The phase structure of the two dimensional lattice CP(1) model in the presence of the θ term is analyzed by tensor network methods. The tensor renormalization group, which is a standard renormalization method of tensor networks, is used for the regions θ = 0 and θ ≠ 0. Loop-TNR, which is more suitable for the analysis of near criticality, is also implemented for the region θ = 0. The application of Loop-TNR for the region θ ≠ 0 is left for future work.
We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.
Abstract
Tensor network is an attractive approach to the field theory with negative sign problem. The complex ϕ4 theory at finite density is a test bed for numerical algorithms to verify their effectiveness. The model shows a characteristic feature called the Silver Blaze phenomenon associated with the sign problem in the large volume limit at low temperature. We analyze the four-dimensional model employing the anisotropic tensor renormalization group algorithm with a parallel computation. We find a clear signal of the Silver Blaze phenomenon on a large volume of V = 10244, which implies that the tensor network approach is effective even for four-dimensional field theory beyond two dimensions.