Green Function Monte Carlo Method Research Articles

Reexamining the relation between the binding energy of finite nuclei and the equation of state of infinite nuclear matter

The energy density is calculated in coordinate space for $^{12}$C, $^{40}$Ca, $^{48}$Ca, and $^{208}$Pb using a dispersive optical model constrained by all relevant data including the corresponding energy of the ground state. The energy density of $^{8}$Be is also calculated using the Green's function Monte-Carlo method employing the Argonne/Urbana two and three-body interactions. The nuclear interior minimally contributes to the total binding energy due to the 4$\pi r^2$ phase space factor. Thus, the volume contribution to the energy in the interior is not well constrained. The dispersive-optical-model energy densities are in good agreement with \textit{ab initio} self-consistent Green's function calculations of infinite nuclear matter restricted to treat only short-range and tensor correlations. These results call into question the degree to which the equation of state for nuclear matter is constrained by the empirical mass formula. In particular, the results in this paper indicate that saturated nuclear matter does not require the canonical value of 16 MeV binding per particle but only about 13-14 MeV when the interior of $^{208}$Pb is considered.

Muon capture in nuclei: An ab initio approach based on Green's function Monte Carlo methods

An ab initio quantum Monte Carlo method is introduced for calculating total rates of muon weak capture in light nuclei with mass number $A \leq 12$. As a first application of the method, we perform a calculation of the rate in $^4$He in a dynamical framework based on realistic two- and three-nucleon interactions and realistic nuclear charge-changing weak currents. The currents include one- and two-body terms induced by $\pi$- and $\rho$-meson exchange, and $N$-to-$\Delta$ excitation, and are constrained to reproduce the empirical value of the Gamow-Teller matrix element in tritium. We investigate the sensitivity of theoretical predictions to current parametrizations of the nucleon axial and induced pseudoscalar form factors as well as to two-body contributions in the weak currents. The large uncertainties in the measured values obtained from bubble-chamber experiments (carried out over 50 years ago) prevent us from drawing any definite conclusions.

Doubly Charmed Baryon Mass and Wave Function through a Random Walks Method

The mass and the wave function of doubly charmed $\Xi^{++}_{cc}$ (ccu) baryon are evaluated using Green Function Monte Carlo method to solve the three-body problem with Cornell potential. The mass of $\Xi^{++}_{cc}$ with spin $1/2$ is in a good agreement with the LHCb value. Simulation of the wave function by random walks resulted in a configuration of the quark-diquark type. The radius of $\Xi^{++}_{cc}$ is much larger than the size needed for a large isospin splitting. The prediction for the $\Omega_{cc}$ mass is presented.

Hardcore bosons on checkerboard lattices near half filling: Geometric frustration, vanishing charge order, and a fractional phase

We study a spinless hardcore boson model on checkerboard lattices by Green function Monte Carlo method. At half filling, the ground state energy is obtained up to a $28\ifmmode\times\else\texttimes\fi{}28$ lattice and extrapolated to infinite size; the staggered pseudospin magnetization is found to vanish in the thermodynamic limit. Thus the $(\ensuremath{\pi},\ensuremath{\pi})$ charge order is absent in this system. Away from half filling, two defects induced by each hole (particle) may carry fractional charge $(\ifmmode\pm\else\textpm\fi{}e∕2)$. For the one hole case, we study how the defect-defect correlation changes with $t∕J$, which is the ratio between the hopping integral and cyclic exchange, and is equal to $V∕2t$ when $V⪢t$. Moreover, we argue that these fractional defects may propagate independently when the concentration of holes (or defects) is large enough.

Spin-orbit correlation energy in neutron matter

We study the relevance of the energy correlation produced by the two-body spin-orbit coupling present in realistic nucleon-nucleon potentials. To this purpose, the neutron matter Equation of State (EoS) is calculated with the realistic two-body Argonne $v_8'$ potential. The shift occuring in the EoS when spin-orbit terms are removed is taken as an estimate of the spin-orbit correlation energy. Results obtained within the Bethe-Brueckner-Goldstone expansion, extended up to three hole-line diagrams, are compared with other many-body calculations recently presented in the literature. In particular, excellent agreement is found with the Green's function Monte-Carlo method. This agreement indicates the present theoretical accuracy in the calculation of the neutron matter EoS.

Dispersion of a single hole in the t-J model

The dispersion of a single hole in the t-J model obtained by the exact result of 32 sites and the results obtained by self-consistent Born approximation and the Green function Monte Carlo method can be simply derived by a mean-field theory with d-wave resonating-valence-bond (d-RVB) and antiferromagnetic order parameters. In addition, it offers a simple explanation for the difference observed between those results. The presence of the extended van Hove region at (\ensuremath{\pi},0) is a consequence of the d-RVB pairing instead of the antiferromagnetic order. Results including ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ and ${\mathrm{t}}^{|\mathrm{I}\mathrm{J}}$ are also presented and explained consistently in a similar way.

Green Function Monte Carlo Method for Excited States of Quantum System

A novel scheme to solve the quantum eigenvalue problem through the imaginary-time Green function Monte Carlo method is presented. This method is applicable to the excited states as well as to the ground state of a generic system. We demonstrate the validity of the method with the numerical examples on three simple systems including a discretized sine-Gordon model.

A Monte Carlo approach for SU(2) Yang - Mills theory in dimensions

The Green function Monte Carlo method of Chin et al is applied to SU(2) Yang - Mills theory in (2 + 1) dimensions. Accurate measurements are obtained for the ground-state energy and mean plaquette value, and for various Wilson loops. The results are compared with series and linked-cluster expansions, and with the Euclidean Monte Carlo results of Teper. The finite-size scaling behaviour of the model is discussed briefly.

Intermediate-coupling theory of the spin polaron in the t-J model.

The spin polaron in the t-J model, i.e., a hole dressed by a cloud of virtual magnons of the antiferromagnetic spin background, is treated within the framework of intermediate-coupling theory. To this end the original t-J model is first reformulated in terms of spinless fermions and bosons by means of the generalized Dyson-Maleev representation (DMR). The latter may be regarded as the natural extension of the ordinary DMR of pure (undoped) spin systems to the case where holes are present, and is similar to the one originally proposed by Schmitt-Rink, Varma, and Ruckenstein. The reformulated t-J model, which is reminiscent of the Fr\"ohlich Hamiltonian, is then subjected to a series of unitary transformations, analogous to those employed by Lee, Low, and Pines in their treatment of the Fr\"ohlich polaron. Our approach yields an approximate quasiparticle energy E(k) as well as the corresponding eigenvector. To explore the range of validity of the theory presented here, the analytic expressions are then further analyzed for intermediate (J/t=0.4) and strong (J/t=0.08) coupling, where special attention is paid to the quasiparticle bandwidth W. The intermediate-coupling result for E(k) is in excellent agreement with the dispersion curve recently obtained by Dagotto and co-workers by means of a Green function Monte Carlo method. Surprisingly, even in the strong-coupling range the bandshape remains qualitatively correct.The bandwidth W is rather accurate for weak coupling (J/t\ensuremath{\gtrsim}3), as expected, and still reasonable in the intermediate range 0.4\ensuremath{\lesssim}J/t\ensuremath{\le}3, where it deviates from the correct values by some 10--20 %. Our theory fails, however, to describe the proper behavior of W in the strong-coupling regime. This shows that the limitations of our approach manifest themselves in the bandwidths rather than in the shapes of the dispersion curves. Our conclusion is that intermediate-coupling theory is appropriate for J/t\ensuremath{\gtrsim}0.4, whereas a genuine strong-coupling theory is required for all other cases. \textcopyright{} 1996 The American Physical Society.

Scattering in the Framework of the Green's Function Monte-Carlo Method

A method based on the Green's Function Monte-Carlo approach is used to solve the integral equation for the scattering wave function. The algorithm employed is explained for two-body scattering and tested for cases where the exact solution is known.

Monte Carlo study of the energy dispersion curve of a mobile hole in a quantum antiferromagnet

The ground state energy dispersion curve of a single mobile hole in the two-dimensional t-J model is computed numerically by means of the Green function Monte Carlo method on square lattices of various sizes up to 16 × 16. Transient estimation results show evidence of a gap of the order of 0.2 t, in the physical region of the model, between the energy value at k = (± 1 2 π, ± 1 2 π), which corresponds to the hole band absolute minimum, and k = (±π, 0).

Ground state of two holes in a quantum antiferromagnet: Green's function Monte Carlo study

We investigate numerically the ground state of two holes in the two-dimensional t-J model, by means of the Green function Monte Carlo method. A critical value of J, J c≈0.3 t, is found such that d-wave binding of holes only occurs for J> J c.

Two-hole d-wave binding in the physical region of the t-J model: A Green's-function Monte Carlo study.

We investigate numerically the ground state of two holes with ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ symmetry in the two-dimensional t-J model, on lattices of significantly larger sizes than the ones studied so far. The Green's-function Monte Carlo method, in conjunction with suitable initial states and guidance functions, is used to estimate the ground-state energy of one and two holes, as well as to compute the two-hole distribution function. Our results show a significant decrease of the two-hole binding energy as the size of the lattice is increased from 4\ifmmode\times\else\texttimes\fi{}4. A critical value ${\mathit{J}}_{\mathit{c}}$\ensuremath{\sim}0.27t is found such that d-wave hole binding no longer occurs for J${\mathit{J}}_{\mathit{c}}$.

Monte-Carlo calculation of bound-state properties

We elaborate on the role of the Green's function Monte-Carlo method in connection with quantum few-body systems and their computational treatment algorithms for calculation of bound-state properties are presented. We report results obtained from Green's function Monte-Carlo calculations both of ground-state properties of the mesic moleculesdtμ,ddμ, andpdμ as well as of excited states of the three-dimensional anharmonic oscillator and thedtμ molecule.

Green's-function Monte Carlo study of the two-dimensional, spin-1/2 XY ferromagnet.

The ground states of quantum spin-1/2 XY ferromagnetic triangular and square systems are investigated with the Green's-function Monte Carlo method. Simulations are performed on L\ifmmode\times\else\texttimes\fi{}L two-dimensional lattices with L up to 19. The long-range orders as well as ground-state energies are obtained for both lattices and are extrapolated to infinite system size. The long-range order (LRO) is computed directly using kernels from the system Hamiltonian and a ``perturbed'' Hamiltonian involving the LRO operator. It is seen that LRO exists in the infinite triangular lattice, contrary to conclusions drawn from previous exact-diagonalization studies. The second-order spin-wave theory again appears to yield very good results on these systems.

Cohesive energy of silicon by the Green's-function Monte Carlo method.

The total energy of diamond-structure silicon is calculated by a fixed-node Green's-function Monte Carlo method using a pseudo-Hamiltonian to eliminate the core electrons. This demonstrates the feasibility of calculating properties of solids with the quantum Monte Carlo method, since the statistical error for a supercell of 64 atoms is 0.02 eV/atom. The agreement with experiment, although good, is limited by the accuracy of the pseudo-Hamiltonian. We find that the correlation energy is improved over a variational pair-product trial function by 0.34 eV/atom in the solid compared with 0.21 eV in the free atom.

Diffusive behavior of states in the Hubbard-Stratonovitch transformation.

The Hubbard-Stratonovitch auxiliary-field approach to projection of the ground state of an interacting fermion system from a trial state is examined. It is shown that the method is equivalent to solving a differential equation with diffusion, drift, and branching terms on the manifold of normalized Slater determinants. Explicit general expressions are given for the coefficients of the diffusion equation in terms of the Hubbard-Stratonovitch fields. The form of the equation is somewhat similar to that obtained in the continuum Green's-function Monte Carlo method, although its interpretation and relation to the physical many-body problem is quite different. The character of the representation of the many-body ground state arising in the auxiliary-field projection approach is discussed within this framework. The diffusion process is normally found to concentrate the states representing the ground state near the classical mean-field solutions. The consequences of this picture of the auxiliary-field approach for the ``fermion minus-sign'' problem in this context are discussed, and some conclusions are reached concerning the range of validity of a recently suggested approximation where minus signs are ignored.

Overcoming the large-Z problem in quantum monte carlo

The Z6 scaling of computational effort in quantum Monte Carlo (QMC) results in part from a type of critical slowing down. This insight leads to an attack based on acceleration algorithms which modify the dynamics in Langevin-based simulations without changing the steady-state. Configuration space is more rapidly sampled and steps are inherently more decorrelated. A second approach uses a hybrid Metropolis–Green function Monte Carlo method which factors the core and valence wave function, and treats the former variationally with accurate trial functions, and the latter exactly in the presence of the variationally evoling core. This allows separation of the relevant core and valence time scales, and has sped up quantum simulations by over three orders of magnitude.

Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study.

We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site ${\mathit{E}}_{0}$/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be ${\mathit{Z}}_{\mathit{c}}$==c/${\mathit{c}}_{\mathit{s}}$=1.14(5).

Multiquark masses and wave functions through a modified green function Monte Carlo method

The Modified Green Function Monte Carlo method (MGFMC) is used to calculate the masses and ground-state wave functions of multiquark systems in the potential model. We present results obtained with the Cornell potential for the masses and the wave functions of light- and heavy-quark baryons and multiquark states ( N = 3, 6, 9, 12) made of light quarks.