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 = 0.4; the exchange term includes −ninj/4. N↑ = 7, N↓ = 7 on 16 sites (2 holes). Cluster: 4 x 4 square lattice, open boundaries (arXiv:1908.10000, lines 79-80); QMBL's diagonalization gives -0.564366793 per site (checks/tj-ed-2026-10/).
- lattice
- square
- sites
- 16
- boundary
- open
- t
- 1
- J
- 0.4
- N_up
- 7
- N_dn
- 7
- nn_term
- true
- energies
- 2
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 (1)
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.
-0.5643668, ED
Checked 2026-10-22 · source text of arXiv:1908.10000 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. -0.56436
text: E_ED of the 4 x 4 lattice with 2 holes, J/t = 0.4, open boundaries (line 94); version read: arXiv text (sources/1908.10000.txt); the journal version was not reached from this session (the publisher returned 403), to be read on the verification pass (RULES.md 8) | Eq. (1): H = -t sum (c+c + H.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1, J/t = 0.4, open boundaries (lines 79-80); energies per site: QMBL's diagonalization of the open 4 x 4 lattice with 2 holes gives -0.564366793 per site, the printed E_ED -0.56436.
-0.56428(2), fPEPS
Checked 2026-10-22 · source text of arXiv:1908.10000 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. -0.56428(2)
journal Supplementary Information, Table S2, row 4×4 (line 64); the arXiv text prints E_fPEPS = -0.56428 without an error bar (sources/1908.10000.txt, line 93); version read: the journal's Supplementary Information (npj Quantum Mater. 5, 28 (2020), ESM of doi:10.1038/s41535-020-0226-4, text in sources/1908.10000-si.txt) and the arXiv text (sources/1908.10000.txt); the journal value is stored where the two differ (RULES.md 8): the error bar (2) is printed only in the SI, and the contraction dimension is Dc = 56 in the journal (Methods; SI Sec. A, line 18) against Dc = 48 in arXiv v1 | Eq. (1): H = -t sum (c+c + H.c.) + J sum (S_i.S_j - n_i n_j / 4), t = 1, J/t = 0.4, open boundaries (lines 79-80); energies per site: QMBL's diagonalization of the open 4 x 4 lattice with 2 holes gives -0.564366793 per site, the printed E_ED -0.56436.