Hubbard square 6x6, U = 4, n ≈ 0.6667

Best values (rules §6)

best variational	-1.1834(1)	VAFQMC Levy et al. (2023) · the lowest unflagged variational energy, a strict variational bound
best projected	-1.18525(4)	CP-AFQMC Shi & Zhang (2013) · the lowest unflagged projected energy, a projected estimate, which depends on its node or constraint

Energies as E/site. Classes are different claims and are not ranked against each other.

The Hamiltonian

H = −t ∑_⟨ij⟩,σ (c^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = 4, N_↑ = N_↓ = 12 on 36 sites.

lattice: square
sites: 36
boundary: periodic
Nf: 12
U: 4
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).

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-1.1834best variational	0	1.0e-4	2.00e-1	1.4e-3	VAFQMC peer reviewed	Levy et al. (2023)		2023

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-1.18525best projected	0	4.0e-5	n/a	n/a	CP-AFQMC	Shi & Zhang (2013)		2013

How these numbers were read

Where each number below was read, and how it was checked.

-1.18525(4), 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. -1.18525(4) per site (E/L, units of t)

Stored energy and sigma equal the printed RC/SYM cell x N exactly. Instance agrees: square 6 x 6, PBC, N_up = N_dn = 12, U = 4, t' = 0, standard Hubbard form (no shift; Einf 16 = 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 c1e8f91 2022-03-17 with this reference, never changed. Venue PRB (peer reviewed). Other rows on the instance: VAFQMC (N_l=4) -42.6024(36) (arXiv:2308.08594), 0.067 above. No duplicate. Same-source values not in the table: CP/UHF -1.18444(3), CP/SYM -1.18625(3) (Table II).

-1.1834(1), 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. −1.1834(1)

Table I | same instance as above | variance reported as 0.2, not stored: convention not verified

Coverage

Newest published number here: 2023. Literature last checked 2026-09-15 (table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance): 6 papers read, 0 of them with an energy on this page.

This instance as data

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