Basis Construction, Symmetry Sectors, and Matrix Elements

A trustworthy truncated matrix is built by enumerating each normalized state once, resolving exact symmetry sectors before diagonalization, removing null directions in the correct inner product, and verifying interaction and observable matrix elements against analytic small-sector cases. Sparse storage and a converged eigensolver come only after these algebraic checks; neither can detect a duplicated bosonic state, a missing combinatorial factor, or a broken constraint.

Required background. Hilbert-Space Truncation as a Regulator supplies $P_\Lambda$ and the omitted subspace. Representations, Intertwiners, Invariants, and Tensor Decomposition supplies irreducible sectors, invariant tensors, and multiplicities.

Helpful background. Multiplets, Invariants, and Selection Rules supplies the group-theoretic rules used to predict zero blocks.

Normalized basis states precede matrix assembly

Basis construction contract. Fix the one-particle modes and their inner product; a canonical representative for every many-particle state; exact momentum, charge, parity, and irrep labels; the cutoff rule; null-state and constraint removal; operator ordering; and the Gram matrix. Record expected block dimensions independently of the Hamiltonian builder.

For a real scalar of mass $m>0$ on a circle of length $L$,

$$k_n=\frac{2\pi n}{L}, \qquad \omega_n=\sqrt{m^2+k_n^2}, \qquad \phi(x)=\sum_{n\in\mathbb Z} \frac{a_ne^{ik_nx}+a_n^\dagger e^{-ik_nx}} {\sqrt{2L\omega_n}},$$

with $[a_n,a_m^\dagger]=\delta_{nm}$. A normalized occupation basis is

$$|\{N_n\}\rangle= \prod_n\frac{(a_n^\dagger)^{N_n}}{\sqrt{N_n!}}|0\rangle.$$

The free-energy, momentum, and field-parity labels are

$$E^{(0)}=\sum_n\omega_nN_n, \qquad P=\sum_n k_nN_n, \qquad \mathcal P_{\mathbb Z_2}=(-1)^{\sum_nN_n}.$$

Thus a zero-momentum even-sector basis at cutoff $E_{\max}$ consists of every integer tuple satisfying $E^{(0)}\le E_{\max}$, $\sum_n nN_n=0$, and $\sum_nN_n$ even. An occupation tuple is already an unordered bosonic state. Generating ordered particle lists and retaining every permutation would duplicate it. This finite-volume scalar basis and its energy-cutoff organization are given explicitly in Rychkov and Vitale 2015, § 2.

When states $|i\rangle$ are not orthonormal—common for composite-operator or conformal bases—the Gram matrix $G_{ij}=\langle i|j\rangle$ is part of the problem. Physical coefficients satisfy

$$Hc=EGc.$$

Null directions of $G$ must be removed or quotiented before solving. Replacing this by an ordinary eigenproblem changes both energies and normalization. A unitary target requires $G$ to be positive definite after the null quotient. A deliberately nonunitary benchmark must instead state the signature of its bilinear form and cannot inherit positivity-based variational bounds. A conformal-truncation treatment of Gram matrices, null-state removal, and the resulting generalized eigenproblem appears in Hogervorst, Rychkov, and van Rees 2015, §§ 3–4.

Exact symmetries determine blocks and zeros

If $U(g)$ is an exact symmetry and both $H$ and $P_\Lambda$ commute with it, the retained space decomposes as

$$P_\Lambda\mathcal H =\bigoplus_{\alpha} \left(\mathbb C^{m_\alpha}\otimes V_\alpha\right),$$

where $V_\alpha$ is an irrep and $m_\alpha$ its multiplicity. Schur’s lemma then fixes the representation-space structure of an invariant Hamiltonian; only multiplicity spaces require nontrivial diagonalization. For an operator $O_R$ transforming in representation $R$, a matrix element vanishes unless $V_\beta$ occurs in $R\otimes V_\alpha$.

This is both a reduction and an exact test. A forbidden nonzero element signals an indexing, phase, tensor, or truncation error. A missing allowed element is not automatically an error, but it deserves an independent calculation.

Gauge theory requires a stronger choice. One may construct only gauge-singlet states, or keep a larger space with Gauss-law projectors and constraints. The method and residual constraint norm must be stated. The link and electric-flux bases are developed in Hamiltonian Lattice Field Theory; light-front momentum partitions and constrained fields are developed in DLCQ and Basis Light-Front Quantization. Neither should be silently replaced by an unconstrained Fock basis.

A scalar φ⁴ matrix has exact selection rules

Consider

$$V=\frac{g}{4!}\int_0^L dx\,{:}\phi^4(x){:}.$$

Expanding the four fields produces monomials with zero through four creation operators. The spatial integral enforces exact integer momentum conservation. Normal ordering with respect to the declared $H_0$ fixes which contractions have already been absorbed into the mass and vacuum energy. The matrix element can change particle number by $0,\pm2,\pm4$ and preserves $\mathbb Z_2$ parity.

For each operator string, compute a canonical final occupation tuple and its ladder factor. For example,

$$a_r^\dagger a_s|\{N_n\}\rangle =\sqrt{(N_r+1)N_s}\, |\ldots,N_r+1,\ldots,N_s-1,\ldots\rangle$$

for $r\ne s$. Repeated mode labels require the corresponding falling and rising factorials; treating four labels as distinct is a common source of factor errors.

Exactly checkable one-mode matrix

Restrict temporarily to the zero mode and define $X=a+a^\dagger$. In the ordered even basis $(|0\rangle,|2\rangle,|4\rangle)$,

$${:}X^4{:} =a^{\dagger4}+4a^{\dagger3}a+6a^{\dagger2}a^2 +4a^\dagger a^3+a^4$$

has the exact matrix

$$\begin{pmatrix} 0&0&\sqrt{24}\\ 0&12&16\sqrt3\\ \sqrt{24}&16\sqrt3&72 \end{pmatrix}.$$

For the zero-mode contribution of the circle normalization above,

$$\frac{g}{4!}\int_0^L dx\,{:}\phi_0^4{:} =\frac{g}{96m^2L}{:}X^4{:}.$$

This matrix checks normalization, Hermiticity, normal ordering, parity, and repeated-mode combinatorics. It is a construction benchmark, not an approximation to the full interacting field theory unless all nonzero modes are deliberately removed from the target model.

Sparse assembly has a seven-step method contract

1. Enumerate independently. Generate canonical basis keys and verify small-cutoff counts with a second implementation or a generating function.
2. Attach exact labels. Momentum, charge, parity, irrep, and constraint labels must be computable from the key without applying $H$.
3. Construct the metric. Confirm $G=I$ for normalized Fock states or find and remove null directions when $G\ne I$.
4. Apply operator monomials. Act on a source state, compute the exact ladder and group-theory factor, canonicalize the target, and look it up once.
5. Assemble each contribution separately. Keep $H_0$, each interaction, each counterterm, and every observable as independently testable matrices.
6. Test identities. For covariant matrices $H_{ij}=\langle i|H|j\rangle$, check $H^\dagger=H$; equivalently, the one-index-raised operator $A=G^{-1}H$ must obey $A^\dagger G=GA$. Then test exact commutators, forbidden blocks, analytic entries, finite traces or sum rules, and parameter derivatives such as $\partial H/\partial g$.
7. Only then solve. Report eigenpair residuals in the $G$ inner product and retain enough eigenvectors for the intended observables.

A sparse matrix can have no stored forbidden entries and still be incomplete. For small cutoffs, also construct a dense reference matrix by a different route and compare every entry.

The convergence map locates basis errors

In the shared map, basis construction controls the symmetry-complete retained branch. It does not eliminate the omitted branch; increasing a correct basis still changes induced interactions and effective observables.

The symmetry-complete basis is one necessary branch of a controlled truncation. Exact block structure and matrix checks must join omitted-state corrections, effective observables, and cross-basis tests; monotone energy convergence alone does not certify a matrix element. The false-plateau branch is schematic and not to scale.

Minimum truncation certification record

Required fields for a truncation result and the test that can falsify each field.

Field	Required declaration	Independent test	Failure signal
Target	Hamiltonian, prior regulator, volume, boundary data, observable	Units and free or exact limit	Changing target across cutoff points
Projectors	PΛ, QΛ, all cutoff axes, limit order	State counts and nestedness	Unidentified omitted states
Basis and sectors	Normalization, Gram matrix, null removal, exact charges	Hermiticity and selection rules	Duplicates or broken constraints
Induced Hamiltonian	Derived operator basis and approximation order	Omitted-state toy model or perturbative coefficient	Drift incompatible with the declared tail
Counterterms	Inputs, running coefficients, and no-double-counting rule	Refit protocol at every cutoff	A fitted datum presented as a prediction
Variational status	Manifold, optimizer, symmetry, bound hypotheses	Residual, variance, and ansatz enlargement	Energy plateau with a large residual
Effective observables	Projected and induced operator terms	Sum rule or matched matrix element	Spectrum stable while the observable drifts
Cutoff sequence	Independent basis, volume, counterterm, time, and state scans	Fixed-axis and cross-term fits	Only one diagonal sequence
Extrapolation	Asymptotic form, fit window, covariance, alternatives	Window and model stability	Exponent chosen from the desired answer
Held-out tests	Unused spectrum, matrix element, dynamics, and second basis	Blind comparison after choices freeze	All tests participated in tuning
Adversarial enlargement	Larger state and operator bases	Repeat the full match and prediction	Former plateau moves beyond its error
Claim	Bound, asymptotic evidence, empirical stability, or unresolved	Error and cost reproduced independently	Precision exceeds the weakest test

Adversarial failure: Hermitian but duplicated

Suppose a two-boson zero-momentum generator stores both ordered lists $(n,-n)$ and $(-n,n)$ for $n\ne0$ as independent states, then fills matrix elements symmetrically. The resulting matrix is Hermitian and its eigensolver residuals can be tiny. Nevertheless, the Hilbert space contains two copies of one physical occupation state, producing spurious degeneracies and incorrect transition strengths.

An independent occupation-tuple count exposes the duplication. Alternatively, construct symmetric normalized states explicitly and compare the small-sector matrix entry by entry. Hermiticity is necessary but cannot establish basis uniqueness.

Observable-level validation checklist

• Compare basis dimensions in every exact sector against an independent enumerator at several small cutoffs.
• Verify that $G$ has the signature required by the declared theory after null removal—positive definite for a unitary Hilbert space. Test $H^\dagger=H$ and $O^\dagger=O$ for covariant matrix elements; for $A=G^{-1}H$, test $A^\dagger G=GA$ instead.
• Check exact charge and momentum commutators and every group-theoretically forbidden block.
• Reproduce the one-mode ${:}X^4{:}$ matrix, then test nonzero-momentum entries with repeated and distinct labels.
• Confirm the free spectrum and degeneracies before switching on interactions.
• Reserve at least one operator insertion that uses different selection rules from $H$; a Hamiltonian-only check cannot validate it.
• Repeat a final prediction in a second complete basis or truncation rule after matching the same physical inputs.

You should now be able to (1) enumerate a normalized symmetry-resolved scalar Fock basis without duplicates or null directions and (2) assemble and verify a sparse interaction or operator matrix using analytic selection rules and small matrices. Conformal and Hamiltonian Truncation applies these steps to UV operator states; Renormalizing a Truncated Hamiltonian supplies the induced matrix terms. Gauge-invariant bases return to Hamiltonian Lattice Field Theory.

Exercises

Enumerate a small scalar sector

Let $m=1$, $L=2\pi$, and $E_{\max}=2$. List the zero-momentum even states made only from modes $n=0,\pm1$ whose free energy does not exceed the cutoff. The vacuum energy is set to zero.

Solution

The mode energies are $\omega_0=1$ and $\omega_{\pm1}=\sqrt2$. Even particle number and energy at most $2$ allow the vacuum and two zero-mode quanta: $|0\rangle$ and $|N_0=2\rangle$. The pair $a_1^\dagger a_{-1}^\dagger|0\rangle$ has energy $2\sqrt2>2$ and is excluded. One-particle states are odd under $\phi\mapsto-\phi$, and mixed zero/nonzero states fail momentum or parity. Thus the sector dimension is exactly two.

Derive the off-diagonal one-mode entry

Show that $\langle2|{:}X^4{:}|4\rangle=16\sqrt3$.

Solution

Only the term $4a^\dagger a^3$ lowers particle number by two. Acting on $|4\rangle$ gives

$$4a^\dagger a^3|4\rangle =4\sqrt{4\cdot3\cdot2}\,a^\dagger|1\rangle =4\sqrt{24}\sqrt2\,|2\rangle =16\sqrt3\,|2\rangle.$$

All other normally ordered monomials change particle number by $-4,0,2$, or $4$, so they do not contribute to this entry.

References

• 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.
• Rychkov, Slava, and Lorenzo G. Vitale. “Hamiltonian Truncation Study of the $\phi^4$ Theory in Two Dimensions.” Physical Review D 91, 085011 (2015). DOI.