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تسجيل مستخدم جديدTensor network approach to real-time path integral
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Shinji Takeda
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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.
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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 converge
nt 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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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 plan
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