and, according to the fermionic wave function, develops a gap in the spinon spectrum, which vanishes for J_D \to 0JD→0. The small value of J_D/JJD/J for the onset of magnetic order is particularly relevant for the interpretation of low-temperature behaviors of kagome antiferromagnets, including ZnCu_{3}3(OH)_{6}6Cl_{2}2. For this reason, we assess the spin dynamical structure factor and the corresponding low-energy spectrum, by using the variational Monte Carlo technique. The existence of a continuum of excitations above the magnon modes is observed within the magnetically ordered phase, with a broad peak above the lowest-energy magnons, similarly to what has been detected by inelastic neutron scattering on Cs_{2}2Cu_{3}3SnF_{12}12.

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<0t′/t<0, the stripe with \lambda=5λ=5 is found to be remarkably stable, while for t^\prime/t>0t′/t>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.

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.

Activities Based on Empirical Data to Determine Coefficients in Semi-Empirical Mass Formula

In this paper, we propose activities based on empirical data to arrive at Coulomb, asymmetry, and pairing energy coefficients in the semi-empirical mass formula (SEMF). The Coloumb energy co-efficient is obtained using mirror nuclei data. Asymmetry energy co-efficient is determined from mass parabolas. The concept of neutron separation energies is used to arrive at the pairing energy co-efficient and its dependence with regards to the power of mass number A. Using these three obtained coefficients in SEMF for binding energies (BE), a plot of BE/A vs A−1/3 has resulted in Volume and surface energy coefficients using linear regression. It has found that the five coefficients obtained using these activities are closely matching with those obtained using Least Square Minimization (LSM) and Variational Monte Carlo (VMC) techniques. The mean percentage error (MPE), calculated between obtained binding energies from the optimized coefficients and the experimental binding energies, using the three approaches are almost equal. These activities enrich the understanding of students by laying importance on the physics involved, as they confirm that alternative strategies from experimental data are able to match those obtained using mathematical optimization procedures such as LSM and VMC.

Triton Scattering Phase-Shifts for S-wave using Morse Potential

In this paper, the phase-shifts for neutron-dueteron (n-d) scattering have been determined using the molecular Morse potential as theoretical model of interaction. The Triton (n-d) 2S1/2 ground state initially has been chosen as -7.61 MeV to determine the model parameters using variational Monte-Carlo technique in combination with matrix methods numerical approach to solving the time independent Schrodinger equation (TISE). The obtained potential is incorporated into the phase function equation, which is solved using Runge-Kutta (RK) 4,5 order technique, to calculate the phaseshifts at various lab energies below 15 MeV, for which experimental data is available. The results have been compared with those obtained using another molecular potential named Manning-Rosen (MR) and have been observed to fare better. Finally, the Triton ground state has been chosen as its binding energy (BE), given by -8.481795 MeV, as determined from experimental atomic mass evaluation data and the calculations are repeated. It has been found that these phase-shifts from BE data are slightly better matched with experimental ones as compared to those obtained using -7.61 MeV ground state for Triton (n-d two-body system) modeled using Morse potential.

Numerical solution of Schrodinger equation for rotating Morse potential using matrix methods with Fourier sine basis and optimization using variational Monte‐Carlo approach

AbstractIn this paper, time independent Schrodinger equation (TISE) for rotating Morse potential is solved numerically using matrix mechanics approach with Fourier sine basis, to obtain the ro‐vibrational spectra for diatomic molecules, here for HCl. Alongside, an optimization procedure for extraction of best model parameters based on variational Monte‐Carlo technique (VMC) is implemented. The first three vibrational frequencies from simulation are compared with experimental data by calculating mean‐square error, called χ2‐value. Minimizing χ2 by VMC technique is akin to extracting inverted potential based on Morse function that best fits experimental data. Finally, these optimized parameters for Morse potential, along with centrifugal term, are utilized in TISE to obtain ro‐vibrational frequencies for HCl. It has been found that mean % error obtained for ro‐vibrational lines from VMC are two orders of magnitude smaller than those obtained from best parameter fits using multiple regression analysis. The programs have been implemented in Scilab, and matrix methods results have been validated with those obtained using analytical solutions from Nikiforov–Uvarov method and asymptotic iteration method.

Electronic Structure of Nanoclusters by Quantum Monte Carlo Methods

The valence electron binding energies of the atomic clusters $$\hbox {XAl}_3^-$$ (X = Si, Ge, and Sn) and $$\hbox {YAl}_4^-$$ (Y = Li, Na, and Cu) are investigated using a combination of the fixed-node diffusion quantum Monte Carlo method (FN-DMC), the density functional theory, and the Hartree–Fock approximation. A brief review of the used theoretical approaches is presented with emphasis on the variational and diffusion Monte Carlo techniques and their applications to study the electronic structures of atomic clusters. The obtained results for the vertical detachment energies (VDE) from the FN-DMC are in excellent agreement with available experimental photoelectron spectroscopy data. A comparison between the FN-DMC results and HF ones allows to quantify the electron correlation effects and their impact on the stability of the clusters. The analysis reveals that the electron correlation enhances the VDE of the atomic clusters significantly and plays an essential role in their stability.

Dynamical properties of Néel and valence-bond phases in the J1–J2 model on the honeycomb lattice

By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic S = 1/2 Heisenberg model on the honeycomb lattice, in presence of first-neighbor (J1) and second-neighbor (J2) couplings, for J2 < 0.5J1. The ground state of the system shows long-range antiferromagnetic order for J2/J1 ≲ 0.23 (Néel phase), plaquette valence-bond order for 0.23 ≲ J2/J1 ≲ 0.36, and columnar dimer order for J2/J1 ≳ 0.36. Within the Néel phase, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for J2 = 0. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the K point (i.e., the corner of the Brillouin zone). This mode softens when J2/J1 is increased and becomes gapless at the transition point, J2/J1 ≈ 0.23. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of J2/J1, valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.

Exact restoration of Galilei invariance in density functional calculations with quantum Monte Carlo

Galilean invariance is usually violated in self-consistent mean-field calculations that employ effective density-dependent nuclear forces. We present a novel approach, based on variational quantum Monte Carlo techniques, suitable to preserve this symmetry and assess the effect of its violation, seldom attempted in the past. To this aim, we generalize the linear optimization method to encompass the density-dependence of effective Hamiltonians, and study 4He, 16O, and 40Ca ground-state properties employing the Gogny interaction.

Pomeranchuk Instability of Composite Fermi Liquids.

Nematicity in quantum Hall systems has been experimentally well established at excited Landau levels. The mechanism of the symmetry breaking, however, is still unknown. Pomeranchuk instability of Fermi liquid parameter F_{ℓ}≤-1 in the angular momentum ℓ=2 channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational MonteCarlo technique, Fermi liquid parameters F_{ℓ} of the composite fermion Fermi liquid with a finite layer width. We consider F_{ℓ} in different Landau levels n=0, 1, 2 as a function of layer width parameter η. We find that unlike the lowest Landau level, which shows no sign of Pomeranchuk instability, higher Landau levels show nematic instability below critical values of η. Furthermore, the critical value η_{c} is higher for the n=2 Landau level, which is consistent with observation of nematic order in ambient conditions only in the n=2 Landau levels. The picture emerging from our work is that approaching the true 2D limit brings half-filled higher Landau-level systems to the brink of nematic Pomeranchuk instability.

Superconductivity, charge-density waves, antiferromagnetism, and phase separation in the Hubbard-Holstein model

By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping $t$, the on-site electron-electron interaction $U$, the phonon energy $\omega_0$, and the electron-phonon coupling $g$. At half filling, the ground state is an antiferromagnetic insulator for $U \gtrsim 2g^2/\omega_0$, while it is a charge-density-wave (or bi-polaronic) insulator for $U \lesssim 2g^2/\omega_0$. In addition to these phases, we find a superconducting phase that intrudes between them. For $\omega_0/t=1$, superconductivity emerges when both $U/t$ and $2g^2/t\omega_0$ are small; then, by increasing the value of the phonon energy $\omega_0$, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.

Charge orders in organic charge-transfer salts

Motivated by recent experimental suggestions of charge-order-driven ferroelectricity in organic charge-transfer salts, such as κ-(BEDT-TTF)2Cu[N(CN)2]Cl, we investigate magnetic and charge-ordered phases that emerge in an extended two-orbital Hubbard model on the anisotropic triangular lattice at 3/4 filling. This model takes into account the presence of two organic BEDT-TTF molecules, which form a dimer on each site of the lattice, and includes short-range intramolecular and intermolecular interactions and hoppings. By using variational wave functions and quantum Monte Carlo techniques, we find two polar states with charge disproportionation inside the dimer, hinting to ferroelectricity. These charge-ordered insulating phases are stabilized in the strongly correlated limit and their actual charge pattern is determined by the relative strength of intradimer to interdimer couplings. Our results suggest that ferroelectricity is not driven by magnetism, since these polar phases can be stabilized also without antiferromagnetic order and provide a possible microscopic explanation of the experimental observations. In addition, a conventional dimer-Mott state (with uniform density and antiferromagnetic order) and a nonpolar charge-ordered state (with charge-rich and charge-poor dimers forming a checkerboard pattern) can be stabilized in the strong-coupling regime. Finally, when electron–electron interactions are weak, metallic states appear, with either uniform charge distribution or a peculiar 12-site periodicity that generates honeycomb-like charge order.

Excited states of helium atom in a strong magnetic field using variational Monte Carlo technique

Variational Monte Carlo method was employed to investigate the total energies of the excited states \(2^{1} \left( 0 \right)^{ + }\), \(3^{1} \left( 0 \right)^{ + }\), \(1^{1} \left( 0 \right)^{ - }\), \(1^{3} \left( 0 \right)^{ + }\), \(1^{3} \left( 0 \right)^{ - }\), \(2^{3} \left( 0 \right)^{ + }\), \(1^{1} \left( { - 1} \right)^{ + }\) and \(1^{3} \left( { - 1} \right)^{ + }\) of the helium atom. Strong magnetic field effect on the energy is investigated and illustrated graphically. The point of transition from the ground state to \(1^{3} \left( { - 1} \right)^{ + }\) state was obtained and comparisons with previous works were introduced. The results concluded comply with the most recent, accurate and the exact values.

Competition between spin liquids and valence-bond order in the frustrated spin- 12 Heisenberg model on the honeycomb lattice

Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor $J_1$ and second-neighbor $J_2$ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio $J_2/J_1$, we find strong evidence for a continuous transition to a nonmagnetic phase at $J_2/J_1 \approx 0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence bond state gives an excellent approximation to the exact ground-state energy. For $0.23 \lesssim J_2/J_1 \lesssim 0.4$, a gapless $Z_2$ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for $J_2/J_1 \gtrsim 0.3$. Finally, for $0.36 \lesssim J_2/J_1 \le 0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.

Quantum spin liquid in a breathing kagome lattice

Motivated by recent experiments on the vanadium oxyfluoride material DQVOF, we examine possible spin liquid phases on a breathing kagome lattice of S=1/2 spins. By performing a projective symmetry group analysis, we determine the possible phases for both fermionic and bosonic $\mathbb{Z}_2$ spin liquids on this lattice, and establish the correspondence between the two. The nature of the ground state of the Heisenberg model on the isotropic kagome lattice is a hotly debated topic, with both $\mathbb{Z}_2$ and U(1) spin liquids argued to be plausible ground states. Using variational Monte Carlo techniques, we show that a gapped $\mathbb{Z}_2$ spin liquid emerges as the clear ground state in the presence of this breathing anisotropy. Our results suggest that the breathing anisotropy helps to stabilize this spin liquid ground state, which may aid us in understanding the results of experiments and help to direct future numerical studies on these systems.

Variational wave functions for theS=12Heisenberg model on the anisotropic triangular lattice: Spin liquids and spiral orders

By using variational wave functions and quantum Monte Carlo techniques, we investigate the complete phase diagram of the Heisenberg model on the anisotropic triangular lattice, where two out of three bonds have super-exchange couplings $J$ and the third one has instead $J^\prime$. This model interpolates between the square lattice and the isotropic triangular one, for $J^\prime/J \le 1$, and between the isotropic triangular lattice and a set of decoupled chains, for $J/J^\prime \le 1$. We consider all the fully-symmetric spin liquids that can be constructed with the fermionic projective-symmetry group classification [Y. Zhou and X.-G. Wen, arXiv:cond-mat/0210662] and we compare them with the spiral magnetic orders that can be accommodated on finite clusters. Our results show that, for $J^\prime/J \le 1$, the phase diagram is dominated by magnetic orderings, even though a spin-liquid state may be possible in a small parameter window, i.e., $0.7 \lesssim J^\prime/J \lesssim 0.8$. In contrast, for $J/J^\prime \le 1$, a large spin-liquid region appears close to the limit of decoupled chains, i.e., for $J/J^\prime \lesssim 0.6$, while magnetically ordered phases with spiral order are stabilized close to the isotropic point.

Variational Monte Carlo study of a gapless spin liquid in the spin-12XXZ antiferromagnetic model on the kagome lattice

By using the variational Monte Carlo technique, we study the spin-$1/2$ XXZ antiferromagnetic model (with easy-plane anisotropy) on the kagome lattice. A class of Gutzwiller projected fermionic states with a spin Jastrow factor is considered to describe either spin liquids (with $U(1)$ or $Z_2$ symmetry) or magnetically ordered phases (with ${\bf q}=(0,0)$ or ${\bf q}=(4\pi/3,0)$). We find that the magnetic states are not stable in the thermodynamic limit. Moreover, there is no energy gain to break the gauge symmetry from $U(1)$ to $Z_2$ within the spin-liquid states, as previously found in the Heisenberg model. The best variational wave function is therefore the $U(1)$ Dirac state, supplemented by the spin Jastrow factor. Furthermore, a vanishing $S=2$ spin gap is obtained at the variational level, in the whole regime from the $XY$ to the Heisenberg model.