# Medley of Monte Carlo

by Nicole Gaynor

*Monte Carlo methods are a common approach to large computational problems. Lucas Wagner and Robert Sugar both use this method to study different aspects of physics on Blue Waters.*

What do superconductors and the intergalactic medium have in common? Scientists study both on Blue Waters using something called Monte Carlo methods. Monte Carlo methods are widely used in computational sciences because they allow researchers to find a solution to a problem that is intractable to answer directly.

For example, climate forecasts are based on imperfect observations and discrete model points that sit at least 25 km apart. Instead of increasing the number of points at which temperature, rainfall, and a slew of other variables are calculated, researchers run the model multiple times with slightly different (but still realistic) conditions and use the results from all the models to determine a range of possible solutions.

In physics, the same methodology helps investigate the properties of and interactions between subatomic particles.

Robert Sugar, a professor emeritus at the University of California in Santa Barbara, and the MILC collaboration use Blue Waters to study quarks, the subatomic particles that make up the better-known protons and neutrons in the atomic nucleus. Lucas Wagner, professor of physics at the University of Illinois at Urbana-Champaign, experiments with the interactions between electrons in strongly correlated materials that change properties depending on temperature and pressure and under certain conditions become superconductors. These are their stories.

#### Robert Sugar: Quantum Chromodynamics

“For me, obtaining a better understanding of the basic laws of nature is a tremendously exciting project,” says Sugar.

Sugar is part of the MILC Collaboration and the principal investigator for the largest Blue Waters allocation thus far—about 90 million node-hours over three years. MILC, a group of scientists who study lattice quantum chromodynamics (QCD), used about half ofthe allocation. QCD is a theory that explains the strong forces that hold together quarks, the fundamental building block of strongly interacting matter, in the atomic nucleus to make strongly interacting particles like protons and neutrons (generically called hadrons). MILC and other physicists use a lattice (or grid) to calculate the properties of QCD in order to make the problem computationally friendly.

##### Tiny particles ≠ tiny calculations

MILC depends on large-scale computing resources to make these calculations. The team has run simulations on Blue Waters that, for the first time, use the actual masses of the lightest of all quarks, the up and down quarks that make up the neutrons and protons inatomic nuclei. To perform simulations using the masses of the up and down quarks is very computationally expensive and requires a powerful system like Blue Waters.

Up and down quarks weigh on the order of 10^{-30} kg. To perform precise calculations using such small particles (or even to estimate these masses) takes a lot of computing power and time when the mass of other types of quarks is at least 27 times that of the up and down quarks.

On top of that, a single lattice simulation cannot represent the continuum that actually exists. Think of a lattice as a screen, where there is information at every point where the threads cross. Scientists must perform several experiments using different lattice spacing, or different screen mesh fineness, and extrapolate to infinitely small spacing to approximate the continuum. Cut the lattice spacing in half and the required computing time increases by a factor of 32.

QCD is part of the Standard Model of particle physics. All experimental and computational data thus far is consistent with this model. Despite that, the Standard Model does not explain some phenomena, neglects dark matter and dark energy, and does not incorporate the theory of gravitation. Physicists believe the Standard Model is a limiting case of a more general theory. Calculations like those MILC conducts aim to find the limits of the Standard Model.

##### Quark in transition

MILC’s recent calculations, highlighted in an October 2014 article in *Physical Review D*, use the MILC Highly Improved Staggered Quark (HISQ) ensembles with four “flavors” of quarks: up, down, strange, and charm (top and bottom quarks are not included in the ensembles because they are too heavy to influence the calculations).

The simulations measure the effects of strong interactions (QCD) on the decay of mesons, which are particles made up of a quark and an anti-quark. Meson decays are mediated (or caused) by weak interactions, which cause a transition from one type of quark toanother with a strength that is given by the elements of a 3×3 matrix known as the CKM matrix. The decay rate is measured experimentally and is equal to the square of the strength, times the square of a decay constant that is measured in the QCD simulations, times some known factors. Therefore, one can calculate a CKM matrix element from an experiment that measures the decay rate coupled with a QCD calculation that measures the decay constant.

The CKM matrix elements are fundamental parameters of the Standard Model, and any inconsistency would signal a problem with the Standard Model. The matrix elements therefore play an important role in the search for a breakdown in the Standard Model that points to the need for a more general theory of the fundamental laws of physics.

Until recently, QCD calculations were less precise than experiments. However, some of the recent MILC calculations of decay constants are two to four times more precise than any prior lattice calculations, making them comparable to experiments. This development will allow scientists to learn more from the experiments carried out at multiple particle accelerators across the globe, all of which are expensive.

The MILC group has also performed high-precision calculations of quark masses and their ratios. These quantities are fundamental parameters of the Standard Model and play an important role in our understanding of sub-atomic physics.

Using the physical masses of the up and down quarks allowed MILC to increase the precision of their measurements and avoid estimating the quark masses and decay constants using chiral extrapolation. Chiral extrapolation can be a substantial source of error inthe calculations. MILC then used four different lattice spacings so they could extrapolate to zero lattice spacing—the continuum.

“We have demonstrated that the tools are at hand to perform high-precision calculations of a wide variety of quantities of importance to the field of high-energy physics,” says Sugar. “There should be a lot more to come.”

#### Lucas Wagner: QWalk and materials physics

Similar to high-energy physics, the basic laws for the physics of materials are known and were written down in the early 20th century—the equations are just too hard to solve. The biggest challenge in modern times is a class of materials, called strongly correlated materials, in which the interactions between electrons determine a material’s properties. These materials are difficult to describe but hold great promise for new technological developments.

##### VO_{2}

Wagner approached the problem through simulation of vanadium dioxide (VO_{2}), a material that changes phase from a metal to an insulator when its temperature drops below 340 Kelvin, or about 67 degrees Celsius. The exact temperature depends on the pressure on the material. In fact, if pressure is applied to part of a chunk of VO_{2} that is at a uniform temperature, part of it becomes a metal while the other part remains an insulator.

On its own merit, VO_{2} holds promise for powerefficient computing, so-called “smart glass” that can be transparent or opaque to ultraviolet radiation at low or high temperatures, respectively, while remaining transparent to visible light, and ultra-fast switches in electronics. It is already used in improved recording and storage media and to strengthen structural alloys. VO_{2} is one of the simpler strongly correlated materials. Scientists hope to use what they learn about VO_{2} to explore the properties of materials that have more numerous or more complex phase transitions.

The way in which the sudden phase transition happens is an old question, says Wagner. Electrons interact in such a way that they cannot be separated. Wagner approached the problem using computer code his group developed called QWalk, which applies Quantum Monte Carlo (QMC) methods to simulate the electron interactions. QMC solves the Schrödinger equation, which describes the probability distribution of electrons in materials, and relies on large simulations to reduce the margin of error in solutions.

“Previous to Blue Waters, we couldn’t even represent the materials in our simulations,” says Wagner. “[Using QWalk,] we were able to represent the material well enough to actually see the metal-insulator phase transition, and describe it in great detail.”

Directly solving the Schrödinger equation means that QWalk may be able to predict how other strongly correlated materials change structural and magnetic properties because the calculations use only established laws of physics (i.e. first principles) and do not rely on experimental data or assumptions. This methodology may even help design new strongly correlated materials.

##### High temperature superconductivity

There are two competing theories about how electrons work in superconducting materials, says Wagner: magnetism and vibration. He says scientists argue over their relative contributions to superconductivity as two distinct effects.

Experimentalists look at the vibrational aspect or magnetism, but not both at the same time. Raman spectroscopy looks at vibrations by bouncing light off the molecules. The wavelength (i.e. the color) that bounces back indicates the frequency of vibration. X-rays reveal the magnetic fields, which are a result of uneven charge distribution. The distribution of charge in the material subjected to x-rays determines how the x-rays diffract. Computer simulations lend insight into both at the same time.

Recent simulations on Blue Waters suggest that these theories are not separate at all—they are tightly coupled to one another. Structural changes move electrons around, which changes the distribution of charge, and hence the magnetic characteristics, of the material. Then the modified magnetic field distorts the molecular structure of the material and the feedback loop continues.

Wagner probes the inner workings of the cuprate (a type of copper-containing compound) La_{1-x}Sr_{x}CuO_{4}—again the first time that someone was able to look at that cuprate using a first-principles calculation. Interaction between structure and magnetism (and electron spin) showed up in this material, too, indicating again that the characteristics are not separate even though experimental observation can only look at one characteristic at a time.

This series of simulations from first principles opens a new avenue of research into high-temperature superconductors.

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