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.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
The Jordan-Wigner transformation permits one to convert spin 1/2 operators into spinless fermion ones, or vice versa. In some cases, it transforms an interacting spin Hamiltonian into a noninteracting fermionic one, which is exactly solved at the mean-field level. Even when the resulting fermionic Hamiltonian is interacting, its mean-field solution can provide surprisingly accurate energies and correlation functions. Jordan-Wigner is, however, only one possible means of interconverting spin and fermionic degrees of freedom. Here, we apply several such techniques to the XXZ and J1-J2 Heisenberg models, as well as to the pairing or reduced Bardeen-Cooper-Schrieffer Hamiltonian, with the aim of discovering which of these mappings is most useful in applying fermionic mean-field theory to the study of spin Hamiltonians.
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.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest
(J1)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state
(T=0) and the finite-
T phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
The variational study of the ground state of the spin
−1/2 anisotropic Heisenberg antiferromagnet has been revisited on a square lattice by improving and correcting past numerical results found in Sol. State. Comm. 165, 33 (2013). The Hamiltonian has been implemented on a square lattice with antiferromagnetic interactions between nearest- and next-nearest neighbors. The nearest-neighbor couplings have different strengths, namely,
J1 and
J1′, for the x and y directions, respectively. These couplings compete with the next-nearest ones denoted by
J2. We obtained a new phase diagram in the
λ−α plane, where
λ=J1′/J1 and
α=J2/J1, whose topology is slightly different of that previously found. There is no direct frontier dividing the collinear (CAF) and the antiferromagnetic order (AF), rather, the quantum paramagnetic phase (QP) separates these two phases for all positive values of
λ and
α. The true nature of the frontiers has been obtained by scanning rigorously the relevant points of the
λ−α plane.