Frank Göhmann, Karol K. Kozlowski, Jesko Sirker, Junji Suzuki
SciPost Phys. 12, 158 (2022) ·
published 12 May 2022

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We present a series representation for the dynamical twopoint function of
the local spin current for the XXZ chain in the antiferromagnetic massive
regime at zero temperature. From this series we can compute the correlation
function with very high accuracy up to very long times and large distances.
Each term in the series corresponds to the contribution of all scattering
states of an even number of excitations. These excitations can be interpreted
in terms of an equal number of particles and holes. The lowest term in the
series comprises all scattering states of one hole and one particle. This term
determines the longtime largedistance asymptotic behaviour which can be
obtained explicitly from a saddlepoint analysis. The spacetime Fourier
transform of the twopoint function of currents at zero momentum gives the
optical spin conductivity of the model. We obtain highly accurate numerical
estimates for this quantity by numerically Fourier transforming our data. For
the oneparticle, onehole contribution, equivalently interpreted as a
twospinon contribution, we obtain an exact and explicit expression in terms of
known special functions. For large enough anisotropy, the twospinon
contribution carries most of the spectral weight, as can be seen by calculating
the fsum rule.
Olivier Babelon, Karol K. Kozlowski, Vincent Pasquier
SciPost Phys. 5, 035 (2018) ·
published 18 October 2018

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We construct a basis of solutions of the scalar $\boldsymbol{ \texttt{t} }
\boldsymbol{ \texttt{Q} }$ equation describing the spectrum of the $q$Toda and
Toda$_2$ chains by using auxiliary nonlinear integral equations. Our
construction allows us to provide quantisation conditions for the spectra of
these models in the form of thermodynamic Bethe Ansatzlike equations.