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arXiv:1502.05846 · International Journal of Modern Physics C

Renormalized multicanonical sampling

David O. Yevick

Published Aug 5, 2015• 6 citations
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Abstract

For a homogeneous system divisible into identical, weakly interacting subsystems, the multicanonical procedure can be accelerated if it is first applied to determine the density of states for a single subsystem. This result is then employed to approximate the state density of a subsystem with twice the size that forms the starting point of a new multicanonical iteration. Since this compound subsystem interacts less on average with its environment, iterating this sequence of steps rapidly generates the state density of the full system.

DOI
10.1142/s0129183116500339

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For a homogeneous system divisible into identical, weakly interacting subsystems, the multicanonical procedure can be accelerated if it is first applied to determine the density of states for a single subsystem. This result is then employed to approximate the state density of a subsystem with twice the size that forms the starting point of a new multicanonical iteration. Since this compound subsystem interacts less on average with its environment, iterating this sequence of steps rapidly generates the state density of the full system.
This paper extends to two and three dimensions the recently proposed renormalized multicanonical sampling procedure [D. Yevick, Int. J. Mod. Phys. C 27, 1650033 (2016)IJMPEO0129-183110.1142/S0129183116500339], which generates the density of states of a system from that of a smaller system. The accuracy and efficiency of the method is additionally enhanced through a transition matrix Monte Carlo method that incorporates multicanonical sampling. The performance of the resulting procedure is investigated through calculations of the specific heat of the standard Ising model for square lattices with up to 128^{2} spins and for cubic lattices with up to 24^{3} spins. A comparison of the scaling properties to those of other algorithms based on the transition matrix Monte Carlo method demonstrates that the relative advantage of the renormalized procedure increases rapidly with system size.
The application of the quantum-chemical density-matrix renormalization group (DMRG) algorithm is cumbersome for complex electronic structures with many active orbitals. The high computational cost is mainly due to the poor convergence of standard DMRG calculations. A factor which affects the convergence behavior of the calculations is the choice of the start-up procedure. In this start-up step matrix representations of operators have to be calculated in a guessed many-electron basis of the DMRG environment block. Different possibilities for the construction of these basis states exist, and we first compare four procedures to approximate the environment states using Slater determinants explicitly. These start-up procedures are applied to DMRG calculations on a sophisticated test system: the chromium dimer. It is found that the converged energies and the rate of convergence depend significantly on the choice of the start-up procedure. However, since already the most simple start-up procedure, which uses only the Hartree-Fock determinant, is comparatively good, Slater determinants, in general, appear not to be a good choice as approximate environment basis states for convergence acceleration. Based on extensive test calculations it is demonstrated that the computational cost can be significantly reduced if the number of total states m is successively increased. This is done in such a way that the environment states are built up stepwise from system states of previous truncated DMRG sweeps for slowly increasing m values.
This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, t eq , of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of t eq =0.27(15) L2.80(13) and with the number of energy levels to explore, k E , of [Formula: see text], in perfect agreement with the above result. In addition, a kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on Markov processes.