Complex Langevin Dynamics in Quantum Field Theory

Complex Langevin dynamics offers a stochastic framework that extends the traditional Langevin equation into the complex plane, enabling the numerical simulation of quantum field theories with complex-valued actions. By complexifying field variables and evolving them under a drift term derived from the action’s gradient, the method aims to circumvent the sign problem encountered at finite chemical potential or in real‐time formulations. Under appropriate conditions, the long‐time distribution of complexified configurations reproduces the expectation values of observables in the original theory. Advances such as gauge cooling and adaptive step‐size control have stabilised the algorithm, while correctness criteria based on boundary‐term analysis and spectral properties have provided diagnostics for convergence failures. Applications range from solvable toy models and heavy‐quark approximations of quantum chromodynamics to explorations of phase transitions, real‐time tunnelling and topological effects, demonstrating the global significance of this approach for nonperturbative studies in high‐energy and condensed‐matter settings.

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Technical terms

Complex Langevin dynamics: A stochastic algorithm that evolves field variables in complexified configuration space under a drift derived from the complex action, aiming to reproduce quantum expectation values despite a nonreal weight.

Sign problem: The difficulty arising when the integrand of a path integral oscillates or attains negative values, preventing efficient sampling by standard Monte Carlo methods.

Holomorphic drift: The requirement that the Langevin drift term be complex‐analytic in the field variables to ensure the formal validity of the evolution and the correctness of resulting averages.

Gauge cooling: A numerical procedure applying gauge transformations during Langevin evolution to suppress large excursions into noncompact gauge directions and stabilise the simulation.

References

1. Complex Langevin: Correctness criteria, boundary terms, and spectrum. Physical Review D (2024).
2. Complex Langevin dynamics and zeroes of the fermion determinant. Journal of High Energy Physics (2017).
3. Simulating full QCD at nonzero density using the complex Langevin equation. Physics Letters B (2014).
4. The QCD phase diagram in the limit of heavy quarks using complex Langevin dynamics. Journal of High Energy Physics (2016).