Bosons in the form of ultra cold alkali atoms can be confined to a one dimensional (1d) domain by the use of harmonic traps. This motivates the study of the ground state occupations λi of effective single particle states φi, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The λi and φi are the eigenvalues and eigenfunctions respectively of the one body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations λi are asymptotically proportional to √ N in both the circular and harmonically trapped cases.

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L = 512, and estimate the percolation thresholds to be pc(bond) = 0.248 811 82(10) and pc(site) = 0.311 607 7(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents 1/ν = 1.141 0(15), β/ν = 0.477 05(15), the leading correction exponent yi = −1.2(2), and the shortest-path exponent dmin = 1.375 6(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent ≈ −2, rather than yi ≈ −1.2.

Abstract:The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

Using a stochastic cellular automaton model for urban traffic flow, we study and compare Macroscopic Fundamental Diagrams (MFDs) of arterial road networks governed by different types of adaptive traffic signal systems, under various boundary conditions. In particular, we simulate realistic signal systems that include signal linking and adaptive cycle times, and compare their performance against a highly adaptive system of self-organizing traffic signals which is designed to uniformly distribute the network density. We find that for networks with time-independent boundary conditions, well-defined stationary MFDs are observed, whose shape depends on the particular signal system used, and also on the level of heterogeneity in the system. We find that the spatial heterogeneity of both density and flow provide important indicators of network performance. We also study networks with time-dependent boundary conditions, containing morning and afternoon peaks. In this case, intricate hysteresis loops are observed in the MFDs which are strongly correlated with the density heterogeneity. Our results show that the MFD of the self-organizing traffic signals lies above the MFD for the realistic systems, suggesting that by adaptively homogenizing the network density, overall better performance and higher capacity can be achieved.

We study the dynamic critical behavior of the worm algorithm for the two-and three-dimensional Ising models, by Monte Carlo simulation. The autocorrelation functions exhibit an unusual three-time-scale behavior. As a practical matter, the worm algorithm is slightly more efficient than the Swendsen-Wang algorithm for simulating the two-point function of the three-dimensional Ising model. [6,7] by passing back and forth between the Potts spin representation and the Fortuin -Kasteleyn bond representation [8,9]. In this Letter we shall study another such algorithm, namely, the worm algorithm [10,11], which simulates the high-temperature graphs of the spin model, considered as a statisticalmechanical model in their own right. Surprisingly, no systematic study of the dynamic critical behavior of the worm algorithm has heretofore been carried out, even in the simplest case of the Ising model. As we shall show, the worm algorithm presents some unusual dynamical features, which combine to make it an extraordinarily efficient algorithm for simulating some (but not all) aspects of the three-dimensional Ising model. Indeed, it is surprising (at least to us) that an algorithm based on local (''worm diffusion'') moves could perform so well.We consider the zero-field ferromagnetic Ising model, with nearest-neighbor coupling J > 0, on a connected finite graph G V; E with vertex set V and edge set E. The high-temperature graphs of this model are subsets A

We address a long-standing debate regarding the finite-size scaling of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behaviour previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behaviour of the corresponding Green's function. We numerically demonstrate the universality of our rigorous findings by extensive Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices with free and periodic boundaries.

We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behaviour for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.Finite-size Scaling (FSS) is a fundamental physical theory within statistical mechanics, describing the asymptotic approach to the thermodynamic limit of finite systems in the neighbourhood of a critical phase transi-It is well-known [3] that models of critical phenomena typically possess an upper critical dimension, d c , such that in dimensions d ≥ d c , their thermodynamic behaviour is governed by critical exponents taking simple mean-field values [4]. In contrast to the simplicity of the thermodynamic behaviour, however, the theory of FSS in dimensions above d c is surprisingly subtle, and remains the subject of ongoing debate [5][6][7][8][9][10][11][12]. We will show here that such subtleties can be explained in a simple way, by taking an appropriate geometric perspective.Perhaps the most important class of models in equilibrium statistical mechanics are the n-vector models [13], describing systems of pairwise-interacting unit-vector spins in R n [14]. The cases n = 1, 2, 3 respectively correspond to the Ising, XY and Heisenberg models of ferromagnetism, while the limiting case n = 0 corresponds to the Self-avoiding Walk (SAW) model of polymers [3].The n-vector model has wide-ranging applications in condensed matter physics, particularly in the theory of superfluidity/superconductivity and quantum magnetism. In addition, the case n = 2 is related to the Bose-Hubbard model [15] which is actively studied in the field of ultra-cold atom physics. In such quantum applications, the quantum system in d spatial dimensions is related to the classical model in d + 1 dimensions. Since [3] d c = 4 for the nearest-neighbour n-vector model, this shows that understanding its FSS when d ≥ d c is of importance not only to the theory of FSS itself, but also in the field of condensed matter physics more generally. We also note that the value of d c can be reduced by the introduction of long-range interactions.

We develop cluster algorithms for a broad class of loop models on two-dimensional lattices, including several standard On loop models at n 1. We show that our algorithm has little or no critical slowingdown when 1 n 2. We use this algorithm to investigate the honeycomb-lattice On loop model, for which we determine several new critical exponents, and a square-lattice On loop model, for which we obtain new information on the phase diagram. DOI: 10.1103/PhysRevLett.98.120601 PACS numbers: 05.10.Ln, 05.50.+q, 64.60.Cn, 64.60.Fr From the beginning of the theory of critical phenomena, two models have played a central role: the q-state Potts model [1,2] and the On spin model [3,4]. The parameter q or n is initially a positive integer, but the FortuinKasteleyn (FK) representation [5] and the loop representation [6] show, respectively, how the models can be extended to arbitrary real or even complex values of q and n [7]. In particular, for q, n > 0 the extended model has a probabilistic interpretation as a model of random geometric objects: clusters [8] or loops [9], respectively. These geometric models play a major role in recent developments of conformal field theory [10] via their connection with stochastic Loewner evolution (SLE) [11,12].Since nontrivial models of statistical mechanics are rarely exactly soluble, Monte Carlo simulations have been an important tool for obtaining information on phase diagrams and critical exponents [13]. Unfortunately, Monte Carlo simulations typically suffer from severe critical slowing-down, so that the computational efficiency tends rapidly to zero as the critical point is approached [14]. An important advance was made in 1987 with the invention of the Swendsen-Wang (SW) algorithm [15] for simulating the ferromagnetic Potts model at positive integer q, based on passing back and forth between the spin and FK representations. The SW algorithm does not eliminate critical slowing-down, but it radically reduces it [16]. Since then, many similar ''cluster algorithms'' have been devised, based on this principle [17] of augmenting the original spin model with auxiliary variables and then passing back and forth. But cluster algorithms have traditionally been limited to integer q, since they make essential use of the spin representation.This limitation was first overcome in 1998 by Chayes and Machta [18], who devised a cluster algorithm for simulating the FK random-cluster model at any real q 1. For loop models, by contrast, efficient simulation at noninteger n has remained out of reach; to our knowledge only two Monte Carlo simulations at n Þ 1 have ever been published [19,20], and they used local algorithms. (Instead, numerical transfer-matrix techniques have been employed [21].) As a result, many open questions remain: for instance, the nature of the phase transition is unclear for the n > 3 2 honeycomb-lattice loop model with vacancies [22]; and the phase diagrams and universality classes of loop models on lattices other than honeycomb are largely unexplored [23].In this Letter we sh...