High-Dimensional Tensor Renormalization Group with Tensor Train Lattice

Abstract The tensor renormalization group (TRG) is a highly effective method for calculating the free energy of lattice models and has demonstrated exceptional accuracy for several low-dimensional systems. However, its application to higher-dimensional models is often hampered by a large memory footprint and high computational cost. In this paper, we propose a new algorithm, “tt-lattice”, which approximates the tensor at each lattice vertex using a tensor train (tt) decomposition. The algorithm is designed to maintain the tt-format throughout the TRG computation, which significantly reduces both memory usage and computational cost. For a uniform lattice model, the computational cost of tt-lattice scales as $\mathcal {O}(d^2\log (L)\chi ^6)$, with the largest intermediate tensor size of $\mathcal {O}(\chi ^4)$, where d is the lattice dimension, L is the linear size of each lattice axis, and $\chi$ is the cutoff of the number of lattice edge states. To validate our algorithm, we conducted numerical experiments on a Gaussian field model, for which the exact free energy is computable in any lattice dimensions. The results demonstrate that the tt-lattice method can calculate the free energy with reasonable accuracy even for a 10D lattice model with modern computational architectures.