Homework Scaffold Generator — Tenax Course Assignments

Open-ended problems requiring multiple steps, judgment, and interpretation.

Example: "Use Tenax to map out the phase diagram of the XXZ model (H = Σ Δ Sz·Sz + (S+S- + S-S+)/2) as a function of Δ ∈ [-1, 2]. Identify the ferromagnetic, XY, and Néel phases using appropriate order parameters and entanglement entropy."

Tier 4: Adversarial

Contains an incorrect premise or subtle error that the student (or AI model) must identify and correct. Tests critical thinking.

Example: "The Heisenberg antiferromagnet on a 1D chain spontaneously breaks SU(2) symmetry, developing Néel order in the ground state. Use Tenax DMRG to compute the staggered magnetization m_s = (1/L) Σ (-1)^i ⟨Sz_i⟩ for L = 20, 40, 80 and show it approaches a finite value as L → ∞."

(The premise is wrong: the 1D Heisenberg chain does NOT have Néel order — the Mermin-Wagner theorem forbids spontaneous breaking of continuous symmetry in 1D. Students should find m_s → 0 as L → ∞ and explain why.)

Problem Templates by Course Topic

Crystal Structure / Lattice Models (Weeks 1–3)

Tier 2: Build a Heisenberg Hamiltonian on a square lattice cylinder using AutoMPO.

# Starter code
import jax
from tenax import AutoMPO, DMRGConfig, FiniteMPS, dmrg

Lx, Ly = 4, 3
N = Lx * Ly
auto = AutoMPO(L=N, d=2)

# TODO: Add Heisenberg bonds for a cylinder geometry
# Hint 1: Within-ring bonds are periodic in y: j = x*Ly + (y+1) % Ly
# Hint 2: Between-ring bonds connect site (x,y) to ((x+1),y)
# Hint 3: AutoMPO requires site indices in ascending order: use min(i,j), max(i,j)

# YOUR CODE HERE

mpo = auto.to_mpo(compress=True)
key = jax.random.PRNGKey(0)
mps = FiniteMPS.random(L=N, d=2, chi=16, key=key)
config = DMRGConfig(max_bond_dim=64, num_sweeps=10, verbose=True)
result = dmrg(mpo, mps, config)
print(f"E/N = {result.energy / N:.8f}")

Band Theory / 1D Systems (Weeks 4–6)

Tier 2: Compare the DMRG ground state energy of the tight-binding model (XX model) with the exact analytic result from band theory.

Tier 3: Add a staggered potential to the tight-binding model and study the metal-insulator transition as a function of potential strength, using entanglement entropy as the diagnostic.

Magnetism (Weeks 7–9)

Tier 2: Use iDMRG to compute the spin-1 Heisenberg chain and verify the Haldane gap by computing the excitation energy.

Tier 3: Map the phase diagram of the bilinear-biquadratic spin-1 chain H = Σ [cos(θ) S·S + sin(θ) (S·S)²] as a function of θ.

Tier 4: "The spin-1/2 Heisenberg antiferromagnet has a gap of approximately 0.41 J (the Haldane gap). Use iDMRG to compute this gap." (Wrong: the Haldane gap exists for integer spin, not half-integer. S=1/2 Heisenberg is gapless.)

Superconductivity / Correlations (Weeks 10–12)

Tier 2: Compute the spin-spin correlation function ⟨S^z_0 S^z_r⟩ for the Heisenberg chain using DMRG. Verify the expected algebraic decay ~ (-1)^r / r.

Tier 3: Study pairing correlations in the t-J model using DMRG. Define custom operators via build_auto_mpo with a site_ops dictionary for fermions.

Phase Transitions (Weeks 13–14)

Tier 2: Use TRG to compute the specific heat of the 2D Ising model near T_c.

Tier 3: Use iPEPS to study the quantum phase transition in the 2D transverse-field Ising model. Map the order parameter (magnetization) as a function of h/J.

2D Systems / Advanced Topics (Weeks 15–16)

Tier 3: Compare cylinder DMRG (via AutoMPO) and iPEPS for the 2D Heisenberg antiferromagnet. Which gives better energy per site? What are the trade-offs?

Tier 4: "iPEPS with D=2 is sufficient to capture the ground state of the 2D Heisenberg model because the entanglement in 2D follows an area law, and D=2 already satisfies the area law bound." (Misleading: while area law holds, the prefactor matters. D=2 gives E ≈ -0.6548 vs exact E ≈ -0.6694. Higher D is needed for quantitative accuracy. The area law guarantees efficiency scaling, not that the smallest D suffices.)

Generating a Problem

When the instructor requests a problem:

1. Identify the topic and appropriate tier.
2. Write the problem statement with clear physics context.
3. Provide starter code with TODO markers and progressive hints.
4. Include a reference solution (hidden from students) with expected output.
5. Add a "what to report" section — guide students on what to include in their writeup (plots, numerical values, physical interpretation).
6. For Tier 4: include the correct resolution of the adversarial premise in the solution.

Benchmark Prompt Metadata

Each problem can be tagged for LLM benchmarking:

{
  "tier": 2,
  "topic": "magnetism",
  "week": 8,
  "tenax_functions": ["idmrg", "build_bulk_mpo_heisenberg", "iDMRGConfig"],
  "physics_concepts": ["Haldane gap", "spin-1 chain", "gapped vs gapless"],
  "expected_difficulty": "medium",
  "adversarial": false
}

This metadata supports the instructor's goal of building a corpus of benchmark prompts for evaluating AI models across complexity tiers.