Tensor Network Formulation of $\mathcal{PT}$-Symmetric Quantum Field Theory

We develop a non-perturbative analytic tensor network formulation of $\mathcal{PT}$-symmetric scalar field theories defined on complex integration contours. Applying this formulation to the two-dimensional $\mathcal{PT}$-symmetric $\phi^4$ theory at negative quartic coupling, we derive an explicit analytic expression for the initial tensor and show that its components separate into even and odd sectors according to the parity of the sum of the tensor indices. We further formulate the theory on an alternative complex contour and analytically establish, in arbitrary dimensions, an exact finite-volume relation between the lattice partition function defined on this contour and the real part of the analytic continuation of the Hermitian lattice $\phi^4$ partition function to negative quartic coupling.