Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
Abstract Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy’s theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
Abstract We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the dynamic part of the path integral which corresponds to the integration over field variables at all later times. This turns the path integral into a two-stage problem where, for each initial condition, there exits one and only one critical point and hence a single thimble in the complex space, whose existence and uniqueness are guaranteed by the characteristics of the initial value problem. In this paper, we test the method for a fully quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We compare the method to solving the Schrödinger equation numerically, and to the classical-statistical approximation, which emerges naturally in a well-defined limit. We find that the Picard-Lefschetz result matches the expectation from quantum mechanics and that, for this application, the classical-statistical approximation does not.
A bstract We present a method to compute real-time path integrals numerically, by Monte-Carlo sampling on near-Lefschetz thimbles. We present a collection of tools based on the Lefschetz thimble methods, which together provide an alternative to existing methods such as the Generalised thimble. These involve a convenient coordinate parameterization of the thimble, direct numerical integration along a radial coordinate into an effective path integral weight and locally deforming the Lefschetz thimble using its Gaussian (non-interacting theory) counterpart in a region about the critical point. We apply this to quantum mechanics, identify possible pitfalls and benefits, and benchmark its efficiency.
Monte Carlo simulations are useful tools for modeling quantum systems, but in some cases they suffer from a sign problem, leading to an exponential slow down in their convergence to a value. While solving the sign problem is generically NP hard, many techniques exist for mitigating the sign problem in specific cases; in particular, the technique of deforming the Monte Carlo simulation's plane of integration onto Lefschetz thimbles (complex hypersurfaces of stationary phase) has seen significant success in the context of quantum field theories. We extend this methodology to spin systems by utilizing spin coherent state path integrals to reexpress the spin system's partition function in terms of continuous variables. Using some toy systems, we demonstrate its effectiveness at lessening the sign problem in this setting, despite the fact that the initial mapping to spin coherent states introduces its own sign problem. The standard formulation of the spin coherent path integral is known to make use of uncontrolled approximations; despite this, for large spins they are typically considered to yield accurate results, so it is somewhat surprising that our results show significant systematic errors. Therefore, possibly of independent interest, our use of Lefschetz thimbles to overcome the intrinsic sign problem in spin coherent state path integral Monte Carlo enables a novel numerical demonstration of a breakdown in the spin coherent path integral.
A bstract We propose first-principle calculations of an open system based on the real-time path integral formalism treating the environment as well as the system of our interest together on a computer. The sign problem that occurs in applying Monte Carlo methods can be overcome in general by using the so-called Lefschetz thimble method, which has been developed over the past decade. Here we focus on the Caldeira-Leggett model, which is well known, in particular, as a model of quantum decoherence. In this case, the calculation simplifies drastically since the path integral becomes Gaussian for typical initial conditions. The relevant saddle point, which is unique and complex, can be determined by solving a linear equation with a huge but sparse coefficient matrix, and the integration over the Lefschetz thimble can be performed analytically. Thus we obtain, without assumptions or approximations, the reduced density matrix after a long-time evolution, tracing out a large number of harmonic oscillators in the environment. In particular, we confirm the dependence of the decoherence time on the coupling constant and the temperature that has been predicted from the master equation in a certain parameter regime.
Real-time path integrals provide a natural formulation of quantum dynamics also in situations where a Hamiltonian description is unavailable or impractical, but their direct evaluation is obstructed by severe phase cancellation from the oscillatory weight.A guiding theme of this thesis is that, after complexifying the integration variables, real-time path integrals are naturally organized by complex saddle points.In favorable cases the original contour can be replaced by steepest-descent manifolds (Lefschetz thimbles) attached to the relevant saddles, providing a controlled saddle-based description of real-time observables.In Part I, we study quantum decoherence in a closed-system real-time path-integral formulation: the total system (subsystem plus environment) evolves unitarily, and the environment is traced out only at the final time to obtain the reduced density matrix.For the Caldeira-Leggett model with Gaussian initial conditions, the discretized effective action becomes a quadratic form, so that the reduced density matrix is characterized by a complex saddle point and the thimble integral reduces to a finite-dimensional Gaussian integral.The complex saddle is obtained by solving a large but sparse linear system, allowing an exact evaluation of the reduced density matrix without relying on the usual master-equation approximations.We then extend this methodology to a three-dimensional cubic lattice toy model by separating center-of-mass and relative coordinates, treating the center-of-mass mode as the subsystem and the relative modes as an effective environment.This enables controlled numerical studies of how the suppression of off-diagonal matrix elements depends on the coupling, inverse temperature, and system size, and allows us to extract the early-time scaling behavior of decoherence diagnostics from the complex-saddle evaluation.In Part II, we investigate spacetime dynamics in the Lorentzian IKKT matrix model.To define a Lorentzian formulation that is not continuously connected to the Euclidean model by the standard contour deformation, we include a Lorentz invariant mass term with coefficient γ, for which the classical equations admit real, smooth, expanding solutions when γ > 0. Since the Lorentz group is noncompact, the path integral can contain a divergent volume from the boost directions; we remove this volume nonperturbatively by implementing a boost "gauge-fixing" that selects the frame minimizing tr(A 0 ) 2 , leading to the condition tr(A 0 A i ) = 0 with the associated Faddeev-Popov determinant.We then apply the complex Langevin method to the resulting complex-weight matrix integral.Time and spatial observables are defined from matrix configurations, with holomorphic time ordering implemented by auxiliary variables, and spatial structure characterized through block observables such as the extent R 2 (t) and the moment-of-inertia tensor.In the bosonic model we find approximately real time and no indication of spontaneous breaking of the spatial symmetry in the i parameter range studied.Including fermions, we control the singular-drift problem induced by the Pfaffian by introducing a fermionic mass regulator m f and an anisotropic mass deformation, which enhances fermionic effects while maintaining numerical stability.In this setup we observe a real, expanding spatial extent and a dynamical dominance of three spatial directions at late times, consistent with a (3 + 1)-dimensional expanding spacetime.We finally discuss the intended limiting procedure-taking the large-N limit first and then approaching the undeformed theory-and comment on future directions, including a comparison with partially supersymmetric deformations that preserve 16 supercharges.