A Wigner-Jordan (WJ) transformation is used to study the one-dimensional (1D) and two-dimensional (2D) quantum antiferromagnetic Heisenberg model. The advantage of using the Wigner-Jordan transformation is that it preserves all spin-commutation relations as well as the spin on-site exclusion principle. In the 1D case a nearest-neighbor covalent-bonding state of the WJ fermions is found to have a ground-state energy (-0.4351J per site) comparable with that from the Bethe-ansatz solution (-0.4431J per site), and a linear energy spectrum at low energies with velocity 1.6366J, in close agreement with the velocity obtained by Haldane for the quantum antiferromagnetic Heisenberg model with 1/
d2 interaction (1.5708J). The method used for studying the 1D model is then applied to the 2D Heisenberg model in a square lattice. The resulting state at finite temperature is the in-phase flux state, i.e., a flux state of the Wigner-Jordan spinless fermions with an in-phase fermion orbital current circulating around each elementary plaquette.The single-particle excitation spectrum, i.e., the energy dispersion of reversing orientation of a spin in the system, of the in-phase flux state is overall similar to that of spin waves, with significant difference near the edge point k=(\ensuremath{\pi},0), at which the excitation energy of the WJ fermion is zero, whereas that of the spin-wave excitation is 2J. The specific heat of the in-phase flux state predicts a correct temperature dependence over the entire temperature range, namely a
T2 dependence at low temperature, a peak near T/J=0.6, and a 1/
T2 decreasing at high temperature. It also gives excellent agreement with the specific heat calculated from numerical methods. In contrast, the spin-wave theory only correctly predicts a
T2 dependence at low temperature. The Raman spectrum of the in-phase flux state is calculated and shows significant improvement over that from spin-wave theories compared with the experimental spectrum of
La2CuO4. The exchange parameter, J, obtained from the comparison is 1060 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, in agreement with that obtained from analyzing the neutron-scattering data.
The excitation spectrum of the quantum antiferromagnetic Heisenberg model for a square lattice, as predicted from the in-phase flux state of Wigner-Jordan fermions, is calculated and compared with that calculated from conventional spin-wave theory. While the two spectra are overall similar, there are significant differences for short-wavelength excitations. The ratio of the excitation energy at wave vector (\ensuremath{\pi}/2,\ensuremath{\pi}/2) to that at (\ensuremath{\pi}/2,0) is predicted to be \ensuremath{\surd}2 for the Wigner-Jordan fermion excitations, and 2/ \ensuremath{\surd}3 for the spin-wave excitations. At long wavelengths, in contrast to the gapped spin-wave spectrum of Auerbach and Arovas, the excitation spectrum of Wigner-Jordan fermions is gapless at any finite temperature. Possible experimental examination of the excitation spectrum is discussed.
A two-dimensional (2D) Wigner-Jordon transformation which maps spin variables into spinless fermion variables is found and is used to study the 2D spin-1/2 antiferromagnetic Heisenberg model. The transformation generates a fictitious gauge field in the XY component of the Heisenberg model, and hence induces an in-phase orbital current in the Ising component flowing around each elementary plaquette of the underlying lattice. The ground state of the Heisenberg model in a 2D square lattice is found to be an in-phase N\'eel flux phase, i.e., a coexisting state of the flux phase with in-phase orbital currents and a long-range antiferromagnetic spin order. The zero-temperature mean-field energy of the in-phase N\'eel flux state,
E0=-0.33J per bond, is only 1% higher than the best-estimated ground-state energy, -0.334J per bond.
Making the assumption that high-energy fermions exist in the two dimensional spin- 1/2 Heisenberg antiferromagnet, we present predictions based on the pi-flux ansatz for the dynamic structure factor when the antiferromagnet is subject to a uniform magnetic field. The main result is the presence of gapped excitations in a momentum region near (pi,pi) with energy lower than that at (pi,pi). This is qualitatively different from spin-wave theory predictions and may be tested by experiments or by quantum Monte Carlo.
A disordered spin state, the nearest-neighbor virtual bonding state of Wigner-Jordan fermions, of the quantum antiferromagnetic Heisenberg model on a triangular lattice is found to have a zero-temperature energy -1.106J per spin, lower than that calculated from spin-wave theories, and in excellent agreement with that from numerical calculations, (-1.094\ifmmode\pm\else\textpm\fi{}0.018)J per spin. The excitations of the state are Wigner-Jordan fermions with a gapless spectrum. The low-temperature specific heat is predicted to be proportional to ${\mathit{T}}^{\ensuremath{\nu}}$, with \ensuremath{\nu} between 2/3 and 1, significantly deviating from the spin-wave prediction,
T2. At higher temperature, a broad peak in the specific heat near T/2J\ensuremath{\approxeq}0.75 is also predicted.
We study the ground state properties of the Heisenberg spin-
21 chain with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions using two approximate methods. One of them is the Jordan-Wigner mean-field theory and another approach is based on the transformation of spin operators to Bose ones and on the variational treatment of bosonic Hamiltonian. Both approaches give close results for the ground state energy and the magnetization curve at
T=0. It is proved that quantum fluctuations change the classical critical exponents in the vicinity of the transition point from the ferromagnetic to the singlet ground state. The magnetization processes display a different behavior in the regions near and far from the transition point. The relation of the obtained results to the experimental magnetization curve in
Rb2Cu2Mo3O12 is discussed.