arXiv subjects

Werner Krauth

Publications and source records attributed to Werner Krauth.

At least 55 records · Page 3Linked to original sources

arXiv2015

Liquid-solid transitions in the three-body hard-core model

We determine the phase diagram for a generalisation of two-and three-dimensional hard spheres: a classical system with three-body interactions realised as a hard cut-off on the mean-square distance for each triplet of particles. Quantum versions of this model are important in the context of the unitary Bose gas, which is currently under close theoretical and experimental scrutiny. In two dimensions, the three-body hard-core model possesses a conventional atomic liquid phase and a peculiar solid phase formed by dimers. These dimers interact effectively as hard disks. In three dimensions, the solid phase consists of isolated atoms that arrange in a simple-hexagonal lattice.

cond-mat.stat-mech↗

arXiv2015

2D Melting: From Liquid-Hexatic Coexistence to Continuous Transitions

The phase diagram of two-dimensional continuous particle systems is studied using Event-Chain Monte Carlo. For soft disks with repulsive power-law interactions $\propto r^{-n}$ with $n \gtrsim 6$, the recently established hard-disk melting scenario ($n \to \infty$) holds: a first-order liquid-hexatic and a continuous hexatic-solid transition are identified. Close to $n = 6$, the coexisting liquid exhibits very long orientational correlations, and positional correlations in the hexatic are extremely short. For $n\lesssim 6$, the liquid-hexatic transition is continuous, with correlations consistent with the Kosterlitz-Thouless-Halperin-Nelson-Yong (KTHNY) scenario. To illustrate the generality of these results, we demonstrate that Yukawa particles likewise may follow either the KTHNY or the hard-disk melting scenario, depending on the Debye-Hückel screening length as well as on the temperature.

cond-mat.stat-mech↗

arXiv2014

Coming home from a MOOC

My ten-week Massive Open Online Course "Statistical Mechanics: Algorithms and Computations", in early 2014, focused on subjects such as Monte Carlo sampling, molecular dynamics, transition phases in hard-sphere liquids, simulated annealing, classical spin models, quantum Monte Carlo algorithms, and Bose-Einstein condensation, etc. It familiarized a huge international crowd of students with cutting-edge subjects in computational physics. Here, I present the topics of the course, its basic design ideas, its scope and challenges, and compare it with earlier attempts in online teaching.

physics.ed-ph↗

arXiv2013

Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps

In this article, we present an event-driven algorithm that generalizes the recent hard-sphere event-chain Monte Carlo method without introducing discretizations in time or in space. A factorization of the Metropolis filter and the concept of infinitesimal Monte Carlo moves are used to design a rejection-free Markov-chain Monte Carlo algorithm for particle systems with arbitrary pairwise interactions. The algorithm breaks detailed balance, but satisfies maximal global balance and performs better than the classic, local Metropolis algorithm in large systems. The new algorithm generates a continuum of samples of the stationary probability density. This allows us to compute the pressure and stress tensor as a byproduct of the simulation without any additional computations.

cond-mat.stat-mech↗

arXiv2013

Efimov-driven phase transitions of the unitary Bose gas

In quantum physics, Efimov trimers are bound states of three particles that fall apart like Borromean rings when one of them is removed. Initially predicted in nuclear physics, these striking bosonic states are hard to observe, but the "unitary" interactions at which they form is commonly realized in current cold atoms experiments. There, they set the stage for a new class of universal physics: Two-body interactions are all but invisible, but three-body effects allow the emergence of a largely uncharted new world of many-particle bound states. Three-particle systems were characterized theoretically, and the ground-state properties of small unitary clusters computed numerically, but the macroscopic many-body behaviour has remained unknown. Here we show, using a Path-Integral Monte Carlo algorithm backed up by theoretical arguments, that the unitary Bose gas presents a first-order phase transition from a normal gas to a superfluid Efimov liquid. The normal gas is very well described by the available virial coefficients. At unitarity, the phase diagram of the bosonic system is universal in rescaled pressure and temperature. A triple point separates the normal gas, the superfluid Efimov liquid, and a third phase, the conventional superfluid gas. These two superfluid phases are separated by a critical line that ends in a critical point at high temperature. This rich phase diagram should allow for a number of experimental protocols that would probe these universal transitions between the normal gas, the superfluid gas, and the superfluid Efimov liquid.

cond-mat.quant-gas↗

arXiv2013

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

We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.

cond-mat.stat-mech↗

arXiv2013

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. Here we 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. 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.

cond-mat.stat-mech↗

arXiv2011

Event-driven Monte Carlo algorithm for general potentials

We extend the event-chain Monte Carlo algorithm from hard-sphere interactions to the micro-canonical ensemble (constant potential energy) for general potentials. This event-driven Monte Carlo algorithm is non-local, rejection-free, and allows for the breaking of detailed balance. The algorithm uses a discretized potential, but its running speed is asymptotically independent of the discretization. We implement the algorithm for the cut-off linear potential, and discuss its possible implementation directly in the continuum limit.

cond-mat.stat-mech↗

arXiv2011

Two-step melting in two dimensions: First-order liquid-hexatic transition

Melting in two spatial dimensions, as realized in thin films or at interfaces, represents one of the most fascinating phase transitions in nature, but it remains poorly understood. Even for the fundamental hard-disk model, the melting mechanism has not been agreed on after fifty years of studies. A recent Monte Carlo algorithm allows us to thermalize systems large enough to access the thermodynamic regime. We show that melting in hard disks proceeds in two steps with a liquid phase, a hexatic phase, and a solid. The hexatic-solid transition is continuous while, surprisingly, the liquid-hexatic transition is of first-order. This melting scenario solves one of the fundamental statistical-physics models, which is at the root of a large body of theoretical, computational and experimental research.

cond-mat.soft↗

arXiv2011

Damage spreading and coupling in Markov chains

In this paper, we relate the coupling of Markov chains, at the basis of perfect sampling methods, with damage spreading, which captures the chaotic nature of stochastic dynamics. For two-dimensional spin glasses and hard spheres we point out that the obstacle to the application of perfect-sampling schemes is posed by damage spreading rather than by the survey problem of the entire configuration space. We find dynamical damage-spreading transitions deeply inside the paramagnetic and liquid phases, and show that critical values of the transition temperatures and densities depend on the coupling scheme. We discuss our findings in the light of a classic proof that for arbitrary Monte Carlo algorithms damage spreading can be avoided through non-Markovian coupling schemes.

cond-mat.stat-mech↗

arXiv2011

Dynamics and Thermodynamics of the Low-Temperature Strongly Interacting Bose Gas

We measure the zero-temperature equation of state of a homogeneous Bose gas of $^7$Li atoms by analyzing the \emph{in-situ} density distributions of trapped samples. For increasing repulsive interactions our data shows a clear departure from mean-field theory and provides a quantitative test of the many-body corrections first predicted in 1957 by Lee, Huang and Yang. We further probe the dynamic response of the Bose gas to a varying interaction strength and compare it to simple theoretical models. We deduce a lower bound for the value of the universal constant $ξ>0.44(8)$ that would characterize the universal Bose gas at the unitary limit.

cond-mat.quant-gas↗

arXiv2010

The equilibrium state of a trapped two-dimensional Bose gas

We study experimentally and numerically the equilibrium density profiles of a trapped two-dimensional $^{87}$Rb Bose gas, and investigate the equation of state of the homogeneous system using the local density approximation. We find a clear discrepancy between in-situ measurements and Quantum Monte Carlo simulations, which we attribute to a non-linear variation of the optical density of the atomic cloud with its spatial density. However, good agreement between experiment and theory is recovered for the density profiles measured after time-of-flight, taking advantage of their self-similarity in a two-dimensional expansion.

cond-mat.quant-gas↗

arXiv2010

Event-chain Monte Carlo algorithms for hard-sphere systems

In this paper we present the event-chain algorithms, which are fast Markov-chain Monte Carlo methods for hard spheres and related systems. In a single move of these rejection-free methods, an arbitrarily long chain of particles is displaced, and long-range coherent motion can be induced. Numerical simulations show that event-chain algorithms clearly outperform the conventional Metropolis method. Irreversible versions of the algorithms, which violate detailed balance, improve the speed of the method even further. We also compare our method with a recent implementations of the molecular-dynamics algorithm.

cond-mat.stat-mech↗

arXiv2009

Universal correlations and coherence in quasi-two-dimensional trapped Bose gases

We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures above the Kosterlitz-Thouless transition, where the gas is in the normal phase. We show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. Comparing with Quantum Monte Carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. Although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. Deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. We further study coherence and pair correlations on a microscopic scale. Finite-size effects in the off-diagonal density matrix allows us to characterize the cross-over from Kosterlitz-Thouless to Bose-Einstein behavior for small particle numbers. Bose-Einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.

cond-mat.quant-gas↗

arXiv2009

Convergence and coupling for spin glasses and hard spheres

We discuss convergence and coupling of Markov chains, and present general relations between the transfer matrices describing these two processes. We then analyze a recently developed local-patch algorithm, which computes rigorous upper bound for the coupling time of a Markov chain for non-trivial statistical-mechanics models. Using the coupling from the past protocol, this allows one to exactly sample the underlying equilibrium distribution. For spin glasses in two and three spatial dimensions, the local-patch algorithm works at lower temperatures than previous exact-sampling methods. We discuss variants of the algorithm which might allow one to reach, in three dimensions, the spin-glass transition temperature. The algorithm can be adapted to hard-sphere models. For two-dimensional hard disks, the algorithm allows us to draw exact samples at higher densities than previously possible.

cond-mat.stat-mech↗

arXiv2009

Creep dynamics of elastic manifolds via exact transition pathways

We study the steady state of driven elastic strings in disordered media below the depinning threshold. In the low-temperature limit, for a fixed sample, the steady state is dominated by a single configuration, which we determine exactly from the transition pathways between metastable states. We obtain the dynamical phase diagram in this limit. At variance with a thermodynamic phase transition, the depinning transition is not associated with a divergent length scale of the steady state below threshold, but only of the transient dynamics. We discuss the distribution of barrier heights, and check the validity of the dynamic phase diagram at small but finite temperatures using Langevin simulations. The phase diagram continues to hold for broken statistical tilt symmetry. We point out the relevance of our results for experiments of creep motion in elastic interfaces.

cond-mat.dis-nn↗

arXiv2009

Four lectures on computational statistical physics

In my lectures at the Les Houches Summer School 2008, I discussed central concepts of computational statistical physics, which I felt would be accessible to the very cross-cultural audience at the school. I started with a discussion of sampling, which lies at the heart of the Monte Carlo approach. I specially emphasized the concept of perfect sampling, which offers a synthesis of the traditional direct and Markov-chain sampling approaches. The second lecture concerned classical hard-sphere systems, which illuminate the foundations of statistical mechanics, but also illustrate the curious difficulties that beset even the most recent simulations. I then moved on, in the third lecture, to quantum Monte Carlo methods, that underly much of the modern work in bosonic systems. Quantum Monte Carlo is an intricate subject. Yet one can discuss it in simplified settings (the single-particle free propagator, ideal bosons) and write direct-sampling algorithms for the two cases in two or three dozen lines of code only. These idealized algorithms illustrate many of the crucial ideas in the field. The fourth lecture attempted to illustrate aspects of the unity of physics as realized in the Ising model simulations of recent years. More details on what I discussed in Les Houches, and wrote up (and somewhat rearranged) here, can be found in my book, "Statistical Mechanics: Algorithms and Computations", as well as in recent papers. Computer programs are available for download and perusal at the book's web site www.smac.lps.ens.fr.

cond-mat.stat-mech↗

arXiv2008

Vacancy diffusion in the triangular lattice dimer model

We study vacancy diffusion on the classical triangular lattice dimer model, sub ject to the kinetic constraint that dimers can only translate, but not rotate. A single vacancy, i.e. a monomer, in an otherwise fully packed lattice, is always localized in a tree-like structure. The distribution of tree sizes is asymptotically exponential and has an average of 8.16 \pm 0.01 sites. A connected pair of monomers has a finite probability of being delocalized. When delocalized, the diffusion of monomers is anomalous:

cond-mat.stat-mech↗