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arXiv:cond-mat/0309247 · Physical Review B

Spectrum and transition rates of the XX chain analyzed via Bethe ansatz

Daniel Biegel, Michael Karbach, Gerhard H. Muller, Klaus Wiele

Published May 6, 2004• 14 citations
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Abstract

As part of a study that investigates the dynamics of the $s=\frac{1}{2}$ $\mathrm{XXZ}$ 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 $\mathrm{XX}$ 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.

DOI
10.1103/physrevb.69.174404

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As part of a study that investigates the dynamics of the s=12s=\frac{1}{2}s=21​ XXZ\mathrm{XXZ}XXZ model in the planar regime ∣\ensuremathΔ∣<1,|\ensuremath{\Delta}|<1,∣\ensuremathΔ∣<1, we discuss the singular nature of the Bethe ansatz equations for the case \ensuremathΔ=0\ensuremath{\Delta}=0\ensuremathΔ=0 (XX(XX(XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX\mathrm{XX}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=4096N=4096N=4096 sites.
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 one-dimensional spin-½ Heisenberg model was solved, in principle, by H. Bethe in 1931 using an ad hoc trial wavefunction, now famously known as the Bethe ansatz. Decades went by before the power and scope of this method of exact analysis became widely known for applications to spectrum and thermodynamics of a select class of completely integrable model systems. Until recently, the most vexing exception to immense progress in the further development of the Bethe ansatz has been the absence of a practical method to use the exactly known and readily available Bethe wave functions for the explicit calculation of transition matrix elements. The knowledge of such transition rates is of paramount importance for an understanding of dynamic correlation functions in relation to the underlying quasiparticles and for the interpretation of experimental probes of quantum fluctuations in quasi-one-dimensional magnetic compounds. It was most remarkable, therefore, when Kitanine, Maillet, and Terras succeeded in reducing matrix elements between Bethe wave functions for local spin operators to determinantal expressions. Here these expressions are used to calculate dynamic spin structure factors of the Heisenberg antiferromagnet (XXX model) and the one-dimensional spin-½ XXZ model in the planar regime (|Δ| < 1), both with periodic boundary conditions and an external magnetic field: Determinantal representations are derived for the zero-temperature spin fluctuations parallel and perpendicular to the external field. The singular nature of the Bethe ansatz equations for the case Δ = 0 (XX model) is discussed and the XX limit is performed in the determinantal expressions for the transition matrix elements. As an application, lineshapes for dynamic structure factors relevant for experiments are calculated. The predominant excitations are identified within the framework of quasiparticles excited from a corresponding vacuum.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles.Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur.The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics.These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set.The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes.In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics.In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics.Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime.In the Ising limit, the solitons become antiferromagnetic domain walls.
We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix elements of local spin operators between Bethe wave functions of the one-dimensional s = ½ Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z 1 ,..., z r of the Bethe ansatz equations, the size of the determinants is of order r × r . We present applications to the zero-temperature spin fluctuations parallel and perpendicular to an external magnetic field.