Transverse-field Ising chain, 10 sites, h = 1

TFIsing/chain_10_P_1

Record

-1.2784906 E/N (Pauli)

DMRG (bond dimension = 28)

run script, no paper cited · lowest eligible strict variational bound (rules §6)

The Hamiltonian

H = −∑_⟨ij⟩ σ^z_iσ^z_j − h ∑_i σ^x_i

Transverse field h = 1. Already in Pauli operators, with no S·S term to rescale, so the per-site energy is E/N.

lattice: chain
sites: 10
boundary: periodic
h: 1
energies: 4

Stored as a total energy in the convention of VarBench; quoted here as E/N (Pauli), which is what the papers report (how the conversion works).

Every published energy

Grouped by what the number is. Ranking happens only inside the first group.

Strict variational upper bounds (3)

The record is the lowest eligible energy in this group, and only this group.

E/N (Pauli)	σ	Var(E)	V-score	method	source	year
-1.278523(5)	4.8e-6	2.46e-3	1.5e-4	RBM (alpha = 1) flagged: below-exact VarBench reference run	run script, no paper cited	n/a
-1.2784906record	n/a	1.10e-13	6.7e-15	DMRG (bond dimension = 28) VarBench reference run	run script, no paper cited	n/a
-1.27725(3)	3.0e-5	9.33e-2	5.7e-3	Jastrow baseline VarBench reference run	run script, no paper cited	n/a

Numerically exact (1)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established: the answer, not a claim about it.

E/N (Pauli)	σ	Var(E)	V-score	method	source	year
-1.2784906	n/a	n/a	n/a	Exact Solution VarBench reference run	run script, no paper cited	n/a

Flagged rows

A flag withholds the record and nothing else: the row stays listed, in rank order. Rules §10 is how a flag is lifted or upheld.

below-exact — RBM (alpha = 1), -1.278523(5)

Finding. 6.8 sigma below the exact ground state, which no variational state can be. Settled by rerun on 2026-09-11 (checks/results-tfising-rbm.json).

Diagnosis. The published value is a minimum of the optimization trace, not a converged measurement. Three seeds each, netket 3.22.4, alpha=1 complex RBM, 2000 SR steps. Evaluating the trained parameters by FULL SUMMATION over all 1024 basis states (zero Monte-Carlo error) puts every converged energy ABOVE the exact value, as the variational principle requires, while every training trace dips 1.5e-3 to 3.5e-3 BELOW it. Each published value lies between the two.

Ruled out. Not an autocorrelation-underestimated error bar: tau_corr <= 0.05 and R_hat = 1.0000 across all six runs.

Evidence. checks/results-tfising-rbm.json, checks/tfising_rbm_check.py

Coverage

No row here carries a publication year. Nothing has been added to this instance since the VarBench snapshot of 2024-10-22.

This instance as data

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