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arXiv:2407.14226 · Physical review. D/Physical review. D.

Initial tensor construction and dependence of the tensor renormalization group on initial tensors

Katsumasa Nakayama, Manuel Schneider

Published Nov 4, 2024• 3 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevd.110.094501

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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
We propose a method to construct the initial tensor representation of partition functions and observables for the tensor renormalization group (TRG). The TRG is a numerical calculation technique that utilizes a tensor network representations of physical quantities to investigate physical properties without encountering the sign problem. To apply the TRG, it is essential to construct a locally connected tensor network suitable for recursive coarse-graining. We present a systematic approach for translating a general tensor representation of the partition function to this form. Furthermore, we show the dependence of TRG algorithms on the choice of the initial tensor network representation and propose an improvement of TRG algorithms in this respect
We propose a method to construct the initial tensor representation of partition functions and observables for the tensor renormalization group (TRG). The TRG is a numerical calculation technique that utilizes a tensor network representations of physical quantities to investigate physical properties without encountering the sign problem. To apply the TRG, it is essential to construct a locally connected tensor network suitable for recursive coarse-graining. We present a systematic approach for translating a general tensor representation of the partition function to this form. Furthermore, we show the dependence of TRG algorithms on the choice of the initial tensor network representation and propose an improvement of TRG algorithms in this respect
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.
Tensor renormalization group (TRG) is a coarse-graining algorithm for approximating the partition function using a tensor network in the field of elementary particle physics.Although the computational cost of TRG can be reduced using a randomized singular value decomposition, its computation time is still large.In this paper, we propose a cost-efficient cutoff method for calculating TRG by truncating small tensor elements.Numerical experiments showed that the proposed method is faster than the conventional one without degrading accuracy.
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.
The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than 10(-5) along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range D) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with D=24, corresponding to 4624-dimensional renormalization-group flows.