Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0303147 · Journal of Physics A Mathematical and General

Transition rates via Bethe ansatz for the spin-1/2 planarXXZantiferromagnet

Daniel Biegel, Michael Karbach, Gerhard M ller

Published May 8, 2003• 30 citations
arXivHTMLPDF

Abstract

A novel determinantal representation for matrix elements of local spin operators between Bethe wavefunctions of the one-dimensional s = 1/2 XXZ model is used to calculate transition rates for dynamic spin structure factors in the planar regime. In a first application, high-precision numerical data are presented for lineshapes and band edge singularities of the in-plane ( xx ) two-spinon dynamic spin structure factor.

DOI
10.1088/0305-4470/36/20/301

Related papers

Determinantendarstellung von Übergangsmatrixelementen für das Spin-½ XXZ-Modell

Daniel Biegel

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.

•Nov 10, 2003•0 citations

Transition rates via Bethe ansatz for the spin-½ Heisenberg chain

Daniel Biegel, Michael Karbach, Gerhard H. Muller

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.

arXiv:cond-mat/0205430•Europhysics Letters (EPL)•Sep 1, 2002•42 citations

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

Daniel Biegel, Michael Karbach, Gerhard H. Muller +1 more

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.

arXiv:cond-mat/0309247•Physical Review B•May 6, 2004•14 citations

Evaluation of Dynamic Spin Structure Factor for the Spin-1/2 XXZ Chain in a Magnetic Field

Jun Sato, Masahiro Shiroishi, Minoru Takahashi

Transition rates and dynamic spin structure factor at zero temperature for the spin-1/2 XXZ chain at critical regime in a magnetic field are numerically evaluated in terms of the exact determinant representations for the form factors and norms of the Bethe eigenstates. We have seen that the transition rates converges toward the constant function with the value 1 in the limit delta -> 0. The observed critical exponent of the singularity at the lower boundary is compared with the one predicted from the comformal field theory. We confirm that they are in good agreement. Further we have discovered that a small peak emerges near the upper boundary in the line shape of S(q,omega) for 0

arXiv:cond-mat/0410102•Journal of the Physical Society of Japan•Nov 1, 2004•34 citations

Two-spinon dynamic structure factor of the one-dimensional s= Heisenberg antiferromagnet

Michael Karbach, Gerhard H. Muller, A. Hamid Bougourzi +2 more

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.

arXiv:cond-mat/9606068•Physical review. B, Condensed matter•May 1, 1997•165 citations

Exact Edge Singularities and Dynamical Correlations in Spin- 1 / 2 Chains

Rodrigo G. Pereira, Steven R. White, Ian K. Affleck

Exact formulas for the singularities of the dynamical structure factor, Szz(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts which are known exactly from the Bethe ansatz solution. We also study the long-time asymptotics of the self-correlation function 0|Sjz(t)Sjz(0)|0. Utilizing these results to supplement very accurate time-dependent density matrix renormalization group, for short to moderate times, we calculate Szz(q,omega) to very high precision.

arXiv:0709.0960•Physical Review Letters•Jan 17, 2008•133 citations
A novel determinantal representation for matrix elements of local spin operators between Bethe wavefunctions of the one-dimensional s = 1/2 XXZ model is used to calculate transition rates for dynamic spin structure factors in the planar regime. In a first application, high-precision numerical data are presented for lineshapes and band edge singularities of the in-plane ( xx ) two-spinon dynamic spin structure factor.
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.
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.
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 $|\ensuremath{\Delta}|<1,$ we discuss the singular nature of the Bethe ansatz equations for the case $\ensuremath{\Delta}=0$ (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.
Transition rates and dynamic spin structure factor at zero temperature for the spin-1/2 XXZ chain at critical regime in a magnetic field are numerically evaluated in terms of the exact determinant representations for the form factors and norms of the Bethe eigenstates. We have seen that the transition rates converges toward the constant function with the value 1 in the limit delta -> 0. The observed critical exponent of the singularity at the lower boundary is compared with the one predicted from the comformal field theory. We confirm that they are in good agreement. Further we have discovered that a small peak emerges near the upper boundary in the line shape of S(q,omega) for 0
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.
Exact formulas for the singularities of the dynamical structure factor, Szz(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts which are known exactly from the Bethe ansatz solution. We also study the long-time asymptotics of the self-correlation function 0|Sjz(t)Sjz(0)|0. Utilizing these results to supplement very accurate time-dependent density matrix renormalization group, for short to moderate times, we calculate Szz(q,omega) to very high precision.