Lattice Calculation

[Adah Orhenkowski]

LatticeQCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered.[1][2]

In lattice QCD, fields representing quarks are defined at lattice sites (which leads to fermion doubling), while the gluon fields are defined on the links connecting neighboring sites. This approximation approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations can increase dramatically as the lattice spacing decreases, results are often extrapolated to a = 0 by repeated calculations at different lattice spacings a that are large enough to be tractable.

Numerical lattice QCD calculations using Monte Carlo methods can be extremely computationally intensive, requiring the use of the largest available supercomputers. To reduce the computational burden, the so-called quenched approximation can be used, in which the quark fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard.[3] These simulations typically utilize algorithms based upon molecular dynamics or microcanonical ensemble algorithms.[4][5]

At present, lattice QCD is primarily applicable at low densities where the numerical sign problem does not interfere with calculations. Monte Carlo methods are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC2D).

Monte-Carlo is a method to pseudo-randomly sample a large space of variables.The importance sampling technique used to select the gauge configurations in the Monte-Carlo simulation imposes the use of Euclidean time, by a Wick rotation of spacetime.

In lattice Monte-Carlo simulations the aim is to calculate correlation functions. This is done by explicitly calculating the action, using field configurations which are chosen according to the distribution function, which depends on the action and the fields. Usually one starts with the gauge bosons part and gauge-fermion interaction part of the action to calculate the gauge configurations, and then uses the simulated gauge configurations to calculate hadronic propagators and correlation functions.

Lattice QCD is a way to solve the theory exactly from first principles, without any assumptions, to the desired precision. However, in practice the calculation power is limited, which requires a smart use of the available resources. One needs to choose an action which gives the best physical description of the system, with minimum errors, using the available computational power. The limited computer resources force one to use approximate physical constants which are different from their true physical values:

In lattice perturbation theory the scattering matrix is expanded in powers of the lattice spacing, a. The results are used primarily to renormalize Lattice QCD Monte-Carlo calculations. In perturbative calculations both the operators of the action and the propagators are calculated on the lattice and expanded in powers of a. When renormalizing a calculation, the coefficients of the expansion need to be matched with a common continuum scheme, such as the MS-bar scheme, otherwise the results cannot be compared. The expansion has to be carried out to the same order in the continuum scheme and the lattice one.

The lattice regularization was initially introduced by Wilson as a framework for studying strongly coupled theories non-perturbatively. However, it was found to be a regularization suitable also for perturbative calculations. Perturbation theory involves an expansion in the coupling constant, and is well-justified in high-energy QCD where the coupling constant is small, while it fails completely when the coupling is large and higher order corrections are larger than lower orders in the perturbative series. In this region non-perturbative methods, such as Monte-Carlo sampling of the correlation function, are necessary.

Lattice perturbation theory can also provide results for condensed matter theory. One can use the lattice to represent the real atomic crystal. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed (a UV regulator), and a quantum field theory can be formulated and solved on the physical lattice.

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The exchange-correlation functionals of the generalized gradient approximation (GGA) are still the most used for the calculations of the geometry and electronic structure of solids. The PBE functional [J. P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], the most common of them, provides excellent results in many cases. However, very recently other GGA functionals have been proposed and compete in accuracy with the PBE functional, in particular for the structure of solids. We have tested these GGA functionals, as well as the local-density approximation (LDA) and TPSS (meta-GGA approximation) functionals, on a large set of solids using an accurate implementation of the Kohn-Sham equations, namely, the full-potential linearized augmented plane-wave and local orbitals method. Often these recently proposed GGA functionals lead to improvement over LDA and PBE, but unfortunately none of them can be considered as good for all investigated solids.

The same as in Fig. 3, but after subtracting the perturbative lattice artifacts evaluated numerically in the free theory, i.e. at order O(αs0), at finite values of the bare quark masses and of the lattice spacing.

Top panel: the quark-loop disconnected contribution to the short time-distance window, aμSD, versus the squared lattice spacing a2 in physical units. Bottom panel: the same as in the top panel, but for the intermediate window aμW. The blue band corresponds to the extrapolation performed using a linear fit ansatz in a2.

a2-scaling behavior of the relative difference between the two determinations of the lattice spacing given in Table 7. The violet and green bands correspond to a linear fit in a2 applied, respectively, to all the four data points and to the three finest ones only. The width of the bands represents one standard deviation.

Effective masses aMηs (top) and aMϕ (bottom) obtained, respectively, from the strange pseudoscalar and vector correlators evaluated in the tm regularization in the case of the cD211.054.96 ensemble. The horizontal bands indicate the results of a constant fit in the plateaux regions, where the ground state dominates.

Effective masses aMηc (top) and aMJ/Ψ (bottom) obtained, respectively, from the charm pseudoscalar and vector correlators evaluated in the tm regularization in the case of the cD211.054.96 ensemble. The horizontal bands indicate the results of a constant fit in the plateaux regions, where the ground state dominates.

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We present a detailed study of the lattice dynamics and of the phase stability of cubic zincblende (c-BN) and hexagonal (h-BN) boron nitrides. The phonon-dispersion relations at different densities are calculated using a first-principles force-constant method. The calculated eigenfrequencies and phonon Grneisen parameters are in good agreement with experimental findings. From the electronic and vibrational energies as a function of volume we calculate the phase (p,T) diagram of boron nitride in a quasiharmonic approximation. At low temperature c-BN is the stable modification; the c-BN/h-BN coexistence line intersects the temperature axis at 1440 K. In experiments this temperature lies between 1200 and 1800 K. Anharmonic corrections improve the agreement between the calculation and experiment for high pressure and temperature.

In the materialized views usage research field there are two fundamental types of problems: finding which views are best to materialize and rewriting the original query to use these views. This article focuses on the first problem.

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