The Hamiltonian
H = −t ∑⟨ij⟩,σ P(c†iσcjσ + h.c.)P + J ∑⟨ij⟩ (Si·Sj − ninj/4)
t-J model: P projects out doubly occupied sites, S = σ/2. t = 1, J = 0.333333; the exchange term includes −ninj/4. N↑ = 11, N↓ = 11 on 24 sites (2 holes). Cluster: 24-site open honeycomb flake of seven hexagons, a central hexagon and its six neighbours, C6 symmetric (arXiv:2406.16865, Fig. 2: 'A 24-site honeycomb lattice', open boundary condition and C6 rotation symmetry).
- lattice
- honeycomb-flake24
- sites
- 24
- boundary
- open
- t
- 1
- J
- 0.3333333333333333
- N_up
- 11
- N_dn
- 11
- nn_term
- true
- energies
- 4
Stored as a total energy; quoted here as
E/site, which is what the papers report
(how the conversion works).
Every published energy
Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (3)
Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.
How these numbers were read
Where each number below was read, and how it was checked.
-0.4143583, ED
Checked 2026-10-22 · source text of arXiv:2406.16865 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)
Reported as. -29.834
Table III, E_tot, row ED, 24-site cluster, t/J = 3 (line 362); version read: arXiv text (sources/2406.16865.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eqs. (1)-(2): H = P(H_t + H_J)P, H_t = -t sum (c+c + H.c.), H_J = J sum (S_i.S_j - n_i n_j / 4), J = 1 and t/J = 3 (lines 104-105): the instance's J/t = 1/3 and a factor 3 (scale). Tables I and III print E_tot of the 24-site cluster. QMBL's diagonalization of the 7-hexagon flake with one hole gives -8.2605976752 in units of t, -24.78179 in units of J, the printed ED -24.782 (checks/tj-ed-2026-10/).
-0.398, VMC (phase string)
Checked 2026-10-22 · source text of arXiv:2406.16865 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; found in the accepted manuscript by the adversarial reading of qmbl-verify 2026-10-23 (group tjP5) and checked against the committed text by this script's line test (2026-10-24)
Reported as. -28.656
accepted manuscript, Table III, E_tot, row |Ψ̃_±1⟩_2h, 24-site cluster, t/J = 3 (line 370); arXiv:2406.16865v1 Table III has no such row; the accepted manuscript constructs it 'as in the single-hole case' (line 359); version read: the accepted manuscript of Phys. Rev. B 110, 155111 made available via CHORUS (dated 23 September 2024; text in sources/2406.16865-am.txt); the publisher's version of record is closed access | Eqs. (1)-(2): H = P(H_t + H_J)P, H_t = -t sum (c+c + H.c.), H_J = J sum (S_i.S_j - n_i n_j / 4), J = 1 and t/J = 3 (lines 104-105): the instance's J/t = 1/3 and a factor 3 (scale). Tables I and III print E_tot of the 24-site cluster. QMBL's diagonalization of the 7-hexagon flake with one hole gives -8.2605976752 in units of t, -24.78179 in units of J, the printed ED -24.782 (checks/tj-ed-2026-10/).
-0.3705, VMC (phase string)
Checked 2026-10-22 · source text of arXiv:2406.16865 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)
Reported as. -26.676
Table III, E_tot, row |Ψ_±1⟩_2h, 24-site cluster, t/J = 3 (line 361); version read: arXiv text (sources/2406.16865.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eqs. (1)-(2): H = P(H_t + H_J)P, H_t = -t sum (c+c + H.c.), H_J = J sum (S_i.S_j - n_i n_j / 4), J = 1 and t/J = 3 (lines 104-105): the instance's J/t = 1/3 and a factor 3 (scale). Tables I and III print E_tot of the 24-site cluster. QMBL's diagonalization of the 7-hexagon flake with one hole gives -8.2605976752 in units of t, -24.78179 in units of J, the printed ED -24.782 (checks/tj-ed-2026-10/).
-0.2985833, VMC (Bloch wave)
Checked 2026-10-22 · source text of arXiv:2406.16865 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; first reading and an adversarial second reading (2026-10-22)
Reported as. -21.498
Table III, E_tot, row |Ψ_0⟩_2h, 24-site cluster, t/J = 3 (line 360); version read: arXiv text (sources/2406.16865.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eqs. (1)-(2): H = P(H_t + H_J)P, H_t = -t sum (c+c + H.c.), H_J = J sum (S_i.S_j - n_i n_j / 4), J = 1 and t/J = 3 (lines 104-105): the instance's J/t = 1/3 and a factor 3 (scale). Tables I and III print E_tot of the 24-site cluster. QMBL's diagonalization of the 7-hexagon flake with one hole gives -8.2605976752 in units of t, -24.78179 in units of J, the printed ED -24.782 (checks/tj-ed-2026-10/).