# PastResearchNotices

### From Werner KRAUTH

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## Efimov-driven phase transitions of the unitary Bose gas

In a recent article with Swann Piatecki, published in Nature Communications 5, 3503 (2014), we discuss Efimov trimers: bound configurations of three quantum particles that fall apart when any one of them is removed. They open a window into a rich quantum world that has become the focus of intense experimental and theoretical research, as the region of ‘unitary’ interactions, where Efimov trimers form, is now accessible in cold-atom experiments. We use a path-integral Monte Carlo algorithm backed up by theoretical arguments to show that unitary bosons undergo a first-order phase transition from a normal gas to a superfluid Efimov liquid, bound by the same effects as Efimov trimers. A triple point separates these two phases and another superfluid phase, the conventional Bose–Einstein condensate, whose coexistence line with the Efimov liquid ends in a critical point. We discuss the prospects of observing the proposed phase transitions in cold-atom systems.

Here you see three bosons, in path integral representation, and with a certain pseudopotential interaction that is described in more detail in the paper. On the left side, the particles are slightly repulsive, on the right side, they are attractive (so that two particles simple get together and bind into a dimer, whereas the third particle just sits around. In the center, you see the particles at the unitary point: pair interactions are very weak, so pairs get together, but unbind. After a little while, another pair forms, etc etc. The final outcome is that ensembles of two particles fall apart, but three particles stay together, just like Borromean rings...

## Generalized event-chain Monte Carlo: Rejection-free global-balance algorithms from infinitesimal steps

Our 2009 event-chain algorithm for hard spheres, with Etienne Bernard and David Wilson, is a new paradigm for Monte Carlo simulations. For a while, it was unclear whether it could be generalized to general potentials, and an earlier attempt was rather awkward. This is what we succeeded in doing, with Manon Michel and Sebastian C. Kapfer, in a 2014 paper in Journal of Chemical Physics. In the paper, we introduce a factorized Metropolis filter and the concept of infinitesimal Monte Carlo moves to design a rejection-free Markov-chain Monte Carlo algorithm for interacting particle systems that breaks detailed balance yet satisfies global balance. This event-driven algorithm generalizes the recent hard-sphere event-chain Monte Carlo method without introducing any discretizations in time or in space. We demonstrate considerable speed-ups of this method with respect to the classic local Metropolis algorithm. The new algorithm generates a continuum of samples of the stationary probability density. This allows us to derive an exact formula for the pressure that is obtained as a byproduct of the simulation without any additional computations. The generalized event-chain algorithm is really simple to implement, and it showed all its usefulness, for example in recent simulations with Sebastian C. Kapfer.

## Sampling from a polytope and hard-disk Monte Carlo

The hard-disk problem, the statics and the dynamics of equal two-dimensional hard spheres in a periodic box, has had a profound influence on statistical and computational physics. Markov-chain Monte Carlo and molecular dynamics were first discussed for this model. In a recent preprint with Sebastian Kapfer,
we were able to reformulate hard-disk Monte Carlo algorithms in terms of another classic problem, namely the sampling from a polytope. Local Markov-chain Monte Carlo, as proposed by Metropolis et al. in 1953, appears as a sequence of random walks in high-dimensional polytopes, while the moves of the more powerful event-chain algorithm correspond to molecular dynamics evolution. In the paper, we determine the convergence properties of Monte Carlo methods in a special invariant polytope associated with hard-disk configurations, and the implications for convergence of hard-disk sampling. Finally, we discuss parallelization strategies for event-chain Monte Carlo and present results for a multicore implementation.

## Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods

Spring and summer of 2012 was partly spent on a project with colleagues Michael Engel, Joshua Anderson, and Sharon Glotzer from the University of Michigan, Masaharu Isobe from the Nagoya Institute of Technology, and Etienne Bernard from MIT. We were interested in checking our earlier results on the melting transition of hard disks in two dimensions which predicted the existence of a first-order liquid-hexatic phase transition - a big surprise after hundreds of other papers had fought over two other scenarios for melting in two dimensions. Our confirmation preprint finally came out in November 2012, and it was published in April 2013 in Physical Review E, which in 2018 recognized it as its milestone paper for that year. Using three completely independent algorithms (massively parallel local Monte Carlo, molecular dynamics, event-chain Monte Carlo), we confirmed our earlier data (see figure to the left). We are all happy about this independent verification, and Etienne Bernard and I are quite relieved, as so much can go wrong with numerical simulations, especially if they take an eternity, almost, to run.The figure to the left shows the equation of state for hard disks (the equilibrium pressure as a function of the density (or the volume), with the characteristic loop which indicates the presence of two phases - a minority phase that forms a bubble inside the majority phase. The three curves stand for the simulation methods: different algorithms, different computers, even continents produce the same equation of state, the one that noone else has produced before! The inset gives the difference between the old data (from last year) and the new ones. Let me note that, if the simulation is nontrivial, the calculation of the pressure is quite tricky also, especially in Monte Carlo calculations, as an extrapolation of the pair-correlation function is involved. Those of us in the team who computed the pressure from Monte Carlo were much relieved to see the nice agreement with the molecular dynamics pressure, obtained with Masaharu Isobe's code, which is computed simply by counting the number of collisions taking place over a months-long simulation. For more details, take a look at the paper.

## Event-driven Monte Carlo algorithm for general potentials

In recent works, as for example on the melting transition in two dimensions, the event-chain algorithm has proven quite helpful. This hard-sphere Monte Carlo method runs a lot faster than earlier methods although the speed-uo remains constant for large system sizes. Nevertheless, gaining a factor of about 100 is not so bad for run-times (with the new algorithm) on the order of a few months... we got our results before it was time to retire.

Recently, in 2012, Etienne Bernard (now at MIT) and I were able to extend the event-chain algorithm to continuous potentials, and we are now quite excited: The algorithm allows to break detailed balance, it is (hopefully) much faster than local Monte Carlo algorithms, and it is extremely easy to program, to parallelize (hopefully), to modify and, why not, to improve. Technically, we work with stepped potentials (see the figure, similar approaches exist for molecular dynamics), but there is no problem going to finer and finer discretizations: the algorithm doesn't even slow down as we crank up the number of steps. This is explained in a section of the wiki page dedicated to our recent paper. But lots of things need to be done to understand this new approach, to check out possible applications, etc, and we are right now extremely busy.

## Two-dimensional melting: First-order liquid-hexatic transition

Here, I show the key figure of a recent paper, published in Physical Review Letters in 2011, with Etienne Bernard, on the melting transition in hard disks. The main picture shows the orientations of a configuration with 1024x1024 disks, and two different regions are clearly visible: To the left, disks have more or less the same orientation, whereas to the right, the orientations vary (and the local densities are lower). This clearly indicates the presence of a first-order transition (for details see the paper). In our paper, we show not only that the transition is of first order, but also that it is between the liquid and a hexatic phase. Our melting scenario differs from what hundreds of earlier papers seemed to indicate, namely that the two-dimensional melting transition either followed the famous KTHNY scenario or was a direct transition from the liquid to the solid state, as in three dimensions. The fight for truth between these two groups raged for several decades. Using much better simulation methods, we could show that both were off, and the scenario adopted by nature is not what was imagined for so long.To produce the picture, we used the
event-chain algorithm, a Monte Carlo method that we developed a few years ago, with David Wilson. This algorithm is really the first one to outperform, by about two orders of magnitude in speed, the classic Metropolis method from 1953. For a long time, I have been interested in the hard-disk melting problem, but an earlier attempt to speed up the extremely slow converge of numerical methods for this problem, the cluster algorithm that I developed with C. Dress, had failed.

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