TRG¶

The Tensor Renormalization Group (TRG) is a coarse-graining algorithm for computing the partition function of 2D classical lattice models.

Background¶

Starting from a single-site tensor \(T_{udlr}\) (up, down, left, right) placed on every site of an infinite 2D square lattice, TRG iteratively reduces the lattice size by a factor of 2 via:

SVD splitting: decompose the tensor into half-tensors along horizontal and vertical directions.
Plaquette contraction: contract four half-tensors around a plaquette to form the coarse-grained tensor.

The partition function estimate is tracked via log normalisation:

\[\frac{\ln Z}{N} = \sum_k \frac{\ln \| T_k \|}{4^{k+1}}\]

Reference: Levin & Nave, PRL 99, 120601 (2007).

Configuration¶

from tenax import TRGConfig

config = TRGConfig(
    max_bond_dim=16,      # maximum chi after each coarse-graining step
    num_steps=20,         # number of RG iterations
    svd_trunc_err=None,   # optional: truncation error threshold
)

Example – 2D Ising model¶

import math
from tenax import TRGConfig, trg, compute_ising_tensor, ising_free_energy_exact

# Critical temperature of the 2D Ising model
beta_c = math.log(1 + math.sqrt(2)) / 2

# Build the initial tensor for the partition function
tensor = compute_ising_tensor(beta_c, J=1.0)

# Run TRG
config = TRGConfig(max_bond_dim=16, num_steps=20)
log_Z_per_site = trg(tensor, config)

# Compare with the exact Onsager solution
exact = ising_free_energy_exact(beta_c)
print(f"TRG  log(Z)/N = {float(log_Z_per_site):.8f}")
print(f"Exact         = {exact:.8f}")

Helper functions¶

`compute_ising_tensor(beta, J=1.0)`¶

Builds the 4-leg transfer-matrix tensor for the 2D Ising model:

\[T_{udlr} = \sum_s \sqrt{Q_{u,s}} \sqrt{Q_{d,s}} \sqrt{Q_{l,s}} \sqrt{Q_{r,s}}\]

where \(Q_{a,b} = \exp(\beta J \, \sigma_a \sigma_b)\).

Returns a DenseTensor with legs ("up", "down", "left", "right").

`ising_free_energy_exact(beta, J=1.0)`¶

Computes the exact 2D Ising free energy per site via numerical integration of the Onsager formula. Useful as a benchmark for TRG convergence.

Convergence¶

TRG accuracy improves with max_bond_dim. Typical values:

`max_bond_dim`	Relative error at \(\beta_c\)
4	~1e-2
8	~1e-3
16	~1e-5
32	~1e-7

For better accuracy at the same bond dimension, consider HOTRG.