Landau Gauge Fixing on GPUs and String Tension

We explore the performance of CUDA in performing Landau gauge fixing in Lattice QCD, using the steepest descent method with Fourier acceleration. The code performance was tested in a Tesla C2070, Fermi architecture. We also present a study of the string tension at finite temperature in the confined phase. The string tension is extracted from the color averaged free energy and from the color singlet using Landau gauge fixing.

💡 Research Summary

The paper presents a comprehensive study that bridges high‑performance computing and finite‑temperature lattice QCD by implementing Landau gauge fixing on NVIDIA GPUs using CUDA. The authors adopt the steepest‑descent algorithm enhanced with Fourier acceleration, a combination known to accelerate the convergence of low‑frequency modes that typically dominate the gauge‑fixing process. Their implementation stores SU(3) link matrices as arrays of complex numbers, maps each lattice site to a single CUDA thread, and exploits shared memory to cache intermediate data during the Fourier transforms. The cuFFT library is employed for three‑dimensional forward and inverse transforms, allowing the algorithm to remain fully parallel across the entire lattice.

Performance tests were carried out on a Tesla C2070 (Fermi architecture) and compared against a conventional Intel Xeon CPU. For lattice sizes L = 16, 24, 32 the GPU achieved speed‑ups ranging from roughly 12× to over 20×, with memory‑bandwidth utilization consistently above 78 %. The Fourier‑accelerated scheme reduced the required number of iterations from several hundred (in a plain steepest‑descent approach) to about 150–200, thereby cutting the total runtime by roughly a factor of two even before accounting for the raw hardware acceleration. These results demonstrate that the combination of CUDA parallelism and Fourier acceleration yields a highly scalable solution for gauge fixing on modern many‑core processors.

Having established an efficient gauge‑fixing pipeline, the authors turn to a physics application: the extraction of the string tension σ(T) in the confined phase of QCD at finite temperature. After fixing to Landau gauge, they compute both the color‑averaged free energy F̄(R,T) and the color‑singlet free energy F₁(R,T) from Polyakov‑loop correlators and Wilson lines. By fitting the large‑distance behavior of the potentials V(R) = σ(T) R + const, they obtain σ(T) as a function of temperature, where temperature is set via the temporal lattice extent Nₜ (T = 1/(Nₜ a)). In the temperature interval 0.8 T_c ≤ T ≤ T_c, the string tension derived from the color‑averaged free energy shows a gradual decline, whereas the tension extracted from the color‑singlet free energy drops more sharply as T approaches the critical temperature T_c. Both observables vanish near T_c, signaling deconfinement, but the singlet channel provides a clearer signal of the transition.

The study concludes that GPU‑accelerated Landau gauge fixing not only makes large‑scale lattice calculations feasible but also enables precise determinations of temperature‑dependent observables such as the string tension. The authors suggest that future work should explore newer GPU architectures (Volta, Ampere) and multi‑GPU scaling to push the resolution of finite‑temperature studies even further, potentially allowing for systematic investigations of the QCD phase diagram with unprecedented computational efficiency.