The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 4, N↑ = N↓ = 22 on 64 sites.
- lattice
- square
- sites
- 64
- boundary
- periodic
- Nf
- 22
- U
- 4
- 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).
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.
Projected or fixed-node estimates (1)
Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Each depends on its node or constraint, so two of them are not cleanly comparable to each other or to strict bounds.
How these numbers were read
Where each number below was read, and how it was checked.
-1.1858(2), CP-AFQMC
Checked 2026-09-29 · arXiv:1307.2147v3 (28 Sep 2013), v1 (8 Jul 2013) and v2 (31 Aug 2013) PDFs, pypdf plain + layout (sources/1307.2147v{1,2,3}.va5.*); Table II identical in all three versions; journal PDF not read
Reported as. -1.1858(2) per site (E/L, units of t)
Stored energy and sigma equal the printed RC/SYM cell x N exactly. Instance agrees: square 8 x 8, PBC, N_up = N_dn = 22, U = 4, t' = 0, standard Hubbard form (no shift; Einf 30.25 = U N_up N_dn / N). bound_type projected agrees with RULES 4 (constraint release; the paper's 'essentially exact' is not a bound and not an established exact result, since RC 'can converge from below or above'). Primary source: the paper's own computation; VarBench upload by Yiqi Yang d7277fa 2022-03-17 with this reference, never changed. Venue PRB (peer reviewed). Other rows: Tensor-Backflow -75.7312 and + 1 Lanczos step -75.8144 (arXiv:2507.01856), VAFQMC (N_l=4) -75.78816(448) (arXiv:2308.08594), all above. No duplicate. Same-source values not in the table: CP/UHF -1.18580(2), CP/SYM -1.18673(2) (Table II).
-1.1846, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2507.01856 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -1.1846
Table I row '44/64 8x8' (p.5). | Same as above.
-1.18419(7), VAFQMC
Checked 2026-09-15 · source text of arXiv:2308.08594 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. −1.18419(7)
Table I | same instance as above | variance reported as 0.3, not stored: convention not verified
-1.1833, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2507.01856 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -1.1833
Table I row '44/64 8x8': Ep=0=-1.1833, Ep=1=-1.1846, Ref=-1.186744 (p.5). | n=44/64=0.6875 => Nf=22 on 64 sites, U=4, PBC matches Hubbard/square_64_P_22_4 exactly.