We study the two-dimensional classical XY model by the large-scale Monte Carlo simulation of the Swendsen-Wang multi-cluster algorithm using multiple GPUs on the open science supercomputer TSUBAME 2.0. Simulating systems up to the linear system size L = 65536, we investigate the Kosterlitz-Thouless (KT) transition. Using the generalized version of the probability-changing cluster algorithm based on the helicity modulus, we locate the KT transition temperature in a self-adapted way. The obtained inverse KT temperature β KT is 1.11996(6). We estimate the exponent to specify the multiplicative logarithmic correction, −2r, and precisely reproduce the theoretical prediction −2r = 1/8.

We present the GPU calculation with the common unified device architecture (CUDA) for the Swendsen-Wang multi-cluster algorithm of two-dimensional classical spin systems. We adjust the two connected component labeling algorithms recently proposed with CUDA for the assignment of the cluster in the Swendsen-Wang algorithm. Starting with the q-state Potts model, we extend our implementation to the system of vector spins, the q-state clock model, with the idea of embedded cluster. We test the performance, and the calculation time on GTX580 is obtained as 2.51 nano sec per a spin flip for the q = 2 Potts model (Ising model) and 2.42 nano sec per a spin flip for the q = 6 clock model with the linear size L = 4096 at the critical temperature, respectively. The computational speed for the q = 2 Potts model on GTX580 is 12.4 times as fast as the calculation speed on a current CPU core. That for the q = 6 clock model on GTX580 is 35.6 times as fast as the calculation speed on a current CPU core.

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We study q-state clock models of regular and Villain types with q = 5, 6 using cluster-spin updates and observed double transitions in each model. We calculate the correlation ratio and size-dependent correlation length as quantities for characterizing the existence of Berezinskii-Kosterlitz-Thouless (BKT) phase and its transitions by large-scale Monte Carlo simulations. We discuss the advantage of correlation ratio in comparison to other commonly used quantities in probing BKT transition. Using finite size scaling of BKT type transition, we estimate transition temperatures and corresponding exponents. The comparison between the results from both types revealed that the existing transitions belong to BKT universality.