the respective quantities of the quantum spin models on the Bethe lattice ($p \to \infty$). In particular, we conjecture that the phase transition field of the TFIM on the Bethe lattice is positioned at $h_t^{(\infty)} = 3.29332$ and the ground-state energies per bond of the XY and the Heisenberg models, respectively, occur at $E_0^{(\infty)} = -1.08083446 $ and $-1.291944$. The three quantum spin models have not yet been considered on the Bethe lattice with the coordination number four.

The support received from the Grants QIMABOS APVV-0808-12 and VEGA-2/0130/15 is acknowledged.

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