Determine the quantum critical point location for the 1D transverse-field Ising model with next-nearest-neighbor interactions at higher spin

Determine the exact region in the phase diagram where the quantum critical point lies for the one-dimensional Ising model under a transverse magnetic field with next-nearest-neighbor interactions when the spin magnitude S exceeds 1/2, by establishing the dependence of the critical field B_c on the coupling ratio J_2/J_1 for S > 1/2.

Background

The paper studies the one-dimensional Ising model in a transverse field (1DIMTF), including next-nearest-neighbor (NNN) interactions for S = 1/2, where the quantum critical behavior depends on the coupling ratio J_2/J_1. For S = 1/2, a specific ratio (−J_2/J_1 = 0.32) is used to locate the critical field B_c.

For higher spin magnitudes (S > 1/2), the authors note the absence of reported values of J_2/J_1 in the literature, which prevents determination of the quantum critical point (CP) location in the phase diagram for those cases. Resolving this requires establishing how B_c varies with J_2/J_1 at higher S, thereby pinpointing the CP region.

References

We stress that we have investigated only NNN for S = 1/2 because the value of B_c depends on the ratio J_2/J_1 . However, this ratio has not yet been reported in the literature for higher values of S, and therefore the exact region in the phase diagram where the CP lies remains to be determined.

— Universal and non-universal facets of quantum critical phenomena unveiled along the Schmidt decomposition theorem  (2512.11093 - Soares et al., 11 Dec 2025) in Subsection 'The 1DIMTF considering NNN interaction for S = 1/2 under the light of S_q'