Exact kinetic propagators for coherent state complex Langevin simulations

Abstract

We introduce and benchmark an improved algorithm for complex Langevin simulations of bosonic coherent state path integrals. Our approach utilizes a Strang splitting of the imaginary-time propagator rather than the conventional linear-order Taylor expansion, allowing us to construct an action that incorporates higher-order terms at negligible computational cost. The resulting algorithm enjoys guaranteed linear stability independent of the imaginary-time discretization, enabling more resource-efficient simulations. We demonstrate this improved performance for single-species bosons and for two-component bosons with Rashba spin-orbit coupling.

Figures (4)

• Figure 1: Demonstration of $N_{\tau}$ convergence on a single component, two-dimensional Bose gas with contact interactions $\bar{g}_{\rm 2D} = 0.0165$ at $\bar{T} = \bar{\beta}^{-1} = 2.0$. Convergence plots for the a) particle number as well as the b) intensive canonical internal energy (filled markers) and grand free energy (open markers), showing the standard first order "primitive" treatment with maroon diamonds and the exact one-body propagator method introduced in this work with gold squares. At lower $N_{\tau}$, the missing primitive data is due to numerical instabilities where data cannot be collected. Simulations used the ETD algorithm with $\Delta t = 0.005$ and were conducted in a square cell with side length $\bar{L} = 9.19$ with $N_{x} = 36$ plane waves in each direction.
• Figure 2: Demonstration of $N_{\tau}$ convergence on a two-component, two-dimensional Rashba spin-orbit coupled Bose gas in the stripe phase with contact interactions $\bar{g}_{2D} = 0.1$ at $\bar{T} = \bar{\beta}^{-1} = 1.0$ in immiscible conditions $(\eta = 1.1)$. Convergence plots for the a) particle number as well as the b) intensive canonical internal energy (filled markers) and grand free energy (open markers), showing the standard first order "primitive" treatment with maroon triangles and the exact one-body propagator method with gold circles. At lower $N_{\tau}$, we note missing primitive data due to numerical instabilities where data cannot be collected. Simulations used the ETD algorithm with $\Delta t = 0.01$ and were conducted in a square cell with side length $\bar{L} = 10\pi$ with $N_{x} = 56$ plane waves in each direction.
• Figure 3: Imaginary time dependence of ideal Bose gas simulations. Ensemble-averaged a) particle number and b) internal energy as a function of imaginary time discretization $N_{\tau}$ for the primitive method (red diamonds) and the new exact propagator method (blue squares) introduced in this manuscript. The black solid line denotes the exact thermodynamic reference for the ideal Bose gas, computed using a sum over wavevector basis states with sufficient momentum cutoff. Data from the primitive data below $N_{\tau} = 80$ are not shown due to numerical instability, which prevents meaningful data collection. Simulations consisted of a two-dimensional ensemble of non-interacting bosons with $\bar{g}_{\rm 2D} = 0$, $\ell = \sqrt{\hbar^2 /2m |\mu|} = 4.29$, $\mu = -0.33$ K, $\beta = 0.50$ K$^{-1}$, $L_{x} / \ell = L_{y} /\ell = 11.67$, and $N_{x} = N_{y} = 80$ grid points in each direction. The exponential-time-differencing (ETD) numerical algorithm was used with a Langevin step discretization $\Delta t = 0.0033$. Error bars depict standard errors of the mean, determined during sample averaging.
• Figure 4: Comparison of the results from the main text with mean-field theory references for both the a), b) single-component homogeneous Bose gas and the c), d) Rashba spin-orbit coupled Bose gas in the stripe phase. The mean-field reference for the corresponding observables are plotted in the black dashed line in all cases.