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Tensor renormalization group with randomized singular value decomposition.

Satoshi Morita1, Ryo Igarashi2, Hui-Hai Zhao3

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Summary

A new tensor renormalization group algorithm, using randomized singular value decomposition, enhances computational efficiency for 2D classical models. This method reduces complexity and memory requirements, even at critical points.

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Area of Science:

  • Computational physics
  • Numerical methods

Background:

  • Tensor renormalization group (TRG) methods are crucial for simulating 2D classical models.
  • Conventional TRG implementations face challenges with computational complexity and memory usage, particularly concerning the bond dimension.
  • Singular value decomposition (SVD) is a key component in TRG algorithms.

Purpose of the Study:

  • To develop a more efficient tensor renormalization group algorithm.
  • To reduce the computational complexity and memory footprint of TRG methods.
  • To explore the applicability of randomized SVD in TRG algorithms.

Main Methods:

  • A novel tensor renormalization group algorithm was developed.
  • The algorithm is based on a randomized singular value decomposition (SVD) approach.
  • The method was applied to various two-dimensional classical models, including the Ising model on a square lattice.

Main Results:

  • The proposed algorithm demonstrates reduced computational complexity (scaling with the fifth power of bond dimension) and memory usage (scaling with the third power) compared to conventional methods (sixth and fourth powers, respectively).
  • An oversampling parameter, larger than the bond dimension, proved sufficient to achieve results comparable to full SVD, even at critical points.
  • The algorithm is broadly applicable to two-dimensional classical models.

Conclusions:

  • The randomized SVD-based TRG algorithm offers significant computational advantages for simulating 2D classical systems.
  • This approach provides a more efficient and scalable method for numerical simulations in statistical mechanics.
  • The findings suggest a promising direction for improving the performance of tensor network algorithms.