arxiv: 2604.06922 · v3 · submitted 2026-04-08 · ❄️ cond-mat.str-el · cond-mat.stat-mech· cs.MS· quant-ph

Recognition: no theorem link

A Practical Introduction to Tensor Network Renormalization with TNRKit.jl

Victor Vanthilt , Adwait Naravane , Chenqi Meng , Atsushi Ueda

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Pith reviewed 2026-05-10 17:57 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechcs.MSquant-ph

keywords tensor network renormalizationJulia softwareconformal data extractionfixed point tensorsstatistical mechanicscritical phenomenaTRG algorithm

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The pith

TNRKit extracts universal conformal data directly from fixed-point tensors in tensor network renormalization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents TNRKit, a Julia package that implements tensor network renormalization methods for two- and three-dimensional models. The package allows users to build tensor representations of partition functions and apply coarse-graining algorithms such as TRG, HOTRG, and LoopTNR while respecting symmetries. From the resulting fixed-point tensors, it extracts conformal data like scaling dimensions and the central charge. Such extraction provides a way to access critical properties without full simulation of the system.

Core claim

TNRKit is a symmetry-aware framework for constructing tensor-network representations of partition functions and coarse-graining them, enabling the extraction of universal conformal data including scaling dimensions and the central charge directly from fixed-point tensors.

What carries the argument

Fixed-point tensors obtained after coarse-graining, whose spectrum encodes the scaling dimensions and central charge of the model.

If this is right

Thermodynamic quantities can be calculated from the same tensor networks used for conformal data.
The framework supports benchmarking of different renormalization algorithms like TRG and HOTRG.
New tensor renormalization methods can be developed and tested within the extensible package structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Extending the package to quantum models could bridge classical and quantum critical phenomena studies.
Fixed-point extraction might help in identifying relevant operators in renormalization group flows for lattice field theories.

Load-bearing premise

The coarse-graining steps must produce tensors that accurately capture the universal properties in their eigenvalues without being dominated by numerical artifacts.

What would settle it

Running the package on the two-dimensional Ising model and finding that the extracted lowest scaling dimension is not close to 0.125 would indicate that the fixed-point data does not reliably encode the conformal spectrum.

Figures

Figures reproduced from arXiv: 2604.06922 by Adwait Naravane, Atsushi Ueda, Chenqi Meng, Victor Vanthilt.

**Figure 1.** Figure 1: (a) Matrix representation of the 1D partition function. (b) Checkerboard encoding of a 2D partition function. The blue circles and red squares denote the Boltzmann tensors and the spin degrees of freedom to be traced out, respectively. networks [27,28], which is the subject of this section, and then later use TRG and TNR methods to approximately – but accurately – evaluate this tensor network. If both of t… view at source ↗

**Figure 2.** Figure 2: Schematic illustration of constructing the initial tensors via character ex [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Block structure of the classical Ising tensor in the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of a two-dimensional tensor network contraction. As more [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Levin–Nave TRG. Each four-leg tensor on the square lattice is split into two [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Combination step of the Levin-Nave TRG algorithm. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Traced partition function tensor used to normalise the tensor and calculate [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: The HOTRG algorithm combines two tensors [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: Diagrams explaining the two ways to generate coarse-graining isometries [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

**Figure 10.** Figure 10: Coarse-graining steps in the HOTRG algorithm [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

**Figure 11.** Figure 11: The CDL tensors after TRG. The local loop, marked by a red square, [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

**Figure 12.** Figure 12: Geometry of the tube. We use double arrows to indicate the identification [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗

**Figure 13.** Figure 13: Accuracy vs inverse temperature for TRG (blue), HOTRG (red) and BTRG [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗

**Figure 14.** Figure 14: CFT spectrum throughout coarse graining for TRG, BTRG, LoopTNR and [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗

**Figure 15.** Figure 15: The central charge and Luttinger parameter of the six-vertex model ob [PITH_FULL_IMAGE:figures/full_fig_p033_15.png] view at source ↗

**Figure 16.** Figure 16: LoopTNR results for the single-flavour Gross-Neveu model. [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗

read the original abstract

We present TNRKit, an open-source Julia package for Tensor Network Renormalization (TNR) of two- and three-dimensional classical statistical models and Euclidean lattice field theories. Built on top of TensorKit, it provides a symmetry-aware framework for constructing tensor-network representations of partition functions and coarse-graining them using methods such as TRG, HOTRG, and LoopTNR. Beyond thermodynamic quantities, the package enables the extraction of universal conformal data -- including scaling dimensions and the central charge -- directly from fixed-point tensors. TNRKit is designed with both usability and extensibility in mind, offering a practical platform for applying, benchmarking, and developing modern tensor renormalization algorithms. This paper also serves as a self-contained introduction to the TNR framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

TNRKit is a practical new Julia package bundling TRG/HOTRG/LoopTNR with TensorKit symmetries and conformal-data extraction, mainly an engineering and documentation effort rather than new physics.

read the letter

TNRKit packages existing tensor-network renormalization routines into an open-source Julia library built on TensorKit. It handles symmetries during coarse-graining and adds a workflow to read scaling dimensions and central charge straight from the fixed-point tensors. The paper also functions as a compact introduction to applying these methods to classical statistical models and Euclidean lattice theories in two and three dimensions. That combination of code, symmetry support, and extraction tools is the actual new artifact here. The authors have put real work into making the framework usable and extensible, which lowers the barrier for people who want to benchmark or extend these algorithms without writing everything from scratch. The emphasis on both thermodynamic quantities and universal data extraction is a sensible design choice for the target users. The core algorithms themselves are not new, but the integrated, symmetry-aware implementation with conformal readout is. On the downside, the extraction step assumes that the retained singular values after repeated truncations still capture the universal operators cleanly. Any residual truncation error or incomplete symmetry enforcement could shift the eigenvalues and spoil the scaling dimensions or central-charge estimates. The paper needs to show explicit benchmarks on models with known exact conformal data to demonstrate that the numbers come out accurate and stable under reasonable bond-dimension choices. Without those checks, the claim that the package directly yields universal data remains plausible but unproven in the text. This work is aimed at condensed-matter researchers who already use or want to try tensor-network methods in Julia. It will be most valuable to practitioners looking for a documented starting point rather than to theorists seeking new analytic insights. The contribution is solid enough on the software side to warrant peer review, even if the numerical validation section will likely need strengthening.

Referee Report

1 major / 2 minor

Summary. The manuscript presents TNRKit.jl, an open-source Julia package built on TensorKit for tensor network renormalization of 2D and 3D classical statistical models and Euclidean lattice field theories. It implements coarse-graining routines including TRG, HOTRG, and LoopTNR to compute thermodynamic quantities from tensor-network representations of partition functions, and claims to enable direct extraction of universal conformal data (scaling dimensions and central charge) from the resulting fixed-point tensors. The paper also functions as a self-contained practical introduction to the TNR framework, with emphasis on usability, extensibility, and symmetry awareness.

Significance. If the numerical claims hold, the package would provide a reproducible, symmetry-aware platform that lowers the barrier for applying modern TNR methods to critical phenomena and conformal data extraction. The open-source release and tutorial-style presentation are strengths that could aid benchmarking of new algorithms and promote reproducible research in condensed-matter and statistical physics.

major comments (1)

[Abstract] Abstract: the central claim that universal conformal data (scaling dimensions and central charge) can be extracted directly from fixed-point tensors produced by the implemented TRG/HOTRG/LoopTNR routines is load-bearing. The manuscript must demonstrate that bond-dimension truncation and any symmetry-breaking errors do not distort the retained eigenvalues/vectors; without explicit benchmarks against exact results (e.g., Ising model) or error analysis of the conformal-data extraction step, the universality of the extracted quantities remains unverified.

minor comments (2)

The manuscript would benefit from including at least one fully worked numerical example (with code) that reproduces a known central charge or scaling dimension, to substantiate the extraction claim.
Ensure consistent notation for tensor indices and singular-value spectra across the description of the coarse-graining routines and the conformal-data extraction procedure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of verifying the conformal data extraction procedure. We address the major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that universal conformal data (scaling dimensions and central charge) can be extracted directly from fixed-point tensors produced by the implemented TRG/HOTRG/LoopTNR routines is load-bearing. The manuscript must demonstrate that bond-dimension truncation and any symmetry-breaking errors do not distort the retained eigenvalues/vectors; without explicit benchmarks against exact results (e.g., Ising model) or error analysis of the conformal-data extraction step, the universality of the extracted quantities remains unverified.

Authors: We agree that the claim in the abstract requires explicit support. The manuscript already contains numerical demonstrations for the 2D Ising model in which scaling dimensions (including the leading magnetic exponent 1/8) and the central charge are extracted from the fixed-point tensors obtained via TRG and LoopTNR and compared against known exact values. To strengthen this further and directly address the referee's concern, we will add a dedicated subsection on error analysis. This will include (i) systematic variation of bond dimension, (ii) quantification of how truncation affects the retained eigenvalues and eigenvectors, and (iii) explicit checks that symmetry sectors remain correctly preserved under the TensorKit implementation. These additions will be placed in the section describing conformal-data extraction and will be accompanied by additional tables and figures. revision: yes

Circularity Check

0 steps flagged

No circularity: software package implements established TNR methods

full rationale

The paper introduces TNRKit.jl as an open-source implementation of standard tensor network renormalization algorithms (TRG, HOTRG, LoopTNR) on top of TensorKit. It enables extraction of conformal data such as scaling dimensions and central charge from fixed-point tensors using well-known procedures in the TNR literature. No new theoretical derivations, predictions, or uniqueness theorems are claimed that reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The package is presented as a computational tool and self-contained introduction, with its functionality depending on external libraries and established methods rather than any self-referential chain within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software-package paper rather than a theoretical derivation; no new physical free parameters, axioms, or invented entities are introduced beyond the standard tensor-network renormalization framework.

pith-pipeline@v0.9.0 · 5443 in / 1034 out tokens · 44283 ms · 2026-05-10T17:57:52.637351+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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