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DMRG sweeps and truncation errors

Interpret local variational optimization and Schmidt truncation while keeping fidelity, observable error, and sweep convergence distinct.

Before you begin
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The question for today

What you’ll understand

  • Describe an environment-based variational sweep through an MPS.
  • Compute discarded Schmidt weight and normalized truncation fidelity.
  • Explain why small discarded weight is not the same as a small observable error.

Density-matrix renormalization group optimizes a many-body state by updating a small part of an MPS while its surrounding tensors define an effective environment. Sweeps revisit sites so that each update benefits from improved neighboring blocks. Its characteristic truncation keeps the most important Schmidt components across a cut. To assess a result, distinguish the local eigensolver tolerance, convergence of repeated sweeps, restriction to a finite bond dimension, and the eventual finite-size limit. Each can remain wrong while another appears exceptionally well converged.

Optimize locally inside a globally restricted family

In mixed canonical form, orthonormal block states make a local one-site or two-site tensor the center of a small effective eigenproblem. Contracting the Hamiltonian with the fixed environments defines that effective action. A sparse iterative solver finds its lowest direction, after which the center is factorized and shifted along the chain. The energy is variational for the represented normalized state when expectations are evaluated consistently. Approximate local solves and truncations can prevent individual practical steps from being strictly monotone.

A one-site update keeps existing virtual dimensions fixed and may have trouble introducing missing symmetry sectors or escaping a poor local optimum. A two-site update temporarily enlarges the local space and truncates after its solve, allowing the bond basis to change more flexibly. More sweeps do not guarantee the global best dimension-D state. Compare initializations, symmetry choices, sweep directions, and larger dimensions, especially near competing phases. Energy convergence alone can be less sensitive than a long-range correlation or an order parameter.

E[ψD]=⟨ψD∣H∣ψD⟩⟨ψD∣ψD⟩≥E0E[\psi_D]=\frac{\langle\psi_D|H|\psi_D\rangle}{\langle\psi_D|\psi_D\rangle}\geq E_0
The inequality concerns the exact ground energy of the same finite Hamiltonian. It applies to the represented state even if optimization is incomplete; approximate tensor contractions or stochastic evaluations can spoil its numerical certification.↔ Scroll wide equations horizontally.

Truncation has a precise one-cut meaning

Sort Schmidt weights from largest to smallest and retain D of them. Their omitted sum is the discarded weight epsilon. The unnormalized retained vector differs from the original by squared Hilbert-space norm epsilon. If it is renormalized, its squared overlap with the original is one minus epsilon. These are different comparisons. Calling epsilon the norm error without specifying squared norm or normalization loses a square root or changes the quantity entirely.

For normalized pure states before and after this one-cut truncation, their trace distance is the square root of epsilon. Hence a bounded observable has an expectation difference no larger than twice its operator norm times that square root. The actual change may be much smaller, and some diagonal observables change linearly with epsilon in special examples. Repeated cuts during many sweeps require additional accumulated-error analysis; one final discarded weight is not the global distance to the exact ground state.

ϵ=∑α>Dλα2,∣⟨ψ∣ψ~D⟩∣2=1−ϵ,∣Δ⟨O⟩∣≤2∥O∥ϵ\epsilon=\sum_{\alpha>D}\lambda_\alpha^2,\qquad |\langle\psi|\widetilde\psi_D\rangle|^2=1-\epsilon,\qquad |\Delta\langle O\rangle|\leq2\|O\|\sqrt\epsilon
The truncated state with a tilde is renormalized and refers to one Schmidt cut of a normalized input state. Epsilon is dimensionless; the bounded-observable inequality compares those two states, not a general DMRG result to the exact ground state.↔ Scroll wide equations horizontally.

Use several diagnostics at fixed physical size

Increase the maximum bond dimension and decrease local solver tolerance until changes in target observables are below the requested scale. Check energy variance or full residual when computationally accessible. A small variance indicates proximity to some eigenstate only with further spectral information, so it cannot alone establish the correct ground sector. Empirical extrapolation of energy against discarded weight can be useful in a demonstrated asymptotic regime, but it is not an exact universal linear law for every observable.

Finite-size effects remain after bond convergence. Open ends can pin an order parameter or change a gap, so central-region measurements and several lengths help diagnose boundaries. Near criticality, a finite bond dimension produces its own correlation-length cutoff, and increasing size without increasing that dimension can saturate an apparently thermodynamic result prematurely. The matrix lab offers a controlled two-singular-value family for hand normalization. It is neither a many-body eigensolver nor a DMRG implementation, and its geometry has no physical tensor interpretation.

Var⁡ψ(H)=⟨H2⟩ψ−⟨H⟩ψ2=∥(H−Eψ)∣ψ⟩∥22\operatorname{Var}_\psi(H)=\langle H^2\rangle_\psi-\langle H\rangle_\psi^2=\|(H-E_\psi)|\psi\rangle\|_2^2
The state is normalized and E_psi is its Rayleigh energy. Variance has energy-squared units, while its square root is a residual norm. Exact eigenstates have zero variance, including excited states.↔ Scroll wide equations horizontally.

Work it through

A single discarded component

A normalized Schmidt spectrum has weights 0.7,0.2,0.10.7,0.2,0.1. Retain two components and renormalize. Find discarded weight, squared fidelity, state distance, and a bound for an observable with operator norm one.

  1. The omitted weight is ϵ=0.1\epsilon=0.1. The unnormalized retained vector has norm squared 0.9 and differs from the original by norm squared 0.1; these quantities answer different questions.

  2. Renormalization divides the retained vector by 0.9\sqrt{0.9}. The overlap with the original is 0.9\sqrt{0.9}, so squared fidelity is 0.9 and pure-state trace distance is 0.1≃0.316\sqrt{0.1}\simeq0.316.

  3. The bounded-observable expectation change is at most 20.1≃0.6322\sqrt{0.1}\simeq0.632. This is a worst-case guarantee for that one truncation, not a prediction that every measured observable changes by that amount.

The result

The discarded weight is 0.1 and squared fidelity 0.9; an observable-error bound scales with its square root, not automatically with 0.1.

The living notebook

Explore the model

Interactive lab

Interactive lab · matrix

Normalize a toy singular spectrum

Reset and set scale a=0.5a=0.5, shear b=0b=0. Predict final-matrix singular values 1 and 0.5; their normalized squared weights are 0.8 and 0.2. Repeat from Reset with both controls a=0.25a=0.25, b=0b=0: weights become 16/17 and 1/17. Readouts describe the selected final triangular matrix; geometry starts at identity and interpolates during playback. This is not a many-body SVD or quantum gate.

Normalize a toy singular spectrumThe selected matrix is A = [[a,b],[0,1]]. Playback interpolates geometry between the identity and A; readouts always describe A. Its determinant is a and its singular values come from AᵀA. Positive a makes every supported matrix invertible. This two-dimensional family is not a general eigensolver or a learned network. Current readings: Area scale · det A: 1.4; Singular values · σmax, σmin: 1.597, 0.876; Condition number · κ₂(A): 1.823. All parameters are adjustable with the labeled controls below.Matrix A with first row 1.4, 0.6 and second row zero, one; display interpolates between identity and A while final determinant remains 1.4.−4−2024−2−1012x · equal geometric scaleyA = [ 1.4 0.6 ; 0 1 ]Display blend: 0% · final values below
Interpolated geometryOriginal unit squareSelected final transform
Area scale · det A
1.4
Singular values · σmax, σmin
1.597, 0.876
Condition number · κ₂(A)
1.823
t = 0 s

The selected matrix is A = [[a,b],[0,1]]. Playback interpolates geometry between the identity and A; readouts always describe A. Its determinant is a and its singular values come from AᵀA. Positive a makes every supported matrix invertible. This two-dimensional family is not a general eigensolver or a learned network. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forRaw singular values require normalization before they can be interpreted as stipulated Schmidt weights.

Pause & reason

Check your understanding

For one Schmidt truncation with discarded weight ε followed by renormalization, what is the squared overlap with the original normalized state?

Your turn

Try it yourself

A Schmidt decomposition has weights 0.9,0.10.9,0.1. An observable is diagonal in these Schmidt components with values +1,−1+1,-1. Keep only the first component and renormalize. Find the actual expectation change and compare with the general norm-one bound.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Compute expectations before and after truncation; the general inequality is a worst-case bound over observables.

Compare with the solution

Initially the expectation is 0.9−0.1=0.80.9-0.1=0.8. The normalized retained state has expectation one, giving change 0.2. The discarded weight is 0.1, so the general bound is 20.1≃0.6322\sqrt{0.1}\simeq0.632, which is respected but not saturated. This particular diagonal example changes by twice the discarded weight; it does not establish linear scaling for arbitrary observables or repeated DMRG sweeps.

Keep the idea

Take it with you

  • Sweeps, local solves, and bond-dimension restriction create distinct convergence questions.
  • Discarded Schmidt weight is a squared unnormalized error and fixes one-cut normalized fidelity.
  • Observable changes require their own analysis; a small final truncation weight is not a global ground-state certificate.

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