The Hamiltonian
H = −t1 ∑⟨ij⟩,σ (c†iσcjσ + h.c.) −t2 ∑⟨⟨ij⟩⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓
t1 = 1, t2 = -0.2, U = 8, N↑ = N↓ = 56 on 128 sites.
This instance is a t12 variant with a next-nearest-neighbour hopping t2, written as in VarBench's Eq. (S6). t2 is the t′ of the sources' H = −t ∑⟨ij⟩ − t′ ∑⟨⟨ij⟩⟩ + U ∑ ni↑ni↓. The couplings are those each source prints (see the rows).
- lattice
- rectangular-8x16
- sites
- 128
- boundary
- periodic
- Nf
- 56
- U
- 8
- t1
- 1
- t2
- -0.2
- energies
- 5
Stored as a total energy; quoted here as
E/site, which is what the papers report
(how the conversion works).
How these numbers were read
Where each number below was read, and how it was checked.
-0.7419, Transformer backflow (MARCH)
Checked 2026-10-22 · source text of arXiv:2507.02644 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. −0.7419
Supplementary Table 5, row 16 × 8, t' = -0.2; dl/gu-si.txt line 195; no error bar | Energy per site in units of t. PBC per the Supplementary Table 5 caption ('Raw data for the energies of Hubbard model under PBC'); U = 8 and hole doping 1/8 per the main text (Fig. 4 caption: 'U = 8 and hole doping δ = 1/8 under PBC'; the t' = -0.2 study: 'Lx × 8 with Lx ranging from 8 to 32, with t' = -0.2 and δ = 1/8'). H has -t' sum_<<ij>>, so t2 = t'. The 16 x 4 and 16 x 16 values at t' = 0 equal the arXiv text's -0.76298 and -0.75630 to the printed digits. No error bar printed. Version read: journal Supplementary Information (41467_2026_74028_MOESM1_ESM.pdf); the arXiv text has no raw-data table.
-0.741, Tensor backflow (NNN backflow, direct optimization, 1...
Checked 2026-10-22 · 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. −0.7410
Table V, row 8 × 16, t' = -0.2, NNN backflow; dl/liang-prb.txt line 493, column Ep=1; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.
-0.7406, Tensor backflow (NN backflow, direct optimization, 1...
Checked 2026-10-22 · 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. −0.7406
Table V, row 8 × 16, t' = -0.2, NN backflow; dl/liang-prb.txt line 492, column Ep=1; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.
-0.7359, Tensor backflow (NNN backflow, direct optimization)
Checked 2026-10-22 · 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. −0.7359
Table V, row 8 × 16, t' = -0.2, NNN backflow; dl/liang-prb.txt line 493, column Ep=0; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.
-0.7354, Tensor backflow (NN backflow, direct optimization)
Checked 2026-10-22 · 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. −0.7354
Table V, row 8 × 16, t' = -0.2, NN backflow; dl/liang-prb.txt line 492, column Ep=0; no error bar | Energy per site in units of t at n = 0.875, U = 8, t' = -0.2 (Table V caption: 'n = 0.875 and U = 8'; Table VI column 'n, U, t′ = 0.875, 8, −0.2'). PBC: the 8 x 16 NNN Ep = 1 value -0.7410 and the 12 x 12 U_UHF = 4 Ep = 1 value -0.7420 equal Table II's PBC rows. Ep = 0 is the tensor-backflow state, Ep = 1 the same state after one Lanczos step; both are variational. No error bar printed. Version read: Phys. Rev. B 113, 155121 (publisher PDF); arXiv:2507.01856v4 (sources/2507.01856.txt) prints the same numbers.