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Physical review. B, Condensed matter

Low-dimensional quantum antiferromagnetic Heisenberg model studied using Wigner-Jordan transformations

Y. R. Wang

Published Jul 1, 1992• 52 citations
Publisher

Abstract

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/${\mathit{d}}^{2}$ 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 ${\mathit{T}}^{2}$ dependence at low temperature, a peak near T/J=0.6, and a 1/${\mathit{T}}^{2}$ 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 ${\mathit{T}}^{2}$ 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 ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$. 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.

DOI
10.1103/physrevb.46.151

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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{\mathit{d}}^{2}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{\mathit{T}}^{2}T2 dependence at low temperature, a peak near T/J=0.6, and a 1/T2{\mathit{T}}^{2}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{\mathit{T}}^{2}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 {\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}. The exchange parameter, J, obtained from the comparison is 1060 cm\ensuremath−1{\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}cm\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 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{\mathit{T}}^{2}T2. At higher temperature, a broad peak in the specific heat near T/2J\ensuremath{\approxeq}0.75 is also predicted.
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{\mathit{E}}_{0}E0​=-0.33J per bond, is only 1% higher than the best-estimated ground-state energy, -0.334J per bond.
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 N´eel and collinear ones) are considered. By calculating the ground-state energies, the phase transitions induced by the interaction frustration were analyzed.
The ground-state properties and spin excitation spectrum of the Heisenberg model are studied within the framework of the two-dimensional Jordan-Wigner theory. The mean-field state with the lowest energy is the magnetized \ensuremath{\pi}-flux state with a gap in the spin excitation spectrum. It is shown that this mean-field gap is decreased significantly by considering amplitude fluctuations in the random-phase approximation (RPA). The RPA ground-state energy and staggered magnetization are E0{\mathit{E}}_{0}E0​=-0.332J per bond and M=0.327, respectively, which are very close to the best numerical estimates. \textcopyright{} 1996 The American Physical Society.
We construct a new mean-field theory for a quantum (spin-1/2) Heisenberg antiferromagnet in one (1D) and two (2D) dimensions using a Hartree-Fock decoupling of the four-point correlation functions. We show that the solution to the self-consistency equations based on two-point correlation functions does not produce any unphysical finite-temperature phase transition, in accord with the Mermin-Wagner theorem, unlike the common approach based on the mean-field equation for the order parameter. The next-neighbor spin-spin correlation functions, calculated within this approach, reproduce closely the strong renormalization by quantum fluctuations obtained via a Bethe ansatz in 1D and a small renormalization of the classical antiferromagnetic state in 2D. The heat capacity approximates with reasonable accuracy the full Bethe ansatz result at all temperatures in 1D. In 2D, we obtain a reduction of the peak height in the heat capacity at a finite temperature that is accessible by high-order 1/T1/T1/T expansions.