The Hamiltonian
H = −t ∑⟨ij⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t = 1, U = 4, N↑ = N↓ = 58 on 144 sites.
- lattice
- square
- sites
- 144
- boundary
- periodic
- Nf
- 58
- U
- 4
- energies
- 3
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 (2)
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.
-1.1089(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. -1.1089(2) per site (E/L, units of t)
Stored energy and sigma equal the printed RC/SYM cell x N exactly. Instance agrees: square 12 x 12, PBC, N_up = N_dn = 58, U = 4, t' = 0, standard Hubbard form (no shift; Einf 93.44444444444444 = 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 bfed1d8 2022-03-17 with this reference, never changed. Venue PRB (peer reviewed). Other rows: Tensor-Backflow -158.9472 and + 1 Lanczos step -159.2784 (arXiv:2507.01856), both above. No duplicate. Same-source values not in the table: CP/UHF -1.10912(3), CP/SYM -1.11015(3) (Table II).
-1.1061, 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. -1.1061
Table I row '116/144 12x12' (p.5). | Same as above.
-1.1038, 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. -1.1038
Table I row '116/144 12x12': Ep=0=-1.1038, Ep=1=-1.1061, Ref=-1.110244 (p.5). | n=116/144=0.80556 => Nf=58 on 144 sites, U=4, PBC matches Hubbard/square_144_P_58_4 exactly.