HOTRG¶

Higher-Order Tensor Renormalization Group (HOTRG) improves upon TRG by using Higher-Order SVD (HOSVD) to compute optimal truncation isometries.

Background¶

Instead of pairwise SVD splits (as in TRG), HOTRG constructs the truncation isometry from the environment tensor – formed by contracting two adjacent tensors over their shared bonds – and computing its HOSVD. This produces a globally better approximation at each coarse-graining step.

Algorithm (horizontal step):

Form \(M[u, u', d, d'] = \sum_{l,r} T[u,d,l,r] \cdot T[u',d',r,l]\)
HOSVD of \(M\): compute truncated isometries \(U_u\), \(U_d\)
Compress \(T\) using the isometries and contract to form \(T_{\text{new}}\)

The vertical step is analogous with left/right bonds.

Reference: Xie et al., PRB 86, 045139 (2012).

Configuration¶

from tenax import HOTRGConfig

config = HOTRGConfig(
    max_bond_dim=16,               # maximum chi
    num_steps=20,                  # RG iterations
    direction_order="alternating", # "alternating" or "horizontal_first"
    svd_trunc_err=None,            # optional truncation error threshold
)

Example – 2D Ising model¶

import math
from tenax import HOTRGConfig, hotrg, compute_ising_tensor, ising_free_energy_exact

beta_c = math.log(1 + math.sqrt(2)) / 2
tensor = compute_ising_tensor(beta_c)

config = HOTRGConfig(max_bond_dim=16, num_steps=20)
log_Z_per_site = hotrg(tensor, config)

exact = ising_free_energy_exact(beta_c)
print(f"HOTRG log(Z)/N = {float(log_Z_per_site):.8f}")
print(f"Exact          = {exact:.8f}")

TRG vs HOTRG¶

At the same bond dimension, HOTRG typically achieves better accuracy because the HOSVD-based isometries account for the full tensor environment rather than a single pairwise split.

Method	`max_bond_dim=16` relative error
TRG	~1e-5
HOTRG	~1e-7

The trade-off is that HOTRG is more expensive per step (additional SVDs for the environment tensor).

Direction order¶

"alternating" (default): alternate horizontal and vertical coarse-graining steps. This preserves the square-lattice symmetry at each step.
"horizontal_first": perform both horizontal and vertical coarse-graining within each step. May converge faster for anisotropic systems.