Cluster Monte Carlo Algorithms for Dissipative Quantum Systems

We review efficient Monte Carlo methods for simulating quantum systems which couple to a dissipative environment. A brief introduction of the Caldeira-Leggett model and the Monte Carlo method will be followed by a detailed discussion of cluster algorithms and the treatment of long-range interactions. Dissipative quantum spins and resistively shunted Josephson junctions will be considered.Comment: to be publushed in Proceedings of the Yukawa Symposium 200

Recent developments in Quantum Monte-Carlo simulations with applications for cold gases

This is a review of recent developments in Monte Carlo methods in the field of ultra cold gases. For bosonic atoms in an optical lattice we discuss path integral Monte Carlo simulations with worm updates and show the excellent agreement with cold atom experiments. We also review recent progress in simulating bosonic systems with long-range interactions, disordered bosons, mixtures of bosons, and spinful bosonic systems. For repulsive fermionic systems determinantal methods at half filling are sign free, but in general no sign-free method exists. We review the developments in diagrammatic Monte Carlo for the Fermi polaron problem and the Hubbard model, and show the connection with dynamical mean-field theory. We end the review with diffusion Monte Carlo for the Stoner problem in cold gases.Comment: 68 pages, 22 figures, review article; replaced with published versio

Existence of a Thermodynamic Spin-Glass Phase in the Zero-Concentration Limit of Anisotropic Dipolar Systems

The nature of ordering in dilute dipolar interacting systems dates back to the work of Debye and is one of the most basic, oldest and as-of-yet unsettled problems in magnetism. While spin-glass order is readily observed in several RKKY-interacting systems, dipolar spin-glasses are subject of controversy and ongoing scrutiny, e.g., in ${{\rm LiHo_xY_{1-x}F_4}}$, a rare-earth randomly diluted uniaxial (Ising) dipolar system. In particular, it is unclear if the spin-glass phase in these paradigmatic materials persists in the limit of zero concentration or not. We study an effective model of ${{\rm LiHo_xY_{1-x}F_4}}$ using large-scale Monte Carlo simulations that combine parallel tempering with a special cluster algorithm tailored to overcome the numerical difficulties that occur at extreme dilutions. We find a paramagnetic to spin-glass phase transition for all Ho ion concentrations down to the smallest concentration numerically accessible of 0.1%, and including Ho ion concentrations which coincide with those studied experimentally up to 16.7%. Our results suggest that randomly-diluted dipolar Ising systems have a spin-glass phase in the limit of vanishing dipole concentration, with a critical temperature vanishing linearly with concentration, in agreement with mean field theory.Comment: 6 pages, 3 figures, 2 table

Computing quantum phase transitions

This article first gives a concise introduction to quantum phase transitions, emphasizing similarities with and differences to classical thermal transitions. After pointing out the computational challenges posed by quantum phase transitions, a number of successful computational approaches is discussed. The focus is on classical and quantum Monte Carlo methods, with the former being based on the quantum-to classical mapping while the latter directly attack the quantum problem. These methods are illustrated by several examples of quantum phase transitions in clean and disordered systems.Comment: 99 pages, 15 figures, submitted to Reviews in Computational Chemistr

Stochastic series expansion method for quantum Ising models with arbitrary interactions

A quantum Monte Carlo algorithm for the transverse Ising model with arbitrary short- or long-range interactions is presented. The algorithm is based on sampling the diagonal matrix elements of the power series expansion of the density matrix (stochastic series expansion), and avoids the interaction summations necessary in conventional methods. In the case of long-range interactions, the scaling of the computation time with the system size N is therefore reduced from N^2 to Nln(N). The method is tested on a one-dimensional ferromagnet in a transverse field, with interactions decaying as 1/r^2.Comment: 9 pages, 5 figure

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions

We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a d-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states and using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical schemes that are accompanied with a posteriori error estimates for coarse-grained lattice systems with short and long-range interactions.Comment: 31 pages, 2 figure

Monte Carlo Simulations of Doped, Diluted Magnetic Semiconductors - a System with Two Length Scales

We describe a Monte Carlo simulation study of the magnetic phase diagram of diluted magnetic semiconductors doped with shallow impurities in the low concentration regime. We show that because of a wide distribution of interaction strengths, the system exhibits strong quantum effects in the magnetically ordered phase. A discrete spin model, found to closely approximate the quantum system, shows long relaxation times, and the need for specialized cluster algorithms for updating spin configurations. Results for a representative system are presented.Comment: 12 pages, latex, 7 figures; submitted to International Journal of Modern Physics C, Proceedings of the U.S.-Japan Bilateral Seminar: Understanding and Conquering Long Time Scales in Computer Simulation