Scattering and effective interactions of ultracold atoms with spin-orbit coupling

We derive an analytical expression for the scattering amplitude of two ultracold atoms of arbitrary spin and with general spin-orbit (SO) coupling, on the basis of our recent work (Phys. Rev. A \textbf{86}, 053608 (2012)). As an application, we demonstrate that SO coupling can induce scattering resonance in the case with finite scattering length. The same approach can be applied to calculate the two-body bound state of SO-coupled ultracold atoms. For the ultracold spin-1/2 fermi gases in three- or two- dimensional systems with SO coupling, we also obtain the renormalization relation of effective contact interaction with momentum cutoff, as well as the applicability of Huang-Yang pseudo-potential.Comment: 12 pages, 2 figure

Finite-size scaling in the canonical ensemble

We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent $y_1$ that depends on the temperature exponent $y_t$ and the dimensionality $d$ as $y_1=-|2y_t-d|$. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.Comment: 3 figures, 1 tabl

Ising-like phase transition of an n-component Eulerian face-cubic model

By means of Monte Carlo simulations and a finite-size scaling analysis, we find a critical line of an n-component Eulerian face-cubic model on the square lattice and the simple cubic lattice in the region v>1, where v is the bond weight. The phase transition belongs to the Ising universality class independent of n. The critical properties of the phase transition can also be captured by the percolation of the complement of the Eulerian graph.Comment: 6 pages, 6 figures, Phys. Rev. E 88, 052125 (2013

Overlap of two topological phases in the antiferromagnetic Potts model

By controlling the vortex core energy, the three-state ferromagnetic Potts model can exhibit two types of topological paradigms, including the quasi-long-range ordered phase and the vortex lattice phase [PRL 116, 097206 (2016)]. Here, by Monte Carlo simulations using an effective worm algorithm, we show that by controlling the vortex core energy, the antiferromagnetic Potts model can also exhibit the two topological phases, more interestingly, the two topological phases can overlap with each other.Comment: 6 pages, 9 figure

Clique percolation in random graphs

As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two $k$-cliques means that they share at least $l<k$ vertices. In this paper, we develop a theoretical approach to study clique percolation in Erd\H{o}s-R\'{e}nyi graphs, which gives not only the exact solutions of the critical point, but also the corresponding order parameter. Based on this, we prove theoretically that the fraction $\psi$ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction $\phi$ of vertices in the giant clique cluster for $l>1$ makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for $l=1$. More interesting, our analysis shows that at the critical point, the order parameter $\phi_c$ for $l>1$ is neither $0$ nor $1$, but a constant depending on $k$ and $l$. All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.Comment: 6 pages, 5 figure

Adaptive Multi-GPU Exchange Monte Carlo for the 3D Random Field Ising Model

We present an adaptive multi-GPU Exchange Monte Carlo method designed for the simulation of the 3D Random Field Model. The algorithm design is based on a two-level parallelization scheme that allows the method to scale its performance in the presence of faster and GPUs as well as multiple GPUs. The set of temperatures is adapted according to the exchange rate observed from short trial runs, leading to an increased exchange rate at zones where the exchange process is sporadic. Performance results show that parallel tempering is an ideal strategy for being implemented on the GPU, and runs between one to two orders of magnitude with respect to a single-core CPU version, with multi-GPU scaling being approximately $99\%$ efficient. The results obtained extend the possibilities of simulation to sizes of $L = 32, 64$ for a workstation with two GPUs.Comment: 15 pages, 10 figure

A worm algorithm for the fully-packed loop model

We present a Markov-chain Monte Carlo algorithm of worm type that correctly simulates the fully-packed loop model on the honeycomb lattice, and we prove that it is ergodic and has uniform stationary distribution. The fully-packed loop model on the honeycomb lattice is equivalent to the zero-temperature triangular-lattice antiferromagnetic Ising model, which is fully frustrated and notoriously difficult to simulate. We test this worm algorithm numerically and estimate the dynamic exponent z = 0.515(8). We also measure several static quantities of interest, including loop-length and face-size moments. It appears numerically that the face-size moments are governed by the magnetic dimension for percolation.Comment: 31 pages, 10 figures. Several new figures added, and some minor typos corrected. Uses the following latex packages: algorithm.sty, algorithmic.sty, elsart-num.bst, elsart1p.cls, elsart.cl

No-enclave percolation corresponds to holes in the cluster backbone

The no-enclave percolation (NEP) model introduced recently by Sheinman et al. can be mapped to a problem of holes within a standard percolation backbone, and numerical measurements of these holes gives the size-distribution exponent $\tau = 1.82(1)$ of the NEP model. An argument is given that $\tau=1 + d_B/2 \approx 1.822$ where $d_B$ is the backbone dimension. On the other hand, a model of simple holes within a percolation cluster implies $\tau = 1 + d_f/2 = 187/96 \approx 1.948$, where $d_f$ is the fractal dimension of the cluster, and this value is consistent with Sheinman et al.'s experimental results of gel collapse which gives $\tau = 1.91(6)$. Both models yield a discontinuous maximum hole size at $p_c$, signifying explosive percolation behavior. At $p_c$, the largest hole fills exactly half the system, due to symmetry. Extensive numerical simulations confirm our results.Comment: 5 pages, 6 figure

Cluster Simulation of the O(N) loop model on the Honeycomb lattice

We study the O(N) loop model on the Honeycomb lattice with real value $N \geq 1$ by means of a cluster algorithm. The formulation of the algorithm is based on the equivalence of the O(N) loop model and the low-temperature graphical representation of a $N$-color Ashkin-Teller model on the triangular lattice. The latter model with integer $N$ can be simulated by means of an embedding Swendsen-Wang-type cluster method. By taking into account the symmetry among loops of different colors, we develop another version of the Swendsen-Wang-type method. This version allows the number of colors $N$ to take any real value $N \geq 1$. As an application, we investigate the $N=1.25, 1.50, 1.75$, and 2 loop model at criticality. The determined values of various critical exponents are in excellent agreement with theoretical predictions. In particular, from quantities associated with half of the loops, we determine some critical exponents that corresponds to those for the tricritical $q=N^2$ Potts model but have not been observed yet. Dynamic scaling behavior of the algorithm is also analyzed. The numerical data strongly suggest that our cluster algorithm {\it hardly} suffers from critical slowing down.Comment: 35 pages, 19 figures, and 5 table

Recursive Percolation

We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and show that it can be repeated recursively any number $n$ of generations. In two dimensions, we determine the percolation thresholds up to $n=5$. The corresponding critical clusters become more and more compact as $n$ increases, and define universal scaling functions of the standard two-dimensional form and critical exponents that are distinct for any $n$. This family of exponents differs from any previously known universality class, and cannot be accommodated by existing analytical methods. We confirm that recursive percolation is well defined also in three dimensions.Comment: 8 pages, 8 figure