Simulating full QCD at nonzero density using the complex Langevin equation (1307.7748v2)

Abstract: The complex Langevin method is extended to full QCD at non-zero chemical potential. The use of gauge cooling stabilizes the simulations at small enough lattice spacings. At large fermion mass the results are compared to the HQCD approach, in which the spatial hoppings of fermionic variables are neglected, and good agreement is found. The method allows simulations also at high densities, all the way up to saturation.

• The paper demonstrates that the complex Langevin equation overcomes the sign problem in full QCD simulations at nonzero chemical potential.
• It employs gauge cooling techniques to stabilize the complexified field space and validate results through comparisons with heavy quark QCD benchmarks.
• The findings provide insights into QCD phase transitions by revealing fermion density saturation and the vanishing chiral condensate at high densities.

Simulating QCD at Nonzero Density Using the Complex Langevin Equation

The paper "Simulating full QCD at nonzero density using the complex Langevin equation" by Denes Sexty represents an insightful advance in the simulation of Quantum Chromodynamics (QCD) under conditions of nonzero chemical potential using the complex Langevin method. This work effectively extends the application of the complex Langevin equation (CLE), traditionally limited by the sign problem, to finite density QCD, facilitating the exploration of phase diagrams critical to understanding fundamental interactions in theoretical physics.

Overview of the Methodology

The complex Langevin method is applied to QCD to address the intractable sign problem that arises when employing traditional lattice QCD simulations at nonzero chemical potential. The core challenge is the complexity of the fermion determinant, which becomes a complex number, rendering standard importance sampling ineffectual. By contrast, the CLE operates within an enlarged manifold, specifically a complexified field space, to circumvent this issue. For QCD simulations, this involves treating $SU(N)$ gauge theories within the complexification space $SL(N,\mathbb{C})$.

The stability and accuracy of the CLE are bolstered through a technique known as gauge cooling. This approach leverages the gauge symmetry inherent in the system to maintain the distribution within a localized region of the complexified field space, thus mitigating the potential for non-physical results. This paper demonstrates that the algorithm maintains reliability and accuracy for simulations of full QCD, particularly on lattices with sufficiently small spacings.

Numerical Metrics and Observations

A noteworthy aspect of the paper is the careful comparison made with heavy quark QCD (HQCD) at larger fermion masses, showing a strong agreement in results, and thereby lending credibility to the complex Langevin simulations. HQCD sidesteps the complications of spatial fermion hopping and simplifies the determinant, providing a reliable reference point for CLE simulations.

The paper reports that finite density simulations yield results up to saturation, revealing expected physical characteristics such as the absence of the Silver-Blaze phenomenon at high temperatures and the restoration of the $Z_3$ symmetry in the saturation region. The simulations display significant numerical results: the fermion density increases as a function of chemical potential until it reaches saturation, while the chiral condensate vanishes concomitantly. The analysis of Polyakov loops further corroborates these findings, exhibiting peaks before decaying to zero in accordance with theoretical predictions.

Implications and Future Directions

From a theoretical standpoint, this paper enhances the understanding of finite density QCD by providing a viable path for simulations at high densities that were previously accessible mostly through indirect or approximate methods. Practically, the methodology offers a more direct simulation approach, avoiding the insurmountable difficulties posed by the sign and overlap problems.

Looking toward the future, the paper opens avenues for exploring QCD phase transitions and thermodynamics under extreme conditions that model those found in early universe cosmology or inside neutron stars. Potential developments may include refining the gauge cooling techniques and extending the applicability of complex Langevin dynamics to more intricate lattice setups or other complex systems.

Conclusion

This work establishes the feasibility and reliability of using the complex Langevin equation, supplemented by gauge cooling, for simulating full QCD at nonzero chemical potential. As such, it represents a significant contribution to contemporary lattice field theory, offering a practical methodology for probing QCD under conditions that were previously difficult to access.