QMBLthe quantum many-body leaderboard

All instances / Hubbard

Hubbard chain, 14 sites, U = 4.64158882, n ≈ 0.5714

Exact energy and best values (rules §6)

exact -0.7783541 ED
QMBL check, checks/hubbard-u-labels/ · the instance is solved
best variational -0.7783541 DMRG
Wu et al. (2024) · 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 (c†iσcjσ + h.c.) + U ∑i ni↑ni↓

t = 1, U = 4.64158882, N↑ = N↓ = 4 on 14 sites.

Holds the rows run at U = 4.64158882, the coupling in VarBench's name, after the instance moved to U = 4.64158883. The name's 4.64158882 misrounds the coupling of this Hamiltonian, 4.6415888336 (10^(6/9)). The exact diagonalization, -10.896957984833337, is the ground state at U = 4.64158883 and 3.03e-9 above the ground state at 4.64158882. The published results for this Hamiltonian are on Hubbard/chain_14_P_4_4.64158883. Evidence: checks/hubbard-u-labels.

lattice
chain
sites
14
boundary
periodic
Nf
4
U
4.64158882
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.

E/sitedistance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.7783541exact 0 n/a n/a n/a ED QMBL check, checks/hubbard-u-labels/ QMBL code n/a

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/sitedistance to class bestσVar(E)V-score methodsourcecodeyearcomment
-0.7783541best variational 0 n/a 8.20e-12 2.5e-13 DMRG
VarBench reference run
 Wu et al. (2024) run script 2024

How these numbers were read

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

-0.7783541, ED

Checked 2026-09-16 · Lanczos ED over the full fixed-(N_up, N_dn) space, no symmetry, t = 1: numpy/scipy (checks/hubbard-u-labels/ed_check.py) and node (ed_lib.mjs, run_fermions.mjs)

Reported as. -10.896957987861226 (ed_check.py) | -10.896957987860604 (ed_lib.mjs)

Two independent codes agree to 6.2e-13. The DMRG (MaxBondDim ~1500) row: its run script sets U = 4.64158882. Its value is the ground state at that U truncated at the ninth decimal (8.6e-10 above it), and its variance, 8.2e-12, places the state within 7e-11 of that ground state (Temple's bound, next level -10.77209441); it sits 2.2e-9 below the ground state at 4.64158883, which no variational energy can.

Coverage

Newest published number here: 2024.

This instance as data

JSON · source file on GitHub · add a result · open an issue about this instance