Anisotropic Tensor Renormalization Group (ATRG)

• ATRG is a tensor-network method that applies anisotropic local factorization to coarse grain classical lattice models across arbitrary dimensions.
• The method reduces computational cost and memory usage compared to HOTRG by introducing an additional truncation and optimized low-rank approximations.
• Benchmarked in 2D, 3D, and 4D models, ATRG has influenced refinements such as triad representations and GPU acceleration for large-scale simulations.

Anisotropic Tensor Renormalization Group (ATRG) is a real-space tensor-network coarse-graining algorithm for classical lattice models on hypercubic lattices in arbitrary spatial dimension $d$. Introduced as a lower-cost alternative to the Higher-Order Tensor Renormalization Group (HOTRG), it preserves the lattice topology after each renormalization step while reducing computational cost and memory usage by performing an anisotropic local factorization before the final merge of neighboring tensors. Its defining tradeoff is explicit: at fixed bond dimension it is generally less accurate than HOTRG because it introduces an additional truncation, but at fixed computation time it can achieve better accuracy, especially in three and higher dimensions (Adachi et al., 2019).

1. Definition, scope, and conceptual setting

ATRG belongs to the broader family of tensor renormalization group methods in which the partition function is written as a tensor-network contraction,

$Z = \mathrm{tTr}\,\prod_i T^{(i)},$

with one local tensor per lattice site and contraction over all connected bond indices. For a $d$-dimensional hypercubic lattice, the local tensor has $2d$ legs: four in two dimensions, six in three dimensions, and, in general, one incoming and one outgoing leg for each lattice direction. As in HOTRG, neighboring tensors are coarse-grained along one lattice direction at a time, and the renormalization direction is alternated from step to step, so the tensor network remains on the same hypercubic lattice topology after each iteration (Adachi et al., 2019).

The term “anisotropic” refers to the algorithmic treatment of tensor legs, not to anisotropy of the underlying physical model. In a renormalization step along a chosen direction, ATRG first decomposes the local tensors and then compresses the transverse structure through a directionally chosen low-rank factorization. This directional handling breaks the more isotropic truncation structure of HOTRG and does not preserve exact axisymmetry of the renormalized network, but it is precisely what lowers the cost (Adachi et al., 2019).

A later review of tensor-network renormalization emphasizes the same higher-dimensional background—directional coarse graining, local SVD-based truncation, and the rapidly growing computational burden in $3$D and $4$D lattice field theory—that forms the conceptual setting in which ATRG is used (Akiyama, 3 Mar 2026).

2. Local renormalization step and anisotropic factorization

In two dimensions, with a local tensor $T_{y_0 y_1 x_0 x_1}$ and coarse graining along the $y$-direction, ATRG begins with a partial singular value decomposition,

$$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$

From this decomposition the algorithm defines four three-leg tensors,

$$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$

$Z = \mathrm{tTr}\,\prod_i T^{(i)},$0

A notable implementation detail is that the singular values are not split symmetrically as $Z = \mathrm{tTr}\,\prod_i T^{(i)},$1; instead, the full singular values are included in $Z = \mathrm{tTr}\,\prod_i T^{(i)},$2 and $Z = \mathrm{tTr}\,\prod_i T^{(i)},$3. Empirically, this gives the smallest final free-energy error, and the authors argue that it corresponds to the best local approximation because, without the initial truncation, this choice is equivalent to a direct partial SVD of the product $Z = \mathrm{tTr}\,\prod_i T^{(i)},$4 (Adachi et al., 2019).

The additional approximation that distinguishes ATRG from HOTRG appears in the bond-swap stage. One first contracts the shared leg between $Z = \mathrm{tTr}\,\prod_i T^{(i)},$5 and $Z = \mathrm{tTr}\,\prod_i T^{(i)},$6,

$Z = \mathrm{tTr}\,\prod_i T^{(i)},$7

and then approximates $Z = \mathrm{tTr}\,\prod_i T^{(i)},$8 itself by a truncated partial SVD before the final coarse-graining contraction. After this bond swap, the horizontal bonds are compressed with “squeezers” $Z = \mathrm{tTr}\,\prod_i T^{(i)},$9 and $d$0, which are not generally isometries. The renormalized halves are then combined into the new tensor

$d$1

This $d$2 becomes the input for the next step, now taken along the orthogonal lattice direction (Adachi et al., 2019).

ATRG also distinguishes between free-energy calculations and more general observable calculations. If only the free energy is required, the explicit forms of the squeezers are unnecessary: the half-tensors $d$3 and $d$4 can be obtained directly through another partial SVD of an intermediate tensor $d$5. If one wants observables such as energy or magnetization, however, the squeezers themselves are needed. They are chosen to minimize the Frobenius-norm error

$d$6

which makes explicit that ATRG uses general local compression maps rather than strictly orthogonal projectors (Adachi et al., 2019).

3. Cost reduction, memory scaling, and implementation refinements

The central quantitative result of ATRG is its reduction of the leading asymptotic cost relative to HOTRG. For $d$7-dimensional hypercubic lattices, HOTRG scales as

$d$8

in computation time and

$d$9

in memory, whereas ATRG scales as

$2d$0

in computation time and

$2d$1

in memory. The underlying reason is that ATRG decomposes tensors early and applies low-rank approximations to anisotropically chosen intermediate objects, thereby avoiding the materialization of the most expensive high-order tensors that dominate HOTRG in higher dimensions (Adachi et al., 2019).

In two dimensions, full SVD yields an ATRG renormalization cost $2d$2, while partial SVD reduces this to $2d$3, the same leading order as TRG with partial or randomized SVD or projective truncation. The naive memory footprint is $2d$4 because of the intermediate tensor $2d$5, but a factorized implementation reduces this to $2d$6. The original paper also gives a workflow optimization: instead of explicitly decomposing the renormalized tensor $2d$7 again in the next step, one can continue from the SVDs of the previously produced $2d$8 and $2d$9, replacing an $3$0 decomposition of $3$1 by an $3$2 procedure involving the SVDs of $3$3, $3$4, and a two-bond intermediate tensor $3$5 (Adachi et al., 2019).

A dedicated follow-up targets the bottleneck “swapping bonds part” of ATRG. In the original scheme, this step consists of a contraction of two tensors and a partial SVD of a matrix, both with cost $3$6. The proposed reorganization factorizes the large bundled anisotropic indices before the swap, reducing the cost of the swapping part to

$3$7

while reducing the memory cost of the whole algorithm from

$3$8

The total asymptotic ATRG cost remains $3$9, because after the swap is accelerated the dominant cost shifts to the subsequent contraction step. In the four-dimensional Ising model, this modified algorithm yields a free-energy density consistent with the original ATRG while significantly reducing elapsed time (Oba, 2019).

The original ATRG paper also notes that the bond-swap stage can be implemented with partial SVD using Arnoldi-type methods. The dominant matrix-vector multiplication can be factorized into two successive tensor-vector multiplications, avoiding construction of a large intermediate tensor. In that implementation, the bond-swap cost is reduced from $4$0 to $4$1, while memory remains $4$2. Squeezer construction, naively $4$3, can likewise be reduced to

$4$4

by preprocessing each tensor with an SVD or QR decomposition (Adachi et al., 2019).

4. Accuracy, convergence behavior, and fixed-point structure

ATRG’s lower cost is inseparable from an additional approximation absent in HOTRG. The truncation of the intermediate tensor $4$5 during the bond-swap stage makes ATRG less accurate than HOTRG at equal bond dimension $4$6. In the two-dimensional Ising benchmarks, the free-energy error of ATRG lies between those of TRG and HOTRG for all tested $4$7: better than TRG at the same cost class, worse than HOTRG at the same $4$8. The method is therefore designed to optimize “accuracy per computational time” rather than “accuracy per bond dimension,” and the original paper explicitly stresses that these are different metrics (Adachi et al., 2019).

The convergence behavior is also qualitatively different from HOTRG. ATRG exhibits larger and nonmonotonic fluctuations, and increasing $4$9 does not necessarily improve the final free energy monotonically. The stated likely reason is that ATRG contains two independent truncations in each renormalization step, each optimized locally rather than globally for the full tensor network. This makes smooth convergence less reliable in practice. ATRG also does not preserve axisymmetry of the renormalized network because of the anisotropic intermediate truncation (Adachi et al., 2019).

A common misconception is to identify ATRG with tensor-network renormalization schemes that explicitly remove short-range correlations. ATRG addresses a different problem: scalable, topology-preserving coarse graining in higher dimensions. By contrast, Gilt defines a graph-independent local truncation based on an environment spectrum $T_{y_0 y_1 x_0 x_1}$0 and the closed-form rule

$T_{y_0 y_1 x_0 x_1}$1

with the aim of removing local correlations while preserving the network graph. In that sense, Gilt is not anisotropic in the ATRG sense but complementary to it, and the paper explicitly presents local UV cleanup and coarse graining as modular tasks that can be combined in different ways (Hauru et al., 2017).

ATRG nonetheless retains one structural feature familiar from TRG and HOTRG: in two dimensions the corner-double-line (CDL) tensor remains a fixed point. This suggests continuity with the broader TRG/HOTRG family rather than a change in the renormalization fixed-point philosophy (Adachi et al., 2019).

5. Benchmarks and physical applications

The original benchmarks focus on the square-lattice and simple-cubic Ising models at criticality. In both cases, each lattice direction is renormalized $T_{y_0 y_1 x_0 x_1}$2 times alternately, corresponding to systems of size $T_{y_0 y_1 x_0 x_1}$3, effectively thermodynamic-limit calculations. For the two-dimensional square-lattice Ising model, the free-energy density is evaluated at

$T_{y_0 y_1 x_0 x_1}$4

The calculations go up to $T_{y_0 y_1 x_0 x_1}$5 for TRG and ATRG, and up to $T_{y_0 y_1 x_0 x_1}$6 for HOTRG. Using the computation-time proxy

$T_{y_0 y_1 x_0 x_1}$7

ATRG gives the smallest free-energy error among the three methods when plotted against $T_{y_0 y_1 x_0 x_1}$8, although the authors add that the observed wall-clock advantage is smaller than idealized $T_{y_0 y_1 x_0 x_1}$9-scaling suggests because partial SVD is practically slower than some contractions in HOTRG. For the three-dimensional simple-cubic Ising model at

$y$0

with

$y$1

ATRG reaches $y$2 versus $y$3 for HOTRG and yields a lower free-energy density at the same leading-order computational time (Adachi et al., 2019).

Subsequent work established ATRG as a practical tool in four dimensions. In the four-dimensional complex $y$4 theory at finite density, ATRG reduces the cost from HOTRG’s

$y$5

to

$y$6

making calculations feasible on volumes up to

$y$7

With $y$8 and $y$9, the study observes a clear Silver Blaze signal: the particle number density remains approximately zero up to $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$0 and then rises, while the average phase factor on $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$1 is already nearly zero for $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$2. The implementation uses randomized SVD, with stable results reported for

$$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$3

in the partial-SVD stage (Akiyama et al., 2020).

ATRG has also been applied to the four-dimensional Ising model. In that study, HOTRG remains the more accurate method at fixed bond dimension, but ATRG reaches much larger $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$4 because of its lower cost. The reported four-dimensional implementation uses randomized SVD with

$$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$5

achieves an execution-time scaling $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$6, and finds that ATRG at $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$7 gives a free energy comparable to HOTRG at $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$8 with almost the same execution time. The transition point extracted from ATRG differs from HOTRG by about $$T_{y_0 y_1 x_0 x_1} \approx \sum_{\alpha=1}^{\chi} S_{\{T\}\alpha\alpha}\, U_{\{T\} y_0 x_0 \alpha}\, V_{\{T\} y_1 x_1 \alpha}.$$9, while the internal-energy signal is regarded as qualitatively consistent with a finite jump but not yet fully converged (Akiyama et al., 2019).

In three-dimensional non-Abelian systems, ATRG has been adapted to the $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$0 principal chiral model on the cubic lattice. There the input is a rank-6 tensor obtained from a character expansion and Haar integration, preserving the global $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$1 symmetry for the retained irreducible representations. The ATRG initialization uses the canonical SVD

$$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$2

truncated to bond dimension $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$3. In one study, ATRG gives a critical coupling closer to the Monte Carlo benchmark $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$4 than tTRG and produces magnetization scaling consistent with the quoted $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$5 exponents

$$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$6

The same work reports smoother magnetization curves and less noise than tTRG near criticality (Akiyama et al., 2024).

6. Later developments, related methods, and broader significance

The main practical appeal of ATRG is clearest in four dimensions, where even ATRG’s $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$7 cost remains substantial. This has motivated a series of triad-based refinements. A four-dimensional “Triad-ATRG” rewrites the post-bond-swapped ATRG unit-cell tensor into a mixed network of four-leg and three-leg tensors, reducing the dominant contraction cost from

$$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$8

while keeping the free-energy convergence close to ATRG. In the four-dimensional Ising benchmark, the free-energy difference between Triad-ATRG and ATRG at $$A_{y_0 x_0 \alpha} = U_{\{T\} y_0 x_0 \alpha}, \qquad B_{y_1 x_1 \alpha} = S_{\{T\}\alpha\alpha} V_{\{T\} y_1 x_1 \alpha},$$9 is reported as $Z = \mathrm{tTr}\,\prod_i T^{(i)},$00, and parallel implementations on multiple GPUs further improve performance (Sugimoto et al., 2024, Sugimoto et al., 29 Jul 2025).

Another line of work addresses a prerequisite that ATRG shares with other TRG-family methods: the construction of the initial locally connected tensor network. A general initialization procedure based on delta insertion and index shifting transforms generic Boltzmann-factor representations into tensor networks in which every summed index appears exactly twice. That work also shows that numerical performance can depend strongly on the chosen initial representation and argues that boundary-type improvements using squeezers can remove or reduce that dependence. This is directly relevant to ATRG because its local compression maps are formulated in terms of squeezers rather than one-sided isometries (Nakayama et al., 21 Jan 2025).

Taken together, these developments position ATRG between standard TRG and HOTRG on one side and more graph-aware or short-range-correlation-removing approaches on the other. Like HOTRG, ATRG preserves lattice topology under coarse graining and applies in arbitrary dimension; like TRG, it relies centrally on matrix and tensor decompositions. Its distinctive feature remains the anisotropic local factorization that avoids the steep tensor growth of HOTRG. A plausible implication of the later literature is that ATRG has become a platform rather than a single fixed algorithm: bond-swapping accelerations, triad representations, GPU parallelization, symmetry-preserving tensor constructions, and graph-preserving preprocessing all modify the same basic anisotropic renormalization logic rather than replacing it (Oba, 2019, Akiyama et al., 2024, Sugimoto et al., 29 Jul 2025).

The method’s practical criterion has remained stable since its introduction. When the bottleneck is computational time or memory—especially in three dimensions and above—ATRG is preferred to HOTRG because its reduced scaling permits larger bond dimensions and more ambitious system sizes. When the objective is maximal accuracy at fixed bond dimension and cost is secondary, HOTRG remains favorable. That division of labor, already explicit in the original formulation, continues to organize the later application literature (Adachi et al., 2019).