Two-Dimensional Spin-1/2 J1 – J'1 – J2 Heisenberg Model within Jordan–Wigner Transformation

Abstract

The Jordan-Wigner transformation is applied to the spatially anisotropic spin-1/2 Heisenberg model on a square lattice with the nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions. The transformed Hamiltonian describes the interacting spinless fermions that hop between neighbor sites in a gauge field. Using the mean-field-type approximation to both the direct interaction between fermions and the phase factors, which represent the gauge field, the problem is reduced to that concerning a free Fermi gas. Two types of antiferromagnetic ordering (the Néel and collinear ones) are considered. By calculating the ground-state energies, the phase transitions induced by the interaction frustration were analyzed. K e y w o r d s: two-dimensional quantum spin models, frustrated models, fermionization.

Магнетокалоричний ефект у спін-1/2 одновимірній XX моделі з двома регулярнозмінними g-факторами

“…Among the set of works, where rigorous results were obtained due to the implementation of the onedimensional Jordan-Wigner transformation, there can be mentioned, for example, those, where the Hamiltonian contains not only two-spin interactions but also three-spin ones (see, e.g., works [17][18][19][20][21][22][23][24]). In particular, papers [23,24] were devoted to the study of one-dimensional magnetoelectrics, where the coupling of localized spins (i.e., magnetic moments) with the electric polarization of the bond connecting those spins is described by the Katsura-Nagaosa-Balatsky mechanism [25].…”

“…Among the set of works, where rigorous results were obtained due to the implementation of the onedimensional Jordan-Wigner transformation, there can be mentioned, for example, those, where the Hamiltonian contains not only two-spin interactions but also three-spin ones (see, e.g., works [17][18][19][20][21][22][23][24]). In particular, papers [23,24] were devoted to the study of one-dimensional magnetoelectrics, where the coupling of localized spins (i.e., magnetic moments) with the electric polarization of the bond connecting those spins is described by the Katsura-Nagaosa-Balatsky mechanism [25]. Menchyshyn et al [23] showed that the additional account for three-spin interactions can result in a non-trivial magnetoelectric effect (the induction of the electric polarization by a magnetic field at the zero electric field and vice versa), which is not realized in 1D magnetoelectrics with only pair exchanges [26][27][28].…”

“…Menchyshyn et al [23] showed that the additional account for three-spin interactions can result in a non-trivial magnetoelectric effect (the induction of the electric polarization by a magnetic field at the zero electric field and vice versa), which is not realized in 1D magnetoelectrics with only pair exchanges [26][27][28]. In work [24], three-spin interactions (𝑋𝑍𝑌 − 𝑌 𝑍𝑋 and 𝑋𝑍𝑋 + 𝑌 𝑍𝑌 ) arose, so to speak, naturally when considering the stationary energy flow on the basis of the Lagrange multiplier method (see works [17,29]) in the one-dimensional spin- 1 2 isotropic 𝑋𝑌 model of the magnetoelectric with only two-spin interactions. It is also worth singling out work [20], where a rigorous result was obtained for the 𝑋𝑋 model not only with three-spin interactions (𝑋𝑍𝑌 − 𝑌 𝑍𝑋), but also with a uniform long-range pair interaction between the spin 𝑧components.…”

“…Using an approximation of the mean-field type for both the direct interaction between fermions and the phase multipliers corresponding to the gauge field, the problem was reduced to a free Fermi gas, and the properties of the ground state were examined [35]. Later, this method was adapted to study other systems, in particular, the anisotropic and isotropic 𝑋𝑌 models on a rectangular lattice [38][39][40], and the frustrated Heisenberg model with interactions between nearest and next-nearest neighbors [41,42]. It should be noted that the flow of the strength vector of the fictitious magnetic field was considered in the cited works to be identical for all rectangular elementary plaques.…”