method-mps

name

method-mps

description

Use when an MPS / matrix-product-state reproduction or calculation needs method-level route and tool selection — ground states, order parameters, correlations, gaps, dynamics, or finite-T of 1D and quasi-1D (narrow-cylinder) Hamiltonians, in either the finite-chain regime (DMRG, TEBD) or the infinite / uniform regime (VUMPS, IDMRG, iTEBD, TDVP). Covers picking the algorithm and routing to the tool.

Method MPS

Overview

A Matrix Product State (MPS) writes the wavefunction of a 1D (or narrow quasi-1D) quantum system as a chain of small tensors, one per site, linked by bond indices. The bond size — the bond dimension D (also written χ) — is the single accuracy knob: it caps how much entanglement the state can carry, so the method is controlled (systematically improvable by raising D).

• What it does. Represent the state compactly, then either variationally minimize the energy (sweeping, DMRG / VUMPS) or evolve in imaginary time (TEBD / TDVP) to reach the ground state; real-time evolution gives dynamics. Observables (energy, order parameters, correlations, gaps, entanglement) are read off the converged tensors.
• Two regimes. Finite — an open or periodic chain of fixed length L (DMRG, TEBD). Infinite / uniform — one small unit cell of tensors repeated forever, giving the thermodynamic limit directly with no finite-size extrapolation (VUMPS, IDMRG, iTEBD).
• Target. Ground-state energy per site, order parameters, correlation functions and the correlation length, the gap, entanglement entropy; with time evolution, dynamics and (via purification / METTS) finite temperature.
• What's approximated. Truncating the bonds to finite D discards the smallest entanglement. Exact for gapped 1D as D grows; for gapless / critical systems the finite D imposes an effective correlation length (finite-entanglement scaling), so D must be pushed and scaled.

Infinite-regime ground states — VUMPS. Among the infinite-MPS ground-state algorithms, VUMPS (Variational Uniform MPS) converges the variational gradient to machine precision — even at criticality, where IDMRG and iTEBD stall — which is exactly the comparison in FIG.7 of the VUMPS paper (arXiv:1701.07035).

When this card is invoked, before any choice, orient the user with this table (interaction principles below), filling the right column with their actual problem — Hamiltonian, geometry, target. If those aren't fixed yet, use the table to elicit them.

Ingredient	What it is	Your setup
Hamiltonian H	the 1D / quasi-1D model and its couplings	(user's model + couplings)
Geometry	finite chain (length L, OBC/PBC) or infinite/uniform (unit-cell length)	(finite L or infinite; cylinder width if quasi-1D)
Bond dimension D	the MPS link size — the accuracy knob	(target D / D-series)
Algorithm	DMRG · TEBD · VUMPS · IDMRG · iTEBD · TDVP	(which, and why)
Target	energy / order parameter / correlations / gap / dynamics / finite-T	(which observable, which states)
What's approximated	finite-D truncation (+ Trotter step for TEBD)	(D-convergence / τ→0 plan)

Interaction principles — all user-facing surfacing in this card. Plain language, no jargon: define every term, symbol, and axis before first use. No walls of words — a few sentences or one compact table per turn. One decision at a time, recommendation-first with one-line pros/cons. Precise and concise; let the user feel each choice, never a silent default.

Sources

• Methodology reference (reproduction-grade algorithm, parameters, validation, gap analysis): references/mps-methodology.md
• Tool skills:
  • /using-mpskit — MPSKit.jl (Julia): infinite MPS (VUMPS, IDMRG, IDMRG2), finite DMRG, TDVP. The canonical route for uniform/infinite ground states in this harness.
  • /using-tenpy — TeNPy (Python): iTEBD (the algorithm MPSKit lacks), iDMRG, VUMPS, finite DMRG/TEBD, finite-T purification.
  • /using-itensors — ITensorMPS.jl (Julia): finite DMRG / TEBD; infinite via ITensorInfiniteMPS.
• Primary literature (rendered in .knowledge/literature/mps-based-algorithm/):
  • 1701.07035 — Zauner-Stauber, Fishman, Verstraete et al., PRB 97, 045145 (2018) — VUMPS; the source for the infinite-regime Details, the gradient-norm stop criterion, and the FIG.7 benchmark.
• Foundational (pull with /download-ref if needed): White PRL 69 (1992) / PRB 48 (1993) — DMRG; Vidal PRL 91 (2003), 93 (2004) — TEBD; McCulloch arXiv:0804.2509 (2008) — IDMRG; Haegeman et al. PRL 107 (2011), PRB 94 (2016) — tangent space / TDVP; Schollwöck Ann. Phys. 326 (2011) — MPS review.

Select method — step 1

Suited for

• Ground states — energy per site, order parameters, correlation functions, gaps, entanglement — of 1D chains and narrow quasi-1D cylinders. The workhorse for these.
• Thermodynamic limit directly (infinite/uniform regime): no finite-size extrapolation — VUMPS / IDMRG / iTEBD give the per-site energy of the infinite system; the correlation length comes from the transfer matrix; gapless vs gapped is diagnosed by finite-entanglement scaling in D.
• Critical / gapless 1D — feasible but D-hungry; VUMPS is the cleanest converger here.
• Dynamics (real-time TEBD / TDVP) and finite temperature (purification / METTS) — supported, but treat as beyond the basic ground-state reproduction unless the target figure is explicitly dynamics or thermodynamics.

Worked examples

Anchor the user's problem to the nearest row and quote its scale as a concrete reference. The four FIG.7 panels of 1701.07035 are an infinite-MPS convergence-speed benchmark (wall time / iterations vs gradient norm) across the three infinite algorithms; D is the bond dimension, the cell is the repeated unit.

Ref	Model	Problem type	Scale	Calculated
1701.07035 FIG.7a	spin-1 XXZ, Δ=1 (Haldane, gapped)	infinite ground state, convergence speed	D=120, 1-site cell, no symmetry	e₀ (≈ −1.401484); VUMPS drives ‖B‖→~1e-12
1701.07035 FIG.7b	spin-½ XXZ, Δ=2 (gapped AFM)	infinite ground state	D=54, 2-site cell	e₀ (no closed form)
1701.07035 FIG.7c	spin-½ XXZ, Δ=1 (isotropic Heisenberg, critical)	infinite ground state — the hard case	D=70, 1-site cell via sublattice rotation	e₀ = ¼ − ln2 ≈ −0.4431; VUMPS reaches ‖B‖~1e-12 while IDMRG / iTEBD stall ~1e-4
1701.07035 FIG.7d	Hubbard chain	infinite ground state (fermionic)	2-site cell	e₀
classic	spin-½ Heisenberg / XXZ, finite chain	finite ground state, finite-size trend	DMRG, L up to 100s, Dfew×100	E(L), order parameter, correlations → 1/L extrapolation

Route elsewhere — when MPS isn't the right tool

MPS fits 1D and narrow quasi-1D. Route away when:

Target	Better tool	Why
Genuinely 2D / wide systems	PEPS (/method-peps)	MPS bond dimension blows up exponentially with cylinder width
Classical stat-mech / finite-T tensor networks	LTRG (/method-ltrg)	different contraction; MPS isn't the natural object
Full spectrum / level statistics, small system	ED (/method-ed)	MPS gives low-lying states, not the whole spectrum
Large-scale unfrustrated 2D/3D ground states	QMC (/method-qmc)	sign-free QMC scales better when there's no sign problem

When the user's goal falls outside 1D/quasi-1D: recognize it before any setup; explain what fits better and why in a short what/why table; stay warm — guide, don't dismiss. (Interaction principles above.)

Options & trade-offs

Algorithm	Regime	Good at	Weak at	When to use
DMRG	finite	robust, mature ground states of gapped 1D / cylinders; order parameters & correlations	finite-size and finite-D error → needs extrapolation; slow/unstable for gapless	finite chains and cylinders; anything needing edge physics or 1/L trends
VUMPS	infinite	variational uniform fixed point; fastest, cleanest convergence — gradient ‖B‖→machine precision even at criticality	needs a translation-invariant setup; per-iteration effective eigensolves	the default for infinite ground states; the FIG.7 winner, especially critical
IDMRG	infinite	thermodynamic limit by growing + translating a DMRG cell; reuses DMRG machinery; MPO Hamiltonians	converges to the true uniform fixed point slower than VUMPS; can stall at criticality	infinite ground states when DMRG infrastructure is already in hand
iTEBD	infinite	simple; imaginary-time projection from a product state; thermodynamic limit	Trotter error (τ→0 needed); slow near criticality; awkward for long-range H	infinite, short-range, gapped; a preparation/evolution route
TEBD (real-time) / TDVP	finite & infinite	dynamics; TDVP handles long-range / MPO Hamiltonians	entanglement growth caps reachable time	dynamics targets (out of basic reproduction scope)

Routing — surface to the user

Present the choice as a table: which algorithm, when to choose it, which fits their case. Make them feel the one consequence that decides it:

• Finite chain / cylinder, or edge physics / 1/L trends needed → DMRG.
• Infinite / thermodynamic-limit ground state → VUMPS (the default): it converges the tangent-space gradient to machine precision even at criticality, where IDMRG and iTEBD stagnate — the exact lesson of FIG.7. Pick IDMRG/iTEBD only to reproduce a paper that used them, or to compare convergence (as FIG.7 does).
• Dynamics → TEBD/TDVP (flag as beyond basic ground-state scope). When reproducing, the deciding fact is usually which algorithm the paper used — confirm that first.

Select software — step 2

Routing rule

If the chosen algorithm is available in MPSKit, use MPSKit; otherwise route to TeNPy or ITensorMPS.jl. MPSKit is the harness's canonical Julia tensor-network stack and covers the whole infinite-MPS family except one — iTEBD, which forces a route to TeNPy.

The packages

• MPSKit.jl (Julia; Verstraete-group / QuantumKitHub, maintained) — VUMPS, IDMRG, IDMRG2, GradientGrassmann, finite DMRG / DMRG2, TDVP. No TEBD of any kind (time evolution is TDVP / WII / TaylorCluster). Pairs with MPSKitModels.jl (ready Hamiltonians: heisenberg_XXZ, heisenberg_XYZ, Hubbard, …) and TensorKit.jl (tensors, abelian/non-abelian symmetries). → /using-mpskit.
• TeNPy (Python; Hauschild & Pollmann, maintained) — iTEBD (TEBDEngine), iDMRG, VUMPS, finite DMRG / TEBD, finite-T purification. The route whenever the target is iTEBD, or when a Python ecosystem is preferred. → /using-tenpy.
• ITensorMPS.jl (Julia; ITensor project) — finite DMRG / TEBD with the observer/sweeps interface; infinite via ITensorInfiniteMPS.jl. → /using-itensors.

Feature matrix — algorithm × package

Algorithm	MPSKit	TeNPy	ITensorMPS
finite DMRG	✓	✓	✓
finite TEBD	✗	✓	✓
iTEBD	✗	✓	(ITensorInfiniteMPS)
IDMRG	✓	✓	ITensorInfiniteMPS
VUMPS	✓	✓	—
TDVP	✓	✓	✓

The one gotcha to surface: MPSKit has no TEBD — a paper's iTEBD curve must come from TeNPy (this is exactly what FIG.7's iTEBD panel required). Everything else infinite (VUMPS, IDMRG) stays in MPSKit.

Surface to the user

Surface the software choice as a short what/why table (interaction principles above): the recommended package for the chosen algorithm (per the routing rule and feature matrix), what it is and who maintains it, and 1–2 real alternatives with their setup state. Offer Search web for official paper code / setup unless forbidden or already verified. Be honest that a custom convergence probe (per-iteration trajectory, gradient norm) may need a package iteration hook or a Pkg.develop clone — see reproduce-paper step 2.

Handoff

Invoke /using-mpskit for infinite (VUMPS/IDMRG) or finite work in the Julia stack; /using-tenpy for iTEBD or the Python route; /using-itensors for ITensorMPS finite work. The tool skill owns bond dimension, sweeps, tolerance, unit-cell construction, symmetry setup, the convergence probe, and the time estimate. This card owns algorithm choice, the conceptual knobs, and verification.

Method setup — step 3

Each knob with its default and a consolidated principle/effect (with scaling where it matters). Package-specific values (constructor keywords, env vars) live in the /using-* cards. (Math unicode/plain.)

Knob	Default	Principle, effect & scaling
Bond dimension D (χ)	problem-dependent (FIG.7: 54–120; cylinders & critical want larger)	the single accuracy lever — caps the entanglement the state carries. Energy decreases monotonically and asymptotes as D grows; gapless/critical systems never fully converge at fixed D (finite-entanglement scaling: effective ξ ~ D^κ) so D must be scaled. Cost ~D³ per local update
Unit cell (infinite)	smallest cell compatible with the ground-state order	the repeated translation period. Must be ≥ the period of the order, or the ansatz literally cannot represent the state — a Néel / AFM ground state needs a 2-site cell, a uniform / Haldane state a 1-site cell. Symmetry-breaking trick: a sublattice rotation can fold an ordered 2-site state back to a 1-site cell — FIG.7c rotates every second spin (π about z), mapping critical XXZ to a 1-site problem
Tolerance / stop criterion	VUMPS/IDMRG: gradient norm ‖B‖ < 1e-10…1e-12; DMRG: ΔE/sweep + 2-site variance; iTEBD: energy plateau	the convergence target. Tangent-space gradient norm ‖B‖ — the norm of the projected energy gradient, →0 at the variational optimum (paper Eq.34) — is the right stop signal for VUMPS/IDMRG; in MPSKit it is calc_galerkin, in TeNPy tangent_projector_test. DMRG uses energy change per sweep and the variance ⟨H²⟩−⟨H⟩². Too loose → biased energy; too tight → wasted sweeps
Imaginary-time step τ (iTEBD)	schedule 0.1 → 5e-4 in stages	Trotter error ~τ² (2nd order). Refine τ→0 in stages, running each to its energy plateau. Pitfall: a small τ alone makes per-step ΔE tiny regardless of convergence — never read that as converged; combine τ-refinement with enough total imaginary time
Max iterations / sweeps	VUMPS/IDMRG: ~100s; DMRG: ~10s of sweeps	a work cap; the stop criterion should fire first. Hitting the cap = not converged — raise it or D, don't report it
Symmetry	none, or U(1) Sz / particle number (SU(2) when available)	exploiting a conserved quantity block-diagonalizes the tensors → smaller, faster, and pins the sector. The VUMPS paper deliberately uses no symmetry to benchmark the bare algorithm; for production, U(1) Sz is a large speedup but fixes the sector — match the paper
Initial state	random, or a product state in the target sector	random explores but can land in a metastable state; a product state (Néel, dimer) in the right sector is more robust and pins the symmetry sector. Restart with a new seed if convergence stalls in a local minimum

Confirm the setup with the user before running — one knob per turn, never batched (interaction principles above).

1. Orient once. One plain hook — e.g. "The bond dimension D is the dial: bigger D = more entanglement captured = closer to exact, at ~D³ cost. For an infinite system we also pick the repeating unit cell, which must be big enough to hold the magnetic order." Then tie each knob to its role.
2. Loop, one knob per turn. Recommended option first, labeled "(Recommended)" only with a technical reason (the paper's value when reproducing), one-line why, then 1–2 alternatives with one-line pro/con.
3. Ask one, STOP, wait. Never batch; never accept a silent default.
4. Lead with the two that decide the result: (1) bond dimension D — the accuracy lever; too small reads as converged but is biased, fix is a D-series until the energy stops moving (and finite-entanglement scaling if gapless); (2) unit cell — too small cannot represent the order at all; match it to the ground-state period (or use the rotation trick). Then tolerance / stop criterion, then the rest.

Cost & resource estimate

Wall time is one measured rate × a fixed amount of work: the per-iteration work is a firm count of D³ tensor operations; the unknown is the per-operation throughput (BLAS rate, threading), settled by one timing probe. Written in four quantities: D = bond dimension, d = physical (local) dimension, L_cell = unit-cell / chain length, n_iter = iterations or sweeps.

Axis	Scaling
Compute	per iteration ∝ L_cell · d · D³ — dominated by the local effective eigensolve (VUMPS/DMRG) or SVD (iTEBD) on the D×D bonds; × n_iter. Iterations differ by algorithm: VUMPS reaches a given ‖B‖ in fewer iterations than IDMRG/iTEBD (the FIG.7 point), and critical systems need both more iterations and larger D
Memory	O(L_cell · d · D²) for the tensors + O(D² · χ_MPO) for the Hamiltonian environments — modest; compute, not memory, is usually the gate
Wall time	(L_cell · d · D³ · n_iter) ÷ throughput; throughput is the 1–2 order-of-magnitude unknown, settled by one short timing probe at the target D

FIG.7 illustration (D=54–120, d=2–3, 1–2-site cell, no symmetry): seconds to a few minutes on a laptop, single-thread. Larger D (cylinders, near-critical, 2D-as-cylinder) scales as D³ → route to a cluster (/using-slurm). Symmetry (U(1)) cuts the effective D substantially.

Surface the cost before any scale choice (reproduce-paper step 4). Plain language: wall time is one measured rate × a firm D³ work count — show the work for their D, name the single unknown (throughput, settled by one probe), and that D is the dial that sets the cost. If the target figure is itself a wall-time benchmark (like FIG.7), threading and warm-up handling are governed by reproduce-paper's performance benchmark rules — core/thread count is a confirmed knob, and JIT/setup/measurement time is excluded.

Details

Generic methodology; paper/model facts live in /reproduce-paper and .knowledge/models/. Math is unicode/plain.

Notation

• MPS: state as a chain of tensors A_i, each carrying a physical index (dimension d) and two bond indices (dimension D).
• Bond dimension D (χ): the link size; the accuracy/entanglement cap.
• Canonical form: a gauge where tensors are left- or right-isometries; the mixed canonical form keeps a single center tensor — the object VUMPS optimizes.
• Transfer matrix: the D²×D² map built from a tensor and its conjugate; its second eigenvalue sets the correlation length.
• Unit cell: for an infinite MPS, the finite set of tensors repeated forever (length L_cell).
• Tangent space: the space of first-order variations of a uniform MPS at fixed D; the energy gradient lives here, and its norm is ‖B‖.

Infinite / uniform MPS — the regime VUMPS lives in

A uniform (translation-invariant) MPS repeats one unit cell of tensors infinitely, representing the thermodynamic limit directly. Working in the mixed canonical form, the state is fixed by a left isometry A_L, a right isometry A_R, a center tensor A_C, and a bond matrix C (Eq.29 convention of 1701.07035). Three algorithms find the ground state:

• VUMPS. The energy is stationary when the tangent-space gradient vanishes. VUMPS solves two coupled effective eigenvalue problems — for the center tensor A_C and the bond matrix C against the Hamiltonian environments — then re-gauges to a consistent (A_L, A_R, A_C, C). Iterating drives the tangent-space gradient norm ‖B‖ → 0 (the variational optimum). Because it optimizes the uniform state directly (not a finite cell extrapolated outward), it converges fastest and cleanest, including at criticality. It exploits no symmetry by default. ‖B‖ — paper Eq.34, the projected gradient — is the convergence diagnostic, computed by calc_galerkin (MPSKit) or tangent_projector_test (TeNPy).
• IDMRG. Run two-site DMRG on a growing chain and translate the converged center cell outward; the bulk tensor approaches the uniform fixed point. Reuses all DMRG machinery and MPO Hamiltonians, but reaches a given ‖B‖ in more iterations than VUMPS and can stall near criticality.
• iTEBD. Apply imaginary-time Trotter gates e^{−τh} to the infinite MPS and re-canonicalize by SVD truncating back to D; imaginary time projects onto the ground state. Simple, but carries Trotter error (τ→0 needed) and converges slowly near criticality.

Finite MPS — DMRG and TEBD

• DMRG: sweep site-to-site, locally minimizing the energy by solving an effective eigenproblem on one or two sites, truncating the updated bond to D. Robust for gapped finite chains and cylinders; carries both finite-size and finite-D error.
• TEBD: split H into bond gates, apply Trotterized e^{−τH} (imaginary time → ground state; real time → dynamics), truncate to D after each layer.
• TDVP: project the (real or imaginary) time evolution onto the tangent space of fixed-D MPS; handles long-range / MPO Hamiltonians where TEBD's gate decomposition is awkward.

The convergence diagnostic ‖B‖

For the variational (tangent-space) methods the right convergence signal is not the energy change but the norm of the projected energy gradient ‖B‖ (Eq.34). It is the size of the best tangent-space improvement to the current state; ‖B‖→0 ⇔ a true variational fixed point. Energy can look settled while ‖B‖ is still large (a near-plateau that is not yet stationary), so stop on ‖B‖, report the energy. MPSKit: calc_galerkin(below, H, above, envs) per site; TeNPy: tangent_projector_test(env_data). A per-site RMS over the cell gives one scalar.

Verification

Intermediate (mid-run)

• Gradient / variance falling. ‖B‖ (VUMPS/IDMRG) or the 2-site variance (DMRG) decreases monotonically toward the tolerance; a stall above tolerance = not converged (raise D or iterations, check the unit cell).
• Energy monotone & flattening. Energy per site decreases and asymptotes. For iTEBD, watch a plateau at each τ stage — and do not mistake a tiny per-step ΔE at small τ for convergence (it shrinks with τ regardless).
• Correlation length finite. From the transfer matrix's second eigenvalue; it should be finite and grow with D (diverging-with-D is the critical signature).

Final verification + expert criticism

• D-convergence. Energy vs D monotone and asymptoting; for gapless/critical, do finite-entanglement scaling in D rather than claiming a converged value at one D.
• Cross-method / exact agreement. VUMPS, IDMRG, iTEBD agree on the energy (FIG.7c: ¼−ln2 for critical Heisenberg; ≈−1.401484 for the spin-1 Haldane chain). Disagreement at the same D flags a setup error.
• Symmetry sector correct. Total Sz / particle number of the result matches the intended sector (see .knowledge/symmetry-cheatsheet.md).

Criticize: unit cell too small for the order (can't represent Néel on a 1-site cell) → wrong state silently; D too small read as converged (no D-series) → biased energy; iTEBD false convergence (small τ → tiny ΔE regardless of truncation — must refine τ and run each stage to its plateau); reading the energy before ‖B‖ has converged; for a wall-time benchmark, leaving JIT/warm-up/setup time in the clock or comparing stacks at different thread counts (see reproduce-paper performance benchmark rules); claiming a critical-system value at fixed D without finite-entanglement scaling.

Citations

Rendered under .knowledge/literature/mps-based-algorithm/:

• 1701.07035_*.md — Zauner-Stauber, Fishman, Verstraete et al., PRB 97, 045145 (2018) — VUMPS; the primary source for the infinite-regime Details, the ‖B‖ stop criterion (Eq.34), the Eq.29 canonical convention, and the FIG.7 benchmark.

Foundational (pull with /download-ref if needed): White PRL 69 (1992), PRB 48 (1993) — DMRG; Vidal PRL 91 (2003), 93 (2004) — TEBD; McCulloch arXiv:0804.2509 (2008) — IDMRG; Haegeman et al. PRL 107 (2011), PRB 94 (2016) — tangent space / TDVP; Schollwöck Ann. Phys. 326 (2011) — MPS review. Tool setup lives in /using-mpskit, /using-tenpy, /using-itensors.