QMBLthe quantum many-body leaderboard

All instances / t-J

t-J square, 18 sites, J = 0.4, 2 holes

Exact energy and best values (rules §6)

exact -0.6590431 ED
10.1007/978-94-010-0771-9_18, · the instance is solved
best variational -0.65904 VMC (Gutzwiller)
10.1007/978-94-010-0771-9_18, · the lowest unflagged variational energy, a strict variational bound

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

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↑ = 8, N↓ = 8 on 18 sites (2 holes). Cluster: square-shaped periodic cluster of 18 sites, edge vectors (3, 3) and (3, -3) (RULES.md 2.1); the periodic boundaries are fixed by QMBL's diagonalization, which reproduces the source's exact energy (checks/tj-ed-2026-10/).

lattice
square
sites
18
boundary
periodic
t
1
J
0.4
N_up
8
N_dn
8
nn_term
true
energies
13

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.

E/sitedistance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.6590431exact 0 n/a n/a n/a ED
proceedings
 10.1007/978-94-010-0771-9_18, 2000

Strict variational upper bounds (12)

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/sitedistance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.65904best variational 0 n/a 5.00e-4 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.65904tied with best variational at 2σ <1e-5 n/a 1.00e-4 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.65902 2.0e-5 n/a 2.00e-3 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.65898 6.0e-5 n/a 5.00e-3 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.65882 2.2e-4 n/a 1.80e-2 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.6583 7.4e-4 1.0e-4 n/a n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.65826 7.8e-4 n/a 6.00e-2 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.65638 2.7e-3 n/a 1.91e-1 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.6542 4.8e-3 n/a n/a n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.6541 4.9e-3 1.0e-4 n/a n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.64559 1.3e-2 n/a 9.40e-1 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000
-0.63847 2.1e-2 n/a 1.17e+0 n/a VMC (Gutzwiller)
proceedings
 10.1007/978-94-010-0771-9_18, 2000

How these numbers were read

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

-0.6590431, ED

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65904

Table 1.2, L = 18, N = 16, J = 0.4, p = ∞ (the exact ground state) (line 305); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65904, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65904

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 6 (line 303); variance: column ΔE_p x L^2 = 0.0005 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65904, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65904

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 7 (line 304); variance: column ΔE_p x L^2 = 0.0001 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65902, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65902

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 5 (line 302); variance: column ΔE_p x L^2 = 0.002 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65898, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65898

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 4 (line 301); variance: column ΔE_p x L^2 = 0.005 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65882, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65882

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 3 (line 300); variance: column ΔE_p x L^2 = 0.018 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.6583(1), VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.6583(1)

Table 1.1, L = 18, N = 16, J = 0.4, p = 2, column FLST (line 237); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65826, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65826

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 2 (line 299); variance: column ΔE_p x L^2 = 0.060 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.65638, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65638

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 1 (line 298); variance: column ΔE_p x L^2 = 0.191 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.6542, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.65420

Table 1.1, L = 18, N = 16, J = 0.4, p = 1, column Exact (line 236); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.6541(1), VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.6541(1)

Table 1.1, L = 18, N = 16, J = 0.4, p = 1, column FLST (line 236); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.64559, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.64559

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.55, p = 0 (line 297); variance: column ΔE_p x L^2 = 0.940 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

-0.63847, VMC (Gutzwiller)

Checked 2026-10-22 · source text of arXiv:cond-mat/0006353 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. -0.63847

Table 1.2, L = 18, N = 16, J = 0.4, Δ_BCS = 0.00, p = 0 (line 290); variance: column ΔE_p x L^2 = 1.174 of the same row, the variance of the total H in units of t^2 (ΔE_p = (⟨H^2⟩ - ⟨H⟩^2)/L^2, line 222); version read: arXiv text (sources/cond-mat-0006353.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. (1.1): H = J sum (S_i.S_j - n_i n_j / 4) - t sum c~+c~, t = 1; Tables 1.1 and 1.2 print E0/L, L = 18, N = 16, J = 0.4. QMBL's diagonalization of the periodic 18-site cluster gives -0.6590431107 per site, the printed exact -0.65904.

Coverage

Newest published number here: 2000.

This instance as data

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