Gapped Hamiltonian¶
A growing local quantum many-body Hamiltonian family whose lowest excitation stays a positive energy above its full ground-state space as system size grows.
Core Idea¶
A gapped Hamiltonian is a family of quantum many-body models on growing systems in which the first energy above the whole ground-state space remains at least some positive amount higher as the system grows. This is a size-uniform condition, not just a visible separation between levels in one finite model. If several states share the lowest energy, they all belong to the ground space; the first excitation lies outside it.[ref-6cfbf85020a8][ref-695a3147b242]
Bravyi and Gosset prove such a bound for certain open spin-½ chains and identify other cases whose gaps shrink with length.[^ref-6cfbf85020a8] Lemm, Sandvik, and Wang report a positive bound for certain periodic hexagonal spin-3/2 AKLT patches using a mathematical size-transfer test and numerical calculations. They explicitly say the numerical error estimates are not a fully rigorous mathematical proof.[^ref-695a3147b242]
What It Is Not¶
- Not a gap in one finite system. A family can have separated levels at every finite size while the spacing tends to zero. Bravyi and Gosset's closing branch has an upper bound
1/(n−1).[^ref-6cfbf85020a8] - Not necessarily a unique ground state. Several states can share the lowest energy; compare them as one sector against the first outside excitation.[^ref-6cfbf85020a8]
- Not a gap found only in a preferred sector. A lower excitation in another sector would control the true gap. The AKLT study checked the relevant excitation sectors of its auxiliary cluster.[^ref-695a3147b242]
- Not a universal claim about correlations, continuum mass, or quantum phase paths. Those questions need their own assumptions and evidence.
Scope of Application¶
Use this entry when a local quantum Hamiltonian has a clear rule for making larger systems, a specified interaction and boundary condition, and a known lower bound from the full ground sector to the lowest excitation. The cited 1D result concerns open, translation-invariant, nearest-neighbor rank-one-projector spin-½ chains.[^ref-6cfbf85020a8] The cited 2D result concerns periodic hexagonal AKLT patches with both size parameters at least 12.[^ref-695a3147b242]
Each Hamiltonian at a specified size is a quantum operator on that system's state space; those operators are necessary parts of the family, while a uniform gap is a further condition across sizes. The two results do not automatically cover other spin systems, other lattices, or changed boundaries. A positive gap on an auxiliary finite cluster also needs an appropriate size-transfer argument before it supports a growing-family claim.
Clarity¶
Ask: What is the model family? Which states count as ground states? How high is the first state outside that space? Does one positive lower bound survive as size grows? Those questions prevent a degenerate ground space from being mistaken for a zero excitation gap and prevent a finite-size spacing from being mistaken for a thermodynamic-family gap.
Boundary conditions belong in the answer. The Bravyi–Gosset theorem is stated for an open chain,[^ref-6cfbf85020a8] while the growing AKLT patches in the other study are periodic.[^ref-695a3147b242]
Manages Complexity¶
The label gapped summarizes many spectra through one size-independent bound on the lowest excitation. Bravyi and Gosset's theorem gives a parameter test that separates gapped cases from cases whose gap closes in their chain class.[^ref-6cfbf85020a8]
The AKLT study uses an analytical test linking a 36-site open auxiliary cluster to arbitrarily large periodic target patches. DMRG supplies the cluster-gap estimate. Its numerical-input limit remains part of the claim even though the transfer criterion is proved mathematically.[^ref-695a3147b242]
Abstract Reasoning¶
Specify the interaction, growth rule, and boundaries. Locate the entire lowest-energy space at each size, then the first state outside it. A gapped-family conclusion needs a positive lower bound that does not shrink as the permitted sizes grow. An upper bound tending to zero shows that this conclusion fails; a single positive finite-size measurement cannot establish it.
When a proof uses a small auxiliary system, identify both the finite-size criterion and the evidence that the auxiliary gap clears its threshold. The AKLT paper combines a mathematical criterion with numerical input and reports that limit openly.[^ref-695a3147b242]
Knowledge Transfer¶
The same recognition questions can be asked of unlike quantum models: a 1D qubit chain or a 2D AKLT lattice. The model-specific proofs cannot be copied between them. The Bravyi–Gosset parameter theorem belongs to its open projector-chain class,[^ref-6cfbf85020a8] and the AKLT bound belongs to the stated periodic hexagonal family with its auxiliary numerical step.[^ref-695a3147b242]
Calling a different spectrum “gapped” is only an analogy until its own states, growth rule, and bound are specified.
Examples¶
Open spin-½ projector chain¶
For certain fixed two-qubit projector choices, Bravyi and Gosset prove that the gap of their open n-qubit chain stays above a positive constant independent of n. Other choices give a gap at most 1/(n−1), showing why a positive finite-chain value is not enough.[^ref-6cfbf85020a8]
Role map: The fixed interaction makes a growing family; its potentially degenerate zero-energy ground space is the baseline; the smallest nonzero eigenvalue is the excitation; the theorem supplies or rules out the uniform bound under open boundaries.
Periodic hexagonal AKLT patches¶
Lemm, Sandvik, and Wang report a lower gap bound of 0.00646 for periodic spin-3/2 hexagonal patches with both size parameters at least 12. Their analytic criterion uses a separate open 36-site cluster whose gap they estimate numerically. The paper does not claim rigorous DMRG error bounds.[^ref-695a3147b242]
Role map: Periodic patches form the growing family; the zero-energy ground sector is the baseline; the first positive eigenvalue is the excitation; the reported lower bound is uniform over the specified patch sizes. The open cluster is evidence used by the criterion, not the target family.
Relationships to Other Abstractions¶
Current abstraction Gapped Hamiltonian Domain-specific
Parents (1) — more general patterns this builds on
-
Gapped Hamiltonian is part of Quantum Operator Domain-specific
Each size-indexed Hamiltonian is a quantum operator whose spectrum supplies the family's ground sector and first excitation.
Hierarchy path (1) — routes to 1 parentless root
- Gapped Hamiltonian → Quantum Operator → Function (Mapping)
Neighborhood in Abstraction Space¶
Gapped Hamiltonian sits in a sparse region of the domain-specific corpus (64th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Statistical Mechanics & Particle Phenomena (16 abstractions)
Nearest neighbors
- Geometrical Frustration — 0.85
- Quantum Walk — 0.84
- Pansu Derivative — 0.84
- Fractionalization — 0.84
- Principal quantum number — 0.84
Computed from structural-signature embeddings · 2026-10-17
References¶
[^ref-6cfbf85020a8]: Bravyi, Sergey, and David Gosset (2015). Gapped and gapless phases of frustration-free spin-½ chains. Original author preprint, arXiv:1503.04035v3, especially Abstract, Introduction Eq. (1), and Theorem 1. https://arxiv.org/html/1503.04035v3. Full original preprint relevant sections inspected. [^ref-695a3147b242]: Lemm, Marius, Anders W. Sandvik, and Ling Wang (2020). Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice. Original author preprint, arXiv:1910.11810v2, especially main result Eq. (4), finite-size criterion, and numerical-caveat paragraphs. https://arxiv.org/html/1910.11810v2. Full original preprint relevant sections inspected; the paper expressly disclaims rigorous DMRG error estimates.