tenax-trg-workflow

name

tenax-trg-workflow

description

Guide graduate students through Tensor Renormalization Group (TRG) and HOTRG calculations for 2D classical statistical mechanics models using Tenax. Covers building the initial tensor from the partition function, running coarse-graining, extracting free energy and thermodynamic quantities, and studying phase transitions. Use this skill when the user mentions TRG, HOTRG, tensor renormalization group, classical partition function, 2D Ising model, transfer matrix, coarse-graining, real-space renormalization, free energy calculation, critical temperature, critical exponents, or "how to study phase transitions with tensor networks". Also trigger for compute_ising_tensor, TRGConfig, or classical stat mech problems.

TRG/HOTRG Workflow — Classical Stat Mech with Tenax

This skill guides students through tensor renormalization group calculations for 2D classical lattice models. TRG is a real-space RG method that systematically coarse-grains the partition function using tensor decompositions.

Background for Students

The partition function of a 2D classical model can be written as a tensor network: one tensor per lattice site, with shared indices representing the Boltzmann weights of neighboring interactions. The free energy per site is:

f = -(1/β) lim_{N→∞} (1/N) ln Z

TRG computes this by iteratively contracting and truncating the tensor network, doubling the effective system size with each step.

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Stage 1: Build the Initial Tensor

2D Ising model

Tenax provides a built-in function:

from tenax.algorithms.trg import compute_ising_tensor
import jax.numpy as jnp

beta = 0.44  # Inverse temperature (near T_c ≈ 2.269, β_c ≈ 0.4407)
T = compute_ising_tensor(beta)
print(T.shape)  # (2, 2, 2, 2) — one index per lattice direction

The tensor T encodes the Boltzmann weights exp(βJ σ_i σ_j) for all spin configurations on the four bonds meeting at a site.

Custom models

For models beyond Ising, build the tensor from the Boltzmann weights manually. The general recipe:

1. Write the energy as a sum over bonds: E = Σ_{⟨ij⟩} ε(σ_i, σ_j)
2. For each site, form a tensor T[u,r,d,l] = Π_{bonds} √(W(σ_bond)) where W(σ_i, σ_j) = exp(-β ε(σ_i, σ_j)) and the square root is split between the two sites sharing each bond.
3. For the Ising model with σ = ±1: W = [[exp(βJ), exp(-βJ)], [exp(-βJ), exp(βJ)]] and T is constructed by SVD of W.

Physics checkpoint

Ask: "What temperature range are you interested in? Is the model exactly solvable (for validation)? What's the expected critical temperature?"

For the 2D Ising model: T_c = 2/ln(1+√2) ≈ 2.269, β_c ≈ 0.4407.

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Stage 2: Run TRG

from tenax.algorithms.trg import trg, TRGConfig

config = TRGConfig(
    max_bond_dim=16,   # Truncation bond dimension χ
    num_steps=20,      # Number of coarse-graining steps
)
free_energy = trg(T, config)
print(f"Free energy per site: {free_energy:.8f}")

What happens at each step

Each TRG step:

1. Decomposes the 4-index tensor T into pairs of 3-index tensors (via SVD).
2. Contracts neighboring pairs to form a new effective tensor T'.
3. Truncates to bond dimension χ.
4. The effective system size doubles: after n steps, the network represents 2^n × 2^n sites.

After 20 steps: 2^20 ≈ 10^6 sites — effectively the thermodynamic limit.

HOTRG (Higher-Order TRG)

HOTRG improves upon TRG by using a higher-order SVD (Tucker decomposition) for the coarse-graining step. This preserves more information and gives better accuracy at the same χ.

from tenax import hotrg, HOTRGConfig  # HOTRG is a separate module

config = HOTRGConfig(max_bond_dim=16, num_steps=20)
free_energy = hotrg(T, config)

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Stage 3: Study the Phase Transition

Temperature scan

import numpy as np

betas = np.linspace(0.3, 0.6, 50)
free_energies = []
config = TRGConfig(max_bond_dim=24, num_steps=20)

for beta in betas:
    T = compute_ising_tensor(beta)
    f = trg(T, config)
    free_energies.append(float(f))

# Plot f(β) — look for the non-analyticity at β_c

Identifying T_c

The free energy itself is continuous at T_c (2D Ising is a second-order transition), but its second derivative (specific heat) diverges.

# Numerical second derivative
f = np.array(free_energies)
d2f = np.gradient(np.gradient(f, betas), betas)
# The peak of |d²f/dβ²| locates T_c

Specific heat

C = -β² ∂²f/∂β² (per site). The 2D Ising model has a logarithmic divergence: C ∼ -ln|T - T_c|.

Magnetization (order parameter)

To extract the magnetization, insert a modified tensor with a spin-weighted trace. This is more involved — guide students through the impurity tensor technique if they need it.

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Stage 4: Convergence Studies

Bond dimension convergence

beta = 0.4407  # At criticality — hardest case
for chi in [4, 8, 16, 32, 64]:
    config = TRGConfig(max_bond_dim=chi, num_steps=20)
    T = compute_ising_tensor(beta)
    f = trg(T, config)
    print(f"χ = {chi:3d}: f = {f:.10f}")

Away from T_c, even χ=4 gives good results. At T_c, large χ is needed because correlations are long-range and the truncation error is largest.

TRG vs HOTRG comparison

for chi in [8, 16, 32]:
    T = compute_ising_tensor(0.4407)
    f_trg = float(trg(T, TRGConfig(max_bond_dim=chi, num_steps=20)))
    f_hotrg = float(hotrg(T, HOTRGConfig(max_bond_dim=chi, num_steps=20)))
    print(f"χ={chi}: TRG={f_trg:.10f}, HOTRG={f_hotrg:.10f}")

HOTRG should be more accurate at the same χ, especially near criticality.

Exact result for validation

The exact free energy of the 2D Ising model (Onsager):

f_exact(β) = -kT [ln(2) + (1/2π²) ∫∫ ln(cosh²(2βJ) - sinh(2βJ)(cos θ₁ + cos θ₂)) dθ₁ dθ₂]

At β_c: f ≈ -2.1009 (per site, J=1). Students should compare their TRG result.

Tenax provides a built-in exact solution for validation:

from tenax import ising_free_energy_exact

f_exact = ising_free_energy_exact(beta=0.4407)
print(f"Exact free energy: {f_exact:.10f}")

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Stage 5: Beyond the Ising Model

Potts models (q states)

Build a q×q Boltzmann weight matrix and decompose into a 4-index tensor. Z_q symmetry can be exploited.

Clock models

Similar to Potts but with continuous-like angular variables discretized to q orientations. Shows Berezinskii-Kosterlitz-Thouless transitions for q ≥ 5.

Classical dimer models

The tensor encodes the constraint that each site is covered by exactly one dimer.

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Common Pitfalls

Symptom	Cause	Fix
Free energy wrong far from T_c	Too few RG steps	Increase num_steps
Free energy wrong near T_c	χ too small	Increase max_bond_dim
NaN or Inf in free energy	Overflow in exp(βJ) at large β	Use log-space arithmetic
Specific heat peak at wrong T	Finite-χ shift	Increase χ and extrapolate

Pedagogical Notes

• TRG is the tensor network realization of Kadanoff's block-spin RG idea. Each step is literally a block-spin transformation.
• The bond dimension χ plays the role of the number of kept states — analogous to χ in DMRG but for 2D classical systems.
• Connect to the course: the 2D classical Ising model is equivalent (via transfer matrix) to the 1D quantum Ising model. TRG on the classical side ↔ DMRG on the quantum side.