One of Feynman's early applications of path integrals was to superfluid $^{4}\mathrm{He}$. He showed that the thermodynamic properties of Bose systems are exactly equivalent to those of a peculiar type of interacting classical "ring polymer." Using this mapping, one can generalize Monte Carlo simulation techniques commonly used for classical systems to simulate boson systems. In this review, the author introduces this picture of a boson superfluid and shows how superfluidity and Bose condensation manifest themselves. He shows the excellent agreement between simulations and experimental measurements on liquid and solid helium for such quantities as pair correlations, the superfluid density, the energy, and the momentum distribution. Major aspects of computational techniques developed for a boson superfluid are discussed: the construction of more accurate approximate density matrices to reduce the number of points on the path integral, sampling techniques to move through the space of exchanges and paths quickly, and the construction of estimators for various properties such as the energy, the momentum distribution, the superfluid density, and the exchange frequency in a quantum crystal. Finally the path-integral Monte Carlo method is compared to other quantum Monte Carlo methods.

Path integrals in the theory of condensed helium

Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

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Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.

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Conformal invariance, the central charge, and universal finite-size amplitudes at criticality.

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

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Percolation, statistical topography, and transport in random media

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one “effectively independent” sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents areυ=0.7496±0.0007 ind=2 andυ= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200⩽N⩽10000 and 200⩽N⩽ 3000, respectively.

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the Ashkin-Teller model, the Coulomb-gas representation has added significantly to the insight in the phase transitions. For other models, notably theXY model and the clock models, the equivalence has been instrumental for almost our entire understanding of the critical behavior. Recently it was shown that theq-state Potts model and then-vector model are equivalent to a Coulomb gas with an asymmetry between positive and negative charges. Fieldlike operators in these spin models transform noninteger charges and magnetic monopoles. With the aid of exactly solved models the Coulombgas representation allows analytic calculation of some critical indices.

Bernard Nienhuis

Papers

Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas