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Journal of Physical Studies 19(4), Article 4701 [6 pages] (2015)
DOI: https://doi.org/10.30970/jps.19.4701 GROUND STATE OF THE TWO-DIMENSIONAL SPIN-1/2 J1-J2 HEISENBERG MODEL WITHIN THE JORDAN–WIGNER FERMIONIZATIONO. R. Baran, T. M. Verkholyak
Institute for Condensed Matter Physics |
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Jordan-Wigner fermionization (Wang, 1991) is applied to the antiferromagnetic spin-1/2 Heisenberg model \[ H =\sum_{i=1}^{\infty} \! \sum_{j=1}^{\infty} \! ( J_1 \;\! {\bf{S}}_{i,j}\cdot{\bf{S}}_{i+1,j} + J_\perp \;\! {\bf{S}}_{i,j}\cdot{\bf{S}}_{i,j+1} +J_2 \;\! {\bf{S}}_{i,j}\cdot{\bf{S}}_{i+1,j+1} +J_2 \;\! {\bf{S}}_{i,j+1}\cdot{\bf{S}}_{i+1,j} ) \] on a rectangular lattice with competing interactions between the nearest neighbors ($J_1>0$, $J_\perp>0$) and next-nearest neighbors ($J_2>0$). The transformed Hamiltonian corresponds to the interacting spinless fermions hopping between neighboring sites in a gauge field. Here we study the case of the possible N\'{e}el ordering which emerges for a weak next-nearest-neighbor interaction. The problem is considered within the mean-field-type approximation for both the direct interaction between fermions and phase factors, which represent the gauge field, neglecting the correlations between the neighboring sites.
For the case of the square lattice ($J_1=J_\perp$) we calculate the dependence of ground-state sublattice magnetization on the frustration parameter $J_2/J_1$, and study the quantum phase transition from the N\'{e}el ordered to a disordered state, which occurs due to the competition between the nearest-neighbor and next-nearest-neighbor interactions. The obtained result is compared with the results of other analytical and numerical methods.
It is shown that the applied approach provides a qualitatively correct result for the magnetization curve. However, it gives a too large value for the N\'{e}el order parameter for the unfrustrated square lattice (at $J_2=0$) and a somewhat reduced value for the critical ratio of $J_2/J_1$. The self-consistent consideration of the correlations between the neighboring sites is expected to improve the precision of the method.
PACS number(s): 75.10.Jm, 75.40.Cx