Population annealing is a promising recent approach for Monte Carlo simulations in statistical physics, in particular for the simulation of systems with complex free-energy landscapes. It is a hybrid method, combining importance sampling through Markov chains with elements of sequential Monte Carlo in the form of population control. While it appears to provide algorithmic capabilities for the simulation of such systems that are roughly comparable to those of more established approaches such as parallel tempering, it is intrinsically much more suitable for massively parallel computing. Here, we tap into this structural advantage and present a highly optimized implementation of the population annealing algorithm on GPUs that promises speed-ups of several orders of magnitude as compared to a serial implementation on CPUs. While the sample code is for simulations of the 2D ferromagnetic Ising model, it should be easily adapted for simulations of other spin models, including disordered systems. Our code includes implementations of some advanced algorithmic features that have only recently been suggested, namely the automatic adaptation of temperature steps and a multi-histogram analysis of the data at different temperatures. The program calculates the internal energy, specific heat, several magnetization moments, entropy and free energy of the 2D Ising model on square lattices of edge length L with periodic boundary conditions as a function of inverse temperature β. Solution method: The code uses population annealing, a hybrid method combining Markov chain updates with population control. The code is implemented for NVIDIA GPUs using the CUDA language and employs advanced techniques such as multi-spin coding, adaptive temperature steps and multi-histogram reweighting. Restrictions: The system size and size of the population of replicas are limited depending on the memory of the GPU device used. Unusual features: Additional comments: Code repository at https://github.com/LevBarash/PAising. Running time: For the default parameter values used in the sample programs, L = 64, θ = 100, β0 = 0, β f = 1, ∆β = 0.005, R = 20 000, a typical run time on an NVIDIA Tesla K80 GPU is 151 seconds for the single spin coded (SSC) and 17 seconds for the multi-spin coded (MSC) program (see Sec. 2 for a description of these parameters). PROGRAM SUMMARY

Magnetization processes and phase transitions in a geometrically frustrated triangular lattice Ising antiferromagnet in the presence of an external magnetic field and a random site dilution are studied by the use of an effective-field theory with correlations. We find that the interplay between the applied field and the frustration-relieving dilution results in peculiar phase diagrams in the temperaturefield-dilution parameter space.

The phase transitions occurring in the frustrated Ising square antiferromagnet with first- (J1<0) and second-nearest-neighbor (J2<0) interactions are studied within the framework of the effective-field theory with correlations based on the different cluster sizes and for a wide range of R=J2/|J1|. Despite the simplicity of the model, it has proved difficult to precisely determine the order of the phase transitions. In contrast to the previous effective-field study, we have found a first-order transition line in the region close to R=-0.5 not only between the superantiferromagnetic and paramagnetic (R<-0.5) but also between antiferromagnetic and paramagnetic (R>-0.5) phases.

We use the effective-field theory with correlations based on different cluster sizes to investigate phase diagrams of the frustrated Ising antiferromagnet on the honeycomb lattice with isotropic interactions of the strength J 1 < 0 between nearestneighbour pairs and J 2 < 0 between next-nearest neighbour pairs of spins. We present results for the ground-state energy as a function of the frustration parameter R = J 2 /|J 1 |. We find that the cluster-size has a considerable effect on the existence and location of a tricritical point in the phase diagram at which the phase transition changes from the second order to the first one.

We employ an effective-field theory with correlations in order to study the phase diagram and ground-state magnetizations of a selectively diluted Ising antiferromagnet on triangular and honeycomb lattices. Dilution of different sublattices with generally unequal probabilities results in a rather intricate phase diagram in the sublattice dilution parameters space. In the case of the frustrated triangular lattice antiferromagnet the selective dilution affects the degree of frustration which can lead to some peculiar phenomena, such as reentrant behavior of long-range order or unsaturated sublattice magnetizations at zero temperature. The selectively diluted Ising antiferromagnet on the honeycomb lattice is obtained as a special case when one sublattice of the triangular lattice is completely removed by dilution.

Abstract. The population annealing algorithm is a novel approach to study systems with rough free-energy landscapes, such as spin glasses. It combines the power of simulated annealing, Boltzmann weighted differential reproduction and sequential Monte Carlo process to bring the population of replicas to the equilibrium even in the low-temperature region. Moreover, it provides a very good estimate of the free energy. The fact that population annealing algorithm is performed over a large number of replicas with many spin updates, makes it a good candidate for massive parallelism. We chose the GPU programming using a CUDA implementation to create a highly optimized simulation. It has been previously shown for the frustrated Ising antiferromagnet on the stacked triangular lattice with a ferromagnetic interlayer coupling, that standard Markov Chain Monte Carlo simulations fail to equilibrate at low temperatures due to the effect of kinetic freezing of the ferromagnetically ordered chains. We applied the population annealing to study the case with the isotropic intra-and interlayer antiferromagnetic coupling (J 2 /|J 1 | = −1). The reached ground states correspond to non-magnetic degenerate states, where chains are antiferromagnetically ordered, but there is no long-range ordering between them, which is analogical with Wannier phase of the 2D triangular Ising antiferromagnet.

The pair-approximation method is modified in order to describe systems with geometrical frustration. The Ising antiferromagnet on a triangular lattice with selective dilution (Kaya-Berker model) is considered and a self-consistent thermodynamic description of this model is obtained. For this purpose, the Gibbs free energy as a function of temperature, concentration of magnetic atoms on the selected sublattice, and external magnetic field is derived. In particular, the phase diagram is constructed and a comparison of different methods is presented. The thermodynamic quantities are discussed in the context of their physical validity, and the improvement in the description introduced by the modified method is emphasized.

We study a geometrically frustrated triangular Ising antiferromagnet in an external magnetic field which is selectively diluted with nonmagnetic impurities employing an effective-field theory with correlations and Monte Carlo simulations. We focus on the frustration-relieving effects of such a selective dilution on the phase diagram and find that it can lead to rather intricate phase diagrams in the dilution-field parameters space. In particular, in a highly (weakly) diluted system the frustration is greatly (little) relieved and such a system is found to display only the second(first)-order phase transitions at any field. On the other hand, for a wide interval of intermediate dilution values the transition remains second order at low fields but it changes to first order at higher fields and the system displays a tricritical behavior. The existence of the first-order transition in the region of intermediate dilution and high fields is verified by Monte Carlo simulations.