The exact one-to-one mapping between (spinless) Jordan–Wigner lattice fermions and (spin-1/2) spinons is established for all eigenstates of the one-dimensional s = 1/2 XX model on a lattice with an even or odd number N of lattice sites and periodic boundary conditions. Exact product formulae for the transition rates derived via Bethe ansatz are used to calculate asymptotic expressions of the 2-spinon and 4-spinon parts (for large even N ) as well as of the 1-spinon and 3-spinon parts (for large odd N ) of the dynamic spin structure factors. The observability of these spectral contributions is assessed for finite N and for N → ∞.
The exact one-to-one mapping between (spinless) Jordan–Wigner lattice fermions and (spin-1/2) spinons is established for all eigenstates of the one-dimensional s = 1/2 XX model on a lattice with an even or odd number N of lattice sites and periodic boundary conditions. Exact product formulae for the transition rates derived via Bethe ansatz are used to calculate asymptotic expressions of the 2-spinon and 4-spinon parts (for large even N ) as well as of the 1-spinon and 3-spinon parts (for large odd N ) of the dynamic spin structure factors. The observability of these spectral contributions is assessed for finite N and for N → ∞.
As part of a study that investigates the dynamics of the s=21XXZ model in the planar regime $|\ensuremath{\Delta}|<1,$ we discuss the singular nature of the Bethe ansatz equations for the case $\ensuremath{\Delta}=0$ (XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin-fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the formulas to two-spinon transition rates for chains with up to N=4096 sites.
The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) For finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N -> infinity the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling, and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane-Shastry model.
The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional s=1/2,XXZ model at T=0 in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center (Q=0). The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.
The Jordan–Wigner transformation plays an important role in spin models. However, the non-locality of the transformation implies that a periodic chain of N spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the N − 1 bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an anti-periodic chain of lattice fermions without redundancy when the Jordan–Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state, but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model ( σ x σ y − σ x σ y ) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.
The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.
The Jordan-Wigner transformation is known as a powerful tool in condensed matter theory, especially in the theory of low-dimensional quantum spin systems. The aim of this chapter is to review the application of the Jordan-Wigner fermionization technique for calculating dynamic quantities of low-dimensional quantum spin models. After a brief introduction of the Jordan-Wigner transformation for one-dimensional spin one-half systems and some of its extensions for higher dimensions and higher spin values we focus on the dynamic properties of several low-dimensional quantum spin models. We start from the famous s=1/2 XX chain. As a first step we recall well-known results for dynamics of the z-spin-component fluctuation operator and then turn to the dynamics of the dimer and trimer fluctuation operators. The dynamics of the trimer fluctuations involves both the two-fermion (one particle and one hole) and the four-fermion (two particles and two holes) excitations. We discuss some properties of the two-fermion and four-fermion excitation continua. The four-fermion dynamic quantities are of intermediate complexity between simple two-fermion (like the zz dynamic structure factor) and enormously complex multi-fermion (like the xx or xy dynamic structure factors) dynamic quantities. Further we discuss the effects of dimerization, anisotropy of XY interaction, and additional Dzyaloshinskii-Moriya interaction on various dynamic quantities. Finally we consider the dynamic transverse spin structure factor Szz(k,ω) for the s=1/2 XX model on a spatially anisotropic square lattice which allows one to trace a one-to-two-dimensional crossover in dynamic quantities.