Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1905.02351 · Physical review. B./Physical review. B

Boundary tensor renormalization group

Shumpei Iino, Satoshi Morita, Naoki Kawashima

Published Jul 31, 2019• 31 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.100.035449

Related papers

High-precision thermodynamic and critical properties from tensor renormalization-group flows

Michael Hinczewski, A. Nihat Berker

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.

arXiv:0709.2803•Physical Review E•Jan 7, 2008•14 citations

Boundary conformal spectrum and surface critical behavior of classical spin systems: A tensor network renormalization study

Shumpei Iino, Satoshi Morita, Naoki Kawashima

We numerically obtain the conformal spectrum of several classical spin models on a two-dimensional lattice with open boundaries for every boundary fixed point obtained by the Cardy's derivation [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)]. In order to extract accurate conformal data, we implement the tensor network renormalization algorithm [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] extended so as to be applicable to a square lattice with open boundaries. We successfully compute the boundary conformal spectrum consistent with the underlying boundary conformal field theories (BCFTs) for the Ising, tricritical Ising, and three-state Potts models on the lattice, which allows us to confirm the validity of the BCFT analyses for the surface critical behaviors of those lattice models.

arXiv:1911.09907•Physical review. B./Physical review. B•Apr 20, 2020•15 citations

Bond-weighted tensor renormalization group

Daiki Adachi, Tsuyoshi Okubo, Synge Todo

We propose an improved tensor renormalization-group (TRG) algorithm, the bond-weighted TRG (BTRG). In BTRG, we generalize the conventional TRG by introducing bond weights on the edges of the tensor network. We show that BTRG outperforms the conventional TRG and the higher-order tensor renormalization group with the same bond dimension, whereas its computation time is almost the same as that of TRG. Furthermore, BTRG can have nontrivial fixed-point tensors at an optimal hyperparameter. We demonstrate that the singular value spectrum obtained by BTRG is invariant under the renormalization procedure in the case of the two-dimensional Ising model at the critical point. This property indicates that BTRG performs the tensor contraction with high accuracy whereas keeping the scale-invariant structure of tensors.

arXiv:2011.01679•Physical review. B./Physical review. B•Feb 3, 2022•39 citations

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

Katsumasa Nakayama, Manuel Schneider

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

arXiv:2407.14226•Physical review. D/Physical review. D.•Nov 4, 2024•3 citations

Fixed-point tensor network for compactified boson conformal field theory

Gong Cheng, Dong-Yu Bao, Zheng‐Cheng Gu

Fixed-point (FP) tensor networks provide a discrete spacetime representation of conformal field theories (CFTs), offering a new route toward understanding holographic duality, generalized symmetries, and even quantum gravity. In this work, we construct FP tensors for the 2D compactified boson theory at a generic compactification radius, an archetypal irrational CFT, using boundary (open-string) data with conformal boundary conditions. We show that the resulting tensors reproduce the closed-string spectrum with high accuracy and generate stable renormalization-group (RG) flows under the tensor complex renormalization algorithm. Moreover, we identify a controllable exactly marginal deformation at the level of a single tensor, enabling flows that move continuously along the $c=1$ moduli space. This framework establishes a concrete lattice-level route toward describing a broad class of 2D irrational CFTs.

arXiv:2607.03534•Jul 3, 2026•0 citations

Entanglement filtering and improved coarse-graining on two dimensional tensor networks including fermions

Ryo Sakai, Muhammad Asaduzzaman, Simon M. Catterall +2 more

Tensor renormalization group (TRG) has attractive features like the absence of sign problems and the accessibility to the thermodynamic limit, and many applications to lattice field theories have been reported so far. However it is known that the TRG has a fictitious fixed point that is called the CDL tensor and that causes less accurate numerical results. There are improved coarse-graining methods that attempt to remove the CDL structure from tensor networks. Such approaches have been shown to be beneficial on two dimensional spin systems. We discuss how to adapt the removal of the CDL structure to tensor networks including fermions, and numerical results that contain some comparisons to the plain TRG, where significant differences are found, will be shown. The detailed discussion of this work is given in ref. [1].

International Symposium on Lattice Field Theory (LATTICE)•Feb 1, 2023•0 citations
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.
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.
We numerically obtain the conformal spectrum of several classical spin models on a two-dimensional lattice with open boundaries for every boundary fixed point obtained by the Cardy's derivation [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)]. In order to extract accurate conformal data, we implement the tensor network renormalization algorithm [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] extended so as to be applicable to a square lattice with open boundaries. We successfully compute the boundary conformal spectrum consistent with the underlying boundary conformal field theories (BCFTs) for the Ising, tricritical Ising, and three-state Potts models on the lattice, which allows us to confirm the validity of the BCFT analyses for the surface critical behaviors of those lattice models.
We propose an improved tensor renormalization-group (TRG) algorithm, the bond-weighted TRG (BTRG). In BTRG, we generalize the conventional TRG by introducing bond weights on the edges of the tensor network. We show that BTRG outperforms the conventional TRG and the higher-order tensor renormalization group with the same bond dimension, whereas its computation time is almost the same as that of TRG. Furthermore, BTRG can have nontrivial fixed-point tensors at an optimal hyperparameter. We demonstrate that the singular value spectrum obtained by BTRG is invariant under the renormalization procedure in the case of the two-dimensional Ising model at the critical point. This property indicates that BTRG performs the tensor contraction with high accuracy whereas keeping the scale-invariant structure of tensors.
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
Fixed-point (FP) tensor networks provide a discrete spacetime representation of conformal field theories (CFTs), offering a new route toward understanding holographic duality, generalized symmetries, and even quantum gravity. In this work, we construct FP tensors for the 2D compactified boson theory at a generic compactification radius, an archetypal irrational CFT, using boundary (open-string) data with conformal boundary conditions. We show that the resulting tensors reproduce the closed-string spectrum with high accuracy and generate stable renormalization-group (RG) flows under the tensor complex renormalization algorithm. Moreover, we identify a controllable exactly marginal deformation at the level of a single tensor, enabling flows that move continuously along the c=1c=1c=1 moduli space. This framework establishes a concrete lattice-level route toward describing a broad class of 2D irrational CFTs.
Tensor renormalization group (TRG) has attractive features like the absence of sign problems and the accessibility to the thermodynamic limit, and many applications to lattice field theories have been reported so far. However it is known that the TRG has a fictitious fixed point that is called the CDL tensor and that causes less accurate numerical results. There are improved coarse-graining methods that attempt to remove the CDL structure from tensor networks. Such approaches have been shown to be beneficial on two dimensional spin systems. We discuss how to adapt the removal of the CDL structure to tensor networks including fermions, and numerical results that contain some comparisons to the plain TRG, where significant differences are found, will be shown. The detailed discussion of this work is given in ref. [1].