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We discuss the detailed balance condition for hybrid Monte Carlo method
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The diagrammatic Monte Carlo (Diag-MC) method is a numerical technique which samples the entire diagrammatic series of the Greens function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte method and we term the improved version Flat Histogram Diagrammatic Monte Carlo method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Greens function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problem
We examine the sources of error in the histogram reweighting method for Monte Carlo data analysis. We demonstrate that, in addition to the standard statistical error which has been studied elsewhere, there are two other sources of error, one arising through correlations in the reweighted samples, and one arising from the finite range of energies sampled by a simulation of finite length. We demonstrate that while the former correction is usually negligible by comparison with statistical fluctuations, the latter may not be, and give criteria for judging the range of validity of histogram extrapolations based on the size of this latter correction.
We present a method for Monte Carlo sampling on systems with discrete variables (focusing in the Ising case), introducing a prior on the candidate moves in a Metropolis-Hastings scheme which can significantly reduce the rejection rate, called the reduced-rejection-rate (RRR) method. The method employs same probability distribution for the choice of the moves as rejection-free schemes such as the method proposed by Bortz, Kalos and Lebowitz (BKL) [Bortz et al. J.Comput.Phys. 1975]; however, it uses it as a prior in an otherwise standard Metropolis scheme: it is thus not fully rejection-free, but in a wide range of scenarios it is nearly so. This allows to extend the method to cases for which rejection-free schemes become inefficient, in particular when the graph connectivity is not sparse, but the energy can nevertheless be expressed as a sum of two components, one of which is computed on a sparse graph and dominates the measure. As examples of such instances, we demonstrate that the method yields excellent results when performing Monte Carlo simulations of quantum spin models in presence of a transverse field in the Suzuki-Trotter formalism, and when exploring the so-called robust ensemble which was recently introduced in Baldassi et al. [PNAS 2016]. Our code for the Ising case is publicly available [https://github.com/carlobaldassi/RRRMC.jl], and extensible to user-defined models: it provides efficient implementations of standard Metropolis, the RRR method, the BKL method (extended to the case of continuous energy specra), and the waiting time method [Dall and Sibani Comput.Phys.Commun. 2001].
We study the thermodynamics of Ising spins on the triangular kagome lattice (TKL) using exact analytic methods as well as Monte Carlo simulations. We present the free energy, internal energy, specific heat, entropy, sublattice magnetizations, and susceptibility. We describe the rich phase diagram of the model as a function of coupling constants, temperature, and applied magnetic field. For frustrated interactions in the absence of applied field, the ground state is a spin liquid phase with integer residual entropy per spin $s_0/k_B={1/9} ln 72approx 0.4752...$. In weak applied field, the system maps to the dimer model on a honeycomb lattice, with irrational residual entropy 0.0359 per spin and quasi-long-range order with power-law spin-spin correlations that should be detectable by neutron scattering. The power-law correlations become exponential at finite temperatures, but the correlation length may still be long.
We introduce an algorithm for treating growth on surfaces which combines important features of continuum methods (such as the level-set method) and Kinetic Monte Carlo (KMC) simulations. We treat the motion of adatoms in continuum theory, but attach them to islands one atom at a time. The technique is borrowed from the Dielectric Breakdown Model. Our method allows us to give a realistic account of fluctuations in island shape, which is lacking in deterministic continuum treatments and which is an important physical effect. Our method should be most important for problems close to equilibrium where KMC becomes impractically slow.
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