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 and its omitted complement to cutoff-dependent Hamiltonians and operators, symmetry-resolved bases, variational diagnostics, multi-axis convergence tests, and benchmarks that can reveal a false plateau.
Enter this chapter
Section titled “Enter this chapter”Begin with Hilbert-Space Truncation as a Regulator to define both the retained subspace and the states it omits.
Topics
Section titled “Topics”- Hilbert-Space Truncation as a Regulator — Treats the Hilbert-space projector as a regulator.
- Basis Construction, Symmetry Sectors, and Matrix Elements — Constructs normalized symmetry-resolved bases.
- Conformal and Hamiltonian Truncation — Uses solvable ultraviolet states to represent an interacting deformation.
- Renormalizing a Truncated Hamiltonian — Accounts for virtual transitions through omitted states.
- Variational Principles and Field-Theory Ansätze — States what variational energy bounds do and do not imply.
- Observables and Dynamics in Truncated Spaces — Renormalizes observables as well as energies.
- Convergence, Extrapolation, and Error Certification — Distinguishes proven bounds from extrapolation diagnostics.
- Benchmark Theories for Truncation Methods — Uses independent models to expose false convergence.
How the ideas fit together
Section titled “How the ideas fit together”Projecting the Hamiltonian removes virtual paths through the omitted subspace. Effective interactions compensate those paths, and observable operators generally need corresponding corrections. Energy stability alone therefore cannot establish accurate matrix elements. A variational upper bound applies to a specified Hamiltonian and admissible state space; it does not bound regulator or continuum-extrapolation errors. Tests must vary the cutoff and basis while checking observables against independent information. Elias Miró and Ingoldby 2023, §§ 2–3.
Where to go next
Section titled “Where to go next”Continue to Tensor-Network QFT to choose the state or partition function and the network geometry that represents it.