Study of tensor renormalization group algorithm toward application to field theory

We develop the tensor renormalization group (TRG) algorithm for statistical systems with open boundaries, which allows us to investigate not only the bulk but also the boundary property, such as the surface magnetization. We demonstrate that the tensors representing the boundary in our algorithm exhibit the fixed-point structures, just as bulk tensors in previous TRG algorithms. At criticality, the scale-invariant boundary fixed-point tensors have the information of the conformal tower, which is described by the underlying boundary conformal field theory.We develop the tensor renormalization group (TRG) algorithm for statistical systems with open boundaries, which allows us to investigate not only the bulk but also the boundary property, such as the surface magnetization. We demonstrate that the tensors representing the boundary in our algorithm exhibit the fixed-point structures, just as bulk tensors in previous TRG algorithms. At criticality, the scale-invariant boundary fixed-point tensors have the information of the conformal tower, which is described by the underlying boundary conformal field theory.

arXiv:1905.02351Physical review. B./Physical review. BJul 31, 201931 citations

Renormalization in tensor Group Field Theory

Fabien Vignes-Tourneret

Mar 17, 20140 citations

Quantum Field Theories with Tensor Renormalization Group

Shinichiro Akiyama, Y. Kuramashi, Yusuke Yoshimura

We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1) $d$ ) complex $\phi^4$ theory at finite density, (1+1) $d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1) $d$ Nambu--Jona-Lasinio model at finite density and (1+1) $d$ and (2+1) $d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)$d$) complex $\phi^4$ theory at finite density, (1+1)$d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1)$d$ Nambu--Jona-Lasinio model at finite density and (1+1)$d$ and (2+1)$d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.

arXiv:2111.04240International Symposium on Lattice Field Theory (LATTICE)May 16, 20224 citations

Renormalization in Tensorial Group Field Theories

Sylvain Carrozza

Frontiers of Fundamental Physics 14 (FFP14)Jan 1, 20140 citations

Density Matrix and Tensor Network Renormalization

Tao Xiang

Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.

Cambridge University Press eBooksAug 31, 202346 citations

Generalization of the tensor renormalization group approach to 3-D or higher dimensions

Peiyuan Teng

arXiv:1605.00062Physica A Statistical Mechanics and its ApplicationsJan 4, 20178 citations

d

) complex

\phi^4

theory at finite density, (1+1)

d

pure U(1) lattice gauge theory with a

\theta

term, (3+1)

d

Nambu--Jona-Lasinio model at finite density and (1+1)

d

and (2+1)

d

Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.