method-ltrg

name	method-ltrg
description	Use when a finite-temperature Linearized Tensor Renormalization Group (LTRG) reproduction needs method-level route and tool selection — Trotterized classical tensor network from a quantum lattice model, layer-by-layer boundary contraction with SVD truncation, thermodynamic observables (free energy, internal energy, specific heat, susceptibility).

Method LTRG

LTRG is the finite-temperature tensor-network method class: map a d-dimensional quantum lattice model to a (d+1)-dimensional classical tensor network by Trotter-Suzuki decomposition, then contract layer by layer while truncating the growing boundary with SVD to bond dimension Dc. This card owns method selection (step 1), software routing (step 2), and method-level setup (step 3, method side). Method internals — including the algorithm — are in ## Details; software parameter values and the ITensors primitives live in /using-itensors; paper- and model-specific facts live in /reproduce-paper and .knowledge/models/.

Sources

• Methodology reference (reproduction-grade algorithm, parameters, validation, gap analysis): references/ltrg-methodology.md
• Tool skill: /using-itensors
• Primary literature: Li, Ran, Gong, Zhao, Xi, Ye, Su, Linearized tensor renormalization group algorithm… PRL (2011) .knowledge/literature/ltrg/1011.0155_linearized-tensor-renormalization-group-algorithm-for-the-ca.md.

Select method — step 1

Suited for

• Finite-temperature thermodynamics of low-dimensional quantum lattice models — 1D / quasi-1D chains and 2D lattices (e.g. honeycomb). Observables: free energy per site, internal energy, specific heat, susceptibility.
• Maps d-dim quantum → (d+1)-dim classical tensor network via Trotter-Suzuki, then decimates iTEBD-style; sign-problem-free even in 2D, a promising alternative to QMC for frustrated/fermionic 2D thermodynamics.
• Sizes reached: XY chain to length 2¹⁰⁰ (thermodynamic limit); temperature down to T/J ≈ 0.008 (β = 120); retained dimension Dc ≤ 150; 2D honeycomb Heisenberg benchmarked against QMC.

Route elsewhere when

• The target is a ground-state property → /method-mps (DMRG) or /method-peps; LTRG is a finite-temperature method.
• An exact/analytic solution exists — use it only as a benchmark after the LTRG calculation, never as a substitute.

Options & trade-offs

Method	Good at	Weak at	Typical reach
LTRG (this card)	finite-T low-D, 2D sign-free, scalable transfer-network contraction	low-T needs large Dc; Trotter + truncation error to control	T/J ~ 0.008, Dc ≤ 150
TMRG (transfer-matrix DMRG)	finite-T 1D, accurate	scales worse to 2D	1D
coarse-graining TRG	2D classical networks	discards O(Dcⁿ)/step → costlier	2D
purification / METTS	finite-T via MPS, good low-T in 1D	2D entanglement cost	1D / quasi-1D
QMC (/method-qmc)	finite-T, large sizes	sign problem (frustrated/fermionic)	sign-free only
XTRG (later Wei Li work)	reaches lower T (logarithmic-in-β cooling)	more involved bookkeeping	very low T

Select software — step 2

Open-source tools

• No official LTRG package ships with the paper — it is an algorithm-only PRL.
• The harness route is ITensors.jl (Julia): typed indices, SVD with truncation to Dc, and gate/transfer-tensor contraction, so the algorithm is expressed directly. The same algorithm is expressible in TeNPy or quimb.
• No reusable LTRG library function exists in-repo; the algorithm is in ## Details below, and /using-itensors carries the ITensors primitives (typed indices, svd to Dc, gate exponentiation, incremental writes) to express it.

Features to confirm

• Typed indices with explicit tags, svd with maxdim/cutoff, gate exponentiation exp(-τ·h), incremental writes of convergence data — owned by /using-itensors.

Options & trade-offs

Tool	Ecosystem / examples	Efficiency	When
ITensors.jl (this route)	Julia; ITensors primitives in /using-itensors, algorithm in ## Details	native SVD/contraction; core op is the O(D⁶·Dc³) SVD step	default
TeNPy / quimb	Python TN ecosystems	comparable; needs a hand-built LTRG loop	if Python-bound

Handoff

Invoke /using-itensors once the LTRG route is fixed — it owns ITensors.jl setup, index mechanics, SVD/truncation keywords, the ITensors primitives that express the algorithm, and runtime troubleshooting. This card owns the Trotter split, contraction order, normalization bookkeeping, and the convergence plan; the model/paper skills own the Hamiltonian and figure facts.

Method setup — step 3 (method side)

Conceptual knobs and the tricks behind them — for each, the intuition for choosing it and how it moves the result; the trick is the guidance where there is no fixed default. Concrete ITensors values/code live in /using-itensors.

Knob	Controls	Trick / how it affects results
τ	Trotter step	decomposition error ∝ τ² (symmetric split); paper uses 0.1/0.05/0.02/0.01; smaller τ → more layers
K	number of imaginary-time steps	fixes β = Kτ; more steps reach lower T but accumulate more truncations
Dc	retained SVD dimension	dominant accuracy/cost lever (like M in TMRG); paper uses 50/100/150
D (q)	local Hilbert dimension	sets transfer-tensor size; do not confuse with Dc
contraction direction / gate order	layer-absorption scheme	two equivalent schemes (Trotter-first or spatial-first); alternate the two projections per full Trotter step
normalization convention	scale bookkeeping	divide each step by the largest singular value (and each trace matrix by its largest element); collect the log factors to rebuild Z and the free energy

Cost: the local evolution (contract + SVD of the transfer tensor) scales as O(D⁶·Dc³) per step — the dominant cost; the spatial trace contracts 2^p matrices in p pairwise steps (logarithmic in chain length); memory is dominated by the Dc-bond boundary tensors plus the transient enlarged tensor. Estimate from intended τ, target β, q, geometry, and the Dc sweep before a full run.

Frontier — gaps, variants, future directions

Baseline LTRG (the algorithm in ## Details) has known weak points; most are closed by a published successor from the same Wei Li / Gang Su lineage. Use this map to decide what to read or try before reinventing the contraction loop. The 2011 PRL is the floor, not the state of the art.

Gap in baseline LTRG	Latest variant that closes it	What it buys	Reference (pull with /download-ref)
Trotter error dominates high-T; needs τ→0 extrapolation	SETTN — series-expansion TTN: Taylor-expand e^{−βH}, express H and Hⁿ as MPOs	Trotter-error-free ("continuous-time")	PRB 95, 161104(R) (2017), arXiv:1609.01263
Linear cooling: K = β/τ truncations accumulate → low-T cost/error	XTRG — exponential cooling ρ(2β) ← ρ(β)·ρ(β)	reaches β in ~log₂(β/τ₀) steps; far fewer truncations → lower T and better accuracy; runs on 2D cylinders	PRX 8, 031082 (2018), arXiv:1801.00142
Single-layer asymmetry; density matrix not manifestly positive	bilayer LTRG (LTRG++) — evolve bra+ket layers together	higher accuracy; infinite-size LTRG++ ≡ TMRG in TN language	PRB 95, 144428 (2017), arXiv:1612.01896
Finite-T fermions / 2D Hubbard out of reach	tanTRG — tangent-space optimization of the thermal MPO, O(D³)	low-T 2D Hubbard on width-8 cylinder / 10×10; matches DQMC at half filling	PRL 130, 226502 (2023), arXiv:2212.11973
2D limited to small width / honeycomb	XTRG on wider cylinders; finite-T PEPS / thermal CTMRG (/method-peps)	larger-area 2D thermodynamics	open direction
No symmetry exploitation → bigger bonds than necessary	U(1)/SU(2) quantum-number indices	block-sparse tensors, lower cost at fixed accuracy	/using-itensors QN indices

Want to improve on the baseline? Identify the need, then act

State the requirement; take the paired action — read the reference (/download-ref <arXiv>) and/or try the primitive change via /using-itensors. Do not bolt a variant on blindly: each one changes the step 1–3 cost model and the primitives handed to the tool skill.

Your need	Recommendation
"Lower T without exploding Dc"	Switch linear → exponential cooling (XTRG): square the MPDO each step instead of absorbing one Trotter layer. Read arXiv:1801.00142; the new ITensors op is an MPO·MPO product + re-truncation.
"Kill the Trotter error / clean high-T"	Replace the gate product with a series expansion of e^{−βH} (SETTN). Read arXiv:1609.01263; needs H and Hⁿ as MPOs.
"Density matrix loses accuracy / positivity"	Go bilayer (LTRG++) — evolve bra and ket together. Read arXiv:1612.01896.
"Finite-T in 2D / Hubbard / fermions"	tanTRG. Read arXiv:2212.11973; budget O(D³) tangent-space sweeps plus Grassmann / Jordan-Wigner bookkeeping.
"Same physics, just cheaper"	Add U(1)/SU(2) symmetry to the indices (/using-itensors) before raising Dc.

When a variant is chosen, re-run method selection (step 1) with its cost model and hand the new primitives — MPO·MPO product, Hⁿ MPOs, QN indices — to /using-itensors. The convergence plan in ## Verification still applies, but the dominant error source shifts (e.g. XTRG removes the linear-cooling truncation pileup; SETTN removes Trotter error).

Details

LTRG maps a d-dimensional quantum lattice model at finite temperature into a (d+1)-dimensional classical tensor network by Trotter-Suzuki decomposition, then contracts it layer by layer while truncating the growing boundary with SVD.

This card is generic methodology. Paper-specific Hamiltonian choices, figure protocols, and target claims belong in /reproduce-paper; model facts belong in .knowledge/models/.

Notation

• d spatial dimension; β inverse temperature; τ Trotter step; K steps with β = Kτ.
• Dc retained SVD dimension; q (a.k.a. D) local Hilbert dimension.
• Boundary tensor network: the partially contracted region.
• Log scale factors: accumulated normalizations needed to recover Z and the free energy.

Algorithm

1. Split the local quantum Hamiltonian into Trotter-Suzuki pieces; approximate Z = Tr e^{−βH} as a product of imaginary-time gates with β = Kτ.
2. Insert complete local bases between layers and read the result as a (d+1)-dim classical tensor network.
3. Build local transfer tensors from the gate matrix elements; SVD-factor them if the geometry requires.
4. Initialize the boundary tensor network; absorb one uncontracted layer.
5. Reshape and SVD; keep the largest Dc singular values; update the boundary.
6. Normalize, store the log scale factor; repeat layer absorption until the full imaginary-time extent is contracted.
7. Contract the remaining boundary; combine with the log factors to obtain Z, the free energy, and derived thermodynamics.

Verification — implementation stage

Intermediate (mid-run)

• Per-step normalization factors (largest singular value) stay finite and vary smoothly on a log scale — a divergence means a missing normalization.
• Discarded singular weight per SVD stays small and saturates as Dc grows.

Final verification + expert criticism

• τ → 0 extrapolation to remove Trotter error (dominant at high T).
• Dc convergence: the observable stops moving as Dc grows (e.g. Dc = 100 and 150 curves coincide); truncation error dominates at low T.
• Analytic/exact limits: benchmark vs the exact XY-chain solution (δf ≈ 7×10⁻⁶ at β = 120, Dc = 150); high- and low-T limits when the caller supplies them; β → large → ground-state energy e₀.
• Cross-check non-integrable models vs TMRG / QMC where available.
• Confirm every log scale factor is counted exactly once in the final quantity.
• Criticize: a single-(τ, Dc) number with no τ→0 and no Dc-convergence study; trusting low-T data at small Dc; no benchmark against an exact / QMC reference; and ignoring that high-T error is Trotter while low-T error is truncation.

Citations

• .knowledge/literature/ltrg/1011.0155_linearized-tensor-renormalization-group-algorithm-for-the-ca.md — Li et al. (2011), original LTRG paper.
• Successors (not yet in .knowledge; pull with /download-ref): bilayer LTRG++ arXiv:1612.01896 (PRB 95, 144428); SETTN arXiv:1609.01263 (PRB 95, 161104(R)); XTRG arXiv:1801.00142 (PRX 8, 031082); tanTRG arXiv:2212.11973 (PRL 130, 226502).
• ITensors.jl setup, API primitives, and runtime live in /using-itensors.