Shadow Wave Function Research Articles

Massive celestial fermions

In an effort to further the study of amplitudes in the celestial CFT (CCFT), we construct conformal primary wavefunctions for massive fermions. Upon explicitly calculating the wavefunctions for Dirac fermions, we deduce the corresponding transformation of momentum space amplitudes to celestial amplitudes. The shadow wavefunctions are shown to have opposite spin and conformal dimension 2 − ∆. The Dirac conformal primary wave- functions are delta function normalizable with respect to the Dirac inner product provided they lie on the principal series with conformal dimension ∆ = 1 + iλ for λ ∈ ℝ. It is shown that there are two choices of a complete basis: single spin J=frac{1}{2} or J=-frac{1}{2} and λ ∈ ℝ or multiple spin J=pm frac{1}{2} and λ ∈ ℝ+∪0. The massless limit of the Dirac conformal primary wavefunctions is shown to agree with previous literature. The momentum generators on the celestial sphere are derived and, along with the Lorentz generators, form a representation of the Poincaré algebra. Finally, we show that the massive spin-1 conformal primary wavefunctions can be constructed from the Dirac conformal primary wavefunctions using the standard Clebsch-Gordan coefficients. We use this procedure to write the massive spin- frac{3}{2} , Rarita-Schwinger, conformal primary wavefunctions. This provides a prescription for constructing all massive fermionic and bosonic conformal primary wavefunctions starting from spin- frac{1}{2} .

Metal-Insulator Transition of Solid Hydrogen by the Antisymmetric Shadow Wave Function

Abstract We revisit the pressure-induced molecular-atomic metal-insulator transition of solid hydrogen by means of variational quantum Monte Carlo simulations based on the antisymmetric shadow wave function. For the purpose of facilitating the study of the electronic structure of large-scale fermionic systems, the shadow wave function formalism is extended by a series of technical advancements as implemented in our HswfQMC code. Among others, these improvements include a revised optimization method for the employed shadow wave function and an enhanced treatment of periodic systems with long-range interactions. It is found that the superior accuracy of the antisymmetric shadow wave function results in a significantly increased transition pressure with respect to previous theoretical estimates.

Nonlinear Network Description for Many-Body Quantum Systems in Continuous Space.

We show that the recently introduced iterative backflow wave function can be interpreted as a general neural network in continuum space with nonlinear functions in the hidden units. Using this wave function in variational MonteCarlo simulations of liquid ^{4}He in two and three dimensions, we typically find a tenfold increase in accuracy over currently used wave functions. Furthermore, subsequent stages of the iteration procedure define a set of increasingly good wave functions, each with its own variational energy and variance of the local energy: extrapolation to zero variance gives energies in close agreement with the exact values. For two dimensional ^{4}He, we also show that the iterative backflow wave function can describe both the liquid and the solid phase with the same functional form-a feature shared with the shadow wave function, but now joined by much higher accuracy. We also achieve significant progress for liquid ^{3}He in three dimensions, improving previous variational and fixed-node energies.

Variational Monte Carlo method for the Baeriswyl wave function: Application to the one-dimensional bosonic Hubbard model

A variational Monte Carlo method for bosonic lattice models is introduced. The method is based on the Baeriswyl projected wavefunction. The Baeriswyl wavefunction consists of a kinetic energy based projection applied to the wavefunction at infinite interaction, and is related to the shadow wavefunction already used in the study of continuous models of bosons. The wavefunction at infinite interaction, and the projector, are represented in coordinate space, leading to an expression for expectation values which can be evaluated via Monte Carlo sampling. We calculate the phase diagram and other properties of the bosonic Hubbard model. The calculated phase diagram is in excellent agreement with known quantum Monte Carlo results. We also analyze correlation functions.

Fermion sign problem in imaginary-time projection continuum quantum Monte Carlo with local interaction.

We use the shadow wave function formalism as a convenient model to study the fermion sign problem affecting all projector quantum Monte Carlo methods in continuum space. We demonstrate that the efficiency of imaginary-time projection algorithms decays exponentially with increasing number of particles and/or imaginary-time propagation. Moreover, we derive an analytical expression that connects the localization of the system with the magnitude of the sign problem, illustrating this behavior through numerical results. Finally, we discuss the computational complexity of the fermion sign problem and methods for alleviating its severity.

On fermionic shadow wave functions for strongly correlated multi-reference systems based on a single Slater determinant

We demonstrate that extending the shadow wave function to fermionic systems facilitates to accurately calculate strongly correlated multi-reference systems such as the stretched molecule. This development considerably extends the scope of electronic-structure calculations and enables to efficiently recover the static correlation energy using just a single Slater determinant.

Sign problem of the fermionic shadow wave function

We present a whole series of methods to alleviate the sign problem of the fermionic shadow wave function in the context of variational Monte Carlo. The effectiveness of our techniques is demonstrated on liquid ^{3}He. We found that although the variance is reduced, the gain in efficiency is restricted by the increased computational cost. Yet, this development not only extends the scope of the fermionic shadow wave function, but also facilitates highly accurate quantum Monte Carlo simulations previously thought not feasible.

Long-range correlations in quantum solids

Nearly 40 years after the admirable paper [Phys. Rev. 155, 88 (1967)] in which Reatto and Chester suggested a ground state wave function for liquid 4He capturing long-range correlations related to the zero-point motion of compressional waves (longitudinal phonons), we investigate the possibility of mimicking their approach in the case of solid 4He. Starting from elasticity theory, via a ‘quantization’ procedure, we propose a way to describe correlations among particles related to transverse long wavelength oscillations of the crystal lattice. The idea of exploring new kinds of long-range correlations in variational models arises from the evidence that some correlations are missing even in the most accurate models, like the Shadow Wave Function, which are not able to describe the long distance behaviour of the one-body density matrix in the solid phase computed in a perfect crystal: no variational model of the ground state is known which is able to reproduce qualitatively the long-range behaviour of this function, which turns out to be an exponential decaying function of the distance when computed via exact finite temperature and zero temperature Path Integral methods. In this paper we derive a functional form for the contribution of the zero-point motion of transverse phonons to the ground state microscopic wave function. Such an approach opens many possibilities of defining variational models, and we propose a first attempt, consisting of a modified Shadow Wave Function.

Zero-point divacancy concentration in the shadow wave function model for solidH4e

We address the issue of interaction between zero-point vacancies in solid 4He as described within the shadow wave-function model. Applying the reversible-work method and taking into account finite-size effects, we obtain a zero-point monovacancy concentration of (2.03 +- 0.02) 10^{-3}, which is slightly higher than the result due to Reatto et al. for the same model. Utilizing the same methodology, we then consider the divacancy, taking into account both the in-plane as well as out-of-plane configurations with respect to the basal plane. We find no significant anisotropy between both conformation. Furthermore, although there is a small binding tendency, the expected divacancy concentration is only ~4-5 times larger than the value expected in the absence of any clustering propensity, 2.5 10^{-5}. This result suggests that, within the employed model description, no vacancy aggregation leading to phase separation is to be expected in the ground state.

Exact ground state Monte Carlo method for Bosons without importance sampling

Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

Fermionic Shadow Wave Function Variational Calculations of the Vacancy Formation Energy inHe3

We present a novel technique well suited for studying the ground state of inhomogeneous fermionic matter in a wide range of different systems. The system is described using a fermionic shadow wave function, and the energy is computed by means of the variational Monte Carlo technique. The general form of the fermionic shadow wave function is useful for describing many-body systems with the coexistence of different phases as well in the presence of defects or impurities, but it requires overcoming a significant sign problem. As an application, we studied the energy to activate vacancies in solid 3He.

Zero-Point Vacancies in Quantum Solids

A Jastrow wave function (JWF) and a shadow wave function (SWF) describe a quantum solid with Bose–Einstein condensate; i.e. a supersolid. It is known that both JWF and SWF describe a quantum solid with also a finite equilibrium concentration of vacancies x v . We outline a route for estimating x v by exploiting the existing formal equivalence between the absolute square of the ground state wave function and the Boltzmann weight of a classical solid. We compute x v for the quantum solids described by JWF and SWF employing very accurate numerical techniques. For JWF we find a very small value for the zero point vacancy concentration, x v =(1.4±0.1)×10−6. For SWF, which presently gives the best variational energy of solid 4He, we find the significantly larger value x v =(1.4±0.1)×10−3 at a density close to melting. We also study two and three vacancies with SWF. We find that there is a strong short range attraction but the vacancies do not form a bound state, at variance with the exact finite temperature PIMC results.

Path-integral ground-state Monte Carlo study of two-dimensional solidHe4

We have studied a two-dimensional (2D) triangular commensurate crystal of $^{4}\mathrm{He}$ with the exact $T=0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ path-integral ground-state (PIGS) Monte Carlo method. We have projected onto the true ground state two qualitatively different wave functions, a Jastrow-Nosanow wave function and a translationally invariant shadow wave function. The PIGS method passes this hard test of validity and applicability by obtaining the convergence to the same properties, both the diagonal ones as well as the off-diagonal one-body density matrix ${\ensuremath{\rho}}_{1}$. Thus, the commensurate 2D $^{4}\mathrm{He}$ crystal at $T=0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ is exactly solved, we find no Bose-Einstein condensation (BEC), and ${\ensuremath{\rho}}_{1}$ shows a dominant exponential decay in the large distance range. The structure found in ${\ensuremath{\rho}}_{1}$ is due to virtual vacancy-interstitial pairs and this shows up in the momentum distribution. Our result indicates that BEC in 2D solid $^{4}\mathrm{He}$ can only arise in the presence of some kind of disorder, either intrinsic or extrinsic.

Distortion effects and clustering of isotopic impurities in solid molecularpara-hydrogen from variational Monte Carlo simulations with shadow wave functions

We employed a fully optimized Shadow Wave Function (SWF) in combination with Variational Monte Carlo techniques to investigate the properties of HD molecules and molecular ortho-deuterium (o-D_2) in bulk solid para-hydrogen (p-H_2). Calculations were performed for different concentrations of impurities ranging from about 1% to 25% at the equilibrium density for the para-hydrogen crystal. By computing the excess energy both for clustered and isolated impurities we tried to determine a limit for the solubility of HD and o-D_2 in p-H_2.

Effect of vacancies on the structure of solid molecular parahydrogen studied with variational Monte Carlo simulations

We performed variational Monte Carlo simulations with a shadow wave function to study the effects of vacancies on the structure of solid molecular $p\text{\ensuremath{-}}{\mathrm{H}}_{2}$. In a quantum crystal, vacancies tend to modify the local order in the crystal. The main mechanism is due to the fact that molecules tend to occupy the empty space in order to lower the kinetic energy. While this effect is expected to be more important at lower than at higher densities, we find that in the latter case the local order is even more affected by the presence of a vacancy. Results on the occurrence of quantum diffusion of the vacancy in the solid are also presented.

Pressurized 4He in Cylindrical and in Hexagonal Pores

We give a variational Monte Carlo description at T = 0 K of 4He filling under pressure a porous glass within the shadow wave function technique. We have considered as confining media two different smooth pores, one with a circular cross-section of radius R = 13 A resembling a Gelsil pore, and the other with a hexagonal cross–section of side S = 14 A resembling a FSM-16 pore. In all the studied cases the density profiles show a strong layering of the 4He atoms. As the density is increased, solidification takes place layer by layer, starting from the pore wall. Computing the one-body density matrix we are able to estimate the Bose–Einstein condensate fraction, which is still non–zero even when the whole system is in the solid phase.

BOSE–EINSTEIN CONDENSATION IN BULK AND CONFINED SOLID HELIUM

We address the question if the ground state of solid 4 He has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). We point out that energy computation from simulation as performed by now cannot be used to decide this question and that the presently best variational wave function, a shadow wave function, gives an incommensurate state. We have extended the calculation of the one–body density matrix ρ1 to the exact Shadow Path Integral Ground State method. Calculation of ρ1 at ρ = 0.031 Å-3 shows that Vacancy–Interstitial pair processes are present also in the exact computation but the simulated system size is too small to infer the presence of off–diagonal long range order. Variational simulations of 4 He confined in a narrow cylindrical pore are also discussed.

Bose-Einstein Condensation of Incommensurate SolidHe4

It is pointed out that the simulation computation of energy performed so far cannot be used to decide if the ground state of solid 4He has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). The best variational wave function, a shadow wave function, gives an incommensurate state, but the equilibrium concentration of vacancies remains to be determined. We have computed the one-body density matrix in solid 4He for the incommensurate state by means of an exact ground state projector method in which incommensurability occurs spontaneously. We find a vacancy induced Bose-Einstein condensation of about 0.23 atoms per vacancy at a pressure of 54 bar. This means that bulk solid 4He is supersolid at low enough temperature if the exact ground state is incommensurate.

Layer by layer solidification ofHe4in narrow porous media

We give a variational Monte Carlo description of $^{4}\mathrm{He}$ filling under pressure a porous material modeled by a smooth cylindrical nanopore. Our trial wave function is a shadow wave function which allows different degrees of correlation between $^{4}\mathrm{He}$ atoms depending on the distance from the pore wall but still preserving the full Bose symmetry. The radial density profile shows a strong layering of the $^{4}\mathrm{He}$ atoms which are located in concentric annuli. This system has a very rich phase diagram with at least four different phases. The layer in contact with the pore is always solid. For our narrow pore radius $(R=13\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}})$, as the density is increased, solidification takes place layer by layer, starting from the pore wall, as is confirmed by the static structure factors. The pore radius is too small to allow a bulklike solid to nucleate in the liquid region at the center of the pore, and in order to have a complete crystalline order in all the layers a pressure greater than $200\phantom{\rule{0.3em}{0ex}}\text{bar}$ is needed. Computing the one body density matrix we are able to estimate the condensate fraction, which is still nonzero even if all the layers are in the solid phase.

Bose-Einstein condensation in solidHe4

We have computed the one--body density matrix rho_1 in solid 4He at T=0 K using the Shadow Wave Function (SWF) variational technique. The accuracy of the SWF has been tested with an exact projector method. We find that off-diagonal long range order is present in rho_1 for a perfect hcp and bcc solid 4He for a range of densities above the melting one, at least up to 54 bars. This is the first microscopic indication that Bose Einstein Condensation (BEC) is present in perfect solid 4He. At melting the condensate fraction in the hcp solid is 5*10^{-6} and it decreases by increasing the density. The key process giving rise to BEC is the formation of vacancy--interstitial pairs. We also present values for Leggett's upper bound on the superfluid fraction deduced from the exact local density.