A novel method to evaluate real-time path integral for scalar $\phi^4$ theory

Abstract

We present a new scheme which numerically evaluates the real-time path integral for ϕ 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 Cauchy's 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 ϕ 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.

Critical endpoint of (3+1)-dimensional finite density ℤ3 gauge-Higgs model with tensor renormalization group

“…by introducing the following impurity tensor, S n = 1 6 S [12] n + S [13] n + S [14] n + S [23] n + S [24] n + S [34] n .…”

“…The last decade was devoted to an initial stage to apply the tensor renormalization group (TRG) method, 1 which was originally proposed to study two-dimensional (2d) classical spin systems in the field of condensed matter physics [1], to quantum field theories consisting of scalar, fermion, and gauge fields [11][12][13]. There were many attempts to confirm or utilize the following expected advantages of the TRG method employing the lower-dimensional models: (i) no sign problem [4,[14][15][16][17][18][19][20][21][22][23][24], (ii) logarithmic computational cost on the system size, (iii) direct manipulation of the Grassmann variables [2,4,5,16,19,[25][26][27], (iv) evaluation of the partition function or the path-integral itself.…”

Metal–insulator transition in the (2+1)-dimensional Hubbard model with the tensor renormalization group

“…The last decade was devoted to an initial stage to apply the tensor renormalization group (TRG) method 1 , which was originally proposed to study two-dimensional (2d) classical spin systems in the field of condensed matter physics [1], to quantum field theories consisting of scalar, fermion, and gauge fields [11][12][13]. There were many attempts to confirm or utilize the following expected advantages of the TRG method employing the lowerdimensional models: (i) no sign problem [4,[14][15][16][17][18][19][20][21][22][23][24], (ii) logarithmic computational cost on the system size, (iii) direct manipulation of the Grassmann variables [2,4,5,16,19,[25][26][27], (iv) evaluation of the partition function or the path-integral itself.…”