Table 1 Parameters of the two-orbital Hubbard model

System	a	c	Δ_eg	t_αα	t_ββ	t_αβ	\({t}_{\alpha \alpha }^{{\prime} }\)	\({t}_{\beta \beta }^{{\prime} }\)	U_αα	U_ββ	U_αβ	J_H	\({J}_{1}^{{{{{{{{\rm{cal}}}}}}}}}\)	\({J}_{2}^{{{{{{{{\rm{cal}}}}}}}}}\)	\({J}_{1}^{{{{{{{{\rm{cal,corr}}}}}}}}}\)
	[Å]	[Å]	[eV]	[eV]	[eV]	[eV]	[meV]	[meV]	[eV]	[eV]	[eV]	[eV]	[meV]	[meV]	[meV]
La₂NiO₄	3.890	12.55	0.48	−0.070	−0.403	−0.161	−8.6	74.9	3.06	3.15	1.97	0.52	60.7	1.57	51.0
LNO/STO	3.905	12.62	0.48	−0.067	−0.395	−0.156	−8.3	74.5	3.01	3.14	1.94	0.52	59.5	1.56	49.7
LNO/LSAT	3.868	12.69	0.55	−0.065	−0.410	−0.156	−7.4	76.3	3.00	3.11	1.92	0.51	62.2	1.65	51.8
LNO/NGO	3.859	12.71	0.58	−0.064	−0.414	−0.155	−7.1	76.7	3.03	3.16	1.97	0.51	62.3	1.65	51.9
LNO/LAO	3.793	12.78	0.74	−0.060	−0.447	−0.156	−5.5	80.5	3.02	3.08	1.93	0.50	71.6	1.83	56.8

Crystal-field splitting Δ_eg, (next-)nearest-neighbor hopping \({t}_{ij}^{({\prime} )}\) between ith and jth Ni orbitals (α and β here denote the z² and x² − y² orbital, respectively), the inter- and intra-orbital Coulomb interaction U_ij, and Hund’s exchange J_H between the two orbital as calculated by DFT and cRPA with the in-plane lattice constant a of the three substrates; note that \({t}_{\alpha \beta }^{{\prime} }=0\) by symmetry. From these ab initio calculated parameters, the spin couplings J₁ and J₂ are calculated from superexchange (second-order perturbation theory), i.e., from Eq. (1) with t and \({t}^{{\prime} }\), respectively. Estimating higher-order terms using a one-orbital analogy, yields the reduced \({J}_{1}^{{{{{{{{\rm{cal,corr}}}}}}}}}\) couplings—see text. The bulk lattice parameters refer to the low-temperature tetragonal polymorph of La₂NiO₄, after ref. ²⁹.

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