Automatic differentiation for second renormalization of tensor networks
Bin-Bin Chen, Yuan Gao, Yibin Guo, Yuzhi Liu, Hui‐Hai Zhao, Haijun Liao, Lei Wang, Tao Xiang, Wei Li, Z. Y. Xie
Abstract
Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG ($\ensuremath{\partial}\mathrm{TRG}$) that can be applied to improve various TRG methods, in an automatic fashion. $\ensuremath{\partial}\mathrm{TRG}$ systematically extends the essential concept of second renormalization [Phys. Rev. Lett. 103, 160601 (2009)] where the tensor environment is computed recursively in the backward iteration. Given the forward TRG process, $\ensuremath{\partial}\mathrm{TRG}$ automatically finds the gradient of local tensors through backpropagation, with which one can deeply ``train'' the tensor networks. We benchmark $\ensuremath{\partial}\mathrm{TRG}$ in solving the square-lattice Ising model, and we demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The global optimization as well as GPU acceleration renders $\ensuremath{\partial}\mathrm{TRG}$ a highly efficient and accurate many-body computation approach.