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arXiv:1912.02780 · Physical review. B./Physical review. B

Automatic differentiation for second renormalization of tensor networks

Bin-Bin Chen, Yuan Gao, Yibin Guo, Yuzhi Liu A, Hui‐Hai Zhao, Haijun Liao, Lei Wang, Tao Xiang, Wei Li, Z. Y. Xie

Published Jun 23, 2020• 32 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.101.220409

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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∂TRG\ensuremath{\partial}\mathrm{TRG}\ensuremath∂TRG) that can be applied to improve various TRG methods, in an automatic fashion. \ensuremath∂TRG\ensuremath{\partial}\mathrm{TRG}\ensuremath∂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∂TRG\ensuremath{\partial}\mathrm{TRG}\ensuremath∂TRG automatically finds the gradient of local tensors through backpropagation, with which one can deeply ``train'' the tensor networks. We benchmark \ensuremath∂TRG\ensuremath{\partial}\mathrm{TRG}\ensuremath∂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∂TRG\ensuremath{\partial}\mathrm{TRG}\ensuremath∂TRG a highly efficient and accurate many-body computation approach.
Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and optimizes them using gradient search.The concept emerges from deep learning but is not limited to training neural networks.We present the theory and practice of programming tensor network algorithms in a fully differentiable way.By formulating the tensor network algorithm as a computation graph, one can compute higher-order derivatives of the program accurately and efficiently using automatic differentiation.We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed-point iterations.As a demonstration, we compute the specific heat of the Ising model directly by taking the second-order derivative of the free energy obtained in the tensor renormalization group calculation.Next, we perform gradient-based variational optimization of infinite projected entangled pair states for the quantum antiferromagnetic Heisenberg model and obtain state-ofthe-art variational energy and magnetization with moderate efforts.Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.
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We propose a real-space renormalization group algorithm for accurately coarse-graining two-dimensional tensor networks. The central innovation of our method lies in utilizing variational boundary tensors as a globally optimized environment for the entire system. Based on this optimized environment, we construct renormalization projectors that significantly enhance accuracy. By leveraging the canonical form of tensors, our algorithm maintains the same computational complexity as the original tensor renormalization group (TRG) method, yet achieves higher accuracy than existing approaches that do not incorporate entanglement filtering. Our work offers a practical pathway for extending TRG methods to higher dimensions while keeping computational costs manageable.
We propose a method to construct a tensor network representation of partition functions without singular value decompositions nor series expansions. The approach is demonstrated for one- and two-dimensional Ising models and we study the dependence of the tensor renormalization group (TRG) on the form of the initial tensors and their symmetries. We further introduce variants of several tensor renormalization algorithms. Our benchmarks reveal a significant dependence of various TRG algorithms on the choice of initial tensors and their symmetries. However, we show that the boundary TRG technique can eliminate the initial tensor dependence for all TRG methods. The numerical results of TRG calculations can thus be made significantly more robust with only a few changes in the code. Furthermore, we study a three-dimensional Z 2 gauge theory without gauge fixing and confirm the applicability of the initial tensor construction. Our method can straightforwardly be applied to systems with longer range and multisite interactions, such as the next-nearest neighbor Ising model. Published by the American Physical Society 2024
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