method-peps

name	method-peps
description	Use when a PEPS, iPEPS, CTMRG, 2D tensor-network, environment dimension, or classical partition-function reproduction needs method-level route and tool selection.

Method PEPS

PEPS is the tensor-network track for environment contraction, classical partition functions, and PEPS optimization. Use it to decide which method card and tool skill own the next step.

Sources

• Methodology reference (reproduction-grade algorithm, parameters, validation, gap analysis): references/peps-methodology.md
• Track README: tracks/peps/README.md
• Tool skill: /using-pepskit

Route

1. Use CTMRG/environment contraction when the figure depends on free energy, magnetization, transfer matrices, correlation length, or PEPS expectation values.
2. Distinguish fixed-tensor contraction from variational PEPS optimization before selecting parameters.
3. If the target is finite-temperature Linearized Tensor Renormalization Group, hand off to /method-ltrg; it owns the LTRG route, method card, and tool selection.
4. Recommend /using-pepskit for PEPSKit.jl / TensorKit.jl setup, CTMRG settings, and timing.
5. If the paper target is a package tutorial or official code, offer official code / web search before reimplementing formulas.

Tool Handoff

Invoke /using-pepskit for PEPS or CTMRG routes. For LTRG, hand off to /method-ltrg.

Details

Projected Entangled Pair States and associated contraction methods (CTMRG, simple/full update, variational optimization) for two-dimensional quantum and classical lattice problems, with CTMRG as the default environment-contraction route.

Scope

Use this card for:

• Infinite 2D classical partition functions represented as local tensors.
• PEPS environment contractions through CTMRG.
• Visual finite-chi convergence studies of free energy, magnetization, energy, or correlation length.
• Simple classical onboarding figures, especially the 2D square-lattice Ising model.

Do not use this card as the full recipe for:

• Optimizing an iPEPS ground state.
• Finite PEPS contraction.
• Time evolution.
• Claiming reproduction of an original CTMRG paper before its primary source has been ingested under .knowledge/literature/peps-based-algorithm/.

Onboarding Reproduction Target

Default visual target: the 2D classical Ising CTMRG example. Sweep temperature, contract the infinite partition function, and plot magnetization or free energy against the exact Onsager/Yang result. This is the cleanest CTMRG onboarding route because it is classical, visual, fast, and has an analytic reference.

For a paper reproduction claim, first ingest the selected CTMRG paper with download-ref into .knowledge/literature/peps-based-algorithm/, then let /reproduce-paper derive the exact figure protocol from that primary source.

Notation

• Local tensor: rank-4 tensor for a square-lattice classical model, or PEPS tensor / double-layer tensor for a quantum state.
• D: PEPS virtual bond dimension when contracting a PEPS.
• chi_env: CTMRG environment bond dimension.
• Corner tensors C and edge tensors T: environment tensors approximating the infinite network boundary.
• CTMRG residual: convergence metric returned by the boundary iteration.
• Transfer-matrix correlation length: diagnostic extracted from the converged environment.
• Free energy per site: for a classical partition function, obtained from the dominant network value and the chosen normalization convention.

Pitfalls

• Finite-chi rounding: CTMRG smooths sharp critical behavior at finite chi_env. Show chi convergence before interpreting critical data.
• Convergence slowdown: near criticality, residuals and correlation lengths converge slowly. Increase both chi_env and maxiter.
• Normalization mismatch: free energy is sensitive to tensor normalization. Keep the partition-function construction and free-energy formula in the same script.
• Symmetry breaking: magnetization below the critical temperature can depend on initialization or explicit bias. Record the symmetry-breaking convention.
• PEPS versus partition-function environments: PEPS double-layer environments and classical partition-function environments have different edge tensor structure. Do not copy observable formulas across them blindly.

Verification

• Residual convergence: record CTMRG residual and iteration count for every grid point.
• Chi convergence: repeat the curve for at least two chi_env values; near criticality, use more.
• Analytic limits: check high-temperature and low-temperature Ising limits before plotting a full curve.
• Exact curve: compare the 2D Ising free energy, magnetization, and/or energy against the analytic Onsager/Yang formulas used in the script.
• Tiny-network cross-check: for a small finite patch, compare the local tensor construction against direct enumeration before trusting the infinite contraction.

Citations

• Nishino and Okunishi, J. Phys. Soc. Jpn. 65, 891 (1996) — original CTMRG development. Ingest the primary source before using it for a reproduction claim.