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Register a new userA novel method to evaluate real-time path integral for scalar $phi^4$ theory
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Authors
Shinji Takeda
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We present a new scheme which numerically evaluates the real-time path integral for $phi^4$ real scalar field theory in a lattice version of the closed-time formalism. First step of the scheme is to rewrite the path integral in an explicitly convergent form by applying Cauchys integral theorem to each scalar field. In the step an integration path for the scalar field is deformed on a complex plane such that the $phi^4$ term becomes a damping factor in the path integral. Secondly the integrations of the complexified scalar fields are discretized by the Gauss-Hermite quadrature and then the path integral turns out to be a multiple sum. Finally in order to efficiently evaluate the summation we apply information compression technique using the singular value decomposition to the discretized path integral, then a tensor network representation for the path integral is obtained after integrating the discretized fields. As a demonstration, by using the resulting tensor network we numerically evaluate the time-correlator in 1+1 dimensional system. For confirmation, we compare our result with the exact one at small spatial volume. Furthermore, we show the correlator in relatively large volume using a coarse-graining scheme and verify that the result is stable against changes of a truncation order for the coarse-graining scheme.
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We present a tensor network representation of the path integral for the one-component real scalar field theory in 1+1 dimensional Minkowski space-time. It is numerically verified by comparing with the exact result in the non-interacting case.
The Wilson action for Euclidean lattice gauge theory defines a positive-definite transfer matrix that corresponds to a unitary lattice gauge theory time-evolution operator if analytically continued to real time. Hoshina, Fujii, and Kikukawa (HFK) recently pointed out that applying the Wilson action discretization to continuum real-time gauge theory does not lead to this, or any other, unitary theory and proposed an alternate real-time lattice gauge theory action that does result in a unitary real-time transfer matrix. The character expansion defining the HFK action is divergent, and in this work we apply a path integral contour deformation to obtain a convergent representation for U(1) HFK path integrals suitable for numerical Monte Carlo calculations. We also introduce a class of real-time lattice gauge theory actions based on analytic continuation of the Euclidean heat-kernel action. Similar divergent sums are involved in defining these actions, but for one action in this class this divergence takes a particularly simple form, allowing construction of a path integral contour deformation that provides absolutely convergent representations for U(1) and SU(N) real-time lattice gauge theory path integrals. We perform proof-of-principle Monte Carlo calculations of real-time U(1) and SU(3) lattice gauge theory and verify that exact results for unitary time evolution of static quark-antiquark pairs in (1 + 1)D are reproduced.
We test an alternative proposal by Bruno and Hansen [1] to extract the scattering length from lattice simulations in a finite volume. For this, we use a scalar $phi^4$ theory with two mass nondegenerate particles and explore various strategies to implement this new method. We find that the results are comparable to those obtained from the Luscher method, with somewhat smaller statistical uncertainties at larger volumes.
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 Cauchys 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.
We present an analysis about the influence of an external magnetic field on renormalons in a self interacting theory $lambda phi ^{4}$. In the weak magnetic field region, using an appropriate expansion for the Schwinger propagators, we find new renormalons as poles on the real axis of the Borel plane, whose position do not depend on the magnetic field, but where the residues acquire in fact a magnetic dependence. In the strong magnetic limit, working in the lowest Landau level approximation (LLLA), these new poles are not longer present. We compare the magnetic scenario with previous results in the literature concerning thermal effects on renormalons in this theory.
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