Hamiltonian Truncation and Variational Methods

Hamiltonian truncation becomes a controlled field-theory method only when the finite subspace, induced interactions, observables, and limiting procedure are specified together. A small matrix or a low variational energy is not itself the result. This chapter develops the complete chain from a projector $P_\Lambda$ and its omitted complement $Q_\Lambda$ to cutoff-dependent Hamiltonians and operators, symmetry-resolved bases, variational diagnostics, multi-axis convergence tests, and benchmarks that can reveal a false plateau.

The recurring example is a real scalar field on a circle. It is simple enough to admit exact free and few-state checks, but interacting enough to expose vacuum-energy, state, operator, finite-volume, and truncation effects. The same logic applies to conformal bases, particle-number cutoffs, local-state truncations, and nonlinear variational manifolds. The wider truncated-spectrum program is reviewed in James et al. 2018, §§ IV–VII, while the general-dimensional conformal construction is developed in Hogervorst, Rychkov, and van Rees 2015, §§ 2–4.

Enter the chapter through the missing contract

Start by asking what is finite and what claim is sought. If the approximation is a projected linear space, begin with the regulator. If the states are not yet enumerated, begin with basis construction. If ultraviolet CFT data define the basis, use the conformal-truncation route. If finite-cutoff values drift, the renormalization page is central. If a nonlinear ansatz replaces a complete basis, use the variational route. Spectrum-only calculations must still pass through observables and convergence before they reach benchmarks.

Routes through Hamiltonian truncation and variational calculations.

Question you cannot yet answer	Start here	Deliverable before moving on
Which states were removed?	Hilbert-Space Truncation as a Regulator	Projectors PΛ, QΛ, retained sectors, cutoff scales, and limit order
Is the finite matrix complete and correctly normalized?	Basis Construction, Symmetry Sectors, and Matrix Elements	Unique basis, Gram matrix, exact symmetry blocks, selection rules, and small-matrix checks
Does a solvable ultraviolet theory organize the states?	Conformal and Hamiltonian Truncation	UV spectrum and matrix elements, deformation, cutoff, volume, and null-state prescription
What compensates for omitted intermediate states?	Renormalizing a Truncated Hamiltonian	Derived or matched effective terms, no-double-counting rule, inputs, and held-out tests
What does a low variational energy actually establish?	Variational Principles and Field-Theory Ansätze	Manifold, symmetry, normalization, optimization protocol, energy bound, residual, and ansatz enlargement
How should a current, correlator, or quench be represented?	Observables and Dynamics in Truncated Spaces	Effective operator, leakage and recurrence tests, sum rule, and joint state–operator improvement
Is the reported uncertainty supported by the data?	Convergence, Extrapolation, and Error Certification	Independent cutoff scans, justified asymptotic models, covariance, residuals, cross-basis and held-out checks
Would the method fail visibly before an unsolved target?	Benchmark Theories for Truncation Methods	A ladder from exact construction tests to blind interacting predictions, with cost and error recorded

Each of these eight routes appears once because each closes a distinct logical gap. Renormalization cannot repair a duplicated basis; a perfect eigensolver cannot repair a missing effective current; and agreement along one diagonal cutoff path cannot separate two regulator errors.

The projector reveals the central difficulty

Let $H$ be a regulated, self-adjoint Hamiltonian bounded below, and split its domain with orthogonal projectors

$$P_\Lambda+Q_\Lambda=1, \qquad P_\Lambda Q_\Lambda=0.$$

For an exact eigenstate $H|\Psi\rangle=E|\Psi\rangle$, eliminating the omitted component gives

$$H_{\mathrm{eff}}(E)P_\Lambda|\Psi\rangle=E P_\Lambda|\Psi\rangle,$$

with

$$H_{\mathrm{eff}}(E) =P_\Lambda H P_\Lambda +P_\Lambda H Q_\Lambda (E-Q_\Lambda H Q_\Lambda)^{-1} Q_\Lambda H P_\Lambda.$$

This Feshbach–Schur identity is exact whenever the displayed resolvent exists. It explains why simply diagonalizing $P_\Lambda H P_\Lambda$ generally drops cutoff-dependent interactions. It also shows why those interactions can be energy dependent, nonlocal, or state dependent before a controlled expansion reduces them to a usable operator basis. Modern Hamiltonian-truncation treatments build systematic approximations to precisely this omitted-state term Elias Miró and Ingoldby 2023, §§ 2–4.

The same elimination dresses an observable. With a wave operator

$$\Omega(E)=P_\Lambda +Q_\Lambda(E-Q_\Lambda H Q_\Lambda)^{-1} Q_\Lambda H P_\Lambda,$$

matrix elements require $O_{\mathrm{eff}}=P_\Lambda\Omega_f^\dagger O\Omega_iP_\Lambda$ (with the appropriate external energies), not merely $P_\Lambda O P_\Lambda$. Therefore a fitted spectrum does not certify a matrix element or a real-time response.

One scalar thread separates the error sources

On a circle of length $L$, take $m>0$ and

$$H=H_0+\frac{g}{4!}\int_0^L dx\,{:}\phi^4(x){:}, \qquad H_0=\sum_{n\in\mathbb Z}\omega_n a_n^\dagger a_n, \qquad \omega_n=\sqrt{m^2+\left(\frac{2\pi n}{L}\right)^2}.$$

A free-energy cutoff retains states satisfying $\sum_n\omega_nN_n\le E_{\max}$. Momentum and the $\mathbb Z_2$ transformation $\phi\mapsto-\phi$ split the basis into exact blocks. In the free theory, Fock energies and matrix elements are analytic construction tests. With $g\ne0$, increasing $E_{\max}$ introduces virtual states and changes the needed effective interactions. Increasing $L$ changes finite-volume physics, not the same ultraviolet approximation. Changing the reference mass changes the basis and normal-ordering convention. These axes must be recorded and varied independently.

A reproducible baseline uses $L=2\pi$, $m=1$, $g=0.5$, zero-momentum even and odd sectors, and $E_{\max}=8,10,12,14,16$. It requires the $g=0$ analytic spectrum first and uses a separate frozen higher-cutoff interacting reference only for its declared benchmark comparisons. Values from that reference must not enter a fit that is later advertised as a prediction.

Four evidence levels must remain distinct

Rigorous bound. A nested Rayleigh–Ritz space gives upper bounds on ordered discrete eigenvalues under the hypotheses of the min–max principle. The bound applies to those energies, not automatically to gaps, matrix elements, or dynamics Reed and Simon 1978, Chapter XIII, § 1.

Asymptotic extrapolation evidence. Several large-cutoff points follow a derived or independently justified expansion, with correlated fit uncertainty and stability under fit-window and model changes. This is evidence for the stated observable and regulator family; it is not a theorem merely because $\chi^2$ is acceptable.

Empirical stability. Results change little over declared basis, cutoff, counterterm, volume, and optimization variations, and held-out observables agree with independent references. This supports a finite accuracy claim over the tested range.

Unsupported convergence. One fitted quantity is flat, one basis is used, or only the eigensolver residual is small. Such tests establish internal numerical consistency, not proximity to the target QFT.

The distinction is essential because the most dangerous failure is a false plateau. A cutoff-dependent counterterm can keep a fitted energy nearly fixed while a transition matrix element drifts. Enlarging the counterterm basis or changing the Hilbert basis can then move the apparent limit. The chapter’s shared convergence map and certification record make this adversarial test explicit on every method page.

Check your preparation

You are ready to work through the chapter if you can do the following without assuming that a finite matrix is the theory:

• distinguish the target Hamiltonian, its prior spacetime regulator, and the additional Hilbert-space truncation;
• use orthogonal projectors and spectral decompositions, including a generalized eigenvalue problem when a basis has Gram matrix $G\ne I$;
• identify exact momentum, internal-symmetry, and representation sectors;
• state which parameters are renormalization inputs and reserve at least one observable as a prediction;
• distinguish an eigenpair residual from a field-theory truncation error; and
• specify the order or joint strategy for ultraviolet, volume, basis, and variational limits.

If the first two items are unfamiliar, review Regulated Hamiltonian Field Theory and Normal Forms, Spectra, and Projectors. For renormalized predictions, use Regulator Removal and Renormalized Predictions.

Chapter synthesis

A complete truncation result is a connected record, not a list of matrix dimensions. It specifies:

1. the target regulated theory, volume, boundary conditions, and observables;
2. $P_\Lambda$ and $Q_\Lambda$, including every independent cutoff and exact sector;
3. basis normalization, null-state removal, selection rules, and verified matrix elements;
4. induced Hamiltonian and observable terms, their matching inputs, and the rule preventing double counting;
5. variational assumptions, optimizer diagnostics, residuals, and ansatz enlargements when a nonlinear manifold is used;
6. cutoff, volume, counterterm, basis, and time-window sequences rather than a single favorable path;
7. held-out spectrum and matrix-element tests, cross-basis comparisons, and benchmark references; and
8. an uncertainty statement labeled as a bound, an asymptotic extrapolation, empirical stability, or an unresolved limitation.

Hamiltonian lattice constructions feed this chapter through Hamiltonian Lattice Field Theory. Tensor-network representations continue in Tensor Networks for QFT and Many-Body Systems, and quantum-computing encodings continue in Quantum Simulation and Quantum Computing for QFT. Dated comparisons of large-scale methods belong to Research: Lattice and Hamiltonian Field Theory, not to this durable chapter.

You should finish able to (1) route a proposed calculation to the missing basis, induced-operator, variational, observable, or certification contract, and (2) classify every convergence claim by the evidence it actually has.

Review the chapter

Why does a variational plateau not certify an observable?

Give a two-part answer using the min–max principle and effective operators.

Answer outline

For nested trial spaces, the ground-state Ritz energy is an upper bound and cannot increase as the space grows. That statement does not order $\langle O\rangle$. Omitted components change both the state and the operator; the exact reduced matrix element uses $P\Omega_f^\dagger O\Omega_iP$, whereas $POP$ neglects the wave-operator tails. A stable energy can therefore coexist with a drifting current or sum rule.

Design a claim that survives a false plateau

Suppose one mass is used to tune a counterterm at every $E_{\max}$. What is the smallest additional evidence package that can expose a spurious limit?

Answer outline

Reserve at least one matrix element or second spectral quantity that was not used in tuning; evaluate it on a multi-point cutoff sequence; repeat after enlarging the counterterm or effective-operator basis; and reproduce the continuum estimate in a second complete Hilbert basis or truncation rule. Report volume and optimization effects separately. The fitted mass checks the renormalization condition, while the held-out and cross-basis results test the prediction.

References

• Elias Miró, Joan, and James Ingoldby. “Effective Hamiltonians and Counterterms for Hamiltonian Truncation.” Journal of High Energy Physics 2023, 052 (2023). DOI. Open PDF.
• Hogervorst, Matthijs, Slava Rychkov, and Balt C. van Rees. “Truncated Conformal Space Approach in $d$ Dimensions: A Cheap Alternative to Lattice Field Theory?” Physical Review D 91, 025005 (2015). DOI.
• James, Andrew J. A., Robert M. Konik, Philippe Lecheminant, Neil J. Robinson, and Alexei M. Tsvelik. “Non-Perturbative Methodologies for Low-Dimensional Strongly-Correlated Systems: From Non-Abelian Bosonization to Truncated Spectrum Methods.” Reports on Progress in Physics 81, 046002 (2018). DOI.
• Reed, Michael, and Barry Simon. Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, 1978. Publisher record.